KILLED proof of input_P2myjY1kZM.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).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), 16 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), 334 ms] (22) CpxRNTS (23) IntTrsBoundProof [UPPER BOUND(ID), 150 ms] (24) CpxRNTS (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (26) CpxRNTS (27) IntTrsBoundProof [UPPER BOUND(ID), 367 ms] (28) CpxRNTS (29) IntTrsBoundProof [UPPER BOUND(ID), 147 ms] (30) CpxRNTS (31) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (32) CpxRNTS (33) IntTrsBoundProof [UPPER BOUND(ID), 112 ms] (34) CpxRNTS (35) IntTrsBoundProof [UPPER BOUND(ID), 4 ms] (36) CpxRNTS (37) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (38) CpxRNTS (39) IntTrsBoundProof [UPPER BOUND(ID), 138 ms] (40) CpxRNTS (41) IntTrsBoundProof [UPPER BOUND(ID), 22 ms] (42) CpxRNTS (43) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (44) CpxRNTS (45) IntTrsBoundProof [UPPER BOUND(ID), 139 ms] (46) CpxRNTS (47) IntTrsBoundProof [UPPER BOUND(ID), 43 ms] (48) CpxRNTS (49) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (50) CpxRNTS (51) IntTrsBoundProof [UPPER BOUND(ID), 182 ms] (52) CpxRNTS (53) IntTrsBoundProof [UPPER BOUND(ID), 15 ms] (54) CpxRNTS (55) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (56) CpxRNTS (57) IntTrsBoundProof [UPPER BOUND(ID), 5112 ms] (58) CpxRNTS (59) IntTrsBoundProof [UPPER BOUND(ID), 1420 ms] (60) CpxRNTS (61) CompletionProof [UPPER BOUND(ID), 0 ms] (62) CpxTypedWeightedCompleteTrs (63) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (64) CpxRNTS (65) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (66) CdtProblem (67) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (68) CdtProblem (69) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (70) CdtProblem (71) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 190 ms] (72) CdtProblem (73) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 1963 ms] (74) CdtProblem (75) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms] (76) CdtProblem (77) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 1936 ms] (78) CdtProblem (79) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 2045 ms] (80) CdtProblem (81) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms] (82) CdtProblem (83) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (84) CdtProblem (85) CdtLeafRemovalProof [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) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (96) CdtProblem (97) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (98) CdtProblem (99) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (100) CdtProblem (101) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (102) CdtProblem (103) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (104) CdtProblem (105) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (106) CdtProblem (107) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (108) CdtProblem (109) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (110) CdtProblem (111) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (112) CdtProblem (113) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (114) CdtProblem (115) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (116) CdtProblem (117) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (118) CdtProblem (119) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (120) CdtProblem (121) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (122) CdtProblem (123) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (124) CdtProblem (125) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (126) CdtProblem (127) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (128) CdtProblem (129) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (130) CdtProblem (131) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (132) CdtProblem (133) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (134) CdtProblem (135) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (136) CdtProblem (137) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (138) CdtProblem (139) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (140) CdtProblem (141) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (142) CdtProblem (143) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (144) CdtProblem (145) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (146) CdtProblem (147) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (148) CdtProblem (149) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (150) CdtProblem (151) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (152) CdtProblem (153) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (154) CdtProblem (155) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 3 ms] (156) CdtProblem (157) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (158) CdtProblem (159) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (160) CdtProblem (161) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (162) CdtProblem (163) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (164) CdtProblem (165) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (166) CdtProblem (167) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (168) CdtProblem (169) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 3 ms] (170) CdtProblem (171) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (172) CdtProblem (173) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (174) CdtProblem (175) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (176) CdtProblem (177) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (178) CdtProblem (179) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (180) CdtProblem (181) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (182) CdtProblem (183) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (184) CdtProblem (185) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (186) CdtProblem (187) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (188) CdtProblem (189) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (190) CdtProblem (191) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (192) CdtProblem (193) CdtRewritingProof [BOTH BOUNDS(ID, ID), 5 ms] (194) CdtProblem (195) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (196) CdtProblem (197) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (198) CdtProblem (199) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (200) CdtProblem (201) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (202) CdtProblem (203) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (204) CdtProblem (205) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (206) CdtProblem (207) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (208) CdtProblem (209) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 108 ms] (210) CdtProblem (211) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (212) 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(nil) -> 0 min(add(n, x)) -> minIter(add(n, x), add(n, x), 0) minIter(nil, add(n, y), m) -> minIter(add(n, y), add(n, y), s(m)) minIter(add(n, x), y, m) -> if_min(le(n, m), x, y, m) if_min(true, x, y, m) -> m if_min(false, x, y, m) -> minIter(x, y, m) 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(nil, nil) -> nil minsort(add(n, x), y) -> if_minsort(eq(n, min(add(n, x))), add(n, x), y) if_minsort(true, add(n, x), y) -> add(n, minsort(app(rm(n, x), y), nil)) if_minsort(false, add(n, x), y) -> minsort(x, add(n, y)) S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (1) RenamingProof (BOTH BOUNDS(ID, ID)) Renamed function symbols to avoid clashes with predefined symbol. ---------------------------------------- (2) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: 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(nil) -> 0' min(add(n, x)) -> minIter(add(n, x), add(n, x), 0') minIter(nil, add(n, y), m) -> minIter(add(n, y), add(n, y), s(m)) minIter(add(n, x), y, m) -> if_min(le(n, m), x, y, m) if_min(true, x, y, m) -> m if_min(false, x, y, m) -> minIter(x, y, m) 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(nil, nil) -> nil minsort(add(n, x), y) -> if_minsort(eq(n, min(add(n, x))), add(n, x), y) if_minsort(true, add(n, x), y) -> add(n, minsort(app(rm(n, x), y), nil)) if_minsort(false, add(n, x), y) -> minsort(x, add(n, y)) S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (3) RelTrsToTrsProof (UPPER BOUND(ID)) transformed relative TRS to TRS ---------------------------------------- (4) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: 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(nil) -> 0 min(add(n, x)) -> minIter(add(n, x), add(n, x), 0) minIter(nil, add(n, y), m) -> minIter(add(n, y), add(n, y), s(m)) minIter(add(n, x), y, m) -> if_min(le(n, m), x, y, m) if_min(true, x, y, m) -> m if_min(false, x, y, m) -> minIter(x, y, m) 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(nil, nil) -> nil minsort(add(n, x), y) -> if_minsort(eq(n, min(add(n, x))), add(n, x), y) if_minsort(true, add(n, x), y) -> add(n, minsort(app(rm(n, x), y), nil)) if_minsort(false, add(n, x), y) -> minsort(x, add(n, y)) S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (5) RelTrsToWeightedTrsProof (UPPER BOUND(ID)) Transformed relative TRS to weighted TRS ---------------------------------------- (6) Obligation: The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: 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(nil) -> 0 [1] min(add(n, x)) -> minIter(add(n, x), add(n, x), 0) [1] minIter(nil, add(n, y), m) -> minIter(add(n, y), add(n, y), s(m)) [1] minIter(add(n, x), y, m) -> if_min(le(n, m), x, y, m) [1] if_min(true, x, y, m) -> m [1] if_min(false, x, y, m) -> minIter(x, y, m) [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(nil, nil) -> nil [1] minsort(add(n, x), y) -> if_minsort(eq(n, min(add(n, x))), add(n, x), y) [1] if_minsort(true, add(n, x), y) -> add(n, minsort(app(rm(n, x), y), nil)) [1] if_minsort(false, add(n, x), y) -> minsort(x, add(n, y)) [1] Rewrite Strategy: INNERMOST ---------------------------------------- (7) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (8) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: 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(nil) -> 0 [1] min(add(n, x)) -> minIter(add(n, x), add(n, x), 0) [1] minIter(nil, add(n, y), m) -> minIter(add(n, y), add(n, y), s(m)) [1] minIter(add(n, x), y, m) -> if_min(le(n, m), x, y, m) [1] if_min(true, x, y, m) -> m [1] if_min(false, x, y, m) -> minIter(x, y, m) [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(nil, nil) -> nil [1] minsort(add(n, x), y) -> if_minsort(eq(n, min(add(n, x))), add(n, x), y) [1] if_minsort(true, add(n, x), y) -> add(n, minsort(app(rm(n, x), y), nil)) [1] if_minsort(false, add(n, x), y) -> minsort(x, add(n, y)) [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 minIter :: nil:add -> nil:add -> 0:s -> 0:s if_min :: true:false -> nil:add -> nil:add -> 0:s -> 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 -> nil:add if_minsort :: true:false -> 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: head_1 tail_1 null_1 minsort_2 if_minsort_3 (c) The following functions are completely defined: app_2 rm_2 le_2 eq_2 min_1 minIter_3 if_rm_3 if_min_4 Due to the following rules being added: minIter(v0, v1, v2) -> 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(nil) -> 0 [1] min(add(n, x)) -> minIter(add(n, x), add(n, x), 0) [1] minIter(nil, add(n, y), m) -> minIter(add(n, y), add(n, y), s(m)) [1] minIter(add(n, x), y, m) -> if_min(le(n, m), x, y, m) [1] if_min(true, x, y, m) -> m [1] if_min(false, x, y, m) -> minIter(x, y, m) [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(nil, nil) -> nil [1] minsort(add(n, x), y) -> if_minsort(eq(n, min(add(n, x))), add(n, x), y) [1] if_minsort(true, add(n, x), y) -> add(n, minsort(app(rm(n, x), y), nil)) [1] if_minsort(false, add(n, x), y) -> minsort(x, add(n, y)) [1] minIter(v0, v1, v2) -> 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 minIter :: nil:add -> nil:add -> 0:s -> 0:s if_min :: true:false -> nil:add -> nil:add -> 0:s -> 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 -> nil:add if_minsort :: true:false -> 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(nil) -> 0 [1] min(add(n, x)) -> minIter(add(n, x), add(n, x), 0) [1] minIter(nil, add(n, y), m) -> minIter(add(n, y), add(n, y), s(m)) [1] minIter(add(0, x), y, m) -> if_min(true, x, y, m) [2] minIter(add(s(x'), x), y, 0) -> if_min(false, x, y, 0) [2] minIter(add(s(x''), x), y, s(y')) -> if_min(le(x'', y'), x, y, s(y')) [2] if_min(true, x, y, m) -> m [1] if_min(false, x, y, m) -> minIter(x, y, m) [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(nil, nil) -> nil [1] minsort(add(n, x), y) -> if_minsort(eq(n, minIter(add(n, x), add(n, x), 0)), add(n, x), y) [2] if_minsort(true, add(n, nil), y) -> add(n, minsort(app(nil, y), nil)) [2] if_minsort(true, add(n, add(m', x4)), y) -> add(n, minsort(app(if_rm(eq(n, m'), n, add(m', x4)), y), nil)) [2] if_minsort(false, add(n, x), y) -> minsort(x, add(n, y)) [1] minIter(v0, v1, v2) -> 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 minIter :: nil:add -> nil:add -> 0:s -> 0:s if_min :: true:false -> nil:add -> nil:add -> 0:s -> 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 -> nil:add if_minsort :: true:false -> 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 :|: y >= 0, z = 0, z' = y app(z, z') -{ 1 }-> 1 + n + app(x, y) :|: n >= 0, x >= 0, y >= 0, z = 1 + n + x, z' = y eq(z, z') -{ 1 }-> eq(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 eq(z, z') -{ 1 }-> 0 :|: z' = 1 + x, x >= 0, z = 0 eq(z, z') -{ 1 }-> 0 :|: x >= 0, z = 1 + x, z' = 0 head(z) -{ 1 }-> n :|: n >= 0, x >= 0, z = 1 + n + x if_min(z, z', z'', z1) -{ 1 }-> m :|: z' = x, z'' = y, z = 1, z1 = m, x >= 0, y >= 0, m >= 0 if_min(z, z', z'', z1) -{ 1 }-> minIter(x, y, m) :|: z' = x, z'' = y, z1 = m, x >= 0, y >= 0, z = 0, m >= 0 if_minsort(z, z', z'') -{ 1 }-> minsort(x, 1 + n + y) :|: n >= 0, z'' = y, z' = 1 + n + x, x >= 0, y >= 0, z = 0 if_minsort(z, z', z'') -{ 2 }-> 1 + n + minsort(app(if_rm(eq(n, m'), n, 1 + m' + x4), y), 0) :|: n >= 0, x4 >= 0, z' = 1 + n + (1 + m' + x4), z'' = y, z = 1, y >= 0, m' >= 0 if_minsort(z, z', z'') -{ 2 }-> 1 + n + minsort(app(0, y), 0) :|: n >= 0, z'' = y, z = 1, y >= 0, z' = 1 + n + 0 if_rm(z, z', z'') -{ 1 }-> rm(n, x) :|: n >= 0, z = 1, z'' = 1 + m + x, x >= 0, z' = n, m >= 0 if_rm(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 if_rm(z, z', z'') -{ 1 }-> 1 + m + rm(n, x) :|: n >= 0, z'' = 1 + m + x, x >= 0, z' = n, z = 0, m >= 0 le(z, z') -{ 1 }-> le(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x le(z, z') -{ 1 }-> 1 :|: y >= 0, z = 0, z' = y le(z, z') -{ 1 }-> 0 :|: x >= 0, z = 1 + x, z' = 0 min(z) -{ 1 }-> minIter(1 + n + x, 1 + n + x, 0) :|: n >= 0, x >= 0, z = 1 + n + x min(z) -{ 1 }-> 0 :|: z = 0 minIter(z, z', z'') -{ 1 }-> minIter(1 + n + y, 1 + n + y, 1 + m) :|: n >= 0, z' = 1 + n + y, y >= 0, z = 0, z'' = m, m >= 0 minIter(z, z', z'') -{ 2 }-> if_min(le(x'', y'), x, y, 1 + y') :|: x >= 0, y >= 0, y' >= 0, z = 1 + (1 + x'') + x, x'' >= 0, z' = y, z'' = 1 + y' minIter(z, z', z'') -{ 2 }-> if_min(1, x, y, m) :|: x >= 0, y >= 0, z = 1 + 0 + x, z' = y, z'' = m, m >= 0 minIter(z, z', z'') -{ 2 }-> if_min(0, x, y, 0) :|: z'' = 0, x' >= 0, x >= 0, y >= 0, z = 1 + (1 + x') + x, z' = y minIter(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 minsort(z, z') -{ 2 }-> if_minsort(eq(n, minIter(1 + n + x, 1 + n + x, 0)), 1 + n + x, y) :|: n >= 0, x >= 0, y >= 0, z = 1 + n + x, z' = y minsort(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0 null(z) -{ 1 }-> 1 :|: z = 0 null(z) -{ 1 }-> 0 :|: n >= 0, x >= 0, z = 1 + n + x rm(z, z') -{ 2 }-> if_rm(eq(x3, y''), 1 + x3, 1 + (1 + y'') + x) :|: x >= 0, y'' >= 0, z' = 1 + (1 + y'') + x, z = 1 + x3, 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' = 1 + (1 + x1) + x, z = 0 rm(z, z') -{ 2 }-> if_rm(0, 1 + x2, 1 + 0 + x) :|: x >= 0, z = 1 + x2, z' = 1 + 0 + x, x2 >= 0 rm(z, z') -{ 1 }-> 0 :|: n >= 0, z = n, z' = 0 tail(z) -{ 1 }-> x :|: n >= 0, x >= 0, z = 1 + n + x 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, x >= 0, z' >= 0, z = 1 + n + x 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 - 1 >= 0, z' = 0 head(z) -{ 1 }-> n :|: n >= 0, x >= 0, z = 1 + n + x if_min(z, z', z'', z1) -{ 1 }-> z1 :|: z = 1, z' >= 0, z'' >= 0, z1 >= 0 if_min(z, z', z'', z1) -{ 1 }-> minIter(z', z'', z1) :|: z' >= 0, z'' >= 0, z = 0, z1 >= 0 if_minsort(z, z', z'') -{ 1 }-> minsort(x, 1 + n + z'') :|: n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0, z = 0 if_minsort(z, z', z'') -{ 2 }-> 1 + n + minsort(app(if_rm(eq(n, m'), n, 1 + m' + x4), z''), 0) :|: n >= 0, x4 >= 0, z' = 1 + n + (1 + m' + x4), z = 1, z'' >= 0, m' >= 0 if_minsort(z, z', z'') -{ 2 }-> 1 + (z' - 1) + minsort(app(0, z''), 0) :|: z' - 1 >= 0, z = 1, z'' >= 0 if_rm(z, z', z'') -{ 1 }-> rm(z', x) :|: z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0 if_rm(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_rm(z, z', z'') -{ 1 }-> 1 + m + rm(z', x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, 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 - 1 >= 0, z' = 0 min(z) -{ 1 }-> minIter(1 + n + x, 1 + n + x, 0) :|: n >= 0, x >= 0, z = 1 + n + x min(z) -{ 1 }-> 0 :|: z = 0 minIter(z, z', z'') -{ 1 }-> minIter(1 + n + y, 1 + n + y, 1 + z'') :|: n >= 0, z' = 1 + n + y, y >= 0, z = 0, z'' >= 0 minIter(z, z', z'') -{ 2 }-> if_min(le(x'', z'' - 1), x, z', 1 + (z'' - 1)) :|: x >= 0, z' >= 0, z'' - 1 >= 0, z = 1 + (1 + x'') + x, x'' >= 0 minIter(z, z', z'') -{ 2 }-> if_min(1, z - 1, z', z'') :|: z - 1 >= 0, z' >= 0, z'' >= 0 minIter(z, z', z'') -{ 2 }-> if_min(0, x, z', 0) :|: z'' = 0, x' >= 0, x >= 0, z' >= 0, z = 1 + (1 + x') + x minIter(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 minsort(z, z') -{ 2 }-> if_minsort(eq(n, minIter(1 + n + x, 1 + n + x, 0)), 1 + n + x, z') :|: n >= 0, x >= 0, z' >= 0, z = 1 + n + x minsort(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0 null(z) -{ 1 }-> 1 :|: z = 0 null(z) -{ 1 }-> 0 :|: n >= 0, x >= 0, z = 1 + n + x rm(z, z') -{ 2 }-> if_rm(eq(z - 1, y''), 1 + (z - 1), 1 + (1 + y'') + x) :|: x >= 0, y'' >= 0, z' = 1 + (1 + y'') + x, 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' = 1 + (1 + x1) + x, z = 0 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, x >= 0, z = 1 + n + x 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 } { minIter, if_min } { if_rm, rm } { min } { minsort, 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, x >= 0, z' >= 0, z = 1 + n + x 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 - 1 >= 0, z' = 0 head(z) -{ 1 }-> n :|: n >= 0, x >= 0, z = 1 + n + x if_min(z, z', z'', z1) -{ 1 }-> z1 :|: z = 1, z' >= 0, z'' >= 0, z1 >= 0 if_min(z, z', z'', z1) -{ 1 }-> minIter(z', z'', z1) :|: z' >= 0, z'' >= 0, z = 0, z1 >= 0 if_minsort(z, z', z'') -{ 1 }-> minsort(x, 1 + n + z'') :|: n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0, z = 0 if_minsort(z, z', z'') -{ 2 }-> 1 + n + minsort(app(if_rm(eq(n, m'), n, 1 + m' + x4), z''), 0) :|: n >= 0, x4 >= 0, z' = 1 + n + (1 + m' + x4), z = 1, z'' >= 0, m' >= 0 if_minsort(z, z', z'') -{ 2 }-> 1 + (z' - 1) + minsort(app(0, z''), 0) :|: z' - 1 >= 0, z = 1, z'' >= 0 if_rm(z, z', z'') -{ 1 }-> rm(z', x) :|: z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0 if_rm(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_rm(z, z', z'') -{ 1 }-> 1 + m + rm(z', x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, 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 - 1 >= 0, z' = 0 min(z) -{ 1 }-> minIter(1 + n + x, 1 + n + x, 0) :|: n >= 0, x >= 0, z = 1 + n + x min(z) -{ 1 }-> 0 :|: z = 0 minIter(z, z', z'') -{ 1 }-> minIter(1 + n + y, 1 + n + y, 1 + z'') :|: n >= 0, z' = 1 + n + y, y >= 0, z = 0, z'' >= 0 minIter(z, z', z'') -{ 2 }-> if_min(le(x'', z'' - 1), x, z', 1 + (z'' - 1)) :|: x >= 0, z' >= 0, z'' - 1 >= 0, z = 1 + (1 + x'') + x, x'' >= 0 minIter(z, z', z'') -{ 2 }-> if_min(1, z - 1, z', z'') :|: z - 1 >= 0, z' >= 0, z'' >= 0 minIter(z, z', z'') -{ 2 }-> if_min(0, x, z', 0) :|: z'' = 0, x' >= 0, x >= 0, z' >= 0, z = 1 + (1 + x') + x minIter(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 minsort(z, z') -{ 2 }-> if_minsort(eq(n, minIter(1 + n + x, 1 + n + x, 0)), 1 + n + x, z') :|: n >= 0, x >= 0, z' >= 0, z = 1 + n + x minsort(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0 null(z) -{ 1 }-> 1 :|: z = 0 null(z) -{ 1 }-> 0 :|: n >= 0, x >= 0, z = 1 + n + x rm(z, z') -{ 2 }-> if_rm(eq(z - 1, y''), 1 + (z - 1), 1 + (1 + y'') + x) :|: x >= 0, y'' >= 0, z' = 1 + (1 + y'') + x, 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' = 1 + (1 + x1) + x, z = 0 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, x >= 0, z = 1 + n + x tail(z) -{ 1 }-> 0 :|: z = 0 Function symbols to be analyzed: {le}, {eq}, {tail}, {head}, {null}, {app}, {minIter,if_min}, {if_rm,rm}, {min}, {minsort,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, x >= 0, z' >= 0, z = 1 + n + x 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 - 1 >= 0, z' = 0 head(z) -{ 1 }-> n :|: n >= 0, x >= 0, z = 1 + n + x if_min(z, z', z'', z1) -{ 1 }-> z1 :|: z = 1, z' >= 0, z'' >= 0, z1 >= 0 if_min(z, z', z'', z1) -{ 1 }-> minIter(z', z'', z1) :|: z' >= 0, z'' >= 0, z = 0, z1 >= 0 if_minsort(z, z', z'') -{ 1 }-> minsort(x, 1 + n + z'') :|: n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0, z = 0 if_minsort(z, z', z'') -{ 2 }-> 1 + n + minsort(app(if_rm(eq(n, m'), n, 1 + m' + x4), z''), 0) :|: n >= 0, x4 >= 0, z' = 1 + n + (1 + m' + x4), z = 1, z'' >= 0, m' >= 0 if_minsort(z, z', z'') -{ 2 }-> 1 + (z' - 1) + minsort(app(0, z''), 0) :|: z' - 1 >= 0, z = 1, z'' >= 0 if_rm(z, z', z'') -{ 1 }-> rm(z', x) :|: z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0 if_rm(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_rm(z, z', z'') -{ 1 }-> 1 + m + rm(z', x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, 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 - 1 >= 0, z' = 0 min(z) -{ 1 }-> minIter(1 + n + x, 1 + n + x, 0) :|: n >= 0, x >= 0, z = 1 + n + x min(z) -{ 1 }-> 0 :|: z = 0 minIter(z, z', z'') -{ 1 }-> minIter(1 + n + y, 1 + n + y, 1 + z'') :|: n >= 0, z' = 1 + n + y, y >= 0, z = 0, z'' >= 0 minIter(z, z', z'') -{ 2 }-> if_min(le(x'', z'' - 1), x, z', 1 + (z'' - 1)) :|: x >= 0, z' >= 0, z'' - 1 >= 0, z = 1 + (1 + x'') + x, x'' >= 0 minIter(z, z', z'') -{ 2 }-> if_min(1, z - 1, z', z'') :|: z - 1 >= 0, z' >= 0, z'' >= 0 minIter(z, z', z'') -{ 2 }-> if_min(0, x, z', 0) :|: z'' = 0, x' >= 0, x >= 0, z' >= 0, z = 1 + (1 + x') + x minIter(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 minsort(z, z') -{ 2 }-> if_minsort(eq(n, minIter(1 + n + x, 1 + n + x, 0)), 1 + n + x, z') :|: n >= 0, x >= 0, z' >= 0, z = 1 + n + x minsort(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0 null(z) -{ 1 }-> 1 :|: z = 0 null(z) -{ 1 }-> 0 :|: n >= 0, x >= 0, z = 1 + n + x rm(z, z') -{ 2 }-> if_rm(eq(z - 1, y''), 1 + (z - 1), 1 + (1 + y'') + x) :|: x >= 0, y'' >= 0, z' = 1 + (1 + y'') + x, 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' = 1 + (1 + x1) + x, z = 0 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, x >= 0, z = 1 + n + x tail(z) -{ 1 }-> 0 :|: z = 0 Function symbols to be analyzed: {le}, {eq}, {tail}, {head}, {null}, {app}, {minIter,if_min}, {if_rm,rm}, {min}, {minsort,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, x >= 0, z' >= 0, z = 1 + n + x 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 - 1 >= 0, z' = 0 head(z) -{ 1 }-> n :|: n >= 0, x >= 0, z = 1 + n + x if_min(z, z', z'', z1) -{ 1 }-> z1 :|: z = 1, z' >= 0, z'' >= 0, z1 >= 0 if_min(z, z', z'', z1) -{ 1 }-> minIter(z', z'', z1) :|: z' >= 0, z'' >= 0, z = 0, z1 >= 0 if_minsort(z, z', z'') -{ 1 }-> minsort(x, 1 + n + z'') :|: n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0, z = 0 if_minsort(z, z', z'') -{ 2 }-> 1 + n + minsort(app(if_rm(eq(n, m'), n, 1 + m' + x4), z''), 0) :|: n >= 0, x4 >= 0, z' = 1 + n + (1 + m' + x4), z = 1, z'' >= 0, m' >= 0 if_minsort(z, z', z'') -{ 2 }-> 1 + (z' - 1) + minsort(app(0, z''), 0) :|: z' - 1 >= 0, z = 1, z'' >= 0 if_rm(z, z', z'') -{ 1 }-> rm(z', x) :|: z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0 if_rm(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_rm(z, z', z'') -{ 1 }-> 1 + m + rm(z', x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, 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 - 1 >= 0, z' = 0 min(z) -{ 1 }-> minIter(1 + n + x, 1 + n + x, 0) :|: n >= 0, x >= 0, z = 1 + n + x min(z) -{ 1 }-> 0 :|: z = 0 minIter(z, z', z'') -{ 1 }-> minIter(1 + n + y, 1 + n + y, 1 + z'') :|: n >= 0, z' = 1 + n + y, y >= 0, z = 0, z'' >= 0 minIter(z, z', z'') -{ 2 }-> if_min(le(x'', z'' - 1), x, z', 1 + (z'' - 1)) :|: x >= 0, z' >= 0, z'' - 1 >= 0, z = 1 + (1 + x'') + x, x'' >= 0 minIter(z, z', z'') -{ 2 }-> if_min(1, z - 1, z', z'') :|: z - 1 >= 0, z' >= 0, z'' >= 0 minIter(z, z', z'') -{ 2 }-> if_min(0, x, z', 0) :|: z'' = 0, x' >= 0, x >= 0, z' >= 0, z = 1 + (1 + x') + x minIter(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 minsort(z, z') -{ 2 }-> if_minsort(eq(n, minIter(1 + n + x, 1 + n + x, 0)), 1 + n + x, z') :|: n >= 0, x >= 0, z' >= 0, z = 1 + n + x minsort(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0 null(z) -{ 1 }-> 1 :|: z = 0 null(z) -{ 1 }-> 0 :|: n >= 0, x >= 0, z = 1 + n + x rm(z, z') -{ 2 }-> if_rm(eq(z - 1, y''), 1 + (z - 1), 1 + (1 + y'') + x) :|: x >= 0, y'' >= 0, z' = 1 + (1 + y'') + x, 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' = 1 + (1 + x1) + x, z = 0 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, x >= 0, z = 1 + n + x tail(z) -{ 1 }-> 0 :|: z = 0 Function symbols to be analyzed: {le}, {eq}, {tail}, {head}, {null}, {app}, {minIter,if_min}, {if_rm,rm}, {min}, {minsort,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, x >= 0, z' >= 0, z = 1 + n + x 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 - 1 >= 0, z' = 0 head(z) -{ 1 }-> n :|: n >= 0, x >= 0, z = 1 + n + x if_min(z, z', z'', z1) -{ 1 }-> z1 :|: z = 1, z' >= 0, z'' >= 0, z1 >= 0 if_min(z, z', z'', z1) -{ 1 }-> minIter(z', z'', z1) :|: z' >= 0, z'' >= 0, z = 0, z1 >= 0 if_minsort(z, z', z'') -{ 1 }-> minsort(x, 1 + n + z'') :|: n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0, z = 0 if_minsort(z, z', z'') -{ 2 }-> 1 + n + minsort(app(if_rm(eq(n, m'), n, 1 + m' + x4), z''), 0) :|: n >= 0, x4 >= 0, z' = 1 + n + (1 + m' + x4), z = 1, z'' >= 0, m' >= 0 if_minsort(z, z', z'') -{ 2 }-> 1 + (z' - 1) + minsort(app(0, z''), 0) :|: z' - 1 >= 0, z = 1, z'' >= 0 if_rm(z, z', z'') -{ 1 }-> rm(z', x) :|: z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0 if_rm(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_rm(z, z', z'') -{ 1 }-> 1 + m + rm(z', x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, 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 - 1 >= 0, z' = 0 min(z) -{ 1 }-> minIter(1 + n + x, 1 + n + x, 0) :|: n >= 0, x >= 0, z = 1 + n + x min(z) -{ 1 }-> 0 :|: z = 0 minIter(z, z', z'') -{ 1 }-> minIter(1 + n + y, 1 + n + y, 1 + z'') :|: n >= 0, z' = 1 + n + y, y >= 0, z = 0, z'' >= 0 minIter(z, z', z'') -{ 2 }-> if_min(le(x'', z'' - 1), x, z', 1 + (z'' - 1)) :|: x >= 0, z' >= 0, z'' - 1 >= 0, z = 1 + (1 + x'') + x, x'' >= 0 minIter(z, z', z'') -{ 2 }-> if_min(1, z - 1, z', z'') :|: z - 1 >= 0, z' >= 0, z'' >= 0 minIter(z, z', z'') -{ 2 }-> if_min(0, x, z', 0) :|: z'' = 0, x' >= 0, x >= 0, z' >= 0, z = 1 + (1 + x') + x minIter(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 minsort(z, z') -{ 2 }-> if_minsort(eq(n, minIter(1 + n + x, 1 + n + x, 0)), 1 + n + x, z') :|: n >= 0, x >= 0, z' >= 0, z = 1 + n + x minsort(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0 null(z) -{ 1 }-> 1 :|: z = 0 null(z) -{ 1 }-> 0 :|: n >= 0, x >= 0, z = 1 + n + x rm(z, z') -{ 2 }-> if_rm(eq(z - 1, y''), 1 + (z - 1), 1 + (1 + y'') + x) :|: x >= 0, y'' >= 0, z' = 1 + (1 + y'') + x, 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' = 1 + (1 + x1) + x, z = 0 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, x >= 0, z = 1 + n + x tail(z) -{ 1 }-> 0 :|: z = 0 Function symbols to be analyzed: {eq}, {tail}, {head}, {null}, {app}, {minIter,if_min}, {if_rm,rm}, {min}, {minsort,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, x >= 0, z' >= 0, z = 1 + n + x 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 - 1 >= 0, z' = 0 head(z) -{ 1 }-> n :|: n >= 0, x >= 0, z = 1 + n + x if_min(z, z', z'', z1) -{ 1 }-> z1 :|: z = 1, z' >= 0, z'' >= 0, z1 >= 0 if_min(z, z', z'', z1) -{ 1 }-> minIter(z', z'', z1) :|: z' >= 0, z'' >= 0, z = 0, z1 >= 0 if_minsort(z, z', z'') -{ 1 }-> minsort(x, 1 + n + z'') :|: n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0, z = 0 if_minsort(z, z', z'') -{ 2 }-> 1 + n + minsort(app(if_rm(eq(n, m'), n, 1 + m' + x4), z''), 0) :|: n >= 0, x4 >= 0, z' = 1 + n + (1 + m' + x4), z = 1, z'' >= 0, m' >= 0 if_minsort(z, z', z'') -{ 2 }-> 1 + (z' - 1) + minsort(app(0, z''), 0) :|: z' - 1 >= 0, z = 1, z'' >= 0 if_rm(z, z', z'') -{ 1 }-> rm(z', x) :|: z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0 if_rm(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_rm(z, z', z'') -{ 1 }-> 1 + m + rm(z', x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, 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 - 1 >= 0, z' = 0 min(z) -{ 1 }-> minIter(1 + n + x, 1 + n + x, 0) :|: n >= 0, x >= 0, z = 1 + n + x min(z) -{ 1 }-> 0 :|: z = 0 minIter(z, z', z'') -{ 1 }-> minIter(1 + n + y, 1 + n + y, 1 + z'') :|: n >= 0, z' = 1 + n + y, y >= 0, z = 0, z'' >= 0 minIter(z, z', z'') -{ 3 + z'' }-> if_min(s', x, z', 1 + (z'' - 1)) :|: s' >= 0, s' <= 1, x >= 0, z' >= 0, z'' - 1 >= 0, z = 1 + (1 + x'') + x, x'' >= 0 minIter(z, z', z'') -{ 2 }-> if_min(1, z - 1, z', z'') :|: z - 1 >= 0, z' >= 0, z'' >= 0 minIter(z, z', z'') -{ 2 }-> if_min(0, x, z', 0) :|: z'' = 0, x' >= 0, x >= 0, z' >= 0, z = 1 + (1 + x') + x minIter(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 minsort(z, z') -{ 2 }-> if_minsort(eq(n, minIter(1 + n + x, 1 + n + x, 0)), 1 + n + x, z') :|: n >= 0, x >= 0, z' >= 0, z = 1 + n + x minsort(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0 null(z) -{ 1 }-> 1 :|: z = 0 null(z) -{ 1 }-> 0 :|: n >= 0, x >= 0, z = 1 + n + x rm(z, z') -{ 2 }-> if_rm(eq(z - 1, y''), 1 + (z - 1), 1 + (1 + y'') + x) :|: x >= 0, y'' >= 0, z' = 1 + (1 + y'') + x, 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' = 1 + (1 + x1) + x, z = 0 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, x >= 0, z = 1 + n + x tail(z) -{ 1 }-> 0 :|: z = 0 Function symbols to be analyzed: {eq}, {tail}, {head}, {null}, {app}, {minIter,if_min}, {if_rm,rm}, {min}, {minsort,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, x >= 0, z' >= 0, z = 1 + n + x 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 - 1 >= 0, z' = 0 head(z) -{ 1 }-> n :|: n >= 0, x >= 0, z = 1 + n + x if_min(z, z', z'', z1) -{ 1 }-> z1 :|: z = 1, z' >= 0, z'' >= 0, z1 >= 0 if_min(z, z', z'', z1) -{ 1 }-> minIter(z', z'', z1) :|: z' >= 0, z'' >= 0, z = 0, z1 >= 0 if_minsort(z, z', z'') -{ 1 }-> minsort(x, 1 + n + z'') :|: n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0, z = 0 if_minsort(z, z', z'') -{ 2 }-> 1 + n + minsort(app(if_rm(eq(n, m'), n, 1 + m' + x4), z''), 0) :|: n >= 0, x4 >= 0, z' = 1 + n + (1 + m' + x4), z = 1, z'' >= 0, m' >= 0 if_minsort(z, z', z'') -{ 2 }-> 1 + (z' - 1) + minsort(app(0, z''), 0) :|: z' - 1 >= 0, z = 1, z'' >= 0 if_rm(z, z', z'') -{ 1 }-> rm(z', x) :|: z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0 if_rm(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_rm(z, z', z'') -{ 1 }-> 1 + m + rm(z', x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, 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 - 1 >= 0, z' = 0 min(z) -{ 1 }-> minIter(1 + n + x, 1 + n + x, 0) :|: n >= 0, x >= 0, z = 1 + n + x min(z) -{ 1 }-> 0 :|: z = 0 minIter(z, z', z'') -{ 1 }-> minIter(1 + n + y, 1 + n + y, 1 + z'') :|: n >= 0, z' = 1 + n + y, y >= 0, z = 0, z'' >= 0 minIter(z, z', z'') -{ 3 + z'' }-> if_min(s', x, z', 1 + (z'' - 1)) :|: s' >= 0, s' <= 1, x >= 0, z' >= 0, z'' - 1 >= 0, z = 1 + (1 + x'') + x, x'' >= 0 minIter(z, z', z'') -{ 2 }-> if_min(1, z - 1, z', z'') :|: z - 1 >= 0, z' >= 0, z'' >= 0 minIter(z, z', z'') -{ 2 }-> if_min(0, x, z', 0) :|: z'' = 0, x' >= 0, x >= 0, z' >= 0, z = 1 + (1 + x') + x minIter(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 minsort(z, z') -{ 2 }-> if_minsort(eq(n, minIter(1 + n + x, 1 + n + x, 0)), 1 + n + x, z') :|: n >= 0, x >= 0, z' >= 0, z = 1 + n + x minsort(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0 null(z) -{ 1 }-> 1 :|: z = 0 null(z) -{ 1 }-> 0 :|: n >= 0, x >= 0, z = 1 + n + x rm(z, z') -{ 2 }-> if_rm(eq(z - 1, y''), 1 + (z - 1), 1 + (1 + y'') + x) :|: x >= 0, y'' >= 0, z' = 1 + (1 + y'') + x, 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' = 1 + (1 + x1) + x, z = 0 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, x >= 0, z = 1 + n + x tail(z) -{ 1 }-> 0 :|: z = 0 Function symbols to be analyzed: {eq}, {tail}, {head}, {null}, {app}, {minIter,if_min}, {if_rm,rm}, {min}, {minsort,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, x >= 0, z' >= 0, z = 1 + n + x 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 - 1 >= 0, z' = 0 head(z) -{ 1 }-> n :|: n >= 0, x >= 0, z = 1 + n + x if_min(z, z', z'', z1) -{ 1 }-> z1 :|: z = 1, z' >= 0, z'' >= 0, z1 >= 0 if_min(z, z', z'', z1) -{ 1 }-> minIter(z', z'', z1) :|: z' >= 0, z'' >= 0, z = 0, z1 >= 0 if_minsort(z, z', z'') -{ 1 }-> minsort(x, 1 + n + z'') :|: n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0, z = 0 if_minsort(z, z', z'') -{ 2 }-> 1 + n + minsort(app(if_rm(eq(n, m'), n, 1 + m' + x4), z''), 0) :|: n >= 0, x4 >= 0, z' = 1 + n + (1 + m' + x4), z = 1, z'' >= 0, m' >= 0 if_minsort(z, z', z'') -{ 2 }-> 1 + (z' - 1) + minsort(app(0, z''), 0) :|: z' - 1 >= 0, z = 1, z'' >= 0 if_rm(z, z', z'') -{ 1 }-> rm(z', x) :|: z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0 if_rm(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_rm(z, z', z'') -{ 1 }-> 1 + m + rm(z', x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, 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 - 1 >= 0, z' = 0 min(z) -{ 1 }-> minIter(1 + n + x, 1 + n + x, 0) :|: n >= 0, x >= 0, z = 1 + n + x min(z) -{ 1 }-> 0 :|: z = 0 minIter(z, z', z'') -{ 1 }-> minIter(1 + n + y, 1 + n + y, 1 + z'') :|: n >= 0, z' = 1 + n + y, y >= 0, z = 0, z'' >= 0 minIter(z, z', z'') -{ 3 + z'' }-> if_min(s', x, z', 1 + (z'' - 1)) :|: s' >= 0, s' <= 1, x >= 0, z' >= 0, z'' - 1 >= 0, z = 1 + (1 + x'') + x, x'' >= 0 minIter(z, z', z'') -{ 2 }-> if_min(1, z - 1, z', z'') :|: z - 1 >= 0, z' >= 0, z'' >= 0 minIter(z, z', z'') -{ 2 }-> if_min(0, x, z', 0) :|: z'' = 0, x' >= 0, x >= 0, z' >= 0, z = 1 + (1 + x') + x minIter(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 minsort(z, z') -{ 2 }-> if_minsort(eq(n, minIter(1 + n + x, 1 + n + x, 0)), 1 + n + x, z') :|: n >= 0, x >= 0, z' >= 0, z = 1 + n + x minsort(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0 null(z) -{ 1 }-> 1 :|: z = 0 null(z) -{ 1 }-> 0 :|: n >= 0, x >= 0, z = 1 + n + x rm(z, z') -{ 2 }-> if_rm(eq(z - 1, y''), 1 + (z - 1), 1 + (1 + y'') + x) :|: x >= 0, y'' >= 0, z' = 1 + (1 + y'') + x, 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' = 1 + (1 + x1) + x, z = 0 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, x >= 0, z = 1 + n + x tail(z) -{ 1 }-> 0 :|: z = 0 Function symbols to be analyzed: {tail}, {head}, {null}, {app}, {minIter,if_min}, {if_rm,rm}, {min}, {minsort,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, x >= 0, z' >= 0, z = 1 + n + x eq(z, z') -{ 3 + z' }-> s'' :|: s'' >= 0, s'' <= 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 - 1 >= 0, z' = 0 head(z) -{ 1 }-> n :|: n >= 0, x >= 0, z = 1 + n + x if_min(z, z', z'', z1) -{ 1 }-> z1 :|: z = 1, z' >= 0, z'' >= 0, z1 >= 0 if_min(z, z', z'', z1) -{ 1 }-> minIter(z', z'', z1) :|: z' >= 0, z'' >= 0, z = 0, z1 >= 0 if_minsort(z, z', z'') -{ 1 }-> minsort(x, 1 + n + z'') :|: n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0, z = 0 if_minsort(z, z', z'') -{ 5 + m' }-> 1 + n + minsort(app(if_rm(s2, n, 1 + m' + x4), z''), 0) :|: s2 >= 0, s2 <= 1, n >= 0, x4 >= 0, z' = 1 + n + (1 + m' + x4), z = 1, z'' >= 0, m' >= 0 if_minsort(z, z', z'') -{ 2 }-> 1 + (z' - 1) + minsort(app(0, z''), 0) :|: z' - 1 >= 0, z = 1, z'' >= 0 if_rm(z, z', z'') -{ 1 }-> rm(z', x) :|: z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0 if_rm(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_rm(z, z', z'') -{ 1 }-> 1 + m + rm(z', x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, 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 - 1 >= 0, z' = 0 min(z) -{ 1 }-> minIter(1 + n + x, 1 + n + x, 0) :|: n >= 0, x >= 0, z = 1 + n + x min(z) -{ 1 }-> 0 :|: z = 0 minIter(z, z', z'') -{ 1 }-> minIter(1 + n + y, 1 + n + y, 1 + z'') :|: n >= 0, z' = 1 + n + y, y >= 0, z = 0, z'' >= 0 minIter(z, z', z'') -{ 3 + z'' }-> if_min(s', x, z', 1 + (z'' - 1)) :|: s' >= 0, s' <= 1, x >= 0, z' >= 0, z'' - 1 >= 0, z = 1 + (1 + x'') + x, x'' >= 0 minIter(z, z', z'') -{ 2 }-> if_min(1, z - 1, z', z'') :|: z - 1 >= 0, z' >= 0, z'' >= 0 minIter(z, z', z'') -{ 2 }-> if_min(0, x, z', 0) :|: z'' = 0, x' >= 0, x >= 0, z' >= 0, z = 1 + (1 + x') + x minIter(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 minsort(z, z') -{ 2 }-> if_minsort(eq(n, minIter(1 + n + x, 1 + n + x, 0)), 1 + n + x, z') :|: n >= 0, x >= 0, z' >= 0, z = 1 + n + x minsort(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0 null(z) -{ 1 }-> 1 :|: z = 0 null(z) -{ 1 }-> 0 :|: n >= 0, x >= 0, z = 1 + n + x rm(z, z') -{ 5 + y'' }-> if_rm(s1, 1 + (z - 1), 1 + (1 + y'') + x) :|: s1 >= 0, s1 <= 1, x >= 0, y'' >= 0, z' = 1 + (1 + y'') + x, 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' = 1 + (1 + x1) + x, z = 0 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, x >= 0, z = 1 + n + x tail(z) -{ 1 }-> 0 :|: z = 0 Function symbols to be analyzed: {tail}, {head}, {null}, {app}, {minIter,if_min}, {if_rm,rm}, {min}, {minsort,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, x >= 0, z' >= 0, z = 1 + n + x eq(z, z') -{ 3 + z' }-> s'' :|: s'' >= 0, s'' <= 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 - 1 >= 0, z' = 0 head(z) -{ 1 }-> n :|: n >= 0, x >= 0, z = 1 + n + x if_min(z, z', z'', z1) -{ 1 }-> z1 :|: z = 1, z' >= 0, z'' >= 0, z1 >= 0 if_min(z, z', z'', z1) -{ 1 }-> minIter(z', z'', z1) :|: z' >= 0, z'' >= 0, z = 0, z1 >= 0 if_minsort(z, z', z'') -{ 1 }-> minsort(x, 1 + n + z'') :|: n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0, z = 0 if_minsort(z, z', z'') -{ 5 + m' }-> 1 + n + minsort(app(if_rm(s2, n, 1 + m' + x4), z''), 0) :|: s2 >= 0, s2 <= 1, n >= 0, x4 >= 0, z' = 1 + n + (1 + m' + x4), z = 1, z'' >= 0, m' >= 0 if_minsort(z, z', z'') -{ 2 }-> 1 + (z' - 1) + minsort(app(0, z''), 0) :|: z' - 1 >= 0, z = 1, z'' >= 0 if_rm(z, z', z'') -{ 1 }-> rm(z', x) :|: z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0 if_rm(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_rm(z, z', z'') -{ 1 }-> 1 + m + rm(z', x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, 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 - 1 >= 0, z' = 0 min(z) -{ 1 }-> minIter(1 + n + x, 1 + n + x, 0) :|: n >= 0, x >= 0, z = 1 + n + x min(z) -{ 1 }-> 0 :|: z = 0 minIter(z, z', z'') -{ 1 }-> minIter(1 + n + y, 1 + n + y, 1 + z'') :|: n >= 0, z' = 1 + n + y, y >= 0, z = 0, z'' >= 0 minIter(z, z', z'') -{ 3 + z'' }-> if_min(s', x, z', 1 + (z'' - 1)) :|: s' >= 0, s' <= 1, x >= 0, z' >= 0, z'' - 1 >= 0, z = 1 + (1 + x'') + x, x'' >= 0 minIter(z, z', z'') -{ 2 }-> if_min(1, z - 1, z', z'') :|: z - 1 >= 0, z' >= 0, z'' >= 0 minIter(z, z', z'') -{ 2 }-> if_min(0, x, z', 0) :|: z'' = 0, x' >= 0, x >= 0, z' >= 0, z = 1 + (1 + x') + x minIter(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 minsort(z, z') -{ 2 }-> if_minsort(eq(n, minIter(1 + n + x, 1 + n + x, 0)), 1 + n + x, z') :|: n >= 0, x >= 0, z' >= 0, z = 1 + n + x minsort(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0 null(z) -{ 1 }-> 1 :|: z = 0 null(z) -{ 1 }-> 0 :|: n >= 0, x >= 0, z = 1 + n + x rm(z, z') -{ 5 + y'' }-> if_rm(s1, 1 + (z - 1), 1 + (1 + y'') + x) :|: s1 >= 0, s1 <= 1, x >= 0, y'' >= 0, z' = 1 + (1 + y'') + x, 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' = 1 + (1 + x1) + x, z = 0 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, x >= 0, z = 1 + n + x tail(z) -{ 1 }-> 0 :|: z = 0 Function symbols to be analyzed: {tail}, {head}, {null}, {app}, {minIter,if_min}, {if_rm,rm}, {min}, {minsort,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, x >= 0, z' >= 0, z = 1 + n + x eq(z, z') -{ 3 + z' }-> s'' :|: s'' >= 0, s'' <= 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 - 1 >= 0, z' = 0 head(z) -{ 1 }-> n :|: n >= 0, x >= 0, z = 1 + n + x if_min(z, z', z'', z1) -{ 1 }-> z1 :|: z = 1, z' >= 0, z'' >= 0, z1 >= 0 if_min(z, z', z'', z1) -{ 1 }-> minIter(z', z'', z1) :|: z' >= 0, z'' >= 0, z = 0, z1 >= 0 if_minsort(z, z', z'') -{ 1 }-> minsort(x, 1 + n + z'') :|: n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0, z = 0 if_minsort(z, z', z'') -{ 5 + m' }-> 1 + n + minsort(app(if_rm(s2, n, 1 + m' + x4), z''), 0) :|: s2 >= 0, s2 <= 1, n >= 0, x4 >= 0, z' = 1 + n + (1 + m' + x4), z = 1, z'' >= 0, m' >= 0 if_minsort(z, z', z'') -{ 2 }-> 1 + (z' - 1) + minsort(app(0, z''), 0) :|: z' - 1 >= 0, z = 1, z'' >= 0 if_rm(z, z', z'') -{ 1 }-> rm(z', x) :|: z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0 if_rm(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_rm(z, z', z'') -{ 1 }-> 1 + m + rm(z', x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, 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 - 1 >= 0, z' = 0 min(z) -{ 1 }-> minIter(1 + n + x, 1 + n + x, 0) :|: n >= 0, x >= 0, z = 1 + n + x min(z) -{ 1 }-> 0 :|: z = 0 minIter(z, z', z'') -{ 1 }-> minIter(1 + n + y, 1 + n + y, 1 + z'') :|: n >= 0, z' = 1 + n + y, y >= 0, z = 0, z'' >= 0 minIter(z, z', z'') -{ 3 + z'' }-> if_min(s', x, z', 1 + (z'' - 1)) :|: s' >= 0, s' <= 1, x >= 0, z' >= 0, z'' - 1 >= 0, z = 1 + (1 + x'') + x, x'' >= 0 minIter(z, z', z'') -{ 2 }-> if_min(1, z - 1, z', z'') :|: z - 1 >= 0, z' >= 0, z'' >= 0 minIter(z, z', z'') -{ 2 }-> if_min(0, x, z', 0) :|: z'' = 0, x' >= 0, x >= 0, z' >= 0, z = 1 + (1 + x') + x minIter(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 minsort(z, z') -{ 2 }-> if_minsort(eq(n, minIter(1 + n + x, 1 + n + x, 0)), 1 + n + x, z') :|: n >= 0, x >= 0, z' >= 0, z = 1 + n + x minsort(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0 null(z) -{ 1 }-> 1 :|: z = 0 null(z) -{ 1 }-> 0 :|: n >= 0, x >= 0, z = 1 + n + x rm(z, z') -{ 5 + y'' }-> if_rm(s1, 1 + (z - 1), 1 + (1 + y'') + x) :|: s1 >= 0, s1 <= 1, x >= 0, y'' >= 0, z' = 1 + (1 + y'') + x, 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' = 1 + (1 + x1) + x, z = 0 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, x >= 0, z = 1 + n + x tail(z) -{ 1 }-> 0 :|: z = 0 Function symbols to be analyzed: {head}, {null}, {app}, {minIter,if_min}, {if_rm,rm}, {min}, {minsort,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, x >= 0, z' >= 0, z = 1 + n + x eq(z, z') -{ 3 + z' }-> s'' :|: s'' >= 0, s'' <= 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 - 1 >= 0, z' = 0 head(z) -{ 1 }-> n :|: n >= 0, x >= 0, z = 1 + n + x if_min(z, z', z'', z1) -{ 1 }-> z1 :|: z = 1, z' >= 0, z'' >= 0, z1 >= 0 if_min(z, z', z'', z1) -{ 1 }-> minIter(z', z'', z1) :|: z' >= 0, z'' >= 0, z = 0, z1 >= 0 if_minsort(z, z', z'') -{ 1 }-> minsort(x, 1 + n + z'') :|: n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0, z = 0 if_minsort(z, z', z'') -{ 5 + m' }-> 1 + n + minsort(app(if_rm(s2, n, 1 + m' + x4), z''), 0) :|: s2 >= 0, s2 <= 1, n >= 0, x4 >= 0, z' = 1 + n + (1 + m' + x4), z = 1, z'' >= 0, m' >= 0 if_minsort(z, z', z'') -{ 2 }-> 1 + (z' - 1) + minsort(app(0, z''), 0) :|: z' - 1 >= 0, z = 1, z'' >= 0 if_rm(z, z', z'') -{ 1 }-> rm(z', x) :|: z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0 if_rm(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_rm(z, z', z'') -{ 1 }-> 1 + m + rm(z', x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, 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 - 1 >= 0, z' = 0 min(z) -{ 1 }-> minIter(1 + n + x, 1 + n + x, 0) :|: n >= 0, x >= 0, z = 1 + n + x min(z) -{ 1 }-> 0 :|: z = 0 minIter(z, z', z'') -{ 1 }-> minIter(1 + n + y, 1 + n + y, 1 + z'') :|: n >= 0, z' = 1 + n + y, y >= 0, z = 0, z'' >= 0 minIter(z, z', z'') -{ 3 + z'' }-> if_min(s', x, z', 1 + (z'' - 1)) :|: s' >= 0, s' <= 1, x >= 0, z' >= 0, z'' - 1 >= 0, z = 1 + (1 + x'') + x, x'' >= 0 minIter(z, z', z'') -{ 2 }-> if_min(1, z - 1, z', z'') :|: z - 1 >= 0, z' >= 0, z'' >= 0 minIter(z, z', z'') -{ 2 }-> if_min(0, x, z', 0) :|: z'' = 0, x' >= 0, x >= 0, z' >= 0, z = 1 + (1 + x') + x minIter(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 minsort(z, z') -{ 2 }-> if_minsort(eq(n, minIter(1 + n + x, 1 + n + x, 0)), 1 + n + x, z') :|: n >= 0, x >= 0, z' >= 0, z = 1 + n + x minsort(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0 null(z) -{ 1 }-> 1 :|: z = 0 null(z) -{ 1 }-> 0 :|: n >= 0, x >= 0, z = 1 + n + x rm(z, z') -{ 5 + y'' }-> if_rm(s1, 1 + (z - 1), 1 + (1 + y'') + x) :|: s1 >= 0, s1 <= 1, x >= 0, y'' >= 0, z' = 1 + (1 + y'') + x, 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' = 1 + (1 + x1) + x, z = 0 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, x >= 0, z = 1 + n + x tail(z) -{ 1 }-> 0 :|: z = 0 Function symbols to be analyzed: {head}, {null}, {app}, {minIter,if_min}, {if_rm,rm}, {min}, {minsort,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, x >= 0, z' >= 0, z = 1 + n + x eq(z, z') -{ 3 + z' }-> s'' :|: s'' >= 0, s'' <= 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 - 1 >= 0, z' = 0 head(z) -{ 1 }-> n :|: n >= 0, x >= 0, z = 1 + n + x if_min(z, z', z'', z1) -{ 1 }-> z1 :|: z = 1, z' >= 0, z'' >= 0, z1 >= 0 if_min(z, z', z'', z1) -{ 1 }-> minIter(z', z'', z1) :|: z' >= 0, z'' >= 0, z = 0, z1 >= 0 if_minsort(z, z', z'') -{ 1 }-> minsort(x, 1 + n + z'') :|: n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0, z = 0 if_minsort(z, z', z'') -{ 5 + m' }-> 1 + n + minsort(app(if_rm(s2, n, 1 + m' + x4), z''), 0) :|: s2 >= 0, s2 <= 1, n >= 0, x4 >= 0, z' = 1 + n + (1 + m' + x4), z = 1, z'' >= 0, m' >= 0 if_minsort(z, z', z'') -{ 2 }-> 1 + (z' - 1) + minsort(app(0, z''), 0) :|: z' - 1 >= 0, z = 1, z'' >= 0 if_rm(z, z', z'') -{ 1 }-> rm(z', x) :|: z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0 if_rm(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_rm(z, z', z'') -{ 1 }-> 1 + m + rm(z', x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, 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 - 1 >= 0, z' = 0 min(z) -{ 1 }-> minIter(1 + n + x, 1 + n + x, 0) :|: n >= 0, x >= 0, z = 1 + n + x min(z) -{ 1 }-> 0 :|: z = 0 minIter(z, z', z'') -{ 1 }-> minIter(1 + n + y, 1 + n + y, 1 + z'') :|: n >= 0, z' = 1 + n + y, y >= 0, z = 0, z'' >= 0 minIter(z, z', z'') -{ 3 + z'' }-> if_min(s', x, z', 1 + (z'' - 1)) :|: s' >= 0, s' <= 1, x >= 0, z' >= 0, z'' - 1 >= 0, z = 1 + (1 + x'') + x, x'' >= 0 minIter(z, z', z'') -{ 2 }-> if_min(1, z - 1, z', z'') :|: z - 1 >= 0, z' >= 0, z'' >= 0 minIter(z, z', z'') -{ 2 }-> if_min(0, x, z', 0) :|: z'' = 0, x' >= 0, x >= 0, z' >= 0, z = 1 + (1 + x') + x minIter(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 minsort(z, z') -{ 2 }-> if_minsort(eq(n, minIter(1 + n + x, 1 + n + x, 0)), 1 + n + x, z') :|: n >= 0, x >= 0, z' >= 0, z = 1 + n + x minsort(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0 null(z) -{ 1 }-> 1 :|: z = 0 null(z) -{ 1 }-> 0 :|: n >= 0, x >= 0, z = 1 + n + x rm(z, z') -{ 5 + y'' }-> if_rm(s1, 1 + (z - 1), 1 + (1 + y'') + x) :|: s1 >= 0, s1 <= 1, x >= 0, y'' >= 0, z' = 1 + (1 + y'') + x, 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' = 1 + (1 + x1) + x, z = 0 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, x >= 0, z = 1 + n + x tail(z) -{ 1 }-> 0 :|: z = 0 Function symbols to be analyzed: {head}, {null}, {app}, {minIter,if_min}, {if_rm,rm}, {min}, {minsort,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, x >= 0, z' >= 0, z = 1 + n + x eq(z, z') -{ 3 + z' }-> s'' :|: s'' >= 0, s'' <= 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 - 1 >= 0, z' = 0 head(z) -{ 1 }-> n :|: n >= 0, x >= 0, z = 1 + n + x if_min(z, z', z'', z1) -{ 1 }-> z1 :|: z = 1, z' >= 0, z'' >= 0, z1 >= 0 if_min(z, z', z'', z1) -{ 1 }-> minIter(z', z'', z1) :|: z' >= 0, z'' >= 0, z = 0, z1 >= 0 if_minsort(z, z', z'') -{ 1 }-> minsort(x, 1 + n + z'') :|: n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0, z = 0 if_minsort(z, z', z'') -{ 5 + m' }-> 1 + n + minsort(app(if_rm(s2, n, 1 + m' + x4), z''), 0) :|: s2 >= 0, s2 <= 1, n >= 0, x4 >= 0, z' = 1 + n + (1 + m' + x4), z = 1, z'' >= 0, m' >= 0 if_minsort(z, z', z'') -{ 2 }-> 1 + (z' - 1) + minsort(app(0, z''), 0) :|: z' - 1 >= 0, z = 1, z'' >= 0 if_rm(z, z', z'') -{ 1 }-> rm(z', x) :|: z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0 if_rm(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_rm(z, z', z'') -{ 1 }-> 1 + m + rm(z', x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, 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 - 1 >= 0, z' = 0 min(z) -{ 1 }-> minIter(1 + n + x, 1 + n + x, 0) :|: n >= 0, x >= 0, z = 1 + n + x min(z) -{ 1 }-> 0 :|: z = 0 minIter(z, z', z'') -{ 1 }-> minIter(1 + n + y, 1 + n + y, 1 + z'') :|: n >= 0, z' = 1 + n + y, y >= 0, z = 0, z'' >= 0 minIter(z, z', z'') -{ 3 + z'' }-> if_min(s', x, z', 1 + (z'' - 1)) :|: s' >= 0, s' <= 1, x >= 0, z' >= 0, z'' - 1 >= 0, z = 1 + (1 + x'') + x, x'' >= 0 minIter(z, z', z'') -{ 2 }-> if_min(1, z - 1, z', z'') :|: z - 1 >= 0, z' >= 0, z'' >= 0 minIter(z, z', z'') -{ 2 }-> if_min(0, x, z', 0) :|: z'' = 0, x' >= 0, x >= 0, z' >= 0, z = 1 + (1 + x') + x minIter(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 minsort(z, z') -{ 2 }-> if_minsort(eq(n, minIter(1 + n + x, 1 + n + x, 0)), 1 + n + x, z') :|: n >= 0, x >= 0, z' >= 0, z = 1 + n + x minsort(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0 null(z) -{ 1 }-> 1 :|: z = 0 null(z) -{ 1 }-> 0 :|: n >= 0, x >= 0, z = 1 + n + x rm(z, z') -{ 5 + y'' }-> if_rm(s1, 1 + (z - 1), 1 + (1 + y'') + x) :|: s1 >= 0, s1 <= 1, x >= 0, y'' >= 0, z' = 1 + (1 + y'') + x, 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' = 1 + (1 + x1) + x, z = 0 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, x >= 0, z = 1 + n + x tail(z) -{ 1 }-> 0 :|: z = 0 Function symbols to be analyzed: {null}, {app}, {minIter,if_min}, {if_rm,rm}, {min}, {minsort,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, x >= 0, z' >= 0, z = 1 + n + x eq(z, z') -{ 3 + z' }-> s'' :|: s'' >= 0, s'' <= 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 - 1 >= 0, z' = 0 head(z) -{ 1 }-> n :|: n >= 0, x >= 0, z = 1 + n + x if_min(z, z', z'', z1) -{ 1 }-> z1 :|: z = 1, z' >= 0, z'' >= 0, z1 >= 0 if_min(z, z', z'', z1) -{ 1 }-> minIter(z', z'', z1) :|: z' >= 0, z'' >= 0, z = 0, z1 >= 0 if_minsort(z, z', z'') -{ 1 }-> minsort(x, 1 + n + z'') :|: n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0, z = 0 if_minsort(z, z', z'') -{ 5 + m' }-> 1 + n + minsort(app(if_rm(s2, n, 1 + m' + x4), z''), 0) :|: s2 >= 0, s2 <= 1, n >= 0, x4 >= 0, z' = 1 + n + (1 + m' + x4), z = 1, z'' >= 0, m' >= 0 if_minsort(z, z', z'') -{ 2 }-> 1 + (z' - 1) + minsort(app(0, z''), 0) :|: z' - 1 >= 0, z = 1, z'' >= 0 if_rm(z, z', z'') -{ 1 }-> rm(z', x) :|: z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0 if_rm(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_rm(z, z', z'') -{ 1 }-> 1 + m + rm(z', x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, 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 - 1 >= 0, z' = 0 min(z) -{ 1 }-> minIter(1 + n + x, 1 + n + x, 0) :|: n >= 0, x >= 0, z = 1 + n + x min(z) -{ 1 }-> 0 :|: z = 0 minIter(z, z', z'') -{ 1 }-> minIter(1 + n + y, 1 + n + y, 1 + z'') :|: n >= 0, z' = 1 + n + y, y >= 0, z = 0, z'' >= 0 minIter(z, z', z'') -{ 3 + z'' }-> if_min(s', x, z', 1 + (z'' - 1)) :|: s' >= 0, s' <= 1, x >= 0, z' >= 0, z'' - 1 >= 0, z = 1 + (1 + x'') + x, x'' >= 0 minIter(z, z', z'') -{ 2 }-> if_min(1, z - 1, z', z'') :|: z - 1 >= 0, z' >= 0, z'' >= 0 minIter(z, z', z'') -{ 2 }-> if_min(0, x, z', 0) :|: z'' = 0, x' >= 0, x >= 0, z' >= 0, z = 1 + (1 + x') + x minIter(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 minsort(z, z') -{ 2 }-> if_minsort(eq(n, minIter(1 + n + x, 1 + n + x, 0)), 1 + n + x, z') :|: n >= 0, x >= 0, z' >= 0, z = 1 + n + x minsort(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0 null(z) -{ 1 }-> 1 :|: z = 0 null(z) -{ 1 }-> 0 :|: n >= 0, x >= 0, z = 1 + n + x rm(z, z') -{ 5 + y'' }-> if_rm(s1, 1 + (z - 1), 1 + (1 + y'') + x) :|: s1 >= 0, s1 <= 1, x >= 0, y'' >= 0, z' = 1 + (1 + y'') + x, 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' = 1 + (1 + x1) + x, z = 0 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, x >= 0, z = 1 + n + x tail(z) -{ 1 }-> 0 :|: z = 0 Function symbols to be analyzed: {null}, {app}, {minIter,if_min}, {if_rm,rm}, {min}, {minsort,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, x >= 0, z' >= 0, z = 1 + n + x eq(z, z') -{ 3 + z' }-> s'' :|: s'' >= 0, s'' <= 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 - 1 >= 0, z' = 0 head(z) -{ 1 }-> n :|: n >= 0, x >= 0, z = 1 + n + x if_min(z, z', z'', z1) -{ 1 }-> z1 :|: z = 1, z' >= 0, z'' >= 0, z1 >= 0 if_min(z, z', z'', z1) -{ 1 }-> minIter(z', z'', z1) :|: z' >= 0, z'' >= 0, z = 0, z1 >= 0 if_minsort(z, z', z'') -{ 1 }-> minsort(x, 1 + n + z'') :|: n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0, z = 0 if_minsort(z, z', z'') -{ 5 + m' }-> 1 + n + minsort(app(if_rm(s2, n, 1 + m' + x4), z''), 0) :|: s2 >= 0, s2 <= 1, n >= 0, x4 >= 0, z' = 1 + n + (1 + m' + x4), z = 1, z'' >= 0, m' >= 0 if_minsort(z, z', z'') -{ 2 }-> 1 + (z' - 1) + minsort(app(0, z''), 0) :|: z' - 1 >= 0, z = 1, z'' >= 0 if_rm(z, z', z'') -{ 1 }-> rm(z', x) :|: z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0 if_rm(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_rm(z, z', z'') -{ 1 }-> 1 + m + rm(z', x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, 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 - 1 >= 0, z' = 0 min(z) -{ 1 }-> minIter(1 + n + x, 1 + n + x, 0) :|: n >= 0, x >= 0, z = 1 + n + x min(z) -{ 1 }-> 0 :|: z = 0 minIter(z, z', z'') -{ 1 }-> minIter(1 + n + y, 1 + n + y, 1 + z'') :|: n >= 0, z' = 1 + n + y, y >= 0, z = 0, z'' >= 0 minIter(z, z', z'') -{ 3 + z'' }-> if_min(s', x, z', 1 + (z'' - 1)) :|: s' >= 0, s' <= 1, x >= 0, z' >= 0, z'' - 1 >= 0, z = 1 + (1 + x'') + x, x'' >= 0 minIter(z, z', z'') -{ 2 }-> if_min(1, z - 1, z', z'') :|: z - 1 >= 0, z' >= 0, z'' >= 0 minIter(z, z', z'') -{ 2 }-> if_min(0, x, z', 0) :|: z'' = 0, x' >= 0, x >= 0, z' >= 0, z = 1 + (1 + x') + x minIter(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 minsort(z, z') -{ 2 }-> if_minsort(eq(n, minIter(1 + n + x, 1 + n + x, 0)), 1 + n + x, z') :|: n >= 0, x >= 0, z' >= 0, z = 1 + n + x minsort(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0 null(z) -{ 1 }-> 1 :|: z = 0 null(z) -{ 1 }-> 0 :|: n >= 0, x >= 0, z = 1 + n + x rm(z, z') -{ 5 + y'' }-> if_rm(s1, 1 + (z - 1), 1 + (1 + y'') + x) :|: s1 >= 0, s1 <= 1, x >= 0, y'' >= 0, z' = 1 + (1 + y'') + x, 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' = 1 + (1 + x1) + x, z = 0 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, x >= 0, z = 1 + n + x tail(z) -{ 1 }-> 0 :|: z = 0 Function symbols to be analyzed: {null}, {app}, {minIter,if_min}, {if_rm,rm}, {min}, {minsort,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, x >= 0, z' >= 0, z = 1 + n + x eq(z, z') -{ 3 + z' }-> s'' :|: s'' >= 0, s'' <= 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 - 1 >= 0, z' = 0 head(z) -{ 1 }-> n :|: n >= 0, x >= 0, z = 1 + n + x if_min(z, z', z'', z1) -{ 1 }-> z1 :|: z = 1, z' >= 0, z'' >= 0, z1 >= 0 if_min(z, z', z'', z1) -{ 1 }-> minIter(z', z'', z1) :|: z' >= 0, z'' >= 0, z = 0, z1 >= 0 if_minsort(z, z', z'') -{ 1 }-> minsort(x, 1 + n + z'') :|: n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0, z = 0 if_minsort(z, z', z'') -{ 5 + m' }-> 1 + n + minsort(app(if_rm(s2, n, 1 + m' + x4), z''), 0) :|: s2 >= 0, s2 <= 1, n >= 0, x4 >= 0, z' = 1 + n + (1 + m' + x4), z = 1, z'' >= 0, m' >= 0 if_minsort(z, z', z'') -{ 2 }-> 1 + (z' - 1) + minsort(app(0, z''), 0) :|: z' - 1 >= 0, z = 1, z'' >= 0 if_rm(z, z', z'') -{ 1 }-> rm(z', x) :|: z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0 if_rm(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_rm(z, z', z'') -{ 1 }-> 1 + m + rm(z', x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, 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 - 1 >= 0, z' = 0 min(z) -{ 1 }-> minIter(1 + n + x, 1 + n + x, 0) :|: n >= 0, x >= 0, z = 1 + n + x min(z) -{ 1 }-> 0 :|: z = 0 minIter(z, z', z'') -{ 1 }-> minIter(1 + n + y, 1 + n + y, 1 + z'') :|: n >= 0, z' = 1 + n + y, y >= 0, z = 0, z'' >= 0 minIter(z, z', z'') -{ 3 + z'' }-> if_min(s', x, z', 1 + (z'' - 1)) :|: s' >= 0, s' <= 1, x >= 0, z' >= 0, z'' - 1 >= 0, z = 1 + (1 + x'') + x, x'' >= 0 minIter(z, z', z'') -{ 2 }-> if_min(1, z - 1, z', z'') :|: z - 1 >= 0, z' >= 0, z'' >= 0 minIter(z, z', z'') -{ 2 }-> if_min(0, x, z', 0) :|: z'' = 0, x' >= 0, x >= 0, z' >= 0, z = 1 + (1 + x') + x minIter(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 minsort(z, z') -{ 2 }-> if_minsort(eq(n, minIter(1 + n + x, 1 + n + x, 0)), 1 + n + x, z') :|: n >= 0, x >= 0, z' >= 0, z = 1 + n + x minsort(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0 null(z) -{ 1 }-> 1 :|: z = 0 null(z) -{ 1 }-> 0 :|: n >= 0, x >= 0, z = 1 + n + x rm(z, z') -{ 5 + y'' }-> if_rm(s1, 1 + (z - 1), 1 + (1 + y'') + x) :|: s1 >= 0, s1 <= 1, x >= 0, y'' >= 0, z' = 1 + (1 + y'') + x, 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' = 1 + (1 + x1) + x, z = 0 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, x >= 0, z = 1 + n + x tail(z) -{ 1 }-> 0 :|: z = 0 Function symbols to be analyzed: {app}, {minIter,if_min}, {if_rm,rm}, {min}, {minsort,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, x >= 0, z' >= 0, z = 1 + n + x eq(z, z') -{ 3 + z' }-> s'' :|: s'' >= 0, s'' <= 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 - 1 >= 0, z' = 0 head(z) -{ 1 }-> n :|: n >= 0, x >= 0, z = 1 + n + x if_min(z, z', z'', z1) -{ 1 }-> z1 :|: z = 1, z' >= 0, z'' >= 0, z1 >= 0 if_min(z, z', z'', z1) -{ 1 }-> minIter(z', z'', z1) :|: z' >= 0, z'' >= 0, z = 0, z1 >= 0 if_minsort(z, z', z'') -{ 1 }-> minsort(x, 1 + n + z'') :|: n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0, z = 0 if_minsort(z, z', z'') -{ 5 + m' }-> 1 + n + minsort(app(if_rm(s2, n, 1 + m' + x4), z''), 0) :|: s2 >= 0, s2 <= 1, n >= 0, x4 >= 0, z' = 1 + n + (1 + m' + x4), z = 1, z'' >= 0, m' >= 0 if_minsort(z, z', z'') -{ 2 }-> 1 + (z' - 1) + minsort(app(0, z''), 0) :|: z' - 1 >= 0, z = 1, z'' >= 0 if_rm(z, z', z'') -{ 1 }-> rm(z', x) :|: z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0 if_rm(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_rm(z, z', z'') -{ 1 }-> 1 + m + rm(z', x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, 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 - 1 >= 0, z' = 0 min(z) -{ 1 }-> minIter(1 + n + x, 1 + n + x, 0) :|: n >= 0, x >= 0, z = 1 + n + x min(z) -{ 1 }-> 0 :|: z = 0 minIter(z, z', z'') -{ 1 }-> minIter(1 + n + y, 1 + n + y, 1 + z'') :|: n >= 0, z' = 1 + n + y, y >= 0, z = 0, z'' >= 0 minIter(z, z', z'') -{ 3 + z'' }-> if_min(s', x, z', 1 + (z'' - 1)) :|: s' >= 0, s' <= 1, x >= 0, z' >= 0, z'' - 1 >= 0, z = 1 + (1 + x'') + x, x'' >= 0 minIter(z, z', z'') -{ 2 }-> if_min(1, z - 1, z', z'') :|: z - 1 >= 0, z' >= 0, z'' >= 0 minIter(z, z', z'') -{ 2 }-> if_min(0, x, z', 0) :|: z'' = 0, x' >= 0, x >= 0, z' >= 0, z = 1 + (1 + x') + x minIter(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 minsort(z, z') -{ 2 }-> if_minsort(eq(n, minIter(1 + n + x, 1 + n + x, 0)), 1 + n + x, z') :|: n >= 0, x >= 0, z' >= 0, z = 1 + n + x minsort(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0 null(z) -{ 1 }-> 1 :|: z = 0 null(z) -{ 1 }-> 0 :|: n >= 0, x >= 0, z = 1 + n + x rm(z, z') -{ 5 + y'' }-> if_rm(s1, 1 + (z - 1), 1 + (1 + y'') + x) :|: s1 >= 0, s1 <= 1, x >= 0, y'' >= 0, z' = 1 + (1 + y'') + x, 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' = 1 + (1 + x1) + x, z = 0 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, x >= 0, z = 1 + n + x tail(z) -{ 1 }-> 0 :|: z = 0 Function symbols to be analyzed: {app}, {minIter,if_min}, {if_rm,rm}, {min}, {minsort,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, x >= 0, z' >= 0, z = 1 + n + x eq(z, z') -{ 3 + z' }-> s'' :|: s'' >= 0, s'' <= 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 - 1 >= 0, z' = 0 head(z) -{ 1 }-> n :|: n >= 0, x >= 0, z = 1 + n + x if_min(z, z', z'', z1) -{ 1 }-> z1 :|: z = 1, z' >= 0, z'' >= 0, z1 >= 0 if_min(z, z', z'', z1) -{ 1 }-> minIter(z', z'', z1) :|: z' >= 0, z'' >= 0, z = 0, z1 >= 0 if_minsort(z, z', z'') -{ 1 }-> minsort(x, 1 + n + z'') :|: n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0, z = 0 if_minsort(z, z', z'') -{ 5 + m' }-> 1 + n + minsort(app(if_rm(s2, n, 1 + m' + x4), z''), 0) :|: s2 >= 0, s2 <= 1, n >= 0, x4 >= 0, z' = 1 + n + (1 + m' + x4), z = 1, z'' >= 0, m' >= 0 if_minsort(z, z', z'') -{ 2 }-> 1 + (z' - 1) + minsort(app(0, z''), 0) :|: z' - 1 >= 0, z = 1, z'' >= 0 if_rm(z, z', z'') -{ 1 }-> rm(z', x) :|: z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0 if_rm(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_rm(z, z', z'') -{ 1 }-> 1 + m + rm(z', x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, 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 - 1 >= 0, z' = 0 min(z) -{ 1 }-> minIter(1 + n + x, 1 + n + x, 0) :|: n >= 0, x >= 0, z = 1 + n + x min(z) -{ 1 }-> 0 :|: z = 0 minIter(z, z', z'') -{ 1 }-> minIter(1 + n + y, 1 + n + y, 1 + z'') :|: n >= 0, z' = 1 + n + y, y >= 0, z = 0, z'' >= 0 minIter(z, z', z'') -{ 3 + z'' }-> if_min(s', x, z', 1 + (z'' - 1)) :|: s' >= 0, s' <= 1, x >= 0, z' >= 0, z'' - 1 >= 0, z = 1 + (1 + x'') + x, x'' >= 0 minIter(z, z', z'') -{ 2 }-> if_min(1, z - 1, z', z'') :|: z - 1 >= 0, z' >= 0, z'' >= 0 minIter(z, z', z'') -{ 2 }-> if_min(0, x, z', 0) :|: z'' = 0, x' >= 0, x >= 0, z' >= 0, z = 1 + (1 + x') + x minIter(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 minsort(z, z') -{ 2 }-> if_minsort(eq(n, minIter(1 + n + x, 1 + n + x, 0)), 1 + n + x, z') :|: n >= 0, x >= 0, z' >= 0, z = 1 + n + x minsort(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0 null(z) -{ 1 }-> 1 :|: z = 0 null(z) -{ 1 }-> 0 :|: n >= 0, x >= 0, z = 1 + n + x rm(z, z') -{ 5 + y'' }-> if_rm(s1, 1 + (z - 1), 1 + (1 + y'') + x) :|: s1 >= 0, s1 <= 1, x >= 0, y'' >= 0, z' = 1 + (1 + y'') + x, 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' = 1 + (1 + x1) + x, z = 0 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, x >= 0, z = 1 + n + x tail(z) -{ 1 }-> 0 :|: z = 0 Function symbols to be analyzed: {app}, {minIter,if_min}, {if_rm,rm}, {min}, {minsort,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, x >= 0, z' >= 0, z = 1 + n + x eq(z, z') -{ 3 + z' }-> s'' :|: s'' >= 0, s'' <= 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 - 1 >= 0, z' = 0 head(z) -{ 1 }-> n :|: n >= 0, x >= 0, z = 1 + n + x if_min(z, z', z'', z1) -{ 1 }-> z1 :|: z = 1, z' >= 0, z'' >= 0, z1 >= 0 if_min(z, z', z'', z1) -{ 1 }-> minIter(z', z'', z1) :|: z' >= 0, z'' >= 0, z = 0, z1 >= 0 if_minsort(z, z', z'') -{ 1 }-> minsort(x, 1 + n + z'') :|: n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0, z = 0 if_minsort(z, z', z'') -{ 5 + m' }-> 1 + n + minsort(app(if_rm(s2, n, 1 + m' + x4), z''), 0) :|: s2 >= 0, s2 <= 1, n >= 0, x4 >= 0, z' = 1 + n + (1 + m' + x4), z = 1, z'' >= 0, m' >= 0 if_minsort(z, z', z'') -{ 2 }-> 1 + (z' - 1) + minsort(app(0, z''), 0) :|: z' - 1 >= 0, z = 1, z'' >= 0 if_rm(z, z', z'') -{ 1 }-> rm(z', x) :|: z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0 if_rm(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_rm(z, z', z'') -{ 1 }-> 1 + m + rm(z', x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, 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 - 1 >= 0, z' = 0 min(z) -{ 1 }-> minIter(1 + n + x, 1 + n + x, 0) :|: n >= 0, x >= 0, z = 1 + n + x min(z) -{ 1 }-> 0 :|: z = 0 minIter(z, z', z'') -{ 1 }-> minIter(1 + n + y, 1 + n + y, 1 + z'') :|: n >= 0, z' = 1 + n + y, y >= 0, z = 0, z'' >= 0 minIter(z, z', z'') -{ 3 + z'' }-> if_min(s', x, z', 1 + (z'' - 1)) :|: s' >= 0, s' <= 1, x >= 0, z' >= 0, z'' - 1 >= 0, z = 1 + (1 + x'') + x, x'' >= 0 minIter(z, z', z'') -{ 2 }-> if_min(1, z - 1, z', z'') :|: z - 1 >= 0, z' >= 0, z'' >= 0 minIter(z, z', z'') -{ 2 }-> if_min(0, x, z', 0) :|: z'' = 0, x' >= 0, x >= 0, z' >= 0, z = 1 + (1 + x') + x minIter(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 minsort(z, z') -{ 2 }-> if_minsort(eq(n, minIter(1 + n + x, 1 + n + x, 0)), 1 + n + x, z') :|: n >= 0, x >= 0, z' >= 0, z = 1 + n + x minsort(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0 null(z) -{ 1 }-> 1 :|: z = 0 null(z) -{ 1 }-> 0 :|: n >= 0, x >= 0, z = 1 + n + x rm(z, z') -{ 5 + y'' }-> if_rm(s1, 1 + (z - 1), 1 + (1 + y'') + x) :|: s1 >= 0, s1 <= 1, x >= 0, y'' >= 0, z' = 1 + (1 + y'') + x, 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' = 1 + (1 + x1) + x, z = 0 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, x >= 0, z = 1 + n + x tail(z) -{ 1 }-> 0 :|: z = 0 Function symbols to be analyzed: {minIter,if_min}, {if_rm,rm}, {min}, {minsort,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 + s3 :|: s3 >= 0, s3 <= x + z', n >= 0, x >= 0, z' >= 0, z = 1 + n + x eq(z, z') -{ 3 + z' }-> s'' :|: s'' >= 0, s'' <= 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 - 1 >= 0, z' = 0 head(z) -{ 1 }-> n :|: n >= 0, x >= 0, z = 1 + n + x if_min(z, z', z'', z1) -{ 1 }-> z1 :|: z = 1, z' >= 0, z'' >= 0, z1 >= 0 if_min(z, z', z'', z1) -{ 1 }-> minIter(z', z'', z1) :|: z' >= 0, z'' >= 0, z = 0, z1 >= 0 if_minsort(z, z', z'') -{ 1 }-> minsort(x, 1 + n + z'') :|: n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0, z = 0 if_minsort(z, z', z'') -{ 5 + m' }-> 1 + n + minsort(app(if_rm(s2, n, 1 + m' + x4), z''), 0) :|: s2 >= 0, s2 <= 1, n >= 0, x4 >= 0, z' = 1 + n + (1 + m' + x4), z = 1, z'' >= 0, m' >= 0 if_minsort(z, z', z'') -{ 3 }-> 1 + (z' - 1) + minsort(s4, 0) :|: s4 >= 0, s4 <= 0 + z'', z' - 1 >= 0, z = 1, z'' >= 0 if_rm(z, z', z'') -{ 1 }-> rm(z', x) :|: z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0 if_rm(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_rm(z, z', z'') -{ 1 }-> 1 + m + rm(z', x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, 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 - 1 >= 0, z' = 0 min(z) -{ 1 }-> minIter(1 + n + x, 1 + n + x, 0) :|: n >= 0, x >= 0, z = 1 + n + x min(z) -{ 1 }-> 0 :|: z = 0 minIter(z, z', z'') -{ 1 }-> minIter(1 + n + y, 1 + n + y, 1 + z'') :|: n >= 0, z' = 1 + n + y, y >= 0, z = 0, z'' >= 0 minIter(z, z', z'') -{ 3 + z'' }-> if_min(s', x, z', 1 + (z'' - 1)) :|: s' >= 0, s' <= 1, x >= 0, z' >= 0, z'' - 1 >= 0, z = 1 + (1 + x'') + x, x'' >= 0 minIter(z, z', z'') -{ 2 }-> if_min(1, z - 1, z', z'') :|: z - 1 >= 0, z' >= 0, z'' >= 0 minIter(z, z', z'') -{ 2 }-> if_min(0, x, z', 0) :|: z'' = 0, x' >= 0, x >= 0, z' >= 0, z = 1 + (1 + x') + x minIter(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 minsort(z, z') -{ 2 }-> if_minsort(eq(n, minIter(1 + n + x, 1 + n + x, 0)), 1 + n + x, z') :|: n >= 0, x >= 0, z' >= 0, z = 1 + n + x minsort(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0 null(z) -{ 1 }-> 1 :|: z = 0 null(z) -{ 1 }-> 0 :|: n >= 0, x >= 0, z = 1 + n + x rm(z, z') -{ 5 + y'' }-> if_rm(s1, 1 + (z - 1), 1 + (1 + y'') + x) :|: s1 >= 0, s1 <= 1, x >= 0, y'' >= 0, z' = 1 + (1 + y'') + x, 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' = 1 + (1 + x1) + x, z = 0 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, x >= 0, z = 1 + n + x tail(z) -{ 1 }-> 0 :|: z = 0 Function symbols to be analyzed: {minIter,if_min}, {if_rm,rm}, {min}, {minsort,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 CoFloCo for: minIter after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? Computed SIZE bound using CoFloCo for: if_min after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (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 + s3 :|: s3 >= 0, s3 <= x + z', n >= 0, x >= 0, z' >= 0, z = 1 + n + x eq(z, z') -{ 3 + z' }-> s'' :|: s'' >= 0, s'' <= 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 - 1 >= 0, z' = 0 head(z) -{ 1 }-> n :|: n >= 0, x >= 0, z = 1 + n + x if_min(z, z', z'', z1) -{ 1 }-> z1 :|: z = 1, z' >= 0, z'' >= 0, z1 >= 0 if_min(z, z', z'', z1) -{ 1 }-> minIter(z', z'', z1) :|: z' >= 0, z'' >= 0, z = 0, z1 >= 0 if_minsort(z, z', z'') -{ 1 }-> minsort(x, 1 + n + z'') :|: n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0, z = 0 if_minsort(z, z', z'') -{ 5 + m' }-> 1 + n + minsort(app(if_rm(s2, n, 1 + m' + x4), z''), 0) :|: s2 >= 0, s2 <= 1, n >= 0, x4 >= 0, z' = 1 + n + (1 + m' + x4), z = 1, z'' >= 0, m' >= 0 if_minsort(z, z', z'') -{ 3 }-> 1 + (z' - 1) + minsort(s4, 0) :|: s4 >= 0, s4 <= 0 + z'', z' - 1 >= 0, z = 1, z'' >= 0 if_rm(z, z', z'') -{ 1 }-> rm(z', x) :|: z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0 if_rm(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_rm(z, z', z'') -{ 1 }-> 1 + m + rm(z', x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, 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 - 1 >= 0, z' = 0 min(z) -{ 1 }-> minIter(1 + n + x, 1 + n + x, 0) :|: n >= 0, x >= 0, z = 1 + n + x min(z) -{ 1 }-> 0 :|: z = 0 minIter(z, z', z'') -{ 1 }-> minIter(1 + n + y, 1 + n + y, 1 + z'') :|: n >= 0, z' = 1 + n + y, y >= 0, z = 0, z'' >= 0 minIter(z, z', z'') -{ 3 + z'' }-> if_min(s', x, z', 1 + (z'' - 1)) :|: s' >= 0, s' <= 1, x >= 0, z' >= 0, z'' - 1 >= 0, z = 1 + (1 + x'') + x, x'' >= 0 minIter(z, z', z'') -{ 2 }-> if_min(1, z - 1, z', z'') :|: z - 1 >= 0, z' >= 0, z'' >= 0 minIter(z, z', z'') -{ 2 }-> if_min(0, x, z', 0) :|: z'' = 0, x' >= 0, x >= 0, z' >= 0, z = 1 + (1 + x') + x minIter(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 minsort(z, z') -{ 2 }-> if_minsort(eq(n, minIter(1 + n + x, 1 + n + x, 0)), 1 + n + x, z') :|: n >= 0, x >= 0, z' >= 0, z = 1 + n + x minsort(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0 null(z) -{ 1 }-> 1 :|: z = 0 null(z) -{ 1 }-> 0 :|: n >= 0, x >= 0, z = 1 + n + x rm(z, z') -{ 5 + y'' }-> if_rm(s1, 1 + (z - 1), 1 + (1 + y'') + x) :|: s1 >= 0, s1 <= 1, x >= 0, y'' >= 0, z' = 1 + (1 + y'') + x, 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' = 1 + (1 + x1) + x, z = 0 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, x >= 0, z = 1 + n + x tail(z) -{ 1 }-> 0 :|: z = 0 Function symbols to be analyzed: {minIter,if_min}, {if_rm,rm}, {min}, {minsort,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'] minIter: runtime: ?, size: INF if_min: runtime: ?, size: INF ---------------------------------------- (59) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: minIter after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (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 + s3 :|: s3 >= 0, s3 <= x + z', n >= 0, x >= 0, z' >= 0, z = 1 + n + x eq(z, z') -{ 3 + z' }-> s'' :|: s'' >= 0, s'' <= 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 - 1 >= 0, z' = 0 head(z) -{ 1 }-> n :|: n >= 0, x >= 0, z = 1 + n + x if_min(z, z', z'', z1) -{ 1 }-> z1 :|: z = 1, z' >= 0, z'' >= 0, z1 >= 0 if_min(z, z', z'', z1) -{ 1 }-> minIter(z', z'', z1) :|: z' >= 0, z'' >= 0, z = 0, z1 >= 0 if_minsort(z, z', z'') -{ 1 }-> minsort(x, 1 + n + z'') :|: n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0, z = 0 if_minsort(z, z', z'') -{ 5 + m' }-> 1 + n + minsort(app(if_rm(s2, n, 1 + m' + x4), z''), 0) :|: s2 >= 0, s2 <= 1, n >= 0, x4 >= 0, z' = 1 + n + (1 + m' + x4), z = 1, z'' >= 0, m' >= 0 if_minsort(z, z', z'') -{ 3 }-> 1 + (z' - 1) + minsort(s4, 0) :|: s4 >= 0, s4 <= 0 + z'', z' - 1 >= 0, z = 1, z'' >= 0 if_rm(z, z', z'') -{ 1 }-> rm(z', x) :|: z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0 if_rm(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_rm(z, z', z'') -{ 1 }-> 1 + m + rm(z', x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, 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 - 1 >= 0, z' = 0 min(z) -{ 1 }-> minIter(1 + n + x, 1 + n + x, 0) :|: n >= 0, x >= 0, z = 1 + n + x min(z) -{ 1 }-> 0 :|: z = 0 minIter(z, z', z'') -{ 1 }-> minIter(1 + n + y, 1 + n + y, 1 + z'') :|: n >= 0, z' = 1 + n + y, y >= 0, z = 0, z'' >= 0 minIter(z, z', z'') -{ 3 + z'' }-> if_min(s', x, z', 1 + (z'' - 1)) :|: s' >= 0, s' <= 1, x >= 0, z' >= 0, z'' - 1 >= 0, z = 1 + (1 + x'') + x, x'' >= 0 minIter(z, z', z'') -{ 2 }-> if_min(1, z - 1, z', z'') :|: z - 1 >= 0, z' >= 0, z'' >= 0 minIter(z, z', z'') -{ 2 }-> if_min(0, x, z', 0) :|: z'' = 0, x' >= 0, x >= 0, z' >= 0, z = 1 + (1 + x') + x minIter(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 minsort(z, z') -{ 2 }-> if_minsort(eq(n, minIter(1 + n + x, 1 + n + x, 0)), 1 + n + x, z') :|: n >= 0, x >= 0, z' >= 0, z = 1 + n + x minsort(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0 null(z) -{ 1 }-> 1 :|: z = 0 null(z) -{ 1 }-> 0 :|: n >= 0, x >= 0, z = 1 + n + x rm(z, z') -{ 5 + y'' }-> if_rm(s1, 1 + (z - 1), 1 + (1 + y'') + x) :|: s1 >= 0, s1 <= 1, x >= 0, y'' >= 0, z' = 1 + (1 + y'') + x, 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' = 1 + (1 + x1) + x, z = 0 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, x >= 0, z = 1 + n + x tail(z) -{ 1 }-> 0 :|: z = 0 Function symbols to be analyzed: {minIter,if_min}, {if_rm,rm}, {min}, {minsort,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'] minIter: runtime: INF, size: INF if_min: runtime: ?, size: INF ---------------------------------------- (61) 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: minIter(v0, v1, v2) -> null_minIter [0] head(v0) -> null_head [0] if_rm(v0, v1, v2) -> null_if_rm [0] minsort(v0, v1) -> null_minsort [0] if_minsort(v0, v1, v2) -> null_if_minsort [0] eq(v0, v1) -> null_eq [0] le(v0, v1) -> null_le [0] app(v0, v1) -> null_app [0] min(v0) -> null_min [0] tail(v0) -> null_tail [0] null(v0) -> null_null [0] rm(v0, v1) -> null_rm [0] if_min(v0, v1, v2, v3) -> null_if_min [0] And the following fresh constants: null_minIter, null_head, null_if_rm, null_minsort, null_if_minsort, null_eq, null_le, null_app, null_min, null_tail, null_null, null_rm, null_if_min ---------------------------------------- (62) 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(nil) -> 0 [1] min(add(n, x)) -> minIter(add(n, x), add(n, x), 0) [1] minIter(nil, add(n, y), m) -> minIter(add(n, y), add(n, y), s(m)) [1] minIter(add(n, x), y, m) -> if_min(le(n, m), x, y, m) [1] if_min(true, x, y, m) -> m [1] if_min(false, x, y, m) -> minIter(x, y, m) [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(nil, nil) -> nil [1] minsort(add(n, x), y) -> if_minsort(eq(n, min(add(n, x))), add(n, x), y) [1] if_minsort(true, add(n, x), y) -> add(n, minsort(app(rm(n, x), y), nil)) [1] if_minsort(false, add(n, x), y) -> minsort(x, add(n, y)) [1] minIter(v0, v1, v2) -> null_minIter [0] head(v0) -> null_head [0] if_rm(v0, v1, v2) -> null_if_rm [0] minsort(v0, v1) -> null_minsort [0] if_minsort(v0, v1, v2) -> null_if_minsort [0] eq(v0, v1) -> null_eq [0] le(v0, v1) -> null_le [0] app(v0, v1) -> null_app [0] min(v0) -> null_min [0] tail(v0) -> null_tail [0] null(v0) -> null_null [0] rm(v0, v1) -> null_rm [0] if_min(v0, v1, v2, v3) -> null_if_min [0] The TRS has the following type information: eq :: 0:s:null_minIter:null_head:null_min:null_if_min -> 0:s:null_minIter:null_head:null_min:null_if_min -> true:false:null_eq:null_le:null_null 0 :: 0:s:null_minIter:null_head:null_min:null_if_min true :: true:false:null_eq:null_le:null_null s :: 0:s:null_minIter:null_head:null_min:null_if_min -> 0:s:null_minIter:null_head:null_min:null_if_min false :: true:false:null_eq:null_le:null_null le :: 0:s:null_minIter:null_head:null_min:null_if_min -> 0:s:null_minIter:null_head:null_min:null_if_min -> true:false:null_eq:null_le:null_null app :: nil:add:null_if_rm:null_minsort:null_if_minsort:null_app:null_tail:null_rm -> nil:add:null_if_rm:null_minsort:null_if_minsort:null_app:null_tail:null_rm -> nil:add:null_if_rm:null_minsort:null_if_minsort:null_app:null_tail:null_rm nil :: nil:add:null_if_rm:null_minsort:null_if_minsort:null_app:null_tail:null_rm add :: 0:s:null_minIter:null_head:null_min:null_if_min -> nil:add:null_if_rm:null_minsort:null_if_minsort:null_app:null_tail:null_rm -> nil:add:null_if_rm:null_minsort:null_if_minsort:null_app:null_tail:null_rm min :: nil:add:null_if_rm:null_minsort:null_if_minsort:null_app:null_tail:null_rm -> 0:s:null_minIter:null_head:null_min:null_if_min minIter :: nil:add:null_if_rm:null_minsort:null_if_minsort:null_app:null_tail:null_rm -> nil:add:null_if_rm:null_minsort:null_if_minsort:null_app:null_tail:null_rm -> 0:s:null_minIter:null_head:null_min:null_if_min -> 0:s:null_minIter:null_head:null_min:null_if_min if_min :: true:false:null_eq:null_le:null_null -> nil:add:null_if_rm:null_minsort:null_if_minsort:null_app:null_tail:null_rm -> nil:add:null_if_rm:null_minsort:null_if_minsort:null_app:null_tail:null_rm -> 0:s:null_minIter:null_head:null_min:null_if_min -> 0:s:null_minIter:null_head:null_min:null_if_min head :: nil:add:null_if_rm:null_minsort:null_if_minsort:null_app:null_tail:null_rm -> 0:s:null_minIter:null_head:null_min:null_if_min tail :: nil:add:null_if_rm:null_minsort:null_if_minsort:null_app:null_tail:null_rm -> nil:add:null_if_rm:null_minsort:null_if_minsort:null_app:null_tail:null_rm null :: nil:add:null_if_rm:null_minsort:null_if_minsort:null_app:null_tail:null_rm -> true:false:null_eq:null_le:null_null rm :: 0:s:null_minIter:null_head:null_min:null_if_min -> nil:add:null_if_rm:null_minsort:null_if_minsort:null_app:null_tail:null_rm -> nil:add:null_if_rm:null_minsort:null_if_minsort:null_app:null_tail:null_rm if_rm :: true:false:null_eq:null_le:null_null -> 0:s:null_minIter:null_head:null_min:null_if_min -> nil:add:null_if_rm:null_minsort:null_if_minsort:null_app:null_tail:null_rm -> nil:add:null_if_rm:null_minsort:null_if_minsort:null_app:null_tail:null_rm minsort :: nil:add:null_if_rm:null_minsort:null_if_minsort:null_app:null_tail:null_rm -> nil:add:null_if_rm:null_minsort:null_if_minsort:null_app:null_tail:null_rm -> nil:add:null_if_rm:null_minsort:null_if_minsort:null_app:null_tail:null_rm if_minsort :: true:false:null_eq:null_le:null_null -> nil:add:null_if_rm:null_minsort:null_if_minsort:null_app:null_tail:null_rm -> nil:add:null_if_rm:null_minsort:null_if_minsort:null_app:null_tail:null_rm -> nil:add:null_if_rm:null_minsort:null_if_minsort:null_app:null_tail:null_rm null_minIter :: 0:s:null_minIter:null_head:null_min:null_if_min null_head :: 0:s:null_minIter:null_head:null_min:null_if_min null_if_rm :: nil:add:null_if_rm:null_minsort:null_if_minsort:null_app:null_tail:null_rm null_minsort :: nil:add:null_if_rm:null_minsort:null_if_minsort:null_app:null_tail:null_rm null_if_minsort :: nil:add:null_if_rm:null_minsort:null_if_minsort:null_app:null_tail:null_rm 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_minsort:null_if_minsort:null_app:null_tail:null_rm null_min :: 0:s:null_minIter:null_head:null_min:null_if_min null_tail :: nil:add:null_if_rm:null_minsort:null_if_minsort:null_app:null_tail:null_rm null_null :: true:false:null_eq:null_le:null_null null_rm :: nil:add:null_if_rm:null_minsort:null_if_minsort:null_app:null_tail:null_rm null_if_min :: 0:s:null_minIter:null_head:null_min:null_if_min Rewrite Strategy: INNERMOST ---------------------------------------- (63) 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_minIter => 0 null_head => 0 null_if_rm => 0 null_minsort => 0 null_if_minsort => 0 null_eq => 0 null_le => 0 null_app => 0 null_min => 0 null_tail => 0 null_null => 0 null_rm => 0 null_if_min => 0 ---------------------------------------- (64) Obligation: Complexity RNTS consisting of the following rules: app(z, z') -{ 1 }-> y :|: y >= 0, z = 0, z' = y app(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 app(z, z') -{ 1 }-> 1 + n + app(x, y) :|: n >= 0, x >= 0, y >= 0, z = 1 + n + x, z' = y eq(z, z') -{ 1 }-> eq(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x eq(z, z') -{ 1 }-> 2 :|: z = 0, z' = 0 eq(z, z') -{ 1 }-> 1 :|: z' = 1 + x, x >= 0, z = 0 eq(z, z') -{ 1 }-> 1 :|: x >= 0, z = 1 + x, z' = 0 eq(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 head(z) -{ 1 }-> n :|: n >= 0, x >= 0, z = 1 + n + x head(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 if_min(z, z', z'', z1) -{ 1 }-> m :|: z = 2, z' = x, z'' = y, z1 = m, x >= 0, y >= 0, m >= 0 if_min(z, z', z'', z1) -{ 1 }-> minIter(x, y, m) :|: z' = x, z'' = y, z = 1, z1 = m, x >= 0, y >= 0, m >= 0 if_min(z, z', z'', z1) -{ 0 }-> 0 :|: z1 = v3, v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0, v3 >= 0 if_minsort(z, z', z'') -{ 1 }-> minsort(x, 1 + n + y) :|: n >= 0, z'' = y, z = 1, z' = 1 + n + x, x >= 0, y >= 0 if_minsort(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 if_minsort(z, z', z'') -{ 1 }-> 1 + n + minsort(app(rm(n, x), y), 0) :|: z = 2, n >= 0, z'' = y, z' = 1 + n + x, x >= 0, y >= 0 if_rm(z, z', z'') -{ 1 }-> rm(n, x) :|: z = 2, n >= 0, z'' = 1 + m + x, x >= 0, z' = n, m >= 0 if_rm(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 if_rm(z, z', z'') -{ 1 }-> 1 + m + rm(n, x) :|: n >= 0, z = 1, z'' = 1 + m + x, x >= 0, z' = n, m >= 0 le(z, z') -{ 1 }-> le(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x le(z, z') -{ 1 }-> 2 :|: y >= 0, z = 0, z' = y le(z, z') -{ 1 }-> 1 :|: x >= 0, z = 1 + x, z' = 0 le(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 min(z) -{ 1 }-> minIter(1 + n + x, 1 + n + x, 0) :|: n >= 0, x >= 0, z = 1 + n + x min(z) -{ 1 }-> 0 :|: z = 0 min(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 minIter(z, z', z'') -{ 1 }-> minIter(1 + n + y, 1 + n + y, 1 + m) :|: n >= 0, z' = 1 + n + y, y >= 0, z = 0, z'' = m, m >= 0 minIter(z, z', z'') -{ 1 }-> if_min(le(n, m), x, y, m) :|: n >= 0, x >= 0, y >= 0, z = 1 + n + x, z' = y, z'' = m, m >= 0 minIter(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 minsort(z, z') -{ 1 }-> if_minsort(eq(n, min(1 + n + x)), 1 + n + x, y) :|: n >= 0, x >= 0, y >= 0, z = 1 + n + x, z' = y minsort(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0 minsort(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 null(z) -{ 1 }-> 2 :|: z = 0 null(z) -{ 1 }-> 1 :|: n >= 0, x >= 0, z = 1 + n + x 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, z = n, x >= 0, m >= 0 rm(z, z') -{ 1 }-> 0 :|: n >= 0, z = n, z' = 0 rm(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 tail(z) -{ 1 }-> x :|: n >= 0, x >= 0, z = 1 + n + x tail(z) -{ 1 }-> 0 :|: z = 0 tail(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (65) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (66) 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(nil) -> 0 min(add(z0, z1)) -> minIter(add(z0, z1), add(z0, z1), 0) minIter(nil, add(z0, z1), z2) -> minIter(add(z0, z1), add(z0, z1), s(z2)) minIter(add(z0, z1), z2, z3) -> if_min(le(z0, z3), z1, z2, z3) if_min(true, z0, z1, z2) -> z2 if_min(false, z0, z1, z2) -> minIter(z0, 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(nil, nil) -> nil minsort(add(z0, z1), z2) -> if_minsort(eq(z0, min(add(z0, z1))), add(z0, z1), z2) if_minsort(true, add(z0, z1), z2) -> add(z0, minsort(app(rm(z0, z1), z2), nil)) if_minsort(false, add(z0, z1), z2) -> minsort(z1, add(z0, 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(nil) -> c9 MIN(add(z0, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) MINITER(add(z0, z1), z2, z3) -> c12(IF_MIN(le(z0, z3), z1, z2, z3), LE(z0, z3)) IF_MIN(true, z0, z1, z2) -> c13 IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) HEAD(add(z0, z1)) -> c15 TAIL(add(z0, z1)) -> c16 TAIL(nil) -> c17 NULL(nil) -> c18 NULL(add(z0, z1)) -> c19 RM(z0, nil) -> c20 RM(z0, add(z1, z2)) -> c21(IF_RM(eq(z0, z1), z0, add(z1, z2)), EQ(z0, z1)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) MINSORT(nil, nil) -> c24 MINSORT(add(z0, z1), z2) -> c25(IF_MINSORT(eq(z0, min(add(z0, z1))), add(z0, z1), z2), EQ(z0, min(add(z0, z1))), MIN(add(z0, z1))) IF_MINSORT(true, add(z0, z1), z2) -> c26(MINSORT(app(rm(z0, z1), z2), nil), APP(rm(z0, z1), z2), RM(z0, z1)) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) 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(nil) -> c9 MIN(add(z0, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) MINITER(add(z0, z1), z2, z3) -> c12(IF_MIN(le(z0, z3), z1, z2, z3), LE(z0, z3)) IF_MIN(true, z0, z1, z2) -> c13 IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) HEAD(add(z0, z1)) -> c15 TAIL(add(z0, z1)) -> c16 TAIL(nil) -> c17 NULL(nil) -> c18 NULL(add(z0, z1)) -> c19 RM(z0, nil) -> c20 RM(z0, add(z1, z2)) -> c21(IF_RM(eq(z0, z1), z0, add(z1, z2)), EQ(z0, z1)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) MINSORT(nil, nil) -> c24 MINSORT(add(z0, z1), z2) -> c25(IF_MINSORT(eq(z0, min(add(z0, z1))), add(z0, z1), z2), EQ(z0, min(add(z0, z1))), MIN(add(z0, z1))) IF_MINSORT(true, add(z0, z1), z2) -> c26(MINSORT(app(rm(z0, z1), z2), nil), APP(rm(z0, z1), z2), RM(z0, z1)) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) K tuples:none Defined Rule Symbols: eq_2, le_2, app_2, min_1, minIter_3, if_min_4, head_1, tail_1, null_1, rm_2, if_rm_3, minsort_2, if_minsort_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, MIN_1, MINITER_3, IF_MIN_4, HEAD_1, TAIL_1, NULL_1, RM_2, IF_RM_3, MINSORT_2, IF_MINSORT_3 Compound Symbols: c, c1, c2, c3_1, c4, c5, c6_1, c7, c8_1, c9, c10_1, c11_1, c12_2, c13, c14_1, c15, c16, c17, c18, c19, c20, c21_2, c22_1, c23_1, c24, c25_3, c26_3, c27_1 ---------------------------------------- (67) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 15 trailing nodes: HEAD(add(z0, z1)) -> c15 APP(nil, z0) -> c7 LE(0, z0) -> c4 TAIL(add(z0, z1)) -> c16 EQ(0, 0) -> c TAIL(nil) -> c17 LE(s(z0), 0) -> c5 MINSORT(nil, nil) -> c24 EQ(s(z0), 0) -> c2 EQ(0, s(z0)) -> c1 NULL(nil) -> c18 IF_MIN(true, z0, z1, z2) -> c13 RM(z0, nil) -> c20 MIN(nil) -> c9 NULL(add(z0, z1)) -> c19 ---------------------------------------- (68) 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(nil) -> 0 min(add(z0, z1)) -> minIter(add(z0, z1), add(z0, z1), 0) minIter(nil, add(z0, z1), z2) -> minIter(add(z0, z1), add(z0, z1), s(z2)) minIter(add(z0, z1), z2, z3) -> if_min(le(z0, z3), z1, z2, z3) if_min(true, z0, z1, z2) -> z2 if_min(false, z0, z1, z2) -> minIter(z0, 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(nil, nil) -> nil minsort(add(z0, z1), z2) -> if_minsort(eq(z0, min(add(z0, z1))), add(z0, z1), z2) if_minsort(true, add(z0, z1), z2) -> add(z0, minsort(app(rm(z0, z1), z2), nil)) if_minsort(false, add(z0, z1), z2) -> minsort(z1, add(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, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) MINITER(add(z0, z1), z2, z3) -> c12(IF_MIN(le(z0, z3), z1, z2, z3), LE(z0, z3)) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) RM(z0, add(z1, z2)) -> c21(IF_RM(eq(z0, z1), z0, add(z1, z2)), EQ(z0, z1)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) MINSORT(add(z0, z1), z2) -> c25(IF_MINSORT(eq(z0, min(add(z0, z1))), add(z0, z1), z2), EQ(z0, min(add(z0, z1))), MIN(add(z0, z1))) IF_MINSORT(true, add(z0, z1), z2) -> c26(MINSORT(app(rm(z0, z1), z2), nil), APP(rm(z0, z1), z2), RM(z0, z1)) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, 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, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) MINITER(add(z0, z1), z2, z3) -> c12(IF_MIN(le(z0, z3), z1, z2, z3), LE(z0, z3)) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) RM(z0, add(z1, z2)) -> c21(IF_RM(eq(z0, z1), z0, add(z1, z2)), EQ(z0, z1)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) MINSORT(add(z0, z1), z2) -> c25(IF_MINSORT(eq(z0, min(add(z0, z1))), add(z0, z1), z2), EQ(z0, min(add(z0, z1))), MIN(add(z0, z1))) IF_MINSORT(true, add(z0, z1), z2) -> c26(MINSORT(app(rm(z0, z1), z2), nil), APP(rm(z0, z1), z2), RM(z0, z1)) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) K tuples:none Defined Rule Symbols: eq_2, le_2, app_2, min_1, minIter_3, if_min_4, head_1, tail_1, null_1, rm_2, if_rm_3, minsort_2, if_minsort_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, MIN_1, MINITER_3, IF_MIN_4, RM_2, IF_RM_3, MINSORT_2, IF_MINSORT_3 Compound Symbols: c3_1, c6_1, c8_1, c10_1, c11_1, c12_2, c14_1, c21_2, c22_1, c23_1, c25_3, c26_3, c27_1 ---------------------------------------- (69) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: min(nil) -> 0 head(add(z0, z1)) -> z0 tail(add(z0, z1)) -> z1 tail(nil) -> nil null(nil) -> true null(add(z0, z1)) -> false minsort(nil, nil) -> nil minsort(add(z0, z1), z2) -> if_minsort(eq(z0, min(add(z0, z1))), add(z0, z1), z2) if_minsort(true, add(z0, z1), z2) -> add(z0, minsort(app(rm(z0, z1), z2), nil)) if_minsort(false, add(z0, z1), z2) -> minsort(z1, add(z0, z2)) ---------------------------------------- (70) 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) min(add(z0, z1)) -> minIter(add(z0, z1), add(z0, z1), 0) minIter(add(z0, z1), z2, z3) -> if_min(le(z0, z3), z1, z2, z3) minIter(nil, add(z0, z1), z2) -> minIter(add(z0, z1), add(z0, z1), s(z2)) if_min(true, z0, z1, z2) -> z2 if_min(false, z0, z1, z2) -> minIter(z0, z1, z2) 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)) 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, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) MINITER(add(z0, z1), z2, z3) -> c12(IF_MIN(le(z0, z3), z1, z2, z3), LE(z0, z3)) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) RM(z0, add(z1, z2)) -> c21(IF_RM(eq(z0, z1), z0, add(z1, z2)), EQ(z0, z1)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) MINSORT(add(z0, z1), z2) -> c25(IF_MINSORT(eq(z0, min(add(z0, z1))), add(z0, z1), z2), EQ(z0, min(add(z0, z1))), MIN(add(z0, z1))) IF_MINSORT(true, add(z0, z1), z2) -> c26(MINSORT(app(rm(z0, z1), z2), nil), APP(rm(z0, z1), z2), RM(z0, z1)) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, 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, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) MINITER(add(z0, z1), z2, z3) -> c12(IF_MIN(le(z0, z3), z1, z2, z3), LE(z0, z3)) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) RM(z0, add(z1, z2)) -> c21(IF_RM(eq(z0, z1), z0, add(z1, z2)), EQ(z0, z1)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) MINSORT(add(z0, z1), z2) -> c25(IF_MINSORT(eq(z0, min(add(z0, z1))), add(z0, z1), z2), EQ(z0, min(add(z0, z1))), MIN(add(z0, z1))) IF_MINSORT(true, add(z0, z1), z2) -> c26(MINSORT(app(rm(z0, z1), z2), nil), APP(rm(z0, z1), z2), RM(z0, z1)) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) K tuples:none Defined Rule Symbols: le_2, eq_2, min_1, minIter_3, if_min_4, app_2, rm_2, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, MIN_1, MINITER_3, IF_MIN_4, RM_2, IF_RM_3, MINSORT_2, IF_MINSORT_3 Compound Symbols: c3_1, c6_1, c8_1, c10_1, c11_1, c12_2, c14_1, c21_2, c22_1, c23_1, c25_3, c26_3, c27_1 ---------------------------------------- (71) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. IF_MINSORT(true, add(z0, z1), z2) -> c26(MINSORT(app(rm(z0, z1), z2), nil), APP(rm(z0, z1), z2), RM(z0, z1)) 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 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)) MIN(add(z0, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) MINITER(add(z0, z1), z2, z3) -> c12(IF_MIN(le(z0, z3), z1, z2, z3), LE(z0, z3)) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) RM(z0, add(z1, z2)) -> c21(IF_RM(eq(z0, z1), z0, add(z1, z2)), EQ(z0, z1)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) MINSORT(add(z0, z1), z2) -> c25(IF_MINSORT(eq(z0, min(add(z0, z1))), add(z0, z1), z2), EQ(z0, min(add(z0, z1))), MIN(add(z0, z1))) IF_MINSORT(true, add(z0, z1), z2) -> c26(MINSORT(app(rm(z0, z1), z2), nil), APP(rm(z0, z1), z2), RM(z0, z1)) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) 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_MIN(x_1, x_2, x_3, x_4)) = 0 POL(IF_MINSORT(x_1, x_2, x_3)) = [1] + x_2 + x_3 POL(IF_RM(x_1, x_2, x_3)) = 0 POL(LE(x_1, x_2)) = 0 POL(MIN(x_1)) = 0 POL(MINITER(x_1, x_2, x_3)) = 0 POL(MINSORT(x_1, x_2)) = [1] + 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(c10(x_1)) = x_1 POL(c11(x_1)) = x_1 POL(c12(x_1, x_2)) = x_1 + x_2 POL(c14(x_1)) = x_1 POL(c21(x_1, x_2)) = x_1 + x_2 POL(c22(x_1)) = x_1 POL(c23(x_1)) = x_1 POL(c25(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c26(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c27(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(eq(x_1, x_2)) = x_1 POL(false) = 0 POL(if_min(x_1, x_2, x_3, x_4)) = [1] + x_1 + x_2 + x_3 + x_4 POL(if_rm(x_1, x_2, x_3)) = x_3 POL(le(x_1, x_2)) = [1] + x_1 + x_2 POL(min(x_1)) = [1] POL(minIter(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3 POL(nil) = 0 POL(rm(x_1, x_2)) = x_2 POL(s(x_1)) = [1] + x_1 POL(true) = 0 ---------------------------------------- (72) 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) min(add(z0, z1)) -> minIter(add(z0, z1), add(z0, z1), 0) minIter(add(z0, z1), z2, z3) -> if_min(le(z0, z3), z1, z2, z3) minIter(nil, add(z0, z1), z2) -> minIter(add(z0, z1), add(z0, z1), s(z2)) if_min(true, z0, z1, z2) -> z2 if_min(false, z0, z1, z2) -> minIter(z0, z1, z2) 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)) 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, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) MINITER(add(z0, z1), z2, z3) -> c12(IF_MIN(le(z0, z3), z1, z2, z3), LE(z0, z3)) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) RM(z0, add(z1, z2)) -> c21(IF_RM(eq(z0, z1), z0, add(z1, z2)), EQ(z0, z1)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) MINSORT(add(z0, z1), z2) -> c25(IF_MINSORT(eq(z0, min(add(z0, z1))), add(z0, z1), z2), EQ(z0, min(add(z0, z1))), MIN(add(z0, z1))) IF_MINSORT(true, add(z0, z1), z2) -> c26(MINSORT(app(rm(z0, z1), z2), nil), APP(rm(z0, z1), z2), RM(z0, z1)) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, 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, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) MINITER(add(z0, z1), z2, z3) -> c12(IF_MIN(le(z0, z3), z1, z2, z3), LE(z0, z3)) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) RM(z0, add(z1, z2)) -> c21(IF_RM(eq(z0, z1), z0, add(z1, z2)), EQ(z0, z1)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) MINSORT(add(z0, z1), z2) -> c25(IF_MINSORT(eq(z0, min(add(z0, z1))), add(z0, z1), z2), EQ(z0, min(add(z0, z1))), MIN(add(z0, z1))) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) K tuples: IF_MINSORT(true, add(z0, z1), z2) -> c26(MINSORT(app(rm(z0, z1), z2), nil), APP(rm(z0, z1), z2), RM(z0, z1)) Defined Rule Symbols: le_2, eq_2, min_1, minIter_3, if_min_4, app_2, rm_2, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, MIN_1, MINITER_3, IF_MIN_4, RM_2, IF_RM_3, MINSORT_2, IF_MINSORT_3 Compound Symbols: c3_1, c6_1, c8_1, c10_1, c11_1, c12_2, c14_1, c21_2, c22_1, c23_1, c25_3, c26_3, c27_1 ---------------------------------------- (73) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. MINSORT(add(z0, z1), z2) -> c25(IF_MINSORT(eq(z0, min(add(z0, z1))), add(z0, z1), z2), EQ(z0, min(add(z0, z1))), MIN(add(z0, z1))) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) 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 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)) MIN(add(z0, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) MINITER(add(z0, z1), z2, z3) -> c12(IF_MIN(le(z0, z3), z1, z2, z3), LE(z0, z3)) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) RM(z0, add(z1, z2)) -> c21(IF_RM(eq(z0, z1), z0, add(z1, z2)), EQ(z0, z1)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) MINSORT(add(z0, z1), z2) -> c25(IF_MINSORT(eq(z0, min(add(z0, z1))), add(z0, z1), z2), EQ(z0, min(add(z0, z1))), MIN(add(z0, z1))) IF_MINSORT(true, add(z0, z1), z2) -> c26(MINSORT(app(rm(z0, z1), z2), nil), APP(rm(z0, z1), z2), RM(z0, z1)) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(APP(x_1, x_2)) = x_2 POL(EQ(x_1, x_2)) = 0 POL(IF_MIN(x_1, x_2, x_3, x_4)) = 0 POL(IF_MINSORT(x_1, x_2, x_3)) = x_2 + x_3^2 + [2]x_2*x_3 + x_2^2 POL(IF_RM(x_1, x_2, x_3)) = 0 POL(LE(x_1, x_2)) = 0 POL(MIN(x_1)) = 0 POL(MINITER(x_1, x_2, x_3)) = 0 POL(MINSORT(x_1, x_2)) = [1] + x_1 + x_2^2 + [2]x_1*x_2 + x_1^2 POL(RM(x_1, x_2)) = 0 POL(add(x_1, x_2)) = [2] + x_2 POL(app(x_1, x_2)) = x_1 + x_2 POL(c10(x_1)) = x_1 POL(c11(x_1)) = x_1 POL(c12(x_1, x_2)) = x_1 + x_2 POL(c14(x_1)) = x_1 POL(c21(x_1, x_2)) = x_1 + x_2 POL(c22(x_1)) = x_1 POL(c23(x_1)) = x_1 POL(c25(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c26(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c27(x_1)) = x_1 POL(c3(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(if_min(x_1, x_2, x_3, x_4)) = [1] + [2]x_2 + x_3 + x_4 + x_4^2 + x_3*x_4 + x_2*x_4 + x_3^2 + [2]x_2*x_3 + [2]x_2^2 POL(if_rm(x_1, x_2, x_3)) = x_3 POL(le(x_1, x_2)) = 0 POL(min(x_1)) = 0 POL(minIter(x_1, x_2, x_3)) = [1] + [2]x_1^2 + [2]x_1*x_2 POL(nil) = 0 POL(rm(x_1, x_2)) = x_2 POL(s(x_1)) = 0 POL(true) = 0 ---------------------------------------- (74) 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) min(add(z0, z1)) -> minIter(add(z0, z1), add(z0, z1), 0) minIter(add(z0, z1), z2, z3) -> if_min(le(z0, z3), z1, z2, z3) minIter(nil, add(z0, z1), z2) -> minIter(add(z0, z1), add(z0, z1), s(z2)) if_min(true, z0, z1, z2) -> z2 if_min(false, z0, z1, z2) -> minIter(z0, z1, z2) 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)) 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, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) MINITER(add(z0, z1), z2, z3) -> c12(IF_MIN(le(z0, z3), z1, z2, z3), LE(z0, z3)) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) RM(z0, add(z1, z2)) -> c21(IF_RM(eq(z0, z1), z0, add(z1, z2)), EQ(z0, z1)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) MINSORT(add(z0, z1), z2) -> c25(IF_MINSORT(eq(z0, min(add(z0, z1))), add(z0, z1), z2), EQ(z0, min(add(z0, z1))), MIN(add(z0, z1))) IF_MINSORT(true, add(z0, z1), z2) -> c26(MINSORT(app(rm(z0, z1), z2), nil), APP(rm(z0, z1), z2), RM(z0, z1)) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, 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, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) MINITER(add(z0, z1), z2, z3) -> c12(IF_MIN(le(z0, z3), z1, z2, z3), LE(z0, z3)) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) RM(z0, add(z1, z2)) -> c21(IF_RM(eq(z0, z1), z0, add(z1, z2)), EQ(z0, z1)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) K tuples: IF_MINSORT(true, add(z0, z1), z2) -> c26(MINSORT(app(rm(z0, z1), z2), nil), APP(rm(z0, z1), z2), RM(z0, z1)) MINSORT(add(z0, z1), z2) -> c25(IF_MINSORT(eq(z0, min(add(z0, z1))), add(z0, z1), z2), EQ(z0, min(add(z0, z1))), MIN(add(z0, z1))) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) Defined Rule Symbols: le_2, eq_2, min_1, minIter_3, if_min_4, app_2, rm_2, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, MIN_1, MINITER_3, IF_MIN_4, RM_2, IF_RM_3, MINSORT_2, IF_MINSORT_3 Compound Symbols: c3_1, c6_1, c8_1, c10_1, c11_1, c12_2, c14_1, c21_2, c22_1, c23_1, c25_3, c26_3, c27_1 ---------------------------------------- (75) CdtKnowledgeProof (BOTH BOUNDS(ID, ID)) The following tuples could be moved from S to K by knowledge propagation: MIN(add(z0, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) ---------------------------------------- (76) 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) min(add(z0, z1)) -> minIter(add(z0, z1), add(z0, z1), 0) minIter(add(z0, z1), z2, z3) -> if_min(le(z0, z3), z1, z2, z3) minIter(nil, add(z0, z1), z2) -> minIter(add(z0, z1), add(z0, z1), s(z2)) if_min(true, z0, z1, z2) -> z2 if_min(false, z0, z1, z2) -> minIter(z0, z1, z2) 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)) 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, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) MINITER(add(z0, z1), z2, z3) -> c12(IF_MIN(le(z0, z3), z1, z2, z3), LE(z0, z3)) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) RM(z0, add(z1, z2)) -> c21(IF_RM(eq(z0, z1), z0, add(z1, z2)), EQ(z0, z1)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) MINSORT(add(z0, z1), z2) -> c25(IF_MINSORT(eq(z0, min(add(z0, z1))), add(z0, z1), z2), EQ(z0, min(add(z0, z1))), MIN(add(z0, z1))) IF_MINSORT(true, add(z0, z1), z2) -> c26(MINSORT(app(rm(z0, z1), z2), nil), APP(rm(z0, z1), z2), RM(z0, z1)) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, 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)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) MINITER(add(z0, z1), z2, z3) -> c12(IF_MIN(le(z0, z3), z1, z2, z3), LE(z0, z3)) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) RM(z0, add(z1, z2)) -> c21(IF_RM(eq(z0, z1), z0, add(z1, z2)), EQ(z0, z1)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) K tuples: IF_MINSORT(true, add(z0, z1), z2) -> c26(MINSORT(app(rm(z0, z1), z2), nil), APP(rm(z0, z1), z2), RM(z0, z1)) MINSORT(add(z0, z1), z2) -> c25(IF_MINSORT(eq(z0, min(add(z0, z1))), add(z0, z1), z2), EQ(z0, min(add(z0, z1))), MIN(add(z0, z1))) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MIN(add(z0, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) Defined Rule Symbols: le_2, eq_2, min_1, minIter_3, if_min_4, app_2, rm_2, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, MIN_1, MINITER_3, IF_MIN_4, RM_2, IF_RM_3, MINSORT_2, IF_MINSORT_3 Compound Symbols: c3_1, c6_1, c8_1, c10_1, c11_1, c12_2, c14_1, c21_2, c22_1, c23_1, c25_3, c26_3, c27_1 ---------------------------------------- (77) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. APP(add(z0, z1), z2) -> c8(APP(z1, z2)) 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 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)) MIN(add(z0, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) MINITER(add(z0, z1), z2, z3) -> c12(IF_MIN(le(z0, z3), z1, z2, z3), LE(z0, z3)) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) RM(z0, add(z1, z2)) -> c21(IF_RM(eq(z0, z1), z0, add(z1, z2)), EQ(z0, z1)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) MINSORT(add(z0, z1), z2) -> c25(IF_MINSORT(eq(z0, min(add(z0, z1))), add(z0, z1), z2), EQ(z0, min(add(z0, z1))), MIN(add(z0, z1))) IF_MINSORT(true, add(z0, z1), z2) -> c26(MINSORT(app(rm(z0, z1), z2), nil), APP(rm(z0, z1), z2), RM(z0, z1)) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(APP(x_1, x_2)) = x_1 POL(EQ(x_1, x_2)) = 0 POL(IF_MIN(x_1, x_2, x_3, x_4)) = 0 POL(IF_MINSORT(x_1, x_2, x_3)) = x_3^2 + [2]x_2*x_3 + x_2^2 POL(IF_RM(x_1, x_2, x_3)) = 0 POL(LE(x_1, x_2)) = 0 POL(MIN(x_1)) = 0 POL(MINITER(x_1, x_2, x_3)) = 0 POL(MINSORT(x_1, x_2)) = x_2^2 + [2]x_1*x_2 + x_1^2 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(c10(x_1)) = x_1 POL(c11(x_1)) = x_1 POL(c12(x_1, x_2)) = x_1 + x_2 POL(c14(x_1)) = x_1 POL(c21(x_1, x_2)) = x_1 + x_2 POL(c22(x_1)) = x_1 POL(c23(x_1)) = x_1 POL(c25(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c26(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c27(x_1)) = x_1 POL(c3(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(if_min(x_1, x_2, x_3, x_4)) = [1] + [2]x_2 + x_3 + x_4 + x_4^2 + x_3*x_4 + x_2*x_4 + x_3^2 + [2]x_2*x_3 + x_2^2 POL(if_rm(x_1, x_2, x_3)) = x_3 POL(le(x_1, x_2)) = 0 POL(min(x_1)) = 0 POL(minIter(x_1, x_2, x_3)) = [1] + [2]x_1^2 + [2]x_1*x_2 POL(nil) = 0 POL(rm(x_1, x_2)) = x_2 POL(s(x_1)) = 0 POL(true) = 0 ---------------------------------------- (78) 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) min(add(z0, z1)) -> minIter(add(z0, z1), add(z0, z1), 0) minIter(add(z0, z1), z2, z3) -> if_min(le(z0, z3), z1, z2, z3) minIter(nil, add(z0, z1), z2) -> minIter(add(z0, z1), add(z0, z1), s(z2)) if_min(true, z0, z1, z2) -> z2 if_min(false, z0, z1, z2) -> minIter(z0, z1, z2) 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)) 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, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) MINITER(add(z0, z1), z2, z3) -> c12(IF_MIN(le(z0, z3), z1, z2, z3), LE(z0, z3)) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) RM(z0, add(z1, z2)) -> c21(IF_RM(eq(z0, z1), z0, add(z1, z2)), EQ(z0, z1)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) MINSORT(add(z0, z1), z2) -> c25(IF_MINSORT(eq(z0, min(add(z0, z1))), add(z0, z1), z2), EQ(z0, min(add(z0, z1))), MIN(add(z0, z1))) IF_MINSORT(true, add(z0, z1), z2) -> c26(MINSORT(app(rm(z0, z1), z2), nil), APP(rm(z0, z1), z2), RM(z0, z1)) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) MINITER(add(z0, z1), z2, z3) -> c12(IF_MIN(le(z0, z3), z1, z2, z3), LE(z0, z3)) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) RM(z0, add(z1, z2)) -> c21(IF_RM(eq(z0, z1), z0, add(z1, z2)), EQ(z0, z1)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) K tuples: IF_MINSORT(true, add(z0, z1), z2) -> c26(MINSORT(app(rm(z0, z1), z2), nil), APP(rm(z0, z1), z2), RM(z0, z1)) MINSORT(add(z0, z1), z2) -> c25(IF_MINSORT(eq(z0, min(add(z0, z1))), add(z0, z1), z2), EQ(z0, min(add(z0, z1))), MIN(add(z0, z1))) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MIN(add(z0, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) Defined Rule Symbols: le_2, eq_2, min_1, minIter_3, if_min_4, app_2, rm_2, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, MIN_1, MINITER_3, IF_MIN_4, RM_2, IF_RM_3, MINSORT_2, IF_MINSORT_3 Compound Symbols: c3_1, c6_1, c8_1, c10_1, c11_1, c12_2, c14_1, c21_2, c22_1, c23_1, c25_3, c26_3, c27_1 ---------------------------------------- (79) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) 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 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)) MIN(add(z0, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) MINITER(add(z0, z1), z2, z3) -> c12(IF_MIN(le(z0, z3), z1, z2, z3), LE(z0, z3)) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) RM(z0, add(z1, z2)) -> c21(IF_RM(eq(z0, z1), z0, add(z1, z2)), EQ(z0, z1)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) MINSORT(add(z0, z1), z2) -> c25(IF_MINSORT(eq(z0, min(add(z0, z1))), add(z0, z1), z2), EQ(z0, min(add(z0, z1))), MIN(add(z0, z1))) IF_MINSORT(true, add(z0, z1), z2) -> c26(MINSORT(app(rm(z0, z1), z2), nil), APP(rm(z0, z1), z2), RM(z0, z1)) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(APP(x_1, x_2)) = [2]x_1 POL(EQ(x_1, x_2)) = 0 POL(IF_MIN(x_1, x_2, x_3, x_4)) = 0 POL(IF_MINSORT(x_1, x_2, x_3)) = x_3^2 + [2]x_2*x_3 + x_2^2 POL(IF_RM(x_1, x_2, x_3)) = [2]x_3 POL(LE(x_1, x_2)) = 0 POL(MIN(x_1)) = 0 POL(MINITER(x_1, x_2, x_3)) = 0 POL(MINSORT(x_1, x_2)) = x_2^2 + [2]x_1*x_2 + x_1^2 POL(RM(x_1, x_2)) = [2]x_2 POL(add(x_1, x_2)) = [2] + x_2 POL(app(x_1, x_2)) = x_1 + x_2 POL(c10(x_1)) = x_1 POL(c11(x_1)) = x_1 POL(c12(x_1, x_2)) = x_1 + x_2 POL(c14(x_1)) = x_1 POL(c21(x_1, x_2)) = x_1 + x_2 POL(c22(x_1)) = x_1 POL(c23(x_1)) = x_1 POL(c25(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c26(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c27(x_1)) = x_1 POL(c3(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(if_min(x_1, x_2, x_3, x_4)) = [1] + [2]x_2 + x_3 + x_4 + x_4^2 + x_3*x_4 + x_2*x_4 + x_3^2 + [2]x_2*x_3 + x_2^2 POL(if_rm(x_1, x_2, x_3)) = x_3 POL(le(x_1, x_2)) = 0 POL(min(x_1)) = 0 POL(minIter(x_1, x_2, x_3)) = [1] + [2]x_1^2 + [2]x_1*x_2 POL(nil) = 0 POL(rm(x_1, x_2)) = x_2 POL(s(x_1)) = 0 POL(true) = 0 ---------------------------------------- (80) 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) min(add(z0, z1)) -> minIter(add(z0, z1), add(z0, z1), 0) minIter(add(z0, z1), z2, z3) -> if_min(le(z0, z3), z1, z2, z3) minIter(nil, add(z0, z1), z2) -> minIter(add(z0, z1), add(z0, z1), s(z2)) if_min(true, z0, z1, z2) -> z2 if_min(false, z0, z1, z2) -> minIter(z0, z1, z2) 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)) 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, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) MINITER(add(z0, z1), z2, z3) -> c12(IF_MIN(le(z0, z3), z1, z2, z3), LE(z0, z3)) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) RM(z0, add(z1, z2)) -> c21(IF_RM(eq(z0, z1), z0, add(z1, z2)), EQ(z0, z1)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) MINSORT(add(z0, z1), z2) -> c25(IF_MINSORT(eq(z0, min(add(z0, z1))), add(z0, z1), z2), EQ(z0, min(add(z0, z1))), MIN(add(z0, z1))) IF_MINSORT(true, add(z0, z1), z2) -> c26(MINSORT(app(rm(z0, z1), z2), nil), APP(rm(z0, z1), z2), RM(z0, z1)) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) MINITER(add(z0, z1), z2, z3) -> c12(IF_MIN(le(z0, z3), z1, z2, z3), LE(z0, z3)) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) RM(z0, add(z1, z2)) -> c21(IF_RM(eq(z0, z1), z0, add(z1, z2)), EQ(z0, z1)) K tuples: IF_MINSORT(true, add(z0, z1), z2) -> c26(MINSORT(app(rm(z0, z1), z2), nil), APP(rm(z0, z1), z2), RM(z0, z1)) MINSORT(add(z0, z1), z2) -> c25(IF_MINSORT(eq(z0, min(add(z0, z1))), add(z0, z1), z2), EQ(z0, min(add(z0, z1))), MIN(add(z0, z1))) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MIN(add(z0, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) Defined Rule Symbols: le_2, eq_2, min_1, minIter_3, if_min_4, app_2, rm_2, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, MIN_1, MINITER_3, IF_MIN_4, RM_2, IF_RM_3, MINSORT_2, IF_MINSORT_3 Compound Symbols: c3_1, c6_1, c8_1, c10_1, c11_1, c12_2, c14_1, c21_2, c22_1, c23_1, c25_3, c26_3, c27_1 ---------------------------------------- (81) CdtKnowledgeProof (BOTH BOUNDS(ID, ID)) The following tuples could be moved from S to K by knowledge propagation: RM(z0, add(z1, z2)) -> c21(IF_RM(eq(z0, z1), z0, add(z1, z2)), EQ(z0, z1)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) ---------------------------------------- (82) 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) min(add(z0, z1)) -> minIter(add(z0, z1), add(z0, z1), 0) minIter(add(z0, z1), z2, z3) -> if_min(le(z0, z3), z1, z2, z3) minIter(nil, add(z0, z1), z2) -> minIter(add(z0, z1), add(z0, z1), s(z2)) if_min(true, z0, z1, z2) -> z2 if_min(false, z0, z1, z2) -> minIter(z0, z1, z2) 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)) 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, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) MINITER(add(z0, z1), z2, z3) -> c12(IF_MIN(le(z0, z3), z1, z2, z3), LE(z0, z3)) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) RM(z0, add(z1, z2)) -> c21(IF_RM(eq(z0, z1), z0, add(z1, z2)), EQ(z0, z1)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) MINSORT(add(z0, z1), z2) -> c25(IF_MINSORT(eq(z0, min(add(z0, z1))), add(z0, z1), z2), EQ(z0, min(add(z0, z1))), MIN(add(z0, z1))) IF_MINSORT(true, add(z0, z1), z2) -> c26(MINSORT(app(rm(z0, z1), z2), nil), APP(rm(z0, z1), z2), RM(z0, z1)) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) MINITER(add(z0, z1), z2, z3) -> c12(IF_MIN(le(z0, z3), z1, z2, z3), LE(z0, z3)) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) K tuples: IF_MINSORT(true, add(z0, z1), z2) -> c26(MINSORT(app(rm(z0, z1), z2), nil), APP(rm(z0, z1), z2), RM(z0, z1)) MINSORT(add(z0, z1), z2) -> c25(IF_MINSORT(eq(z0, min(add(z0, z1))), add(z0, z1), z2), EQ(z0, min(add(z0, z1))), MIN(add(z0, z1))) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MIN(add(z0, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) RM(z0, add(z1, z2)) -> c21(IF_RM(eq(z0, z1), z0, add(z1, z2)), EQ(z0, z1)) Defined Rule Symbols: le_2, eq_2, min_1, minIter_3, if_min_4, app_2, rm_2, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, MIN_1, MINITER_3, IF_MIN_4, RM_2, IF_RM_3, MINSORT_2, IF_MINSORT_3 Compound Symbols: c3_1, c6_1, c8_1, c10_1, c11_1, c12_2, c14_1, c21_2, c22_1, c23_1, c25_3, c26_3, c27_1 ---------------------------------------- (83) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MINITER(add(z0, z1), z2, z3) -> c12(IF_MIN(le(z0, z3), z1, z2, z3), LE(z0, z3)) by MINITER(add(0, x1), x2, z0) -> c12(IF_MIN(true, x1, x2, z0), LE(0, z0)) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0), LE(s(z0), 0)) MINITER(add(s(z0), x1), x2, s(z1)) -> c12(IF_MIN(le(z0, z1), x1, x2, s(z1)), LE(s(z0), s(z1))) ---------------------------------------- (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) min(add(z0, z1)) -> minIter(add(z0, z1), add(z0, z1), 0) minIter(add(z0, z1), z2, z3) -> if_min(le(z0, z3), z1, z2, z3) minIter(nil, add(z0, z1), z2) -> minIter(add(z0, z1), add(z0, z1), s(z2)) if_min(true, z0, z1, z2) -> z2 if_min(false, z0, z1, z2) -> minIter(z0, z1, z2) 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)) 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, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) RM(z0, add(z1, z2)) -> c21(IF_RM(eq(z0, z1), z0, add(z1, z2)), EQ(z0, z1)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) MINSORT(add(z0, z1), z2) -> c25(IF_MINSORT(eq(z0, min(add(z0, z1))), add(z0, z1), z2), EQ(z0, min(add(z0, z1))), MIN(add(z0, z1))) IF_MINSORT(true, add(z0, z1), z2) -> c26(MINSORT(app(rm(z0, z1), z2), nil), APP(rm(z0, z1), z2), RM(z0, z1)) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MINITER(add(0, x1), x2, z0) -> c12(IF_MIN(true, x1, x2, z0), LE(0, z0)) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0), LE(s(z0), 0)) MINITER(add(s(z0), x1), x2, s(z1)) -> c12(IF_MIN(le(z0, z1), x1, x2, s(z1)), 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)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) MINITER(add(0, x1), x2, z0) -> c12(IF_MIN(true, x1, x2, z0), LE(0, z0)) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0), LE(s(z0), 0)) MINITER(add(s(z0), x1), x2, s(z1)) -> c12(IF_MIN(le(z0, z1), x1, x2, s(z1)), LE(s(z0), s(z1))) K tuples: IF_MINSORT(true, add(z0, z1), z2) -> c26(MINSORT(app(rm(z0, z1), z2), nil), APP(rm(z0, z1), z2), RM(z0, z1)) MINSORT(add(z0, z1), z2) -> c25(IF_MINSORT(eq(z0, min(add(z0, z1))), add(z0, z1), z2), EQ(z0, min(add(z0, z1))), MIN(add(z0, z1))) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MIN(add(z0, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) RM(z0, add(z1, z2)) -> c21(IF_RM(eq(z0, z1), z0, add(z1, z2)), EQ(z0, z1)) Defined Rule Symbols: le_2, eq_2, min_1, minIter_3, if_min_4, app_2, rm_2, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, MIN_1, MINITER_3, IF_MIN_4, RM_2, IF_RM_3, MINSORT_2, IF_MINSORT_3 Compound Symbols: c3_1, c6_1, c8_1, c10_1, c11_1, c14_1, c21_2, c22_1, c23_1, c25_3, c26_3, c27_1, c12_2 ---------------------------------------- (85) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MINITER(add(0, x1), x2, z0) -> c12(IF_MIN(true, x1, x2, z0), LE(0, z0)) ---------------------------------------- (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) min(add(z0, z1)) -> minIter(add(z0, z1), add(z0, z1), 0) minIter(add(z0, z1), z2, z3) -> if_min(le(z0, z3), z1, z2, z3) minIter(nil, add(z0, z1), z2) -> minIter(add(z0, z1), add(z0, z1), s(z2)) if_min(true, z0, z1, z2) -> z2 if_min(false, z0, z1, z2) -> minIter(z0, z1, z2) 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)) 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, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) RM(z0, add(z1, z2)) -> c21(IF_RM(eq(z0, z1), z0, add(z1, z2)), EQ(z0, z1)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) MINSORT(add(z0, z1), z2) -> c25(IF_MINSORT(eq(z0, min(add(z0, z1))), add(z0, z1), z2), EQ(z0, min(add(z0, z1))), MIN(add(z0, z1))) IF_MINSORT(true, add(z0, z1), z2) -> c26(MINSORT(app(rm(z0, z1), z2), nil), APP(rm(z0, z1), z2), RM(z0, z1)) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0), LE(s(z0), 0)) MINITER(add(s(z0), x1), x2, s(z1)) -> c12(IF_MIN(le(z0, z1), x1, x2, s(z1)), 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)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0), LE(s(z0), 0)) MINITER(add(s(z0), x1), x2, s(z1)) -> c12(IF_MIN(le(z0, z1), x1, x2, s(z1)), LE(s(z0), s(z1))) K tuples: IF_MINSORT(true, add(z0, z1), z2) -> c26(MINSORT(app(rm(z0, z1), z2), nil), APP(rm(z0, z1), z2), RM(z0, z1)) MINSORT(add(z0, z1), z2) -> c25(IF_MINSORT(eq(z0, min(add(z0, z1))), add(z0, z1), z2), EQ(z0, min(add(z0, z1))), MIN(add(z0, z1))) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MIN(add(z0, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) RM(z0, add(z1, z2)) -> c21(IF_RM(eq(z0, z1), z0, add(z1, z2)), EQ(z0, z1)) Defined Rule Symbols: le_2, eq_2, min_1, minIter_3, if_min_4, app_2, rm_2, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, MIN_1, MINITER_3, IF_MIN_4, RM_2, IF_RM_3, MINSORT_2, IF_MINSORT_3 Compound Symbols: c3_1, c6_1, c8_1, c10_1, c11_1, c14_1, c21_2, c22_1, c23_1, c25_3, c26_3, c27_1, c12_2 ---------------------------------------- (87) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 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) min(add(z0, z1)) -> minIter(add(z0, z1), add(z0, z1), 0) minIter(add(z0, z1), z2, z3) -> if_min(le(z0, z3), z1, z2, z3) minIter(nil, add(z0, z1), z2) -> minIter(add(z0, z1), add(z0, z1), s(z2)) if_min(true, z0, z1, z2) -> z2 if_min(false, z0, z1, z2) -> minIter(z0, z1, z2) 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)) 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, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) RM(z0, add(z1, z2)) -> c21(IF_RM(eq(z0, z1), z0, add(z1, z2)), EQ(z0, z1)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) MINSORT(add(z0, z1), z2) -> c25(IF_MINSORT(eq(z0, min(add(z0, z1))), add(z0, z1), z2), EQ(z0, min(add(z0, z1))), MIN(add(z0, z1))) IF_MINSORT(true, add(z0, z1), z2) -> c26(MINSORT(app(rm(z0, z1), z2), nil), APP(rm(z0, z1), z2), RM(z0, z1)) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MINITER(add(s(z0), x1), x2, s(z1)) -> c12(IF_MIN(le(z0, z1), x1, x2, s(z1)), LE(s(z0), s(z1))) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) MINITER(add(s(z0), x1), x2, s(z1)) -> c12(IF_MIN(le(z0, z1), x1, x2, s(z1)), LE(s(z0), s(z1))) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) K tuples: IF_MINSORT(true, add(z0, z1), z2) -> c26(MINSORT(app(rm(z0, z1), z2), nil), APP(rm(z0, z1), z2), RM(z0, z1)) MINSORT(add(z0, z1), z2) -> c25(IF_MINSORT(eq(z0, min(add(z0, z1))), add(z0, z1), z2), EQ(z0, min(add(z0, z1))), MIN(add(z0, z1))) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MIN(add(z0, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) RM(z0, add(z1, z2)) -> c21(IF_RM(eq(z0, z1), z0, add(z1, z2)), EQ(z0, z1)) Defined Rule Symbols: le_2, eq_2, min_1, minIter_3, if_min_4, app_2, rm_2, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, MIN_1, MINITER_3, IF_MIN_4, RM_2, IF_RM_3, MINSORT_2, IF_MINSORT_3 Compound Symbols: c3_1, c6_1, c8_1, c10_1, c11_1, c14_1, c21_2, c22_1, c23_1, c25_3, c26_3, c27_1, c12_2, c12_1 ---------------------------------------- (89) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace RM(z0, add(z1, z2)) -> c21(IF_RM(eq(z0, z1), z0, add(z1, z2)), EQ(z0, z1)) by RM(0, add(0, x2)) -> c21(IF_RM(true, 0, add(0, x2)), EQ(0, 0)) RM(0, add(s(z0), x2)) -> c21(IF_RM(false, 0, add(s(z0), x2)), EQ(0, s(z0))) RM(s(z0), add(0, x2)) -> c21(IF_RM(false, s(z0), add(0, x2)), EQ(s(z0), 0)) RM(s(z0), add(s(z1), x2)) -> c21(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) min(add(z0, z1)) -> minIter(add(z0, z1), add(z0, z1), 0) minIter(add(z0, z1), z2, z3) -> if_min(le(z0, z3), z1, z2, z3) minIter(nil, add(z0, z1), z2) -> minIter(add(z0, z1), add(z0, z1), s(z2)) if_min(true, z0, z1, z2) -> z2 if_min(false, z0, z1, z2) -> minIter(z0, z1, z2) 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)) 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, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) MINSORT(add(z0, z1), z2) -> c25(IF_MINSORT(eq(z0, min(add(z0, z1))), add(z0, z1), z2), EQ(z0, min(add(z0, z1))), MIN(add(z0, z1))) IF_MINSORT(true, add(z0, z1), z2) -> c26(MINSORT(app(rm(z0, z1), z2), nil), APP(rm(z0, z1), z2), RM(z0, z1)) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MINITER(add(s(z0), x1), x2, s(z1)) -> c12(IF_MIN(le(z0, z1), x1, x2, s(z1)), LE(s(z0), s(z1))) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) RM(0, add(0, x2)) -> c21(IF_RM(true, 0, add(0, x2)), EQ(0, 0)) RM(0, add(s(z0), x2)) -> c21(IF_RM(false, 0, add(s(z0), x2)), EQ(0, s(z0))) RM(s(z0), add(0, x2)) -> c21(IF_RM(false, s(z0), add(0, x2)), EQ(s(z0), 0)) RM(s(z0), add(s(z1), x2)) -> c21(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)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) MINITER(add(s(z0), x1), x2, s(z1)) -> c12(IF_MIN(le(z0, z1), x1, x2, s(z1)), LE(s(z0), s(z1))) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) K tuples: IF_MINSORT(true, add(z0, z1), z2) -> c26(MINSORT(app(rm(z0, z1), z2), nil), APP(rm(z0, z1), z2), RM(z0, z1)) MINSORT(add(z0, z1), z2) -> c25(IF_MINSORT(eq(z0, min(add(z0, z1))), add(z0, z1), z2), EQ(z0, min(add(z0, z1))), MIN(add(z0, z1))) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MIN(add(z0, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) RM(z0, add(z1, z2)) -> c21(IF_RM(eq(z0, z1), z0, add(z1, z2)), EQ(z0, z1)) Defined Rule Symbols: le_2, eq_2, min_1, minIter_3, if_min_4, app_2, rm_2, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, MIN_1, MINITER_3, IF_MIN_4, IF_RM_3, MINSORT_2, IF_MINSORT_3, RM_2 Compound Symbols: c3_1, c6_1, c8_1, c10_1, c11_1, c14_1, c22_1, c23_1, c25_3, c26_3, c27_1, c12_2, c12_1, c21_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) min(add(z0, z1)) -> minIter(add(z0, z1), add(z0, z1), 0) minIter(add(z0, z1), z2, z3) -> if_min(le(z0, z3), z1, z2, z3) minIter(nil, add(z0, z1), z2) -> minIter(add(z0, z1), add(z0, z1), s(z2)) if_min(true, z0, z1, z2) -> z2 if_min(false, z0, z1, z2) -> minIter(z0, z1, z2) 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)) 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, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) MINSORT(add(z0, z1), z2) -> c25(IF_MINSORT(eq(z0, min(add(z0, z1))), add(z0, z1), z2), EQ(z0, min(add(z0, z1))), MIN(add(z0, z1))) IF_MINSORT(true, add(z0, z1), z2) -> c26(MINSORT(app(rm(z0, z1), z2), nil), APP(rm(z0, z1), z2), RM(z0, z1)) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MINITER(add(s(z0), x1), x2, s(z1)) -> c12(IF_MIN(le(z0, z1), x1, x2, s(z1)), LE(s(z0), s(z1))) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) RM(s(z0), add(s(z1), x2)) -> c21(IF_RM(eq(z0, z1), s(z0), add(s(z1), x2)), EQ(s(z0), s(z1))) RM(0, add(0, x2)) -> c21(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c21(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c21(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)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) MINITER(add(s(z0), x1), x2, s(z1)) -> c12(IF_MIN(le(z0, z1), x1, x2, s(z1)), LE(s(z0), s(z1))) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) K tuples: IF_MINSORT(true, add(z0, z1), z2) -> c26(MINSORT(app(rm(z0, z1), z2), nil), APP(rm(z0, z1), z2), RM(z0, z1)) MINSORT(add(z0, z1), z2) -> c25(IF_MINSORT(eq(z0, min(add(z0, z1))), add(z0, z1), z2), EQ(z0, min(add(z0, z1))), MIN(add(z0, z1))) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MIN(add(z0, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) RM(z0, add(z1, z2)) -> c21(IF_RM(eq(z0, z1), z0, add(z1, z2)), EQ(z0, z1)) Defined Rule Symbols: le_2, eq_2, min_1, minIter_3, if_min_4, app_2, rm_2, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, MIN_1, MINITER_3, IF_MIN_4, IF_RM_3, MINSORT_2, IF_MINSORT_3, RM_2 Compound Symbols: c3_1, c6_1, c8_1, c10_1, c11_1, c14_1, c22_1, c23_1, c25_3, c26_3, c27_1, c12_2, c12_1, c21_2, c21_1 ---------------------------------------- (93) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MINSORT(add(z0, z1), z2) -> c25(IF_MINSORT(eq(z0, min(add(z0, z1))), add(z0, z1), z2), EQ(z0, min(add(z0, z1))), MIN(add(z0, z1))) by MINSORT(add(z0, z1), x2) -> c25(IF_MINSORT(eq(z0, minIter(add(z0, z1), add(z0, z1), 0)), add(z0, z1), x2), EQ(z0, 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) min(add(z0, z1)) -> minIter(add(z0, z1), add(z0, z1), 0) minIter(add(z0, z1), z2, z3) -> if_min(le(z0, z3), z1, z2, z3) minIter(nil, add(z0, z1), z2) -> minIter(add(z0, z1), add(z0, z1), s(z2)) if_min(true, z0, z1, z2) -> z2 if_min(false, z0, z1, z2) -> minIter(z0, z1, z2) 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)) 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, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) IF_MINSORT(true, add(z0, z1), z2) -> c26(MINSORT(app(rm(z0, z1), z2), nil), APP(rm(z0, z1), z2), RM(z0, z1)) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MINITER(add(s(z0), x1), x2, s(z1)) -> c12(IF_MIN(le(z0, z1), x1, x2, s(z1)), LE(s(z0), s(z1))) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) RM(s(z0), add(s(z1), x2)) -> c21(IF_RM(eq(z0, z1), s(z0), add(s(z1), x2)), EQ(s(z0), s(z1))) RM(0, add(0, x2)) -> c21(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c21(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c21(IF_RM(false, s(z0), add(0, x2))) MINSORT(add(z0, z1), x2) -> c25(IF_MINSORT(eq(z0, minIter(add(z0, z1), add(z0, z1), 0)), add(z0, z1), x2), EQ(z0, 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)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) MINITER(add(s(z0), x1), x2, s(z1)) -> c12(IF_MIN(le(z0, z1), x1, x2, s(z1)), LE(s(z0), s(z1))) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) K tuples: IF_MINSORT(true, add(z0, z1), z2) -> c26(MINSORT(app(rm(z0, z1), z2), nil), APP(rm(z0, z1), z2), RM(z0, z1)) MINSORT(add(z0, z1), z2) -> c25(IF_MINSORT(eq(z0, min(add(z0, z1))), add(z0, z1), z2), EQ(z0, min(add(z0, z1))), MIN(add(z0, z1))) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MIN(add(z0, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) RM(z0, add(z1, z2)) -> c21(IF_RM(eq(z0, z1), z0, add(z1, z2)), EQ(z0, z1)) Defined Rule Symbols: le_2, eq_2, min_1, minIter_3, if_min_4, app_2, rm_2, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, MIN_1, MINITER_3, IF_MIN_4, IF_RM_3, IF_MINSORT_3, RM_2, MINSORT_2 Compound Symbols: c3_1, c6_1, c8_1, c10_1, c11_1, c14_1, c22_1, c23_1, c26_3, c27_1, c12_2, c12_1, c21_2, c21_1, c25_3 ---------------------------------------- (95) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF_MINSORT(true, add(z0, z1), z2) -> c26(MINSORT(app(rm(z0, z1), z2), nil), APP(rm(z0, z1), z2), RM(z0, z1)) by IF_MINSORT(true, add(z0, nil), x2) -> c26(MINSORT(app(nil, x2), nil), APP(rm(z0, nil), x2), RM(z0, nil)) IF_MINSORT(true, add(z0, add(z1, z2)), x2) -> c26(MINSORT(app(if_rm(eq(z0, z1), z0, add(z1, z2)), x2), nil), APP(rm(z0, add(z1, z2)), x2), RM(z0, add(z1, z2))) ---------------------------------------- (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) min(add(z0, z1)) -> minIter(add(z0, z1), add(z0, z1), 0) minIter(add(z0, z1), z2, z3) -> if_min(le(z0, z3), z1, z2, z3) minIter(nil, add(z0, z1), z2) -> minIter(add(z0, z1), add(z0, z1), s(z2)) if_min(true, z0, z1, z2) -> z2 if_min(false, z0, z1, z2) -> minIter(z0, z1, z2) 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)) 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, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MINITER(add(s(z0), x1), x2, s(z1)) -> c12(IF_MIN(le(z0, z1), x1, x2, s(z1)), LE(s(z0), s(z1))) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) RM(s(z0), add(s(z1), x2)) -> c21(IF_RM(eq(z0, z1), s(z0), add(s(z1), x2)), EQ(s(z0), s(z1))) RM(0, add(0, x2)) -> c21(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c21(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c21(IF_RM(false, s(z0), add(0, x2))) MINSORT(add(z0, z1), x2) -> c25(IF_MINSORT(eq(z0, minIter(add(z0, z1), add(z0, z1), 0)), add(z0, z1), x2), EQ(z0, min(add(z0, z1))), MIN(add(z0, z1))) IF_MINSORT(true, add(z0, nil), x2) -> c26(MINSORT(app(nil, x2), nil), APP(rm(z0, nil), x2), RM(z0, nil)) IF_MINSORT(true, add(z0, add(z1, z2)), x2) -> c26(MINSORT(app(if_rm(eq(z0, z1), z0, add(z1, z2)), x2), nil), APP(rm(z0, add(z1, z2)), x2), RM(z0, add(z1, z2))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) MINITER(add(s(z0), x1), x2, s(z1)) -> c12(IF_MIN(le(z0, z1), x1, x2, s(z1)), LE(s(z0), s(z1))) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) K tuples: IF_MINSORT(true, add(z0, z1), z2) -> c26(MINSORT(app(rm(z0, z1), z2), nil), APP(rm(z0, z1), z2), RM(z0, z1)) MINSORT(add(z0, z1), z2) -> c25(IF_MINSORT(eq(z0, min(add(z0, z1))), add(z0, z1), z2), EQ(z0, min(add(z0, z1))), MIN(add(z0, z1))) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MIN(add(z0, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) RM(z0, add(z1, z2)) -> c21(IF_RM(eq(z0, z1), z0, add(z1, z2)), EQ(z0, z1)) Defined Rule Symbols: le_2, eq_2, min_1, minIter_3, if_min_4, app_2, rm_2, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, MIN_1, MINITER_3, IF_MIN_4, IF_RM_3, IF_MINSORT_3, RM_2, MINSORT_2 Compound Symbols: c3_1, c6_1, c8_1, c10_1, c11_1, c14_1, c22_1, c23_1, c27_1, c12_2, c12_1, c21_2, c21_1, c25_3, c26_3 ---------------------------------------- (97) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (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) min(add(z0, z1)) -> minIter(add(z0, z1), add(z0, z1), 0) minIter(add(z0, z1), z2, z3) -> if_min(le(z0, z3), z1, z2, z3) minIter(nil, add(z0, z1), z2) -> minIter(add(z0, z1), add(z0, z1), s(z2)) if_min(true, z0, z1, z2) -> z2 if_min(false, z0, z1, z2) -> minIter(z0, z1, z2) 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)) 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, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MINITER(add(s(z0), x1), x2, s(z1)) -> c12(IF_MIN(le(z0, z1), x1, x2, s(z1)), LE(s(z0), s(z1))) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) RM(s(z0), add(s(z1), x2)) -> c21(IF_RM(eq(z0, z1), s(z0), add(s(z1), x2)), EQ(s(z0), s(z1))) RM(0, add(0, x2)) -> c21(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c21(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c21(IF_RM(false, s(z0), add(0, x2))) MINSORT(add(z0, z1), x2) -> c25(IF_MINSORT(eq(z0, minIter(add(z0, z1), add(z0, z1), 0)), add(z0, z1), x2), EQ(z0, min(add(z0, z1))), MIN(add(z0, z1))) IF_MINSORT(true, add(z0, add(z1, z2)), x2) -> c26(MINSORT(app(if_rm(eq(z0, z1), z0, add(z1, z2)), x2), nil), APP(rm(z0, add(z1, z2)), x2), RM(z0, add(z1, z2))) IF_MINSORT(true, add(z0, nil), x2) -> c26(MINSORT(app(nil, x2), nil), APP(rm(z0, nil), x2)) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) MINITER(add(s(z0), x1), x2, s(z1)) -> c12(IF_MIN(le(z0, z1), x1, x2, s(z1)), LE(s(z0), s(z1))) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) K tuples: IF_MINSORT(true, add(z0, z1), z2) -> c26(MINSORT(app(rm(z0, z1), z2), nil), APP(rm(z0, z1), z2), RM(z0, z1)) MINSORT(add(z0, z1), z2) -> c25(IF_MINSORT(eq(z0, min(add(z0, z1))), add(z0, z1), z2), EQ(z0, min(add(z0, z1))), MIN(add(z0, z1))) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MIN(add(z0, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) RM(z0, add(z1, z2)) -> c21(IF_RM(eq(z0, z1), z0, add(z1, z2)), EQ(z0, z1)) Defined Rule Symbols: le_2, eq_2, min_1, minIter_3, if_min_4, app_2, rm_2, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, MIN_1, MINITER_3, IF_MIN_4, IF_RM_3, IF_MINSORT_3, RM_2, MINSORT_2 Compound Symbols: c3_1, c6_1, c8_1, c10_1, c11_1, c14_1, c22_1, c23_1, c27_1, c12_2, c12_1, c21_2, c21_1, c25_3, c26_3, c26_2 ---------------------------------------- (99) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MINITER(add(s(z0), x1), x2, s(z1)) -> c12(IF_MIN(le(z0, z1), x1, x2, s(z1)), LE(s(z0), s(z1))) by MINITER(add(s(0), x1), x2, s(z0)) -> c12(IF_MIN(true, x1, x2, s(z0)), LE(s(0), s(z0))) MINITER(add(s(s(z0)), x1), x2, s(0)) -> c12(IF_MIN(false, x1, x2, s(0)), LE(s(s(z0)), s(0))) MINITER(add(s(s(z0)), x1), x2, s(s(z1))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(z1))), LE(s(s(z0)), s(s(z1)))) MINITER(add(s(x0), x1), x2, s(x3)) -> c12(LE(s(x0), s(x3))) ---------------------------------------- (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) min(add(z0, z1)) -> minIter(add(z0, z1), add(z0, z1), 0) minIter(add(z0, z1), z2, z3) -> if_min(le(z0, z3), z1, z2, z3) minIter(nil, add(z0, z1), z2) -> minIter(add(z0, z1), add(z0, z1), s(z2)) if_min(true, z0, z1, z2) -> z2 if_min(false, z0, z1, z2) -> minIter(z0, z1, z2) 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)) 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, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) RM(s(z0), add(s(z1), x2)) -> c21(IF_RM(eq(z0, z1), s(z0), add(s(z1), x2)), EQ(s(z0), s(z1))) RM(0, add(0, x2)) -> c21(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c21(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c21(IF_RM(false, s(z0), add(0, x2))) MINSORT(add(z0, z1), x2) -> c25(IF_MINSORT(eq(z0, minIter(add(z0, z1), add(z0, z1), 0)), add(z0, z1), x2), EQ(z0, min(add(z0, z1))), MIN(add(z0, z1))) IF_MINSORT(true, add(z0, add(z1, z2)), x2) -> c26(MINSORT(app(if_rm(eq(z0, z1), z0, add(z1, z2)), x2), nil), APP(rm(z0, add(z1, z2)), x2), RM(z0, add(z1, z2))) IF_MINSORT(true, add(z0, nil), x2) -> c26(MINSORT(app(nil, x2), nil), APP(rm(z0, nil), x2)) MINITER(add(s(0), x1), x2, s(z0)) -> c12(IF_MIN(true, x1, x2, s(z0)), LE(s(0), s(z0))) MINITER(add(s(s(z0)), x1), x2, s(0)) -> c12(IF_MIN(false, x1, x2, s(0)), LE(s(s(z0)), s(0))) MINITER(add(s(s(z0)), x1), x2, s(s(z1))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(z1))), LE(s(s(z0)), s(s(z1)))) MINITER(add(s(x0), x1), x2, s(x3)) -> c12(LE(s(x0), s(x3))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) MINITER(add(s(0), x1), x2, s(z0)) -> c12(IF_MIN(true, x1, x2, s(z0)), LE(s(0), s(z0))) MINITER(add(s(s(z0)), x1), x2, s(0)) -> c12(IF_MIN(false, x1, x2, s(0)), LE(s(s(z0)), s(0))) MINITER(add(s(s(z0)), x1), x2, s(s(z1))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(z1))), LE(s(s(z0)), s(s(z1)))) MINITER(add(s(x0), x1), x2, s(x3)) -> c12(LE(s(x0), s(x3))) K tuples: IF_MINSORT(true, add(z0, z1), z2) -> c26(MINSORT(app(rm(z0, z1), z2), nil), APP(rm(z0, z1), z2), RM(z0, z1)) MINSORT(add(z0, z1), z2) -> c25(IF_MINSORT(eq(z0, min(add(z0, z1))), add(z0, z1), z2), EQ(z0, min(add(z0, z1))), MIN(add(z0, z1))) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MIN(add(z0, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) RM(z0, add(z1, z2)) -> c21(IF_RM(eq(z0, z1), z0, add(z1, z2)), EQ(z0, z1)) Defined Rule Symbols: le_2, eq_2, min_1, minIter_3, if_min_4, app_2, rm_2, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, MIN_1, MINITER_3, IF_MIN_4, IF_RM_3, IF_MINSORT_3, RM_2, MINSORT_2 Compound Symbols: c3_1, c6_1, c8_1, c10_1, c11_1, c14_1, c22_1, c23_1, c27_1, c12_1, c21_2, c21_1, c25_3, c26_3, c26_2, c12_2 ---------------------------------------- (101) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (102) 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) min(add(z0, z1)) -> minIter(add(z0, z1), add(z0, z1), 0) minIter(add(z0, z1), z2, z3) -> if_min(le(z0, z3), z1, z2, z3) minIter(nil, add(z0, z1), z2) -> minIter(add(z0, z1), add(z0, z1), s(z2)) if_min(true, z0, z1, z2) -> z2 if_min(false, z0, z1, z2) -> minIter(z0, z1, z2) 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)) 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, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) RM(s(z0), add(s(z1), x2)) -> c21(IF_RM(eq(z0, z1), s(z0), add(s(z1), x2)), EQ(s(z0), s(z1))) RM(0, add(0, x2)) -> c21(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c21(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c21(IF_RM(false, s(z0), add(0, x2))) MINSORT(add(z0, z1), x2) -> c25(IF_MINSORT(eq(z0, minIter(add(z0, z1), add(z0, z1), 0)), add(z0, z1), x2), EQ(z0, min(add(z0, z1))), MIN(add(z0, z1))) IF_MINSORT(true, add(z0, add(z1, z2)), x2) -> c26(MINSORT(app(if_rm(eq(z0, z1), z0, add(z1, z2)), x2), nil), APP(rm(z0, add(z1, z2)), x2), RM(z0, add(z1, z2))) IF_MINSORT(true, add(z0, nil), x2) -> c26(MINSORT(app(nil, x2), nil), APP(rm(z0, nil), x2)) MINITER(add(s(s(z0)), x1), x2, s(0)) -> c12(IF_MIN(false, x1, x2, s(0)), LE(s(s(z0)), s(0))) MINITER(add(s(s(z0)), x1), x2, s(s(z1))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(z1))), LE(s(s(z0)), s(s(z1)))) MINITER(add(s(x0), x1), x2, s(x3)) -> c12(LE(s(x0), s(x3))) MINITER(add(s(0), x1), x2, s(z0)) -> c12(LE(s(0), s(z0))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) MINITER(add(s(s(z0)), x1), x2, s(0)) -> c12(IF_MIN(false, x1, x2, s(0)), LE(s(s(z0)), s(0))) MINITER(add(s(s(z0)), x1), x2, s(s(z1))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(z1))), LE(s(s(z0)), s(s(z1)))) MINITER(add(s(x0), x1), x2, s(x3)) -> c12(LE(s(x0), s(x3))) MINITER(add(s(0), x1), x2, s(z0)) -> c12(LE(s(0), s(z0))) K tuples: IF_MINSORT(true, add(z0, z1), z2) -> c26(MINSORT(app(rm(z0, z1), z2), nil), APP(rm(z0, z1), z2), RM(z0, z1)) MINSORT(add(z0, z1), z2) -> c25(IF_MINSORT(eq(z0, min(add(z0, z1))), add(z0, z1), z2), EQ(z0, min(add(z0, z1))), MIN(add(z0, z1))) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MIN(add(z0, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) RM(z0, add(z1, z2)) -> c21(IF_RM(eq(z0, z1), z0, add(z1, z2)), EQ(z0, z1)) Defined Rule Symbols: le_2, eq_2, min_1, minIter_3, if_min_4, app_2, rm_2, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, MIN_1, MINITER_3, IF_MIN_4, IF_RM_3, IF_MINSORT_3, RM_2, MINSORT_2 Compound Symbols: c3_1, c6_1, c8_1, c10_1, c11_1, c14_1, c22_1, c23_1, c27_1, c12_1, c21_2, c21_1, c25_3, c26_3, c26_2, c12_2 ---------------------------------------- (103) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace RM(s(z0), add(s(z1), x2)) -> c21(IF_RM(eq(z0, z1), s(z0), add(s(z1), x2)), EQ(s(z0), s(z1))) by RM(s(0), add(s(0), x2)) -> c21(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c21(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c21(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)) -> c21(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)) -> c21(EQ(s(x0), s(x1))) ---------------------------------------- (104) 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) min(add(z0, z1)) -> minIter(add(z0, z1), add(z0, z1), 0) minIter(add(z0, z1), z2, z3) -> if_min(le(z0, z3), z1, z2, z3) minIter(nil, add(z0, z1), z2) -> minIter(add(z0, z1), add(z0, z1), s(z2)) if_min(true, z0, z1, z2) -> z2 if_min(false, z0, z1, z2) -> minIter(z0, z1, z2) 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)) 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, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) RM(0, add(0, x2)) -> c21(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c21(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c21(IF_RM(false, s(z0), add(0, x2))) MINSORT(add(z0, z1), x2) -> c25(IF_MINSORT(eq(z0, minIter(add(z0, z1), add(z0, z1), 0)), add(z0, z1), x2), EQ(z0, min(add(z0, z1))), MIN(add(z0, z1))) IF_MINSORT(true, add(z0, add(z1, z2)), x2) -> c26(MINSORT(app(if_rm(eq(z0, z1), z0, add(z1, z2)), x2), nil), APP(rm(z0, add(z1, z2)), x2), RM(z0, add(z1, z2))) IF_MINSORT(true, add(z0, nil), x2) -> c26(MINSORT(app(nil, x2), nil), APP(rm(z0, nil), x2)) MINITER(add(s(s(z0)), x1), x2, s(0)) -> c12(IF_MIN(false, x1, x2, s(0)), LE(s(s(z0)), s(0))) MINITER(add(s(s(z0)), x1), x2, s(s(z1))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(z1))), LE(s(s(z0)), s(s(z1)))) MINITER(add(s(x0), x1), x2, s(x3)) -> c12(LE(s(x0), s(x3))) MINITER(add(s(0), x1), x2, s(z0)) -> c12(LE(s(0), s(z0))) RM(s(0), add(s(0), x2)) -> c21(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c21(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c21(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)) -> c21(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)) -> c21(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)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) MINITER(add(s(s(z0)), x1), x2, s(0)) -> c12(IF_MIN(false, x1, x2, s(0)), LE(s(s(z0)), s(0))) MINITER(add(s(s(z0)), x1), x2, s(s(z1))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(z1))), LE(s(s(z0)), s(s(z1)))) MINITER(add(s(x0), x1), x2, s(x3)) -> c12(LE(s(x0), s(x3))) MINITER(add(s(0), x1), x2, s(z0)) -> c12(LE(s(0), s(z0))) K tuples: IF_MINSORT(true, add(z0, z1), z2) -> c26(MINSORT(app(rm(z0, z1), z2), nil), APP(rm(z0, z1), z2), RM(z0, z1)) MINSORT(add(z0, z1), z2) -> c25(IF_MINSORT(eq(z0, min(add(z0, z1))), add(z0, z1), z2), EQ(z0, min(add(z0, z1))), MIN(add(z0, z1))) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MIN(add(z0, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) RM(z0, add(z1, z2)) -> c21(IF_RM(eq(z0, z1), z0, add(z1, z2)), EQ(z0, z1)) Defined Rule Symbols: le_2, eq_2, min_1, minIter_3, if_min_4, app_2, rm_2, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, MIN_1, MINITER_3, IF_MIN_4, IF_RM_3, IF_MINSORT_3, RM_2, MINSORT_2 Compound Symbols: c3_1, c6_1, c8_1, c10_1, c11_1, c14_1, c22_1, c23_1, c27_1, c12_1, c21_1, c25_3, c26_3, c26_2, c12_2, c21_2 ---------------------------------------- (105) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MINSORT(add(z0, z1), x2) -> c25(IF_MINSORT(eq(z0, minIter(add(z0, z1), add(z0, z1), 0)), add(z0, z1), x2), EQ(z0, min(add(z0, z1))), MIN(add(z0, z1))) by MINSORT(add(z0, z1), x2) -> c25(IF_MINSORT(eq(z0, if_min(le(z0, 0), z1, add(z0, z1), 0)), add(z0, z1), x2), EQ(z0, min(add(z0, z1))), MIN(add(z0, z1))) MINSORT(add(x0, x1), x2) -> c25(EQ(x0, min(add(x0, x1))), MIN(add(x0, x1))) ---------------------------------------- (106) 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) min(add(z0, z1)) -> minIter(add(z0, z1), add(z0, z1), 0) minIter(add(z0, z1), z2, z3) -> if_min(le(z0, z3), z1, z2, z3) minIter(nil, add(z0, z1), z2) -> minIter(add(z0, z1), add(z0, z1), s(z2)) if_min(true, z0, z1, z2) -> z2 if_min(false, z0, z1, z2) -> minIter(z0, z1, z2) 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)) 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, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) RM(0, add(0, x2)) -> c21(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c21(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c21(IF_RM(false, s(z0), add(0, x2))) IF_MINSORT(true, add(z0, add(z1, z2)), x2) -> c26(MINSORT(app(if_rm(eq(z0, z1), z0, add(z1, z2)), x2), nil), APP(rm(z0, add(z1, z2)), x2), RM(z0, add(z1, z2))) IF_MINSORT(true, add(z0, nil), x2) -> c26(MINSORT(app(nil, x2), nil), APP(rm(z0, nil), x2)) MINITER(add(s(s(z0)), x1), x2, s(0)) -> c12(IF_MIN(false, x1, x2, s(0)), LE(s(s(z0)), s(0))) MINITER(add(s(s(z0)), x1), x2, s(s(z1))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(z1))), LE(s(s(z0)), s(s(z1)))) MINITER(add(s(x0), x1), x2, s(x3)) -> c12(LE(s(x0), s(x3))) MINITER(add(s(0), x1), x2, s(z0)) -> c12(LE(s(0), s(z0))) RM(s(0), add(s(0), x2)) -> c21(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c21(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c21(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)) -> c21(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)) -> c21(EQ(s(x0), s(x1))) MINSORT(add(z0, z1), x2) -> c25(IF_MINSORT(eq(z0, if_min(le(z0, 0), z1, add(z0, z1), 0)), add(z0, z1), x2), EQ(z0, min(add(z0, z1))), MIN(add(z0, z1))) MINSORT(add(x0, x1), x2) -> c25(EQ(x0, min(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)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) MINITER(add(s(s(z0)), x1), x2, s(0)) -> c12(IF_MIN(false, x1, x2, s(0)), LE(s(s(z0)), s(0))) MINITER(add(s(s(z0)), x1), x2, s(s(z1))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(z1))), LE(s(s(z0)), s(s(z1)))) MINITER(add(s(x0), x1), x2, s(x3)) -> c12(LE(s(x0), s(x3))) MINITER(add(s(0), x1), x2, s(z0)) -> c12(LE(s(0), s(z0))) K tuples: IF_MINSORT(true, add(z0, z1), z2) -> c26(MINSORT(app(rm(z0, z1), z2), nil), APP(rm(z0, z1), z2), RM(z0, z1)) MINSORT(add(z0, z1), z2) -> c25(IF_MINSORT(eq(z0, min(add(z0, z1))), add(z0, z1), z2), EQ(z0, min(add(z0, z1))), MIN(add(z0, z1))) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MIN(add(z0, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) RM(z0, add(z1, z2)) -> c21(IF_RM(eq(z0, z1), z0, add(z1, z2)), EQ(z0, z1)) Defined Rule Symbols: le_2, eq_2, min_1, minIter_3, if_min_4, app_2, rm_2, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, MIN_1, MINITER_3, IF_MIN_4, IF_RM_3, IF_MINSORT_3, RM_2, MINSORT_2 Compound Symbols: c3_1, c6_1, c8_1, c10_1, c11_1, c14_1, c22_1, c23_1, c27_1, c12_1, c21_1, c26_3, c26_2, c12_2, c21_2, c25_3, c25_2 ---------------------------------------- (107) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (108) 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) min(add(z0, z1)) -> minIter(add(z0, z1), add(z0, z1), 0) minIter(add(z0, z1), z2, z3) -> if_min(le(z0, z3), z1, z2, z3) minIter(nil, add(z0, z1), z2) -> minIter(add(z0, z1), add(z0, z1), s(z2)) if_min(true, z0, z1, z2) -> z2 if_min(false, z0, z1, z2) -> minIter(z0, z1, z2) 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)) 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, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) RM(0, add(0, x2)) -> c21(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c21(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c21(IF_RM(false, s(z0), add(0, x2))) IF_MINSORT(true, add(z0, add(z1, z2)), x2) -> c26(MINSORT(app(if_rm(eq(z0, z1), z0, add(z1, z2)), x2), nil), APP(rm(z0, add(z1, z2)), x2), RM(z0, add(z1, z2))) IF_MINSORT(true, add(z0, nil), x2) -> c26(MINSORT(app(nil, x2), nil), APP(rm(z0, nil), x2)) MINITER(add(s(s(z0)), x1), x2, s(0)) -> c12(IF_MIN(false, x1, x2, s(0)), LE(s(s(z0)), s(0))) MINITER(add(s(s(z0)), x1), x2, s(s(z1))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(z1))), LE(s(s(z0)), s(s(z1)))) MINITER(add(s(x0), x1), x2, s(x3)) -> c12(LE(s(x0), s(x3))) MINITER(add(s(0), x1), x2, s(z0)) -> c12(LE(s(0), s(z0))) RM(s(0), add(s(0), x2)) -> c21(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c21(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c21(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)) -> c21(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)) -> c21(EQ(s(x0), s(x1))) MINSORT(add(z0, z1), x2) -> c25(IF_MINSORT(eq(z0, if_min(le(z0, 0), z1, add(z0, z1), 0)), add(z0, z1), x2), EQ(z0, min(add(z0, z1))), MIN(add(z0, z1))) MINSORT(add(x0, x1), x2) -> c(EQ(x0, min(add(x0, x1)))) MINSORT(add(x0, x1), x2) -> c(MIN(add(x0, x1))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) MINITER(add(s(s(z0)), x1), x2, s(0)) -> c12(IF_MIN(false, x1, x2, s(0)), LE(s(s(z0)), s(0))) MINITER(add(s(s(z0)), x1), x2, s(s(z1))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(z1))), LE(s(s(z0)), s(s(z1)))) MINITER(add(s(x0), x1), x2, s(x3)) -> c12(LE(s(x0), s(x3))) MINITER(add(s(0), x1), x2, s(z0)) -> c12(LE(s(0), s(z0))) K tuples: IF_MINSORT(true, add(z0, z1), z2) -> c26(MINSORT(app(rm(z0, z1), z2), nil), APP(rm(z0, z1), z2), RM(z0, z1)) MINSORT(add(z0, z1), z2) -> c25(IF_MINSORT(eq(z0, min(add(z0, z1))), add(z0, z1), z2), EQ(z0, min(add(z0, z1))), MIN(add(z0, z1))) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MIN(add(z0, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) RM(z0, add(z1, z2)) -> c21(IF_RM(eq(z0, z1), z0, add(z1, z2)), EQ(z0, z1)) Defined Rule Symbols: le_2, eq_2, min_1, minIter_3, if_min_4, app_2, rm_2, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, MIN_1, MINITER_3, IF_MIN_4, IF_RM_3, IF_MINSORT_3, RM_2, MINSORT_2 Compound Symbols: c3_1, c6_1, c8_1, c10_1, c11_1, c14_1, c22_1, c23_1, c27_1, c12_1, c21_1, c26_3, c26_2, c12_2, c21_2, c25_3, c_1 ---------------------------------------- (109) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF_MINSORT(true, add(z0, add(z1, z2)), x2) -> c26(MINSORT(app(if_rm(eq(z0, z1), z0, add(z1, z2)), x2), nil), APP(rm(z0, add(z1, z2)), x2), RM(z0, add(z1, z2))) by IF_MINSORT(true, add(0, add(0, x2)), x3) -> c26(MINSORT(app(if_rm(true, 0, add(0, x2)), x3), nil), APP(rm(0, add(0, x2)), x3), RM(0, add(0, x2))) IF_MINSORT(true, add(0, add(s(z0), x2)), x3) -> c26(MINSORT(app(if_rm(false, 0, add(s(z0), x2)), x3), nil), APP(rm(0, add(s(z0), x2)), x3), RM(0, add(s(z0), x2))) IF_MINSORT(true, add(s(z0), add(0, x2)), x3) -> c26(MINSORT(app(if_rm(false, s(z0), add(0, x2)), x3), nil), APP(rm(s(z0), add(0, x2)), x3), RM(s(z0), add(0, x2))) IF_MINSORT(true, add(s(z0), add(s(z1), x2)), x3) -> c26(MINSORT(app(if_rm(eq(z0, z1), s(z0), add(s(z1), x2)), x3), nil), APP(rm(s(z0), add(s(z1), x2)), x3), RM(s(z0), add(s(z1), x2))) IF_MINSORT(true, add(x0, add(x1, x2)), x3) -> c26(APP(rm(x0, add(x1, x2)), x3)) ---------------------------------------- (110) 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) min(add(z0, z1)) -> minIter(add(z0, z1), add(z0, z1), 0) minIter(add(z0, z1), z2, z3) -> if_min(le(z0, z3), z1, z2, z3) minIter(nil, add(z0, z1), z2) -> minIter(add(z0, z1), add(z0, z1), s(z2)) if_min(true, z0, z1, z2) -> z2 if_min(false, z0, z1, z2) -> minIter(z0, z1, z2) 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)) 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, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) RM(0, add(0, x2)) -> c21(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c21(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c21(IF_RM(false, s(z0), add(0, x2))) IF_MINSORT(true, add(z0, nil), x2) -> c26(MINSORT(app(nil, x2), nil), APP(rm(z0, nil), x2)) MINITER(add(s(s(z0)), x1), x2, s(0)) -> c12(IF_MIN(false, x1, x2, s(0)), LE(s(s(z0)), s(0))) MINITER(add(s(s(z0)), x1), x2, s(s(z1))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(z1))), LE(s(s(z0)), s(s(z1)))) MINITER(add(s(x0), x1), x2, s(x3)) -> c12(LE(s(x0), s(x3))) MINITER(add(s(0), x1), x2, s(z0)) -> c12(LE(s(0), s(z0))) RM(s(0), add(s(0), x2)) -> c21(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c21(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c21(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)) -> c21(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)) -> c21(EQ(s(x0), s(x1))) MINSORT(add(z0, z1), x2) -> c25(IF_MINSORT(eq(z0, if_min(le(z0, 0), z1, add(z0, z1), 0)), add(z0, z1), x2), EQ(z0, min(add(z0, z1))), MIN(add(z0, z1))) MINSORT(add(x0, x1), x2) -> c(EQ(x0, min(add(x0, x1)))) MINSORT(add(x0, x1), x2) -> c(MIN(add(x0, x1))) IF_MINSORT(true, add(0, add(0, x2)), x3) -> c26(MINSORT(app(if_rm(true, 0, add(0, x2)), x3), nil), APP(rm(0, add(0, x2)), x3), RM(0, add(0, x2))) IF_MINSORT(true, add(0, add(s(z0), x2)), x3) -> c26(MINSORT(app(if_rm(false, 0, add(s(z0), x2)), x3), nil), APP(rm(0, add(s(z0), x2)), x3), RM(0, add(s(z0), x2))) IF_MINSORT(true, add(s(z0), add(0, x2)), x3) -> c26(MINSORT(app(if_rm(false, s(z0), add(0, x2)), x3), nil), APP(rm(s(z0), add(0, x2)), x3), RM(s(z0), add(0, x2))) IF_MINSORT(true, add(s(z0), add(s(z1), x2)), x3) -> c26(MINSORT(app(if_rm(eq(z0, z1), s(z0), add(s(z1), x2)), x3), nil), APP(rm(s(z0), add(s(z1), x2)), x3), RM(s(z0), add(s(z1), x2))) IF_MINSORT(true, add(x0, add(x1, x2)), x3) -> c26(APP(rm(x0, add(x1, x2)), x3)) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) MINITER(add(s(s(z0)), x1), x2, s(0)) -> c12(IF_MIN(false, x1, x2, s(0)), LE(s(s(z0)), s(0))) MINITER(add(s(s(z0)), x1), x2, s(s(z1))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(z1))), LE(s(s(z0)), s(s(z1)))) MINITER(add(s(x0), x1), x2, s(x3)) -> c12(LE(s(x0), s(x3))) MINITER(add(s(0), x1), x2, s(z0)) -> c12(LE(s(0), s(z0))) K tuples: IF_MINSORT(true, add(z0, z1), z2) -> c26(MINSORT(app(rm(z0, z1), z2), nil), APP(rm(z0, z1), z2), RM(z0, z1)) MINSORT(add(z0, z1), z2) -> c25(IF_MINSORT(eq(z0, min(add(z0, z1))), add(z0, z1), z2), EQ(z0, min(add(z0, z1))), MIN(add(z0, z1))) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MIN(add(z0, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) RM(z0, add(z1, z2)) -> c21(IF_RM(eq(z0, z1), z0, add(z1, z2)), EQ(z0, z1)) Defined Rule Symbols: le_2, eq_2, min_1, minIter_3, if_min_4, app_2, rm_2, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, MIN_1, MINITER_3, IF_MIN_4, IF_RM_3, IF_MINSORT_3, RM_2, MINSORT_2 Compound Symbols: c3_1, c6_1, c8_1, c10_1, c11_1, c14_1, c22_1, c23_1, c27_1, c12_1, c21_1, c26_2, c12_2, c21_2, c25_3, c_1, c26_3, c26_1 ---------------------------------------- (111) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF_MINSORT(true, add(z0, nil), x2) -> c26(MINSORT(app(nil, x2), nil), APP(rm(z0, nil), x2)) by IF_MINSORT(true, add(x0, nil), z0) -> c26(MINSORT(z0, nil), APP(rm(x0, nil), z0)) ---------------------------------------- (112) 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) min(add(z0, z1)) -> minIter(add(z0, z1), add(z0, z1), 0) minIter(add(z0, z1), z2, z3) -> if_min(le(z0, z3), z1, z2, z3) minIter(nil, add(z0, z1), z2) -> minIter(add(z0, z1), add(z0, z1), s(z2)) if_min(true, z0, z1, z2) -> z2 if_min(false, z0, z1, z2) -> minIter(z0, z1, z2) 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)) 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, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) RM(0, add(0, x2)) -> c21(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c21(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c21(IF_RM(false, s(z0), add(0, x2))) MINITER(add(s(s(z0)), x1), x2, s(0)) -> c12(IF_MIN(false, x1, x2, s(0)), LE(s(s(z0)), s(0))) MINITER(add(s(s(z0)), x1), x2, s(s(z1))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(z1))), LE(s(s(z0)), s(s(z1)))) MINITER(add(s(x0), x1), x2, s(x3)) -> c12(LE(s(x0), s(x3))) MINITER(add(s(0), x1), x2, s(z0)) -> c12(LE(s(0), s(z0))) RM(s(0), add(s(0), x2)) -> c21(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c21(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c21(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)) -> c21(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)) -> c21(EQ(s(x0), s(x1))) MINSORT(add(z0, z1), x2) -> c25(IF_MINSORT(eq(z0, if_min(le(z0, 0), z1, add(z0, z1), 0)), add(z0, z1), x2), EQ(z0, min(add(z0, z1))), MIN(add(z0, z1))) MINSORT(add(x0, x1), x2) -> c(EQ(x0, min(add(x0, x1)))) MINSORT(add(x0, x1), x2) -> c(MIN(add(x0, x1))) IF_MINSORT(true, add(0, add(0, x2)), x3) -> c26(MINSORT(app(if_rm(true, 0, add(0, x2)), x3), nil), APP(rm(0, add(0, x2)), x3), RM(0, add(0, x2))) IF_MINSORT(true, add(0, add(s(z0), x2)), x3) -> c26(MINSORT(app(if_rm(false, 0, add(s(z0), x2)), x3), nil), APP(rm(0, add(s(z0), x2)), x3), RM(0, add(s(z0), x2))) IF_MINSORT(true, add(s(z0), add(0, x2)), x3) -> c26(MINSORT(app(if_rm(false, s(z0), add(0, x2)), x3), nil), APP(rm(s(z0), add(0, x2)), x3), RM(s(z0), add(0, x2))) IF_MINSORT(true, add(s(z0), add(s(z1), x2)), x3) -> c26(MINSORT(app(if_rm(eq(z0, z1), s(z0), add(s(z1), x2)), x3), nil), APP(rm(s(z0), add(s(z1), x2)), x3), RM(s(z0), add(s(z1), x2))) IF_MINSORT(true, add(x0, add(x1, x2)), x3) -> c26(APP(rm(x0, add(x1, x2)), x3)) IF_MINSORT(true, add(x0, nil), z0) -> c26(MINSORT(z0, nil), APP(rm(x0, nil), z0)) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) MINITER(add(s(s(z0)), x1), x2, s(0)) -> c12(IF_MIN(false, x1, x2, s(0)), LE(s(s(z0)), s(0))) MINITER(add(s(s(z0)), x1), x2, s(s(z1))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(z1))), LE(s(s(z0)), s(s(z1)))) MINITER(add(s(x0), x1), x2, s(x3)) -> c12(LE(s(x0), s(x3))) MINITER(add(s(0), x1), x2, s(z0)) -> c12(LE(s(0), s(z0))) K tuples: IF_MINSORT(true, add(z0, z1), z2) -> c26(MINSORT(app(rm(z0, z1), z2), nil), APP(rm(z0, z1), z2), RM(z0, z1)) MINSORT(add(z0, z1), z2) -> c25(IF_MINSORT(eq(z0, min(add(z0, z1))), add(z0, z1), z2), EQ(z0, min(add(z0, z1))), MIN(add(z0, z1))) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MIN(add(z0, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) RM(z0, add(z1, z2)) -> c21(IF_RM(eq(z0, z1), z0, add(z1, z2)), EQ(z0, z1)) Defined Rule Symbols: le_2, eq_2, min_1, minIter_3, if_min_4, app_2, rm_2, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, MIN_1, MINITER_3, IF_MIN_4, IF_RM_3, IF_MINSORT_3, RM_2, MINSORT_2 Compound Symbols: c3_1, c6_1, c8_1, c10_1, c11_1, c14_1, c22_1, c23_1, c27_1, c12_1, c21_1, c12_2, c21_2, c25_3, c_1, c26_3, c26_1, c26_2 ---------------------------------------- (113) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace MINSORT(add(z0, z1), x2) -> c25(IF_MINSORT(eq(z0, if_min(le(z0, 0), z1, add(z0, z1), 0)), add(z0, z1), x2), EQ(z0, min(add(z0, z1))), MIN(add(z0, z1))) by MINSORT(add(z0, z1), z2) -> c25(IF_MINSORT(eq(z0, if_min(le(z0, 0), z1, add(z0, z1), 0)), add(z0, z1), z2), EQ(z0, minIter(add(z0, z1), add(z0, z1), 0)), MIN(add(z0, z1))) ---------------------------------------- (114) 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) min(add(z0, z1)) -> minIter(add(z0, z1), add(z0, z1), 0) minIter(add(z0, z1), z2, z3) -> if_min(le(z0, z3), z1, z2, z3) minIter(nil, add(z0, z1), z2) -> minIter(add(z0, z1), add(z0, z1), s(z2)) if_min(true, z0, z1, z2) -> z2 if_min(false, z0, z1, z2) -> minIter(z0, z1, z2) 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)) 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, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) RM(0, add(0, x2)) -> c21(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c21(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c21(IF_RM(false, s(z0), add(0, x2))) MINITER(add(s(s(z0)), x1), x2, s(0)) -> c12(IF_MIN(false, x1, x2, s(0)), LE(s(s(z0)), s(0))) MINITER(add(s(s(z0)), x1), x2, s(s(z1))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(z1))), LE(s(s(z0)), s(s(z1)))) MINITER(add(s(x0), x1), x2, s(x3)) -> c12(LE(s(x0), s(x3))) MINITER(add(s(0), x1), x2, s(z0)) -> c12(LE(s(0), s(z0))) RM(s(0), add(s(0), x2)) -> c21(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c21(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c21(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)) -> c21(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)) -> c21(EQ(s(x0), s(x1))) MINSORT(add(x0, x1), x2) -> c(EQ(x0, min(add(x0, x1)))) MINSORT(add(x0, x1), x2) -> c(MIN(add(x0, x1))) IF_MINSORT(true, add(0, add(0, x2)), x3) -> c26(MINSORT(app(if_rm(true, 0, add(0, x2)), x3), nil), APP(rm(0, add(0, x2)), x3), RM(0, add(0, x2))) IF_MINSORT(true, add(0, add(s(z0), x2)), x3) -> c26(MINSORT(app(if_rm(false, 0, add(s(z0), x2)), x3), nil), APP(rm(0, add(s(z0), x2)), x3), RM(0, add(s(z0), x2))) IF_MINSORT(true, add(s(z0), add(0, x2)), x3) -> c26(MINSORT(app(if_rm(false, s(z0), add(0, x2)), x3), nil), APP(rm(s(z0), add(0, x2)), x3), RM(s(z0), add(0, x2))) IF_MINSORT(true, add(s(z0), add(s(z1), x2)), x3) -> c26(MINSORT(app(if_rm(eq(z0, z1), s(z0), add(s(z1), x2)), x3), nil), APP(rm(s(z0), add(s(z1), x2)), x3), RM(s(z0), add(s(z1), x2))) IF_MINSORT(true, add(x0, add(x1, x2)), x3) -> c26(APP(rm(x0, add(x1, x2)), x3)) IF_MINSORT(true, add(x0, nil), z0) -> c26(MINSORT(z0, nil), APP(rm(x0, nil), z0)) MINSORT(add(z0, z1), z2) -> c25(IF_MINSORT(eq(z0, if_min(le(z0, 0), z1, add(z0, z1), 0)), add(z0, z1), z2), EQ(z0, minIter(add(z0, z1), add(z0, z1), 0)), MIN(add(z0, z1))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) MINITER(add(s(s(z0)), x1), x2, s(0)) -> c12(IF_MIN(false, x1, x2, s(0)), LE(s(s(z0)), s(0))) MINITER(add(s(s(z0)), x1), x2, s(s(z1))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(z1))), LE(s(s(z0)), s(s(z1)))) MINITER(add(s(x0), x1), x2, s(x3)) -> c12(LE(s(x0), s(x3))) MINITER(add(s(0), x1), x2, s(z0)) -> c12(LE(s(0), s(z0))) K tuples: IF_MINSORT(true, add(z0, z1), z2) -> c26(MINSORT(app(rm(z0, z1), z2), nil), APP(rm(z0, z1), z2), RM(z0, z1)) MINSORT(add(z0, z1), z2) -> c25(IF_MINSORT(eq(z0, min(add(z0, z1))), add(z0, z1), z2), EQ(z0, min(add(z0, z1))), MIN(add(z0, z1))) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MIN(add(z0, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) RM(z0, add(z1, z2)) -> c21(IF_RM(eq(z0, z1), z0, add(z1, z2)), EQ(z0, z1)) Defined Rule Symbols: le_2, eq_2, min_1, minIter_3, if_min_4, app_2, rm_2, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, MIN_1, MINITER_3, IF_MIN_4, IF_RM_3, IF_MINSORT_3, RM_2, MINSORT_2 Compound Symbols: c3_1, c6_1, c8_1, c10_1, c11_1, c14_1, c22_1, c23_1, c27_1, c12_1, c21_1, c12_2, c21_2, c_1, c26_3, c26_1, c26_2, c25_3 ---------------------------------------- (115) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace MINSORT(add(x0, x1), x2) -> c(EQ(x0, min(add(x0, x1)))) by MINSORT(add(z0, z1), z2) -> c(EQ(z0, minIter(add(z0, z1), add(z0, z1), 0))) ---------------------------------------- (116) 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) min(add(z0, z1)) -> minIter(add(z0, z1), add(z0, z1), 0) minIter(add(z0, z1), z2, z3) -> if_min(le(z0, z3), z1, z2, z3) minIter(nil, add(z0, z1), z2) -> minIter(add(z0, z1), add(z0, z1), s(z2)) if_min(true, z0, z1, z2) -> z2 if_min(false, z0, z1, z2) -> minIter(z0, z1, z2) 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)) 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, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) RM(0, add(0, x2)) -> c21(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c21(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c21(IF_RM(false, s(z0), add(0, x2))) MINITER(add(s(s(z0)), x1), x2, s(0)) -> c12(IF_MIN(false, x1, x2, s(0)), LE(s(s(z0)), s(0))) MINITER(add(s(s(z0)), x1), x2, s(s(z1))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(z1))), LE(s(s(z0)), s(s(z1)))) MINITER(add(s(x0), x1), x2, s(x3)) -> c12(LE(s(x0), s(x3))) MINITER(add(s(0), x1), x2, s(z0)) -> c12(LE(s(0), s(z0))) RM(s(0), add(s(0), x2)) -> c21(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c21(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c21(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)) -> c21(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)) -> c21(EQ(s(x0), s(x1))) MINSORT(add(x0, x1), x2) -> c(MIN(add(x0, x1))) IF_MINSORT(true, add(0, add(0, x2)), x3) -> c26(MINSORT(app(if_rm(true, 0, add(0, x2)), x3), nil), APP(rm(0, add(0, x2)), x3), RM(0, add(0, x2))) IF_MINSORT(true, add(0, add(s(z0), x2)), x3) -> c26(MINSORT(app(if_rm(false, 0, add(s(z0), x2)), x3), nil), APP(rm(0, add(s(z0), x2)), x3), RM(0, add(s(z0), x2))) IF_MINSORT(true, add(s(z0), add(0, x2)), x3) -> c26(MINSORT(app(if_rm(false, s(z0), add(0, x2)), x3), nil), APP(rm(s(z0), add(0, x2)), x3), RM(s(z0), add(0, x2))) IF_MINSORT(true, add(s(z0), add(s(z1), x2)), x3) -> c26(MINSORT(app(if_rm(eq(z0, z1), s(z0), add(s(z1), x2)), x3), nil), APP(rm(s(z0), add(s(z1), x2)), x3), RM(s(z0), add(s(z1), x2))) IF_MINSORT(true, add(x0, add(x1, x2)), x3) -> c26(APP(rm(x0, add(x1, x2)), x3)) IF_MINSORT(true, add(x0, nil), z0) -> c26(MINSORT(z0, nil), APP(rm(x0, nil), z0)) MINSORT(add(z0, z1), z2) -> c25(IF_MINSORT(eq(z0, if_min(le(z0, 0), z1, add(z0, z1), 0)), add(z0, z1), z2), EQ(z0, minIter(add(z0, z1), add(z0, z1), 0)), MIN(add(z0, z1))) MINSORT(add(z0, z1), z2) -> c(EQ(z0, minIter(add(z0, z1), add(z0, z1), 0))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) MINITER(add(s(s(z0)), x1), x2, s(0)) -> c12(IF_MIN(false, x1, x2, s(0)), LE(s(s(z0)), s(0))) MINITER(add(s(s(z0)), x1), x2, s(s(z1))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(z1))), LE(s(s(z0)), s(s(z1)))) MINITER(add(s(x0), x1), x2, s(x3)) -> c12(LE(s(x0), s(x3))) MINITER(add(s(0), x1), x2, s(z0)) -> c12(LE(s(0), s(z0))) K tuples: IF_MINSORT(true, add(z0, z1), z2) -> c26(MINSORT(app(rm(z0, z1), z2), nil), APP(rm(z0, z1), z2), RM(z0, z1)) MINSORT(add(z0, z1), z2) -> c25(IF_MINSORT(eq(z0, min(add(z0, z1))), add(z0, z1), z2), EQ(z0, min(add(z0, z1))), MIN(add(z0, z1))) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MIN(add(z0, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) RM(z0, add(z1, z2)) -> c21(IF_RM(eq(z0, z1), z0, add(z1, z2)), EQ(z0, z1)) Defined Rule Symbols: le_2, eq_2, min_1, minIter_3, if_min_4, app_2, rm_2, if_rm_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, MIN_1, MINITER_3, IF_MIN_4, IF_RM_3, IF_MINSORT_3, RM_2, MINSORT_2 Compound Symbols: c3_1, c6_1, c8_1, c10_1, c11_1, c14_1, c22_1, c23_1, c27_1, c12_1, c21_1, c12_2, c21_2, c_1, c26_3, c26_1, c26_2, c25_3 ---------------------------------------- (117) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: min(add(z0, z1)) -> minIter(add(z0, z1), add(z0, z1), 0) ---------------------------------------- (118) 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) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) if_min(true, z0, z1, z2) -> z2 if_min(false, z0, z1, z2) -> minIter(z0, z1, z2) minIter(add(z0, z1), z2, z3) -> if_min(le(z0, z3), z1, z2, z3) minIter(nil, add(z0, z1), z2) -> minIter(add(z0, z1), add(z0, z1), s(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, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) RM(0, add(0, x2)) -> c21(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c21(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c21(IF_RM(false, s(z0), add(0, x2))) MINITER(add(s(s(z0)), x1), x2, s(0)) -> c12(IF_MIN(false, x1, x2, s(0)), LE(s(s(z0)), s(0))) MINITER(add(s(s(z0)), x1), x2, s(s(z1))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(z1))), LE(s(s(z0)), s(s(z1)))) MINITER(add(s(x0), x1), x2, s(x3)) -> c12(LE(s(x0), s(x3))) MINITER(add(s(0), x1), x2, s(z0)) -> c12(LE(s(0), s(z0))) RM(s(0), add(s(0), x2)) -> c21(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c21(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c21(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)) -> c21(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)) -> c21(EQ(s(x0), s(x1))) MINSORT(add(x0, x1), x2) -> c(MIN(add(x0, x1))) IF_MINSORT(true, add(0, add(0, x2)), x3) -> c26(MINSORT(app(if_rm(true, 0, add(0, x2)), x3), nil), APP(rm(0, add(0, x2)), x3), RM(0, add(0, x2))) IF_MINSORT(true, add(0, add(s(z0), x2)), x3) -> c26(MINSORT(app(if_rm(false, 0, add(s(z0), x2)), x3), nil), APP(rm(0, add(s(z0), x2)), x3), RM(0, add(s(z0), x2))) IF_MINSORT(true, add(s(z0), add(0, x2)), x3) -> c26(MINSORT(app(if_rm(false, s(z0), add(0, x2)), x3), nil), APP(rm(s(z0), add(0, x2)), x3), RM(s(z0), add(0, x2))) IF_MINSORT(true, add(s(z0), add(s(z1), x2)), x3) -> c26(MINSORT(app(if_rm(eq(z0, z1), s(z0), add(s(z1), x2)), x3), nil), APP(rm(s(z0), add(s(z1), x2)), x3), RM(s(z0), add(s(z1), x2))) IF_MINSORT(true, add(x0, add(x1, x2)), x3) -> c26(APP(rm(x0, add(x1, x2)), x3)) IF_MINSORT(true, add(x0, nil), z0) -> c26(MINSORT(z0, nil), APP(rm(x0, nil), z0)) MINSORT(add(z0, z1), z2) -> c25(IF_MINSORT(eq(z0, if_min(le(z0, 0), z1, add(z0, z1), 0)), add(z0, z1), z2), EQ(z0, minIter(add(z0, z1), add(z0, z1), 0)), MIN(add(z0, z1))) MINSORT(add(z0, z1), z2) -> c(EQ(z0, minIter(add(z0, z1), add(z0, z1), 0))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) MINITER(add(s(s(z0)), x1), x2, s(0)) -> c12(IF_MIN(false, x1, x2, s(0)), LE(s(s(z0)), s(0))) MINITER(add(s(s(z0)), x1), x2, s(s(z1))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(z1))), LE(s(s(z0)), s(s(z1)))) MINITER(add(s(x0), x1), x2, s(x3)) -> c12(LE(s(x0), s(x3))) MINITER(add(s(0), x1), x2, s(z0)) -> c12(LE(s(0), s(z0))) K tuples: IF_MINSORT(true, add(z0, z1), z2) -> c26(MINSORT(app(rm(z0, z1), z2), nil), APP(rm(z0, z1), z2), RM(z0, z1)) MINSORT(add(z0, z1), z2) -> c25(IF_MINSORT(eq(z0, min(add(z0, z1))), add(z0, z1), z2), EQ(z0, min(add(z0, z1))), MIN(add(z0, z1))) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MIN(add(z0, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) RM(z0, add(z1, z2)) -> c21(IF_RM(eq(z0, z1), z0, add(z1, z2)), EQ(z0, z1)) Defined Rule Symbols: le_2, eq_2, app_2, if_rm_3, rm_2, if_min_4, minIter_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, MIN_1, MINITER_3, IF_MIN_4, IF_RM_3, IF_MINSORT_3, RM_2, MINSORT_2 Compound Symbols: c3_1, c6_1, c8_1, c10_1, c11_1, c14_1, c22_1, c23_1, c27_1, c12_1, c21_1, c12_2, c21_2, c_1, c26_3, c26_1, c26_2, c25_3 ---------------------------------------- (119) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MINITER(add(s(s(z0)), x1), x2, s(s(z1))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(z1))), LE(s(s(z0)), s(s(z1)))) by MINITER(add(s(s(0)), x1), x2, s(s(z0))) -> c12(IF_MIN(true, x1, x2, s(s(z0))), LE(s(s(0)), s(s(z0)))) MINITER(add(s(s(s(z0))), x1), x2, s(s(0))) -> c12(IF_MIN(false, x1, x2, s(s(0))), LE(s(s(s(z0))), s(s(0)))) MINITER(add(s(s(s(z0))), x1), x2, s(s(s(z1)))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(s(z1)))), LE(s(s(s(z0))), s(s(s(z1))))) MINITER(add(s(s(x0)), x1), x2, s(s(x3))) -> c12(LE(s(s(x0)), s(s(x3)))) ---------------------------------------- (120) 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) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) if_min(true, z0, z1, z2) -> z2 if_min(false, z0, z1, z2) -> minIter(z0, z1, z2) minIter(add(z0, z1), z2, z3) -> if_min(le(z0, z3), z1, z2, z3) minIter(nil, add(z0, z1), z2) -> minIter(add(z0, z1), add(z0, z1), s(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, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) RM(0, add(0, x2)) -> c21(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c21(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c21(IF_RM(false, s(z0), add(0, x2))) MINITER(add(s(s(z0)), x1), x2, s(0)) -> c12(IF_MIN(false, x1, x2, s(0)), LE(s(s(z0)), s(0))) MINITER(add(s(x0), x1), x2, s(x3)) -> c12(LE(s(x0), s(x3))) MINITER(add(s(0), x1), x2, s(z0)) -> c12(LE(s(0), s(z0))) RM(s(0), add(s(0), x2)) -> c21(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c21(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c21(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)) -> c21(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)) -> c21(EQ(s(x0), s(x1))) MINSORT(add(x0, x1), x2) -> c(MIN(add(x0, x1))) IF_MINSORT(true, add(0, add(0, x2)), x3) -> c26(MINSORT(app(if_rm(true, 0, add(0, x2)), x3), nil), APP(rm(0, add(0, x2)), x3), RM(0, add(0, x2))) IF_MINSORT(true, add(0, add(s(z0), x2)), x3) -> c26(MINSORT(app(if_rm(false, 0, add(s(z0), x2)), x3), nil), APP(rm(0, add(s(z0), x2)), x3), RM(0, add(s(z0), x2))) IF_MINSORT(true, add(s(z0), add(0, x2)), x3) -> c26(MINSORT(app(if_rm(false, s(z0), add(0, x2)), x3), nil), APP(rm(s(z0), add(0, x2)), x3), RM(s(z0), add(0, x2))) IF_MINSORT(true, add(s(z0), add(s(z1), x2)), x3) -> c26(MINSORT(app(if_rm(eq(z0, z1), s(z0), add(s(z1), x2)), x3), nil), APP(rm(s(z0), add(s(z1), x2)), x3), RM(s(z0), add(s(z1), x2))) IF_MINSORT(true, add(x0, add(x1, x2)), x3) -> c26(APP(rm(x0, add(x1, x2)), x3)) IF_MINSORT(true, add(x0, nil), z0) -> c26(MINSORT(z0, nil), APP(rm(x0, nil), z0)) MINSORT(add(z0, z1), z2) -> c25(IF_MINSORT(eq(z0, if_min(le(z0, 0), z1, add(z0, z1), 0)), add(z0, z1), z2), EQ(z0, minIter(add(z0, z1), add(z0, z1), 0)), MIN(add(z0, z1))) MINSORT(add(z0, z1), z2) -> c(EQ(z0, minIter(add(z0, z1), add(z0, z1), 0))) MINITER(add(s(s(0)), x1), x2, s(s(z0))) -> c12(IF_MIN(true, x1, x2, s(s(z0))), LE(s(s(0)), s(s(z0)))) MINITER(add(s(s(s(z0))), x1), x2, s(s(0))) -> c12(IF_MIN(false, x1, x2, s(s(0))), LE(s(s(s(z0))), s(s(0)))) MINITER(add(s(s(s(z0))), x1), x2, s(s(s(z1)))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(s(z1)))), LE(s(s(s(z0))), s(s(s(z1))))) MINITER(add(s(s(x0)), x1), x2, s(s(x3))) -> c12(LE(s(s(x0)), s(s(x3)))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) MINITER(add(s(s(z0)), x1), x2, s(0)) -> c12(IF_MIN(false, x1, x2, s(0)), LE(s(s(z0)), s(0))) MINITER(add(s(x0), x1), x2, s(x3)) -> c12(LE(s(x0), s(x3))) MINITER(add(s(0), x1), x2, s(z0)) -> c12(LE(s(0), s(z0))) MINITER(add(s(s(0)), x1), x2, s(s(z0))) -> c12(IF_MIN(true, x1, x2, s(s(z0))), LE(s(s(0)), s(s(z0)))) MINITER(add(s(s(s(z0))), x1), x2, s(s(0))) -> c12(IF_MIN(false, x1, x2, s(s(0))), LE(s(s(s(z0))), s(s(0)))) MINITER(add(s(s(s(z0))), x1), x2, s(s(s(z1)))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(s(z1)))), LE(s(s(s(z0))), s(s(s(z1))))) MINITER(add(s(s(x0)), x1), x2, s(s(x3))) -> c12(LE(s(s(x0)), s(s(x3)))) K tuples: IF_MINSORT(true, add(z0, z1), z2) -> c26(MINSORT(app(rm(z0, z1), z2), nil), APP(rm(z0, z1), z2), RM(z0, z1)) MINSORT(add(z0, z1), z2) -> c25(IF_MINSORT(eq(z0, min(add(z0, z1))), add(z0, z1), z2), EQ(z0, min(add(z0, z1))), MIN(add(z0, z1))) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MIN(add(z0, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) RM(z0, add(z1, z2)) -> c21(IF_RM(eq(z0, z1), z0, add(z1, z2)), EQ(z0, z1)) Defined Rule Symbols: le_2, eq_2, app_2, if_rm_3, rm_2, if_min_4, minIter_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, MIN_1, MINITER_3, IF_MIN_4, IF_RM_3, IF_MINSORT_3, RM_2, MINSORT_2 Compound Symbols: c3_1, c6_1, c8_1, c10_1, c11_1, c14_1, c22_1, c23_1, c27_1, c12_1, c21_1, c12_2, c21_2, c_1, c26_3, c26_1, c26_2, c25_3 ---------------------------------------- (121) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (122) 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) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) if_min(true, z0, z1, z2) -> z2 if_min(false, z0, z1, z2) -> minIter(z0, z1, z2) minIter(add(z0, z1), z2, z3) -> if_min(le(z0, z3), z1, z2, z3) minIter(nil, add(z0, z1), z2) -> minIter(add(z0, z1), add(z0, z1), s(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, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) RM(0, add(0, x2)) -> c21(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c21(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c21(IF_RM(false, s(z0), add(0, x2))) MINITER(add(s(s(z0)), x1), x2, s(0)) -> c12(IF_MIN(false, x1, x2, s(0)), LE(s(s(z0)), s(0))) MINITER(add(s(x0), x1), x2, s(x3)) -> c12(LE(s(x0), s(x3))) MINITER(add(s(0), x1), x2, s(z0)) -> c12(LE(s(0), s(z0))) RM(s(0), add(s(0), x2)) -> c21(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c21(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c21(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)) -> c21(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)) -> c21(EQ(s(x0), s(x1))) MINSORT(add(x0, x1), x2) -> c(MIN(add(x0, x1))) IF_MINSORT(true, add(0, add(0, x2)), x3) -> c26(MINSORT(app(if_rm(true, 0, add(0, x2)), x3), nil), APP(rm(0, add(0, x2)), x3), RM(0, add(0, x2))) IF_MINSORT(true, add(0, add(s(z0), x2)), x3) -> c26(MINSORT(app(if_rm(false, 0, add(s(z0), x2)), x3), nil), APP(rm(0, add(s(z0), x2)), x3), RM(0, add(s(z0), x2))) IF_MINSORT(true, add(s(z0), add(0, x2)), x3) -> c26(MINSORT(app(if_rm(false, s(z0), add(0, x2)), x3), nil), APP(rm(s(z0), add(0, x2)), x3), RM(s(z0), add(0, x2))) IF_MINSORT(true, add(s(z0), add(s(z1), x2)), x3) -> c26(MINSORT(app(if_rm(eq(z0, z1), s(z0), add(s(z1), x2)), x3), nil), APP(rm(s(z0), add(s(z1), x2)), x3), RM(s(z0), add(s(z1), x2))) IF_MINSORT(true, add(x0, add(x1, x2)), x3) -> c26(APP(rm(x0, add(x1, x2)), x3)) IF_MINSORT(true, add(x0, nil), z0) -> c26(MINSORT(z0, nil), APP(rm(x0, nil), z0)) MINSORT(add(z0, z1), z2) -> c25(IF_MINSORT(eq(z0, if_min(le(z0, 0), z1, add(z0, z1), 0)), add(z0, z1), z2), EQ(z0, minIter(add(z0, z1), add(z0, z1), 0)), MIN(add(z0, z1))) MINSORT(add(z0, z1), z2) -> c(EQ(z0, minIter(add(z0, z1), add(z0, z1), 0))) MINITER(add(s(s(s(z0))), x1), x2, s(s(0))) -> c12(IF_MIN(false, x1, x2, s(s(0))), LE(s(s(s(z0))), s(s(0)))) MINITER(add(s(s(s(z0))), x1), x2, s(s(s(z1)))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(s(z1)))), LE(s(s(s(z0))), s(s(s(z1))))) MINITER(add(s(s(x0)), x1), x2, s(s(x3))) -> c12(LE(s(s(x0)), s(s(x3)))) MINITER(add(s(s(0)), x1), x2, s(s(z0))) -> c12(LE(s(s(0)), s(s(z0)))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) MINITER(add(s(s(z0)), x1), x2, s(0)) -> c12(IF_MIN(false, x1, x2, s(0)), LE(s(s(z0)), s(0))) MINITER(add(s(x0), x1), x2, s(x3)) -> c12(LE(s(x0), s(x3))) MINITER(add(s(0), x1), x2, s(z0)) -> c12(LE(s(0), s(z0))) MINITER(add(s(s(s(z0))), x1), x2, s(s(0))) -> c12(IF_MIN(false, x1, x2, s(s(0))), LE(s(s(s(z0))), s(s(0)))) MINITER(add(s(s(s(z0))), x1), x2, s(s(s(z1)))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(s(z1)))), LE(s(s(s(z0))), s(s(s(z1))))) MINITER(add(s(s(x0)), x1), x2, s(s(x3))) -> c12(LE(s(s(x0)), s(s(x3)))) MINITER(add(s(s(0)), x1), x2, s(s(z0))) -> c12(LE(s(s(0)), s(s(z0)))) K tuples: IF_MINSORT(true, add(z0, z1), z2) -> c26(MINSORT(app(rm(z0, z1), z2), nil), APP(rm(z0, z1), z2), RM(z0, z1)) MINSORT(add(z0, z1), z2) -> c25(IF_MINSORT(eq(z0, min(add(z0, z1))), add(z0, z1), z2), EQ(z0, min(add(z0, z1))), MIN(add(z0, z1))) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MIN(add(z0, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) RM(z0, add(z1, z2)) -> c21(IF_RM(eq(z0, z1), z0, add(z1, z2)), EQ(z0, z1)) Defined Rule Symbols: le_2, eq_2, app_2, if_rm_3, rm_2, if_min_4, minIter_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, MIN_1, MINITER_3, IF_MIN_4, IF_RM_3, IF_MINSORT_3, RM_2, MINSORT_2 Compound Symbols: c3_1, c6_1, c8_1, c10_1, c11_1, c14_1, c22_1, c23_1, c27_1, c12_1, c21_1, c12_2, c21_2, c_1, c26_3, c26_1, c26_2, c25_3 ---------------------------------------- (123) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace RM(s(s(z0)), add(s(s(z1)), x2)) -> c21(IF_RM(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), EQ(s(s(z0)), s(s(z1)))) by RM(s(s(0)), add(s(s(0)), x2)) -> c21(IF_RM(true, s(s(0)), add(s(s(0)), x2)), EQ(s(s(0)), s(s(0)))) RM(s(s(0)), add(s(s(s(z0))), x2)) -> c21(IF_RM(false, s(s(0)), add(s(s(s(z0))), x2)), EQ(s(s(0)), s(s(s(z0))))) RM(s(s(s(z0))), add(s(s(0)), x2)) -> c21(IF_RM(false, s(s(s(z0))), add(s(s(0)), x2)), EQ(s(s(s(z0))), s(s(0)))) RM(s(s(s(z0))), add(s(s(s(z1))), x2)) -> c21(IF_RM(eq(z0, z1), s(s(s(z0))), add(s(s(s(z1))), x2)), EQ(s(s(s(z0))), s(s(s(z1))))) RM(s(s(x0)), add(s(s(x1)), x2)) -> c21(EQ(s(s(x0)), s(s(x1)))) ---------------------------------------- (124) 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) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) if_min(true, z0, z1, z2) -> z2 if_min(false, z0, z1, z2) -> minIter(z0, z1, z2) minIter(add(z0, z1), z2, z3) -> if_min(le(z0, z3), z1, z2, z3) minIter(nil, add(z0, z1), z2) -> minIter(add(z0, z1), add(z0, z1), s(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, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) RM(0, add(0, x2)) -> c21(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c21(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c21(IF_RM(false, s(z0), add(0, x2))) MINITER(add(s(s(z0)), x1), x2, s(0)) -> c12(IF_MIN(false, x1, x2, s(0)), LE(s(s(z0)), s(0))) MINITER(add(s(x0), x1), x2, s(x3)) -> c12(LE(s(x0), s(x3))) MINITER(add(s(0), x1), x2, s(z0)) -> c12(LE(s(0), s(z0))) RM(s(0), add(s(0), x2)) -> c21(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c21(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c21(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(x0), add(s(x1), x2)) -> c21(EQ(s(x0), s(x1))) MINSORT(add(x0, x1), x2) -> c(MIN(add(x0, x1))) IF_MINSORT(true, add(0, add(0, x2)), x3) -> c26(MINSORT(app(if_rm(true, 0, add(0, x2)), x3), nil), APP(rm(0, add(0, x2)), x3), RM(0, add(0, x2))) IF_MINSORT(true, add(0, add(s(z0), x2)), x3) -> c26(MINSORT(app(if_rm(false, 0, add(s(z0), x2)), x3), nil), APP(rm(0, add(s(z0), x2)), x3), RM(0, add(s(z0), x2))) IF_MINSORT(true, add(s(z0), add(0, x2)), x3) -> c26(MINSORT(app(if_rm(false, s(z0), add(0, x2)), x3), nil), APP(rm(s(z0), add(0, x2)), x3), RM(s(z0), add(0, x2))) IF_MINSORT(true, add(s(z0), add(s(z1), x2)), x3) -> c26(MINSORT(app(if_rm(eq(z0, z1), s(z0), add(s(z1), x2)), x3), nil), APP(rm(s(z0), add(s(z1), x2)), x3), RM(s(z0), add(s(z1), x2))) IF_MINSORT(true, add(x0, add(x1, x2)), x3) -> c26(APP(rm(x0, add(x1, x2)), x3)) IF_MINSORT(true, add(x0, nil), z0) -> c26(MINSORT(z0, nil), APP(rm(x0, nil), z0)) MINSORT(add(z0, z1), z2) -> c25(IF_MINSORT(eq(z0, if_min(le(z0, 0), z1, add(z0, z1), 0)), add(z0, z1), z2), EQ(z0, minIter(add(z0, z1), add(z0, z1), 0)), MIN(add(z0, z1))) MINSORT(add(z0, z1), z2) -> c(EQ(z0, minIter(add(z0, z1), add(z0, z1), 0))) MINITER(add(s(s(s(z0))), x1), x2, s(s(0))) -> c12(IF_MIN(false, x1, x2, s(s(0))), LE(s(s(s(z0))), s(s(0)))) MINITER(add(s(s(s(z0))), x1), x2, s(s(s(z1)))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(s(z1)))), LE(s(s(s(z0))), s(s(s(z1))))) MINITER(add(s(s(x0)), x1), x2, s(s(x3))) -> c12(LE(s(s(x0)), s(s(x3)))) MINITER(add(s(s(0)), x1), x2, s(s(z0))) -> c12(LE(s(s(0)), s(s(z0)))) RM(s(s(0)), add(s(s(0)), x2)) -> c21(IF_RM(true, s(s(0)), add(s(s(0)), x2)), EQ(s(s(0)), s(s(0)))) RM(s(s(0)), add(s(s(s(z0))), x2)) -> c21(IF_RM(false, s(s(0)), add(s(s(s(z0))), x2)), EQ(s(s(0)), s(s(s(z0))))) RM(s(s(s(z0))), add(s(s(0)), x2)) -> c21(IF_RM(false, s(s(s(z0))), add(s(s(0)), x2)), EQ(s(s(s(z0))), s(s(0)))) RM(s(s(s(z0))), add(s(s(s(z1))), x2)) -> c21(IF_RM(eq(z0, z1), s(s(s(z0))), add(s(s(s(z1))), x2)), EQ(s(s(s(z0))), s(s(s(z1))))) RM(s(s(x0)), add(s(s(x1)), x2)) -> c21(EQ(s(s(x0)), s(s(x1)))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) MINITER(add(s(s(z0)), x1), x2, s(0)) -> c12(IF_MIN(false, x1, x2, s(0)), LE(s(s(z0)), s(0))) MINITER(add(s(x0), x1), x2, s(x3)) -> c12(LE(s(x0), s(x3))) MINITER(add(s(0), x1), x2, s(z0)) -> c12(LE(s(0), s(z0))) MINITER(add(s(s(s(z0))), x1), x2, s(s(0))) -> c12(IF_MIN(false, x1, x2, s(s(0))), LE(s(s(s(z0))), s(s(0)))) MINITER(add(s(s(s(z0))), x1), x2, s(s(s(z1)))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(s(z1)))), LE(s(s(s(z0))), s(s(s(z1))))) MINITER(add(s(s(x0)), x1), x2, s(s(x3))) -> c12(LE(s(s(x0)), s(s(x3)))) MINITER(add(s(s(0)), x1), x2, s(s(z0))) -> c12(LE(s(s(0)), s(s(z0)))) K tuples: IF_MINSORT(true, add(z0, z1), z2) -> c26(MINSORT(app(rm(z0, z1), z2), nil), APP(rm(z0, z1), z2), RM(z0, z1)) MINSORT(add(z0, z1), z2) -> c25(IF_MINSORT(eq(z0, min(add(z0, z1))), add(z0, z1), z2), EQ(z0, min(add(z0, z1))), MIN(add(z0, z1))) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MIN(add(z0, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) RM(z0, add(z1, z2)) -> c21(IF_RM(eq(z0, z1), z0, add(z1, z2)), EQ(z0, z1)) Defined Rule Symbols: le_2, eq_2, app_2, if_rm_3, rm_2, if_min_4, minIter_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, MIN_1, MINITER_3, IF_MIN_4, IF_RM_3, IF_MINSORT_3, RM_2, MINSORT_2 Compound Symbols: c3_1, c6_1, c8_1, c10_1, c11_1, c14_1, c22_1, c23_1, c27_1, c12_1, c21_1, c12_2, c21_2, c_1, c26_3, c26_1, c26_2, c25_3 ---------------------------------------- (125) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF_MINSORT(true, add(0, add(0, x2)), x3) -> c26(MINSORT(app(if_rm(true, 0, add(0, x2)), x3), nil), APP(rm(0, add(0, x2)), x3), RM(0, add(0, x2))) by IF_MINSORT(true, add(0, add(0, z2)), x1) -> c26(MINSORT(app(rm(0, z2), x1), nil), APP(rm(0, add(0, z2)), x1), RM(0, add(0, z2))) IF_MINSORT(true, add(0, add(0, x0)), x1) -> c26(RM(0, add(0, x0))) ---------------------------------------- (126) 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) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) if_min(true, z0, z1, z2) -> z2 if_min(false, z0, z1, z2) -> minIter(z0, z1, z2) minIter(add(z0, z1), z2, z3) -> if_min(le(z0, z3), z1, z2, z3) minIter(nil, add(z0, z1), z2) -> minIter(add(z0, z1), add(z0, z1), s(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, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) RM(0, add(0, x2)) -> c21(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c21(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c21(IF_RM(false, s(z0), add(0, x2))) MINITER(add(s(s(z0)), x1), x2, s(0)) -> c12(IF_MIN(false, x1, x2, s(0)), LE(s(s(z0)), s(0))) MINITER(add(s(x0), x1), x2, s(x3)) -> c12(LE(s(x0), s(x3))) MINITER(add(s(0), x1), x2, s(z0)) -> c12(LE(s(0), s(z0))) RM(s(0), add(s(0), x2)) -> c21(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c21(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c21(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(x0), add(s(x1), x2)) -> c21(EQ(s(x0), s(x1))) MINSORT(add(x0, x1), x2) -> c(MIN(add(x0, x1))) IF_MINSORT(true, add(0, add(s(z0), x2)), x3) -> c26(MINSORT(app(if_rm(false, 0, add(s(z0), x2)), x3), nil), APP(rm(0, add(s(z0), x2)), x3), RM(0, add(s(z0), x2))) IF_MINSORT(true, add(s(z0), add(0, x2)), x3) -> c26(MINSORT(app(if_rm(false, s(z0), add(0, x2)), x3), nil), APP(rm(s(z0), add(0, x2)), x3), RM(s(z0), add(0, x2))) IF_MINSORT(true, add(s(z0), add(s(z1), x2)), x3) -> c26(MINSORT(app(if_rm(eq(z0, z1), s(z0), add(s(z1), x2)), x3), nil), APP(rm(s(z0), add(s(z1), x2)), x3), RM(s(z0), add(s(z1), x2))) IF_MINSORT(true, add(x0, add(x1, x2)), x3) -> c26(APP(rm(x0, add(x1, x2)), x3)) IF_MINSORT(true, add(x0, nil), z0) -> c26(MINSORT(z0, nil), APP(rm(x0, nil), z0)) MINSORT(add(z0, z1), z2) -> c25(IF_MINSORT(eq(z0, if_min(le(z0, 0), z1, add(z0, z1), 0)), add(z0, z1), z2), EQ(z0, minIter(add(z0, z1), add(z0, z1), 0)), MIN(add(z0, z1))) MINSORT(add(z0, z1), z2) -> c(EQ(z0, minIter(add(z0, z1), add(z0, z1), 0))) MINITER(add(s(s(s(z0))), x1), x2, s(s(0))) -> c12(IF_MIN(false, x1, x2, s(s(0))), LE(s(s(s(z0))), s(s(0)))) MINITER(add(s(s(s(z0))), x1), x2, s(s(s(z1)))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(s(z1)))), LE(s(s(s(z0))), s(s(s(z1))))) MINITER(add(s(s(x0)), x1), x2, s(s(x3))) -> c12(LE(s(s(x0)), s(s(x3)))) MINITER(add(s(s(0)), x1), x2, s(s(z0))) -> c12(LE(s(s(0)), s(s(z0)))) RM(s(s(0)), add(s(s(0)), x2)) -> c21(IF_RM(true, s(s(0)), add(s(s(0)), x2)), EQ(s(s(0)), s(s(0)))) RM(s(s(0)), add(s(s(s(z0))), x2)) -> c21(IF_RM(false, s(s(0)), add(s(s(s(z0))), x2)), EQ(s(s(0)), s(s(s(z0))))) RM(s(s(s(z0))), add(s(s(0)), x2)) -> c21(IF_RM(false, s(s(s(z0))), add(s(s(0)), x2)), EQ(s(s(s(z0))), s(s(0)))) RM(s(s(s(z0))), add(s(s(s(z1))), x2)) -> c21(IF_RM(eq(z0, z1), s(s(s(z0))), add(s(s(s(z1))), x2)), EQ(s(s(s(z0))), s(s(s(z1))))) RM(s(s(x0)), add(s(s(x1)), x2)) -> c21(EQ(s(s(x0)), s(s(x1)))) IF_MINSORT(true, add(0, add(0, z2)), x1) -> c26(MINSORT(app(rm(0, z2), x1), nil), APP(rm(0, add(0, z2)), x1), RM(0, add(0, z2))) IF_MINSORT(true, add(0, add(0, x0)), x1) -> c26(RM(0, add(0, x0))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) MINITER(add(s(s(z0)), x1), x2, s(0)) -> c12(IF_MIN(false, x1, x2, s(0)), LE(s(s(z0)), s(0))) MINITER(add(s(x0), x1), x2, s(x3)) -> c12(LE(s(x0), s(x3))) MINITER(add(s(0), x1), x2, s(z0)) -> c12(LE(s(0), s(z0))) MINITER(add(s(s(s(z0))), x1), x2, s(s(0))) -> c12(IF_MIN(false, x1, x2, s(s(0))), LE(s(s(s(z0))), s(s(0)))) MINITER(add(s(s(s(z0))), x1), x2, s(s(s(z1)))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(s(z1)))), LE(s(s(s(z0))), s(s(s(z1))))) MINITER(add(s(s(x0)), x1), x2, s(s(x3))) -> c12(LE(s(s(x0)), s(s(x3)))) MINITER(add(s(s(0)), x1), x2, s(s(z0))) -> c12(LE(s(s(0)), s(s(z0)))) K tuples: IF_MINSORT(true, add(z0, z1), z2) -> c26(MINSORT(app(rm(z0, z1), z2), nil), APP(rm(z0, z1), z2), RM(z0, z1)) MINSORT(add(z0, z1), z2) -> c25(IF_MINSORT(eq(z0, min(add(z0, z1))), add(z0, z1), z2), EQ(z0, min(add(z0, z1))), MIN(add(z0, z1))) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MIN(add(z0, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) RM(z0, add(z1, z2)) -> c21(IF_RM(eq(z0, z1), z0, add(z1, z2)), EQ(z0, z1)) Defined Rule Symbols: le_2, eq_2, app_2, if_rm_3, rm_2, if_min_4, minIter_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, MIN_1, MINITER_3, IF_MIN_4, IF_RM_3, IF_MINSORT_3, RM_2, MINSORT_2 Compound Symbols: c3_1, c6_1, c8_1, c10_1, c11_1, c14_1, c22_1, c23_1, c27_1, c12_1, c21_1, c12_2, c21_2, c_1, c26_3, c26_1, c26_2, c25_3 ---------------------------------------- (127) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF_MINSORT(true, add(0, add(s(z0), x2)), x3) -> c26(MINSORT(app(if_rm(false, 0, add(s(z0), x2)), x3), nil), APP(rm(0, add(s(z0), x2)), x3), RM(0, add(s(z0), x2))) by IF_MINSORT(true, add(0, add(s(x0), z2)), x2) -> c26(MINSORT(app(add(s(x0), rm(0, z2)), x2), nil), APP(rm(0, add(s(x0), z2)), x2), RM(0, add(s(x0), z2))) IF_MINSORT(true, add(0, add(s(x0), x1)), x2) -> c26(RM(0, add(s(x0), x1))) ---------------------------------------- (128) 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) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) if_min(true, z0, z1, z2) -> z2 if_min(false, z0, z1, z2) -> minIter(z0, z1, z2) minIter(add(z0, z1), z2, z3) -> if_min(le(z0, z3), z1, z2, z3) minIter(nil, add(z0, z1), z2) -> minIter(add(z0, z1), add(z0, z1), s(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, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) RM(0, add(0, x2)) -> c21(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c21(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c21(IF_RM(false, s(z0), add(0, x2))) MINITER(add(s(s(z0)), x1), x2, s(0)) -> c12(IF_MIN(false, x1, x2, s(0)), LE(s(s(z0)), s(0))) MINITER(add(s(x0), x1), x2, s(x3)) -> c12(LE(s(x0), s(x3))) MINITER(add(s(0), x1), x2, s(z0)) -> c12(LE(s(0), s(z0))) RM(s(0), add(s(0), x2)) -> c21(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c21(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c21(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(x0), add(s(x1), x2)) -> c21(EQ(s(x0), s(x1))) MINSORT(add(x0, x1), x2) -> c(MIN(add(x0, x1))) IF_MINSORT(true, add(s(z0), add(0, x2)), x3) -> c26(MINSORT(app(if_rm(false, s(z0), add(0, x2)), x3), nil), APP(rm(s(z0), add(0, x2)), x3), RM(s(z0), add(0, x2))) IF_MINSORT(true, add(s(z0), add(s(z1), x2)), x3) -> c26(MINSORT(app(if_rm(eq(z0, z1), s(z0), add(s(z1), x2)), x3), nil), APP(rm(s(z0), add(s(z1), x2)), x3), RM(s(z0), add(s(z1), x2))) IF_MINSORT(true, add(x0, add(x1, x2)), x3) -> c26(APP(rm(x0, add(x1, x2)), x3)) IF_MINSORT(true, add(x0, nil), z0) -> c26(MINSORT(z0, nil), APP(rm(x0, nil), z0)) MINSORT(add(z0, z1), z2) -> c25(IF_MINSORT(eq(z0, if_min(le(z0, 0), z1, add(z0, z1), 0)), add(z0, z1), z2), EQ(z0, minIter(add(z0, z1), add(z0, z1), 0)), MIN(add(z0, z1))) MINSORT(add(z0, z1), z2) -> c(EQ(z0, minIter(add(z0, z1), add(z0, z1), 0))) MINITER(add(s(s(s(z0))), x1), x2, s(s(0))) -> c12(IF_MIN(false, x1, x2, s(s(0))), LE(s(s(s(z0))), s(s(0)))) MINITER(add(s(s(s(z0))), x1), x2, s(s(s(z1)))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(s(z1)))), LE(s(s(s(z0))), s(s(s(z1))))) MINITER(add(s(s(x0)), x1), x2, s(s(x3))) -> c12(LE(s(s(x0)), s(s(x3)))) MINITER(add(s(s(0)), x1), x2, s(s(z0))) -> c12(LE(s(s(0)), s(s(z0)))) RM(s(s(0)), add(s(s(0)), x2)) -> c21(IF_RM(true, s(s(0)), add(s(s(0)), x2)), EQ(s(s(0)), s(s(0)))) RM(s(s(0)), add(s(s(s(z0))), x2)) -> c21(IF_RM(false, s(s(0)), add(s(s(s(z0))), x2)), EQ(s(s(0)), s(s(s(z0))))) RM(s(s(s(z0))), add(s(s(0)), x2)) -> c21(IF_RM(false, s(s(s(z0))), add(s(s(0)), x2)), EQ(s(s(s(z0))), s(s(0)))) RM(s(s(s(z0))), add(s(s(s(z1))), x2)) -> c21(IF_RM(eq(z0, z1), s(s(s(z0))), add(s(s(s(z1))), x2)), EQ(s(s(s(z0))), s(s(s(z1))))) RM(s(s(x0)), add(s(s(x1)), x2)) -> c21(EQ(s(s(x0)), s(s(x1)))) IF_MINSORT(true, add(0, add(0, z2)), x1) -> c26(MINSORT(app(rm(0, z2), x1), nil), APP(rm(0, add(0, z2)), x1), RM(0, add(0, z2))) IF_MINSORT(true, add(0, add(0, x0)), x1) -> c26(RM(0, add(0, x0))) IF_MINSORT(true, add(0, add(s(x0), z2)), x2) -> c26(MINSORT(app(add(s(x0), rm(0, z2)), x2), nil), APP(rm(0, add(s(x0), z2)), x2), RM(0, add(s(x0), z2))) IF_MINSORT(true, add(0, add(s(x0), x1)), x2) -> c26(RM(0, add(s(x0), x1))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) MINITER(add(s(s(z0)), x1), x2, s(0)) -> c12(IF_MIN(false, x1, x2, s(0)), LE(s(s(z0)), s(0))) MINITER(add(s(x0), x1), x2, s(x3)) -> c12(LE(s(x0), s(x3))) MINITER(add(s(0), x1), x2, s(z0)) -> c12(LE(s(0), s(z0))) MINITER(add(s(s(s(z0))), x1), x2, s(s(0))) -> c12(IF_MIN(false, x1, x2, s(s(0))), LE(s(s(s(z0))), s(s(0)))) MINITER(add(s(s(s(z0))), x1), x2, s(s(s(z1)))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(s(z1)))), LE(s(s(s(z0))), s(s(s(z1))))) MINITER(add(s(s(x0)), x1), x2, s(s(x3))) -> c12(LE(s(s(x0)), s(s(x3)))) MINITER(add(s(s(0)), x1), x2, s(s(z0))) -> c12(LE(s(s(0)), s(s(z0)))) K tuples: IF_MINSORT(true, add(z0, z1), z2) -> c26(MINSORT(app(rm(z0, z1), z2), nil), APP(rm(z0, z1), z2), RM(z0, z1)) MINSORT(add(z0, z1), z2) -> c25(IF_MINSORT(eq(z0, min(add(z0, z1))), add(z0, z1), z2), EQ(z0, min(add(z0, z1))), MIN(add(z0, z1))) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MIN(add(z0, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) RM(z0, add(z1, z2)) -> c21(IF_RM(eq(z0, z1), z0, add(z1, z2)), EQ(z0, z1)) Defined Rule Symbols: le_2, eq_2, app_2, if_rm_3, rm_2, if_min_4, minIter_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, MIN_1, MINITER_3, IF_MIN_4, IF_RM_3, IF_MINSORT_3, RM_2, MINSORT_2 Compound Symbols: c3_1, c6_1, c8_1, c10_1, c11_1, c14_1, c22_1, c23_1, c27_1, c12_1, c21_1, c12_2, c21_2, c_1, c26_3, c26_1, c26_2, c25_3 ---------------------------------------- (129) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF_MINSORT(true, add(s(z0), add(0, x2)), x3) -> c26(MINSORT(app(if_rm(false, s(z0), add(0, x2)), x3), nil), APP(rm(s(z0), add(0, x2)), x3), RM(s(z0), add(0, x2))) by IF_MINSORT(true, add(s(x0), add(0, z2)), x2) -> c26(MINSORT(app(add(0, rm(s(x0), z2)), x2), nil), APP(rm(s(x0), add(0, z2)), x2), RM(s(x0), add(0, z2))) IF_MINSORT(true, add(s(x0), add(0, x1)), x2) -> c26(RM(s(x0), add(0, x1))) ---------------------------------------- (130) 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) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) if_min(true, z0, z1, z2) -> z2 if_min(false, z0, z1, z2) -> minIter(z0, z1, z2) minIter(add(z0, z1), z2, z3) -> if_min(le(z0, z3), z1, z2, z3) minIter(nil, add(z0, z1), z2) -> minIter(add(z0, z1), add(z0, z1), s(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, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) RM(0, add(0, x2)) -> c21(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c21(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c21(IF_RM(false, s(z0), add(0, x2))) MINITER(add(s(s(z0)), x1), x2, s(0)) -> c12(IF_MIN(false, x1, x2, s(0)), LE(s(s(z0)), s(0))) MINITER(add(s(x0), x1), x2, s(x3)) -> c12(LE(s(x0), s(x3))) MINITER(add(s(0), x1), x2, s(z0)) -> c12(LE(s(0), s(z0))) RM(s(0), add(s(0), x2)) -> c21(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c21(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c21(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(x0), add(s(x1), x2)) -> c21(EQ(s(x0), s(x1))) MINSORT(add(x0, x1), x2) -> c(MIN(add(x0, x1))) IF_MINSORT(true, add(s(z0), add(s(z1), x2)), x3) -> c26(MINSORT(app(if_rm(eq(z0, z1), s(z0), add(s(z1), x2)), x3), nil), APP(rm(s(z0), add(s(z1), x2)), x3), RM(s(z0), add(s(z1), x2))) IF_MINSORT(true, add(x0, add(x1, x2)), x3) -> c26(APP(rm(x0, add(x1, x2)), x3)) IF_MINSORT(true, add(x0, nil), z0) -> c26(MINSORT(z0, nil), APP(rm(x0, nil), z0)) MINSORT(add(z0, z1), z2) -> c25(IF_MINSORT(eq(z0, if_min(le(z0, 0), z1, add(z0, z1), 0)), add(z0, z1), z2), EQ(z0, minIter(add(z0, z1), add(z0, z1), 0)), MIN(add(z0, z1))) MINSORT(add(z0, z1), z2) -> c(EQ(z0, minIter(add(z0, z1), add(z0, z1), 0))) MINITER(add(s(s(s(z0))), x1), x2, s(s(0))) -> c12(IF_MIN(false, x1, x2, s(s(0))), LE(s(s(s(z0))), s(s(0)))) MINITER(add(s(s(s(z0))), x1), x2, s(s(s(z1)))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(s(z1)))), LE(s(s(s(z0))), s(s(s(z1))))) MINITER(add(s(s(x0)), x1), x2, s(s(x3))) -> c12(LE(s(s(x0)), s(s(x3)))) MINITER(add(s(s(0)), x1), x2, s(s(z0))) -> c12(LE(s(s(0)), s(s(z0)))) RM(s(s(0)), add(s(s(0)), x2)) -> c21(IF_RM(true, s(s(0)), add(s(s(0)), x2)), EQ(s(s(0)), s(s(0)))) RM(s(s(0)), add(s(s(s(z0))), x2)) -> c21(IF_RM(false, s(s(0)), add(s(s(s(z0))), x2)), EQ(s(s(0)), s(s(s(z0))))) RM(s(s(s(z0))), add(s(s(0)), x2)) -> c21(IF_RM(false, s(s(s(z0))), add(s(s(0)), x2)), EQ(s(s(s(z0))), s(s(0)))) RM(s(s(s(z0))), add(s(s(s(z1))), x2)) -> c21(IF_RM(eq(z0, z1), s(s(s(z0))), add(s(s(s(z1))), x2)), EQ(s(s(s(z0))), s(s(s(z1))))) RM(s(s(x0)), add(s(s(x1)), x2)) -> c21(EQ(s(s(x0)), s(s(x1)))) IF_MINSORT(true, add(0, add(0, z2)), x1) -> c26(MINSORT(app(rm(0, z2), x1), nil), APP(rm(0, add(0, z2)), x1), RM(0, add(0, z2))) IF_MINSORT(true, add(0, add(0, x0)), x1) -> c26(RM(0, add(0, x0))) IF_MINSORT(true, add(0, add(s(x0), z2)), x2) -> c26(MINSORT(app(add(s(x0), rm(0, z2)), x2), nil), APP(rm(0, add(s(x0), z2)), x2), RM(0, add(s(x0), z2))) IF_MINSORT(true, add(0, add(s(x0), x1)), x2) -> c26(RM(0, add(s(x0), x1))) IF_MINSORT(true, add(s(x0), add(0, z2)), x2) -> c26(MINSORT(app(add(0, rm(s(x0), z2)), x2), nil), APP(rm(s(x0), add(0, z2)), x2), RM(s(x0), add(0, z2))) IF_MINSORT(true, add(s(x0), add(0, x1)), x2) -> c26(RM(s(x0), add(0, x1))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) MINITER(add(s(s(z0)), x1), x2, s(0)) -> c12(IF_MIN(false, x1, x2, s(0)), LE(s(s(z0)), s(0))) MINITER(add(s(x0), x1), x2, s(x3)) -> c12(LE(s(x0), s(x3))) MINITER(add(s(0), x1), x2, s(z0)) -> c12(LE(s(0), s(z0))) MINITER(add(s(s(s(z0))), x1), x2, s(s(0))) -> c12(IF_MIN(false, x1, x2, s(s(0))), LE(s(s(s(z0))), s(s(0)))) MINITER(add(s(s(s(z0))), x1), x2, s(s(s(z1)))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(s(z1)))), LE(s(s(s(z0))), s(s(s(z1))))) MINITER(add(s(s(x0)), x1), x2, s(s(x3))) -> c12(LE(s(s(x0)), s(s(x3)))) MINITER(add(s(s(0)), x1), x2, s(s(z0))) -> c12(LE(s(s(0)), s(s(z0)))) K tuples: IF_MINSORT(true, add(z0, z1), z2) -> c26(MINSORT(app(rm(z0, z1), z2), nil), APP(rm(z0, z1), z2), RM(z0, z1)) MINSORT(add(z0, z1), z2) -> c25(IF_MINSORT(eq(z0, min(add(z0, z1))), add(z0, z1), z2), EQ(z0, min(add(z0, z1))), MIN(add(z0, z1))) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MIN(add(z0, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) RM(z0, add(z1, z2)) -> c21(IF_RM(eq(z0, z1), z0, add(z1, z2)), EQ(z0, z1)) Defined Rule Symbols: le_2, eq_2, app_2, if_rm_3, rm_2, if_min_4, minIter_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, MIN_1, MINITER_3, IF_MIN_4, IF_RM_3, IF_MINSORT_3, RM_2, MINSORT_2 Compound Symbols: c3_1, c6_1, c8_1, c10_1, c11_1, c14_1, c22_1, c23_1, c27_1, c12_1, c21_1, c12_2, c21_2, c_1, c26_3, c26_1, c26_2, c25_3 ---------------------------------------- (131) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF_MINSORT(true, add(s(z0), add(s(z1), x2)), x3) -> c26(MINSORT(app(if_rm(eq(z0, z1), s(z0), add(s(z1), x2)), x3), nil), APP(rm(s(z0), add(s(z1), x2)), x3), RM(s(z0), add(s(z1), x2))) by IF_MINSORT(true, add(s(0), add(s(0), x2)), x3) -> c26(MINSORT(app(if_rm(true, s(0), add(s(0), x2)), x3), nil), APP(rm(s(0), add(s(0), x2)), x3), RM(s(0), add(s(0), x2))) IF_MINSORT(true, add(s(0), add(s(s(z0)), x2)), x3) -> c26(MINSORT(app(if_rm(false, s(0), add(s(s(z0)), x2)), x3), nil), APP(rm(s(0), add(s(s(z0)), x2)), x3), RM(s(0), add(s(s(z0)), x2))) IF_MINSORT(true, add(s(s(z0)), add(s(0), x2)), x3) -> c26(MINSORT(app(if_rm(false, s(s(z0)), add(s(0), x2)), x3), nil), APP(rm(s(s(z0)), add(s(0), x2)), x3), RM(s(s(z0)), add(s(0), x2))) IF_MINSORT(true, add(s(s(z0)), add(s(s(z1)), x2)), x3) -> c26(MINSORT(app(if_rm(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), x3), nil), APP(rm(s(s(z0)), add(s(s(z1)), x2)), x3), RM(s(s(z0)), add(s(s(z1)), x2))) IF_MINSORT(true, add(s(x0), add(s(x1), x2)), x3) -> c26(APP(rm(s(x0), add(s(x1), x2)), x3), RM(s(x0), add(s(x1), x2))) ---------------------------------------- (132) 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) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) if_min(true, z0, z1, z2) -> z2 if_min(false, z0, z1, z2) -> minIter(z0, z1, z2) minIter(add(z0, z1), z2, z3) -> if_min(le(z0, z3), z1, z2, z3) minIter(nil, add(z0, z1), z2) -> minIter(add(z0, z1), add(z0, z1), s(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, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) RM(0, add(0, x2)) -> c21(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c21(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c21(IF_RM(false, s(z0), add(0, x2))) MINITER(add(s(s(z0)), x1), x2, s(0)) -> c12(IF_MIN(false, x1, x2, s(0)), LE(s(s(z0)), s(0))) MINITER(add(s(x0), x1), x2, s(x3)) -> c12(LE(s(x0), s(x3))) MINITER(add(s(0), x1), x2, s(z0)) -> c12(LE(s(0), s(z0))) RM(s(0), add(s(0), x2)) -> c21(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c21(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c21(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(x0), add(s(x1), x2)) -> c21(EQ(s(x0), s(x1))) MINSORT(add(x0, x1), x2) -> c(MIN(add(x0, x1))) IF_MINSORT(true, add(x0, add(x1, x2)), x3) -> c26(APP(rm(x0, add(x1, x2)), x3)) IF_MINSORT(true, add(x0, nil), z0) -> c26(MINSORT(z0, nil), APP(rm(x0, nil), z0)) MINSORT(add(z0, z1), z2) -> c25(IF_MINSORT(eq(z0, if_min(le(z0, 0), z1, add(z0, z1), 0)), add(z0, z1), z2), EQ(z0, minIter(add(z0, z1), add(z0, z1), 0)), MIN(add(z0, z1))) MINSORT(add(z0, z1), z2) -> c(EQ(z0, minIter(add(z0, z1), add(z0, z1), 0))) MINITER(add(s(s(s(z0))), x1), x2, s(s(0))) -> c12(IF_MIN(false, x1, x2, s(s(0))), LE(s(s(s(z0))), s(s(0)))) MINITER(add(s(s(s(z0))), x1), x2, s(s(s(z1)))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(s(z1)))), LE(s(s(s(z0))), s(s(s(z1))))) MINITER(add(s(s(x0)), x1), x2, s(s(x3))) -> c12(LE(s(s(x0)), s(s(x3)))) MINITER(add(s(s(0)), x1), x2, s(s(z0))) -> c12(LE(s(s(0)), s(s(z0)))) RM(s(s(0)), add(s(s(0)), x2)) -> c21(IF_RM(true, s(s(0)), add(s(s(0)), x2)), EQ(s(s(0)), s(s(0)))) RM(s(s(0)), add(s(s(s(z0))), x2)) -> c21(IF_RM(false, s(s(0)), add(s(s(s(z0))), x2)), EQ(s(s(0)), s(s(s(z0))))) RM(s(s(s(z0))), add(s(s(0)), x2)) -> c21(IF_RM(false, s(s(s(z0))), add(s(s(0)), x2)), EQ(s(s(s(z0))), s(s(0)))) RM(s(s(s(z0))), add(s(s(s(z1))), x2)) -> c21(IF_RM(eq(z0, z1), s(s(s(z0))), add(s(s(s(z1))), x2)), EQ(s(s(s(z0))), s(s(s(z1))))) RM(s(s(x0)), add(s(s(x1)), x2)) -> c21(EQ(s(s(x0)), s(s(x1)))) IF_MINSORT(true, add(0, add(0, z2)), x1) -> c26(MINSORT(app(rm(0, z2), x1), nil), APP(rm(0, add(0, z2)), x1), RM(0, add(0, z2))) IF_MINSORT(true, add(0, add(0, x0)), x1) -> c26(RM(0, add(0, x0))) IF_MINSORT(true, add(0, add(s(x0), z2)), x2) -> c26(MINSORT(app(add(s(x0), rm(0, z2)), x2), nil), APP(rm(0, add(s(x0), z2)), x2), RM(0, add(s(x0), z2))) IF_MINSORT(true, add(0, add(s(x0), x1)), x2) -> c26(RM(0, add(s(x0), x1))) IF_MINSORT(true, add(s(x0), add(0, z2)), x2) -> c26(MINSORT(app(add(0, rm(s(x0), z2)), x2), nil), APP(rm(s(x0), add(0, z2)), x2), RM(s(x0), add(0, z2))) IF_MINSORT(true, add(s(x0), add(0, x1)), x2) -> c26(RM(s(x0), add(0, x1))) IF_MINSORT(true, add(s(0), add(s(0), x2)), x3) -> c26(MINSORT(app(if_rm(true, s(0), add(s(0), x2)), x3), nil), APP(rm(s(0), add(s(0), x2)), x3), RM(s(0), add(s(0), x2))) IF_MINSORT(true, add(s(0), add(s(s(z0)), x2)), x3) -> c26(MINSORT(app(if_rm(false, s(0), add(s(s(z0)), x2)), x3), nil), APP(rm(s(0), add(s(s(z0)), x2)), x3), RM(s(0), add(s(s(z0)), x2))) IF_MINSORT(true, add(s(s(z0)), add(s(0), x2)), x3) -> c26(MINSORT(app(if_rm(false, s(s(z0)), add(s(0), x2)), x3), nil), APP(rm(s(s(z0)), add(s(0), x2)), x3), RM(s(s(z0)), add(s(0), x2))) IF_MINSORT(true, add(s(s(z0)), add(s(s(z1)), x2)), x3) -> c26(MINSORT(app(if_rm(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), x3), nil), APP(rm(s(s(z0)), add(s(s(z1)), x2)), x3), RM(s(s(z0)), add(s(s(z1)), x2))) IF_MINSORT(true, add(s(x0), add(s(x1), x2)), x3) -> c26(APP(rm(s(x0), add(s(x1), x2)), x3), RM(s(x0), add(s(x1), x2))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) MINITER(add(s(s(z0)), x1), x2, s(0)) -> c12(IF_MIN(false, x1, x2, s(0)), LE(s(s(z0)), s(0))) MINITER(add(s(x0), x1), x2, s(x3)) -> c12(LE(s(x0), s(x3))) MINITER(add(s(0), x1), x2, s(z0)) -> c12(LE(s(0), s(z0))) MINITER(add(s(s(s(z0))), x1), x2, s(s(0))) -> c12(IF_MIN(false, x1, x2, s(s(0))), LE(s(s(s(z0))), s(s(0)))) MINITER(add(s(s(s(z0))), x1), x2, s(s(s(z1)))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(s(z1)))), LE(s(s(s(z0))), s(s(s(z1))))) MINITER(add(s(s(x0)), x1), x2, s(s(x3))) -> c12(LE(s(s(x0)), s(s(x3)))) MINITER(add(s(s(0)), x1), x2, s(s(z0))) -> c12(LE(s(s(0)), s(s(z0)))) K tuples: IF_MINSORT(true, add(z0, z1), z2) -> c26(MINSORT(app(rm(z0, z1), z2), nil), APP(rm(z0, z1), z2), RM(z0, z1)) MINSORT(add(z0, z1), z2) -> c25(IF_MINSORT(eq(z0, min(add(z0, z1))), add(z0, z1), z2), EQ(z0, min(add(z0, z1))), MIN(add(z0, z1))) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MIN(add(z0, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) RM(z0, add(z1, z2)) -> c21(IF_RM(eq(z0, z1), z0, add(z1, z2)), EQ(z0, z1)) Defined Rule Symbols: le_2, eq_2, app_2, if_rm_3, rm_2, if_min_4, minIter_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, MIN_1, MINITER_3, IF_MIN_4, IF_RM_3, IF_MINSORT_3, RM_2, MINSORT_2 Compound Symbols: c3_1, c6_1, c8_1, c10_1, c11_1, c14_1, c22_1, c23_1, c27_1, c12_1, c21_1, c12_2, c21_2, c_1, c26_1, c26_2, c25_3, c26_3 ---------------------------------------- (133) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (134) 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) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) if_min(true, z0, z1, z2) -> z2 if_min(false, z0, z1, z2) -> minIter(z0, z1, z2) minIter(add(z0, z1), z2, z3) -> if_min(le(z0, z3), z1, z2, z3) minIter(nil, add(z0, z1), z2) -> minIter(add(z0, z1), add(z0, z1), s(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, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) RM(0, add(0, x2)) -> c21(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c21(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c21(IF_RM(false, s(z0), add(0, x2))) MINITER(add(s(s(z0)), x1), x2, s(0)) -> c12(IF_MIN(false, x1, x2, s(0)), LE(s(s(z0)), s(0))) MINITER(add(s(x0), x1), x2, s(x3)) -> c12(LE(s(x0), s(x3))) MINITER(add(s(0), x1), x2, s(z0)) -> c12(LE(s(0), s(z0))) RM(s(0), add(s(0), x2)) -> c21(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c21(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c21(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(x0), add(s(x1), x2)) -> c21(EQ(s(x0), s(x1))) MINSORT(add(x0, x1), x2) -> c(MIN(add(x0, x1))) IF_MINSORT(true, add(x0, add(x1, x2)), x3) -> c26(APP(rm(x0, add(x1, x2)), x3)) IF_MINSORT(true, add(x0, nil), z0) -> c26(MINSORT(z0, nil), APP(rm(x0, nil), z0)) MINSORT(add(z0, z1), z2) -> c25(IF_MINSORT(eq(z0, if_min(le(z0, 0), z1, add(z0, z1), 0)), add(z0, z1), z2), EQ(z0, minIter(add(z0, z1), add(z0, z1), 0)), MIN(add(z0, z1))) MINSORT(add(z0, z1), z2) -> c(EQ(z0, minIter(add(z0, z1), add(z0, z1), 0))) MINITER(add(s(s(s(z0))), x1), x2, s(s(0))) -> c12(IF_MIN(false, x1, x2, s(s(0))), LE(s(s(s(z0))), s(s(0)))) MINITER(add(s(s(s(z0))), x1), x2, s(s(s(z1)))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(s(z1)))), LE(s(s(s(z0))), s(s(s(z1))))) MINITER(add(s(s(x0)), x1), x2, s(s(x3))) -> c12(LE(s(s(x0)), s(s(x3)))) MINITER(add(s(s(0)), x1), x2, s(s(z0))) -> c12(LE(s(s(0)), s(s(z0)))) RM(s(s(0)), add(s(s(0)), x2)) -> c21(IF_RM(true, s(s(0)), add(s(s(0)), x2)), EQ(s(s(0)), s(s(0)))) RM(s(s(0)), add(s(s(s(z0))), x2)) -> c21(IF_RM(false, s(s(0)), add(s(s(s(z0))), x2)), EQ(s(s(0)), s(s(s(z0))))) RM(s(s(s(z0))), add(s(s(0)), x2)) -> c21(IF_RM(false, s(s(s(z0))), add(s(s(0)), x2)), EQ(s(s(s(z0))), s(s(0)))) RM(s(s(s(z0))), add(s(s(s(z1))), x2)) -> c21(IF_RM(eq(z0, z1), s(s(s(z0))), add(s(s(s(z1))), x2)), EQ(s(s(s(z0))), s(s(s(z1))))) RM(s(s(x0)), add(s(s(x1)), x2)) -> c21(EQ(s(s(x0)), s(s(x1)))) IF_MINSORT(true, add(0, add(0, z2)), x1) -> c26(MINSORT(app(rm(0, z2), x1), nil), APP(rm(0, add(0, z2)), x1), RM(0, add(0, z2))) IF_MINSORT(true, add(0, add(0, x0)), x1) -> c26(RM(0, add(0, x0))) IF_MINSORT(true, add(0, add(s(x0), z2)), x2) -> c26(MINSORT(app(add(s(x0), rm(0, z2)), x2), nil), APP(rm(0, add(s(x0), z2)), x2), RM(0, add(s(x0), z2))) IF_MINSORT(true, add(0, add(s(x0), x1)), x2) -> c26(RM(0, add(s(x0), x1))) IF_MINSORT(true, add(s(x0), add(0, z2)), x2) -> c26(MINSORT(app(add(0, rm(s(x0), z2)), x2), nil), APP(rm(s(x0), add(0, z2)), x2), RM(s(x0), add(0, z2))) IF_MINSORT(true, add(s(x0), add(0, x1)), x2) -> c26(RM(s(x0), add(0, x1))) IF_MINSORT(true, add(s(0), add(s(0), x2)), x3) -> c26(MINSORT(app(if_rm(true, s(0), add(s(0), x2)), x3), nil), APP(rm(s(0), add(s(0), x2)), x3), RM(s(0), add(s(0), x2))) IF_MINSORT(true, add(s(0), add(s(s(z0)), x2)), x3) -> c26(MINSORT(app(if_rm(false, s(0), add(s(s(z0)), x2)), x3), nil), APP(rm(s(0), add(s(s(z0)), x2)), x3), RM(s(0), add(s(s(z0)), x2))) IF_MINSORT(true, add(s(s(z0)), add(s(0), x2)), x3) -> c26(MINSORT(app(if_rm(false, s(s(z0)), add(s(0), x2)), x3), nil), APP(rm(s(s(z0)), add(s(0), x2)), x3), RM(s(s(z0)), add(s(0), x2))) IF_MINSORT(true, add(s(s(z0)), add(s(s(z1)), x2)), x3) -> c26(MINSORT(app(if_rm(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), x3), nil), APP(rm(s(s(z0)), add(s(s(z1)), x2)), x3), RM(s(s(z0)), add(s(s(z1)), x2))) IF_MINSORT(true, add(s(x0), add(s(x1), x2)), x3) -> c1(APP(rm(s(x0), add(s(x1), x2)), x3)) IF_MINSORT(true, add(s(x0), add(s(x1), x2)), x3) -> c1(RM(s(x0), add(s(x1), x2))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) MINITER(add(s(s(z0)), x1), x2, s(0)) -> c12(IF_MIN(false, x1, x2, s(0)), LE(s(s(z0)), s(0))) MINITER(add(s(x0), x1), x2, s(x3)) -> c12(LE(s(x0), s(x3))) MINITER(add(s(0), x1), x2, s(z0)) -> c12(LE(s(0), s(z0))) MINITER(add(s(s(s(z0))), x1), x2, s(s(0))) -> c12(IF_MIN(false, x1, x2, s(s(0))), LE(s(s(s(z0))), s(s(0)))) MINITER(add(s(s(s(z0))), x1), x2, s(s(s(z1)))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(s(z1)))), LE(s(s(s(z0))), s(s(s(z1))))) MINITER(add(s(s(x0)), x1), x2, s(s(x3))) -> c12(LE(s(s(x0)), s(s(x3)))) MINITER(add(s(s(0)), x1), x2, s(s(z0))) -> c12(LE(s(s(0)), s(s(z0)))) K tuples: IF_MINSORT(true, add(z0, z1), z2) -> c26(MINSORT(app(rm(z0, z1), z2), nil), APP(rm(z0, z1), z2), RM(z0, z1)) MINSORT(add(z0, z1), z2) -> c25(IF_MINSORT(eq(z0, min(add(z0, z1))), add(z0, z1), z2), EQ(z0, min(add(z0, z1))), MIN(add(z0, z1))) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MIN(add(z0, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) RM(z0, add(z1, z2)) -> c21(IF_RM(eq(z0, z1), z0, add(z1, z2)), EQ(z0, z1)) Defined Rule Symbols: le_2, eq_2, app_2, if_rm_3, rm_2, if_min_4, minIter_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, MIN_1, MINITER_3, IF_MIN_4, IF_RM_3, IF_MINSORT_3, RM_2, MINSORT_2 Compound Symbols: c3_1, c6_1, c8_1, c10_1, c11_1, c14_1, c22_1, c23_1, c27_1, c12_1, c21_1, c12_2, c21_2, c_1, c26_1, c26_2, c25_3, c26_3, c1_1 ---------------------------------------- (135) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF_MINSORT(true, add(x0, add(x1, x2)), x3) -> c26(APP(rm(x0, add(x1, x2)), x3)) by IF_MINSORT(true, add(z0, add(z1, z2)), x3) -> c26(APP(if_rm(eq(z0, z1), z0, add(z1, z2)), x3)) ---------------------------------------- (136) 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) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) if_min(true, z0, z1, z2) -> z2 if_min(false, z0, z1, z2) -> minIter(z0, z1, z2) minIter(add(z0, z1), z2, z3) -> if_min(le(z0, z3), z1, z2, z3) minIter(nil, add(z0, z1), z2) -> minIter(add(z0, z1), add(z0, z1), s(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, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) RM(0, add(0, x2)) -> c21(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c21(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c21(IF_RM(false, s(z0), add(0, x2))) MINITER(add(s(s(z0)), x1), x2, s(0)) -> c12(IF_MIN(false, x1, x2, s(0)), LE(s(s(z0)), s(0))) MINITER(add(s(x0), x1), x2, s(x3)) -> c12(LE(s(x0), s(x3))) MINITER(add(s(0), x1), x2, s(z0)) -> c12(LE(s(0), s(z0))) RM(s(0), add(s(0), x2)) -> c21(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c21(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c21(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(x0), add(s(x1), x2)) -> c21(EQ(s(x0), s(x1))) MINSORT(add(x0, x1), x2) -> c(MIN(add(x0, x1))) IF_MINSORT(true, add(x0, nil), z0) -> c26(MINSORT(z0, nil), APP(rm(x0, nil), z0)) MINSORT(add(z0, z1), z2) -> c25(IF_MINSORT(eq(z0, if_min(le(z0, 0), z1, add(z0, z1), 0)), add(z0, z1), z2), EQ(z0, minIter(add(z0, z1), add(z0, z1), 0)), MIN(add(z0, z1))) MINSORT(add(z0, z1), z2) -> c(EQ(z0, minIter(add(z0, z1), add(z0, z1), 0))) MINITER(add(s(s(s(z0))), x1), x2, s(s(0))) -> c12(IF_MIN(false, x1, x2, s(s(0))), LE(s(s(s(z0))), s(s(0)))) MINITER(add(s(s(s(z0))), x1), x2, s(s(s(z1)))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(s(z1)))), LE(s(s(s(z0))), s(s(s(z1))))) MINITER(add(s(s(x0)), x1), x2, s(s(x3))) -> c12(LE(s(s(x0)), s(s(x3)))) MINITER(add(s(s(0)), x1), x2, s(s(z0))) -> c12(LE(s(s(0)), s(s(z0)))) RM(s(s(0)), add(s(s(0)), x2)) -> c21(IF_RM(true, s(s(0)), add(s(s(0)), x2)), EQ(s(s(0)), s(s(0)))) RM(s(s(0)), add(s(s(s(z0))), x2)) -> c21(IF_RM(false, s(s(0)), add(s(s(s(z0))), x2)), EQ(s(s(0)), s(s(s(z0))))) RM(s(s(s(z0))), add(s(s(0)), x2)) -> c21(IF_RM(false, s(s(s(z0))), add(s(s(0)), x2)), EQ(s(s(s(z0))), s(s(0)))) RM(s(s(s(z0))), add(s(s(s(z1))), x2)) -> c21(IF_RM(eq(z0, z1), s(s(s(z0))), add(s(s(s(z1))), x2)), EQ(s(s(s(z0))), s(s(s(z1))))) RM(s(s(x0)), add(s(s(x1)), x2)) -> c21(EQ(s(s(x0)), s(s(x1)))) IF_MINSORT(true, add(0, add(0, z2)), x1) -> c26(MINSORT(app(rm(0, z2), x1), nil), APP(rm(0, add(0, z2)), x1), RM(0, add(0, z2))) IF_MINSORT(true, add(0, add(0, x0)), x1) -> c26(RM(0, add(0, x0))) IF_MINSORT(true, add(0, add(s(x0), z2)), x2) -> c26(MINSORT(app(add(s(x0), rm(0, z2)), x2), nil), APP(rm(0, add(s(x0), z2)), x2), RM(0, add(s(x0), z2))) IF_MINSORT(true, add(0, add(s(x0), x1)), x2) -> c26(RM(0, add(s(x0), x1))) IF_MINSORT(true, add(s(x0), add(0, z2)), x2) -> c26(MINSORT(app(add(0, rm(s(x0), z2)), x2), nil), APP(rm(s(x0), add(0, z2)), x2), RM(s(x0), add(0, z2))) IF_MINSORT(true, add(s(x0), add(0, x1)), x2) -> c26(RM(s(x0), add(0, x1))) IF_MINSORT(true, add(s(0), add(s(0), x2)), x3) -> c26(MINSORT(app(if_rm(true, s(0), add(s(0), x2)), x3), nil), APP(rm(s(0), add(s(0), x2)), x3), RM(s(0), add(s(0), x2))) IF_MINSORT(true, add(s(0), add(s(s(z0)), x2)), x3) -> c26(MINSORT(app(if_rm(false, s(0), add(s(s(z0)), x2)), x3), nil), APP(rm(s(0), add(s(s(z0)), x2)), x3), RM(s(0), add(s(s(z0)), x2))) IF_MINSORT(true, add(s(s(z0)), add(s(0), x2)), x3) -> c26(MINSORT(app(if_rm(false, s(s(z0)), add(s(0), x2)), x3), nil), APP(rm(s(s(z0)), add(s(0), x2)), x3), RM(s(s(z0)), add(s(0), x2))) IF_MINSORT(true, add(s(s(z0)), add(s(s(z1)), x2)), x3) -> c26(MINSORT(app(if_rm(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), x3), nil), APP(rm(s(s(z0)), add(s(s(z1)), x2)), x3), RM(s(s(z0)), add(s(s(z1)), x2))) IF_MINSORT(true, add(s(x0), add(s(x1), x2)), x3) -> c1(APP(rm(s(x0), add(s(x1), x2)), x3)) IF_MINSORT(true, add(s(x0), add(s(x1), x2)), x3) -> c1(RM(s(x0), add(s(x1), x2))) IF_MINSORT(true, add(z0, add(z1, z2)), x3) -> c26(APP(if_rm(eq(z0, z1), z0, add(z1, z2)), x3)) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) MINITER(add(s(s(z0)), x1), x2, s(0)) -> c12(IF_MIN(false, x1, x2, s(0)), LE(s(s(z0)), s(0))) MINITER(add(s(x0), x1), x2, s(x3)) -> c12(LE(s(x0), s(x3))) MINITER(add(s(0), x1), x2, s(z0)) -> c12(LE(s(0), s(z0))) MINITER(add(s(s(s(z0))), x1), x2, s(s(0))) -> c12(IF_MIN(false, x1, x2, s(s(0))), LE(s(s(s(z0))), s(s(0)))) MINITER(add(s(s(s(z0))), x1), x2, s(s(s(z1)))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(s(z1)))), LE(s(s(s(z0))), s(s(s(z1))))) MINITER(add(s(s(x0)), x1), x2, s(s(x3))) -> c12(LE(s(s(x0)), s(s(x3)))) MINITER(add(s(s(0)), x1), x2, s(s(z0))) -> c12(LE(s(s(0)), s(s(z0)))) K tuples: IF_MINSORT(true, add(z0, z1), z2) -> c26(MINSORT(app(rm(z0, z1), z2), nil), APP(rm(z0, z1), z2), RM(z0, z1)) MINSORT(add(z0, z1), z2) -> c25(IF_MINSORT(eq(z0, min(add(z0, z1))), add(z0, z1), z2), EQ(z0, min(add(z0, z1))), MIN(add(z0, z1))) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MIN(add(z0, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) RM(z0, add(z1, z2)) -> c21(IF_RM(eq(z0, z1), z0, add(z1, z2)), EQ(z0, z1)) Defined Rule Symbols: le_2, eq_2, app_2, if_rm_3, rm_2, if_min_4, minIter_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, MIN_1, MINITER_3, IF_MIN_4, IF_RM_3, IF_MINSORT_3, RM_2, MINSORT_2 Compound Symbols: c3_1, c6_1, c8_1, c10_1, c11_1, c14_1, c22_1, c23_1, c27_1, c12_1, c21_1, c12_2, c21_2, c_1, c26_2, c25_3, c26_3, c26_1, c1_1 ---------------------------------------- (137) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF_MINSORT(true, add(x0, nil), z0) -> c26(MINSORT(z0, nil), APP(rm(x0, nil), z0)) by IF_MINSORT(true, add(z0, nil), x1) -> c26(MINSORT(x1, nil), APP(nil, x1)) ---------------------------------------- (138) 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) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) if_min(true, z0, z1, z2) -> z2 if_min(false, z0, z1, z2) -> minIter(z0, z1, z2) minIter(add(z0, z1), z2, z3) -> if_min(le(z0, z3), z1, z2, z3) minIter(nil, add(z0, z1), z2) -> minIter(add(z0, z1), add(z0, z1), s(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, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) RM(0, add(0, x2)) -> c21(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c21(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c21(IF_RM(false, s(z0), add(0, x2))) MINITER(add(s(s(z0)), x1), x2, s(0)) -> c12(IF_MIN(false, x1, x2, s(0)), LE(s(s(z0)), s(0))) MINITER(add(s(x0), x1), x2, s(x3)) -> c12(LE(s(x0), s(x3))) MINITER(add(s(0), x1), x2, s(z0)) -> c12(LE(s(0), s(z0))) RM(s(0), add(s(0), x2)) -> c21(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c21(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c21(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(x0), add(s(x1), x2)) -> c21(EQ(s(x0), s(x1))) MINSORT(add(x0, x1), x2) -> c(MIN(add(x0, x1))) MINSORT(add(z0, z1), z2) -> c25(IF_MINSORT(eq(z0, if_min(le(z0, 0), z1, add(z0, z1), 0)), add(z0, z1), z2), EQ(z0, minIter(add(z0, z1), add(z0, z1), 0)), MIN(add(z0, z1))) MINSORT(add(z0, z1), z2) -> c(EQ(z0, minIter(add(z0, z1), add(z0, z1), 0))) MINITER(add(s(s(s(z0))), x1), x2, s(s(0))) -> c12(IF_MIN(false, x1, x2, s(s(0))), LE(s(s(s(z0))), s(s(0)))) MINITER(add(s(s(s(z0))), x1), x2, s(s(s(z1)))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(s(z1)))), LE(s(s(s(z0))), s(s(s(z1))))) MINITER(add(s(s(x0)), x1), x2, s(s(x3))) -> c12(LE(s(s(x0)), s(s(x3)))) MINITER(add(s(s(0)), x1), x2, s(s(z0))) -> c12(LE(s(s(0)), s(s(z0)))) RM(s(s(0)), add(s(s(0)), x2)) -> c21(IF_RM(true, s(s(0)), add(s(s(0)), x2)), EQ(s(s(0)), s(s(0)))) RM(s(s(0)), add(s(s(s(z0))), x2)) -> c21(IF_RM(false, s(s(0)), add(s(s(s(z0))), x2)), EQ(s(s(0)), s(s(s(z0))))) RM(s(s(s(z0))), add(s(s(0)), x2)) -> c21(IF_RM(false, s(s(s(z0))), add(s(s(0)), x2)), EQ(s(s(s(z0))), s(s(0)))) RM(s(s(s(z0))), add(s(s(s(z1))), x2)) -> c21(IF_RM(eq(z0, z1), s(s(s(z0))), add(s(s(s(z1))), x2)), EQ(s(s(s(z0))), s(s(s(z1))))) RM(s(s(x0)), add(s(s(x1)), x2)) -> c21(EQ(s(s(x0)), s(s(x1)))) IF_MINSORT(true, add(0, add(0, z2)), x1) -> c26(MINSORT(app(rm(0, z2), x1), nil), APP(rm(0, add(0, z2)), x1), RM(0, add(0, z2))) IF_MINSORT(true, add(0, add(0, x0)), x1) -> c26(RM(0, add(0, x0))) IF_MINSORT(true, add(0, add(s(x0), z2)), x2) -> c26(MINSORT(app(add(s(x0), rm(0, z2)), x2), nil), APP(rm(0, add(s(x0), z2)), x2), RM(0, add(s(x0), z2))) IF_MINSORT(true, add(0, add(s(x0), x1)), x2) -> c26(RM(0, add(s(x0), x1))) IF_MINSORT(true, add(s(x0), add(0, z2)), x2) -> c26(MINSORT(app(add(0, rm(s(x0), z2)), x2), nil), APP(rm(s(x0), add(0, z2)), x2), RM(s(x0), add(0, z2))) IF_MINSORT(true, add(s(x0), add(0, x1)), x2) -> c26(RM(s(x0), add(0, x1))) IF_MINSORT(true, add(s(0), add(s(0), x2)), x3) -> c26(MINSORT(app(if_rm(true, s(0), add(s(0), x2)), x3), nil), APP(rm(s(0), add(s(0), x2)), x3), RM(s(0), add(s(0), x2))) IF_MINSORT(true, add(s(0), add(s(s(z0)), x2)), x3) -> c26(MINSORT(app(if_rm(false, s(0), add(s(s(z0)), x2)), x3), nil), APP(rm(s(0), add(s(s(z0)), x2)), x3), RM(s(0), add(s(s(z0)), x2))) IF_MINSORT(true, add(s(s(z0)), add(s(0), x2)), x3) -> c26(MINSORT(app(if_rm(false, s(s(z0)), add(s(0), x2)), x3), nil), APP(rm(s(s(z0)), add(s(0), x2)), x3), RM(s(s(z0)), add(s(0), x2))) IF_MINSORT(true, add(s(s(z0)), add(s(s(z1)), x2)), x3) -> c26(MINSORT(app(if_rm(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), x3), nil), APP(rm(s(s(z0)), add(s(s(z1)), x2)), x3), RM(s(s(z0)), add(s(s(z1)), x2))) IF_MINSORT(true, add(s(x0), add(s(x1), x2)), x3) -> c1(APP(rm(s(x0), add(s(x1), x2)), x3)) IF_MINSORT(true, add(s(x0), add(s(x1), x2)), x3) -> c1(RM(s(x0), add(s(x1), x2))) IF_MINSORT(true, add(z0, add(z1, z2)), x3) -> c26(APP(if_rm(eq(z0, z1), z0, add(z1, z2)), x3)) IF_MINSORT(true, add(z0, nil), x1) -> c26(MINSORT(x1, nil), APP(nil, x1)) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) MINITER(add(s(s(z0)), x1), x2, s(0)) -> c12(IF_MIN(false, x1, x2, s(0)), LE(s(s(z0)), s(0))) MINITER(add(s(x0), x1), x2, s(x3)) -> c12(LE(s(x0), s(x3))) MINITER(add(s(0), x1), x2, s(z0)) -> c12(LE(s(0), s(z0))) MINITER(add(s(s(s(z0))), x1), x2, s(s(0))) -> c12(IF_MIN(false, x1, x2, s(s(0))), LE(s(s(s(z0))), s(s(0)))) MINITER(add(s(s(s(z0))), x1), x2, s(s(s(z1)))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(s(z1)))), LE(s(s(s(z0))), s(s(s(z1))))) MINITER(add(s(s(x0)), x1), x2, s(s(x3))) -> c12(LE(s(s(x0)), s(s(x3)))) MINITER(add(s(s(0)), x1), x2, s(s(z0))) -> c12(LE(s(s(0)), s(s(z0)))) K tuples: IF_MINSORT(true, add(z0, z1), z2) -> c26(MINSORT(app(rm(z0, z1), z2), nil), APP(rm(z0, z1), z2), RM(z0, z1)) MINSORT(add(z0, z1), z2) -> c25(IF_MINSORT(eq(z0, min(add(z0, z1))), add(z0, z1), z2), EQ(z0, min(add(z0, z1))), MIN(add(z0, z1))) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MIN(add(z0, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) RM(z0, add(z1, z2)) -> c21(IF_RM(eq(z0, z1), z0, add(z1, z2)), EQ(z0, z1)) Defined Rule Symbols: le_2, eq_2, app_2, if_rm_3, rm_2, if_min_4, minIter_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, MIN_1, MINITER_3, IF_MIN_4, IF_RM_3, IF_MINSORT_3, RM_2, MINSORT_2 Compound Symbols: c3_1, c6_1, c8_1, c10_1, c11_1, c14_1, c22_1, c23_1, c27_1, c12_1, c21_1, c12_2, c21_2, c_1, c25_3, c26_3, c26_1, c1_1, c26_2 ---------------------------------------- (139) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (140) 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) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) if_min(true, z0, z1, z2) -> z2 if_min(false, z0, z1, z2) -> minIter(z0, z1, z2) minIter(add(z0, z1), z2, z3) -> if_min(le(z0, z3), z1, z2, z3) minIter(nil, add(z0, z1), z2) -> minIter(add(z0, z1), add(z0, z1), s(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, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) RM(0, add(0, x2)) -> c21(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c21(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c21(IF_RM(false, s(z0), add(0, x2))) MINITER(add(s(s(z0)), x1), x2, s(0)) -> c12(IF_MIN(false, x1, x2, s(0)), LE(s(s(z0)), s(0))) MINITER(add(s(x0), x1), x2, s(x3)) -> c12(LE(s(x0), s(x3))) MINITER(add(s(0), x1), x2, s(z0)) -> c12(LE(s(0), s(z0))) RM(s(0), add(s(0), x2)) -> c21(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c21(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c21(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(x0), add(s(x1), x2)) -> c21(EQ(s(x0), s(x1))) MINSORT(add(x0, x1), x2) -> c(MIN(add(x0, x1))) MINSORT(add(z0, z1), z2) -> c25(IF_MINSORT(eq(z0, if_min(le(z0, 0), z1, add(z0, z1), 0)), add(z0, z1), z2), EQ(z0, minIter(add(z0, z1), add(z0, z1), 0)), MIN(add(z0, z1))) MINSORT(add(z0, z1), z2) -> c(EQ(z0, minIter(add(z0, z1), add(z0, z1), 0))) MINITER(add(s(s(s(z0))), x1), x2, s(s(0))) -> c12(IF_MIN(false, x1, x2, s(s(0))), LE(s(s(s(z0))), s(s(0)))) MINITER(add(s(s(s(z0))), x1), x2, s(s(s(z1)))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(s(z1)))), LE(s(s(s(z0))), s(s(s(z1))))) MINITER(add(s(s(x0)), x1), x2, s(s(x3))) -> c12(LE(s(s(x0)), s(s(x3)))) MINITER(add(s(s(0)), x1), x2, s(s(z0))) -> c12(LE(s(s(0)), s(s(z0)))) RM(s(s(0)), add(s(s(0)), x2)) -> c21(IF_RM(true, s(s(0)), add(s(s(0)), x2)), EQ(s(s(0)), s(s(0)))) RM(s(s(0)), add(s(s(s(z0))), x2)) -> c21(IF_RM(false, s(s(0)), add(s(s(s(z0))), x2)), EQ(s(s(0)), s(s(s(z0))))) RM(s(s(s(z0))), add(s(s(0)), x2)) -> c21(IF_RM(false, s(s(s(z0))), add(s(s(0)), x2)), EQ(s(s(s(z0))), s(s(0)))) RM(s(s(s(z0))), add(s(s(s(z1))), x2)) -> c21(IF_RM(eq(z0, z1), s(s(s(z0))), add(s(s(s(z1))), x2)), EQ(s(s(s(z0))), s(s(s(z1))))) RM(s(s(x0)), add(s(s(x1)), x2)) -> c21(EQ(s(s(x0)), s(s(x1)))) IF_MINSORT(true, add(0, add(0, z2)), x1) -> c26(MINSORT(app(rm(0, z2), x1), nil), APP(rm(0, add(0, z2)), x1), RM(0, add(0, z2))) IF_MINSORT(true, add(0, add(0, x0)), x1) -> c26(RM(0, add(0, x0))) IF_MINSORT(true, add(0, add(s(x0), z2)), x2) -> c26(MINSORT(app(add(s(x0), rm(0, z2)), x2), nil), APP(rm(0, add(s(x0), z2)), x2), RM(0, add(s(x0), z2))) IF_MINSORT(true, add(0, add(s(x0), x1)), x2) -> c26(RM(0, add(s(x0), x1))) IF_MINSORT(true, add(s(x0), add(0, z2)), x2) -> c26(MINSORT(app(add(0, rm(s(x0), z2)), x2), nil), APP(rm(s(x0), add(0, z2)), x2), RM(s(x0), add(0, z2))) IF_MINSORT(true, add(s(x0), add(0, x1)), x2) -> c26(RM(s(x0), add(0, x1))) IF_MINSORT(true, add(s(0), add(s(0), x2)), x3) -> c26(MINSORT(app(if_rm(true, s(0), add(s(0), x2)), x3), nil), APP(rm(s(0), add(s(0), x2)), x3), RM(s(0), add(s(0), x2))) IF_MINSORT(true, add(s(0), add(s(s(z0)), x2)), x3) -> c26(MINSORT(app(if_rm(false, s(0), add(s(s(z0)), x2)), x3), nil), APP(rm(s(0), add(s(s(z0)), x2)), x3), RM(s(0), add(s(s(z0)), x2))) IF_MINSORT(true, add(s(s(z0)), add(s(0), x2)), x3) -> c26(MINSORT(app(if_rm(false, s(s(z0)), add(s(0), x2)), x3), nil), APP(rm(s(s(z0)), add(s(0), x2)), x3), RM(s(s(z0)), add(s(0), x2))) IF_MINSORT(true, add(s(s(z0)), add(s(s(z1)), x2)), x3) -> c26(MINSORT(app(if_rm(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), x3), nil), APP(rm(s(s(z0)), add(s(s(z1)), x2)), x3), RM(s(s(z0)), add(s(s(z1)), x2))) IF_MINSORT(true, add(s(x0), add(s(x1), x2)), x3) -> c1(APP(rm(s(x0), add(s(x1), x2)), x3)) IF_MINSORT(true, add(s(x0), add(s(x1), x2)), x3) -> c1(RM(s(x0), add(s(x1), x2))) IF_MINSORT(true, add(z0, add(z1, z2)), x3) -> c26(APP(if_rm(eq(z0, z1), z0, add(z1, z2)), x3)) IF_MINSORT(true, add(z0, nil), x1) -> c26(MINSORT(x1, nil)) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) MINITER(add(s(s(z0)), x1), x2, s(0)) -> c12(IF_MIN(false, x1, x2, s(0)), LE(s(s(z0)), s(0))) MINITER(add(s(x0), x1), x2, s(x3)) -> c12(LE(s(x0), s(x3))) MINITER(add(s(0), x1), x2, s(z0)) -> c12(LE(s(0), s(z0))) MINITER(add(s(s(s(z0))), x1), x2, s(s(0))) -> c12(IF_MIN(false, x1, x2, s(s(0))), LE(s(s(s(z0))), s(s(0)))) MINITER(add(s(s(s(z0))), x1), x2, s(s(s(z1)))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(s(z1)))), LE(s(s(s(z0))), s(s(s(z1))))) MINITER(add(s(s(x0)), x1), x2, s(s(x3))) -> c12(LE(s(s(x0)), s(s(x3)))) MINITER(add(s(s(0)), x1), x2, s(s(z0))) -> c12(LE(s(s(0)), s(s(z0)))) K tuples: IF_MINSORT(true, add(z0, z1), z2) -> c26(MINSORT(app(rm(z0, z1), z2), nil), APP(rm(z0, z1), z2), RM(z0, z1)) MINSORT(add(z0, z1), z2) -> c25(IF_MINSORT(eq(z0, min(add(z0, z1))), add(z0, z1), z2), EQ(z0, min(add(z0, z1))), MIN(add(z0, z1))) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MIN(add(z0, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) RM(z0, add(z1, z2)) -> c21(IF_RM(eq(z0, z1), z0, add(z1, z2)), EQ(z0, z1)) Defined Rule Symbols: le_2, eq_2, app_2, if_rm_3, rm_2, if_min_4, minIter_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, MIN_1, MINITER_3, IF_MIN_4, IF_RM_3, IF_MINSORT_3, RM_2, MINSORT_2 Compound Symbols: c3_1, c6_1, c8_1, c10_1, c11_1, c14_1, c22_1, c23_1, c27_1, c12_1, c21_1, c12_2, c21_2, c_1, c25_3, c26_3, c26_1, c1_1 ---------------------------------------- (141) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MINSORT(add(z0, z1), z2) -> c25(IF_MINSORT(eq(z0, if_min(le(z0, 0), z1, add(z0, z1), 0)), add(z0, z1), z2), EQ(z0, minIter(add(z0, z1), add(z0, z1), 0)), MIN(add(z0, z1))) by MINSORT(add(0, x1), x2) -> c25(IF_MINSORT(eq(0, if_min(true, x1, add(0, x1), 0)), add(0, x1), x2), EQ(0, minIter(add(0, x1), add(0, x1), 0)), MIN(add(0, x1))) MINSORT(add(s(z0), x1), x2) -> c25(IF_MINSORT(eq(s(z0), if_min(false, x1, add(s(z0), x1), 0)), add(s(z0), x1), x2), EQ(s(z0), minIter(add(s(z0), x1), add(s(z0), x1), 0)), MIN(add(s(z0), x1))) MINSORT(add(x0, x1), x2) -> c25(EQ(x0, minIter(add(x0, x1), add(x0, x1), 0)), MIN(add(x0, x1))) ---------------------------------------- (142) 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) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) if_min(true, z0, z1, z2) -> z2 if_min(false, z0, z1, z2) -> minIter(z0, z1, z2) minIter(add(z0, z1), z2, z3) -> if_min(le(z0, z3), z1, z2, z3) minIter(nil, add(z0, z1), z2) -> minIter(add(z0, z1), add(z0, z1), s(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, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) RM(0, add(0, x2)) -> c21(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c21(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c21(IF_RM(false, s(z0), add(0, x2))) MINITER(add(s(s(z0)), x1), x2, s(0)) -> c12(IF_MIN(false, x1, x2, s(0)), LE(s(s(z0)), s(0))) MINITER(add(s(x0), x1), x2, s(x3)) -> c12(LE(s(x0), s(x3))) MINITER(add(s(0), x1), x2, s(z0)) -> c12(LE(s(0), s(z0))) RM(s(0), add(s(0), x2)) -> c21(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c21(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c21(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(x0), add(s(x1), x2)) -> c21(EQ(s(x0), s(x1))) MINSORT(add(x0, x1), x2) -> c(MIN(add(x0, x1))) MINSORT(add(z0, z1), z2) -> c(EQ(z0, minIter(add(z0, z1), add(z0, z1), 0))) MINITER(add(s(s(s(z0))), x1), x2, s(s(0))) -> c12(IF_MIN(false, x1, x2, s(s(0))), LE(s(s(s(z0))), s(s(0)))) MINITER(add(s(s(s(z0))), x1), x2, s(s(s(z1)))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(s(z1)))), LE(s(s(s(z0))), s(s(s(z1))))) MINITER(add(s(s(x0)), x1), x2, s(s(x3))) -> c12(LE(s(s(x0)), s(s(x3)))) MINITER(add(s(s(0)), x1), x2, s(s(z0))) -> c12(LE(s(s(0)), s(s(z0)))) RM(s(s(0)), add(s(s(0)), x2)) -> c21(IF_RM(true, s(s(0)), add(s(s(0)), x2)), EQ(s(s(0)), s(s(0)))) RM(s(s(0)), add(s(s(s(z0))), x2)) -> c21(IF_RM(false, s(s(0)), add(s(s(s(z0))), x2)), EQ(s(s(0)), s(s(s(z0))))) RM(s(s(s(z0))), add(s(s(0)), x2)) -> c21(IF_RM(false, s(s(s(z0))), add(s(s(0)), x2)), EQ(s(s(s(z0))), s(s(0)))) RM(s(s(s(z0))), add(s(s(s(z1))), x2)) -> c21(IF_RM(eq(z0, z1), s(s(s(z0))), add(s(s(s(z1))), x2)), EQ(s(s(s(z0))), s(s(s(z1))))) RM(s(s(x0)), add(s(s(x1)), x2)) -> c21(EQ(s(s(x0)), s(s(x1)))) IF_MINSORT(true, add(0, add(0, z2)), x1) -> c26(MINSORT(app(rm(0, z2), x1), nil), APP(rm(0, add(0, z2)), x1), RM(0, add(0, z2))) IF_MINSORT(true, add(0, add(0, x0)), x1) -> c26(RM(0, add(0, x0))) IF_MINSORT(true, add(0, add(s(x0), z2)), x2) -> c26(MINSORT(app(add(s(x0), rm(0, z2)), x2), nil), APP(rm(0, add(s(x0), z2)), x2), RM(0, add(s(x0), z2))) IF_MINSORT(true, add(0, add(s(x0), x1)), x2) -> c26(RM(0, add(s(x0), x1))) IF_MINSORT(true, add(s(x0), add(0, z2)), x2) -> c26(MINSORT(app(add(0, rm(s(x0), z2)), x2), nil), APP(rm(s(x0), add(0, z2)), x2), RM(s(x0), add(0, z2))) IF_MINSORT(true, add(s(x0), add(0, x1)), x2) -> c26(RM(s(x0), add(0, x1))) IF_MINSORT(true, add(s(0), add(s(0), x2)), x3) -> c26(MINSORT(app(if_rm(true, s(0), add(s(0), x2)), x3), nil), APP(rm(s(0), add(s(0), x2)), x3), RM(s(0), add(s(0), x2))) IF_MINSORT(true, add(s(0), add(s(s(z0)), x2)), x3) -> c26(MINSORT(app(if_rm(false, s(0), add(s(s(z0)), x2)), x3), nil), APP(rm(s(0), add(s(s(z0)), x2)), x3), RM(s(0), add(s(s(z0)), x2))) IF_MINSORT(true, add(s(s(z0)), add(s(0), x2)), x3) -> c26(MINSORT(app(if_rm(false, s(s(z0)), add(s(0), x2)), x3), nil), APP(rm(s(s(z0)), add(s(0), x2)), x3), RM(s(s(z0)), add(s(0), x2))) IF_MINSORT(true, add(s(s(z0)), add(s(s(z1)), x2)), x3) -> c26(MINSORT(app(if_rm(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), x3), nil), APP(rm(s(s(z0)), add(s(s(z1)), x2)), x3), RM(s(s(z0)), add(s(s(z1)), x2))) IF_MINSORT(true, add(s(x0), add(s(x1), x2)), x3) -> c1(APP(rm(s(x0), add(s(x1), x2)), x3)) IF_MINSORT(true, add(s(x0), add(s(x1), x2)), x3) -> c1(RM(s(x0), add(s(x1), x2))) IF_MINSORT(true, add(z0, add(z1, z2)), x3) -> c26(APP(if_rm(eq(z0, z1), z0, add(z1, z2)), x3)) IF_MINSORT(true, add(z0, nil), x1) -> c26(MINSORT(x1, nil)) MINSORT(add(0, x1), x2) -> c25(IF_MINSORT(eq(0, if_min(true, x1, add(0, x1), 0)), add(0, x1), x2), EQ(0, minIter(add(0, x1), add(0, x1), 0)), MIN(add(0, x1))) MINSORT(add(s(z0), x1), x2) -> c25(IF_MINSORT(eq(s(z0), if_min(false, x1, add(s(z0), x1), 0)), add(s(z0), x1), x2), EQ(s(z0), minIter(add(s(z0), x1), add(s(z0), x1), 0)), MIN(add(s(z0), x1))) MINSORT(add(x0, x1), x2) -> c25(EQ(x0, minIter(add(x0, x1), add(x0, x1), 0)), MIN(add(x0, x1))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) MINITER(add(s(s(z0)), x1), x2, s(0)) -> c12(IF_MIN(false, x1, x2, s(0)), LE(s(s(z0)), s(0))) MINITER(add(s(x0), x1), x2, s(x3)) -> c12(LE(s(x0), s(x3))) MINITER(add(s(0), x1), x2, s(z0)) -> c12(LE(s(0), s(z0))) MINITER(add(s(s(s(z0))), x1), x2, s(s(0))) -> c12(IF_MIN(false, x1, x2, s(s(0))), LE(s(s(s(z0))), s(s(0)))) MINITER(add(s(s(s(z0))), x1), x2, s(s(s(z1)))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(s(z1)))), LE(s(s(s(z0))), s(s(s(z1))))) MINITER(add(s(s(x0)), x1), x2, s(s(x3))) -> c12(LE(s(s(x0)), s(s(x3)))) MINITER(add(s(s(0)), x1), x2, s(s(z0))) -> c12(LE(s(s(0)), s(s(z0)))) K tuples: IF_MINSORT(true, add(z0, z1), z2) -> c26(MINSORT(app(rm(z0, z1), z2), nil), APP(rm(z0, z1), z2), RM(z0, z1)) MINSORT(add(z0, z1), z2) -> c25(IF_MINSORT(eq(z0, min(add(z0, z1))), add(z0, z1), z2), EQ(z0, min(add(z0, z1))), MIN(add(z0, z1))) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MIN(add(z0, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) RM(z0, add(z1, z2)) -> c21(IF_RM(eq(z0, z1), z0, add(z1, z2)), EQ(z0, z1)) Defined Rule Symbols: le_2, eq_2, app_2, if_rm_3, rm_2, if_min_4, minIter_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, MIN_1, MINITER_3, IF_MIN_4, IF_RM_3, IF_MINSORT_3, RM_2, MINSORT_2 Compound Symbols: c3_1, c6_1, c8_1, c10_1, c11_1, c14_1, c22_1, c23_1, c27_1, c12_1, c21_1, c12_2, c21_2, c_1, c26_3, c26_1, c1_1, c25_3, c25_2 ---------------------------------------- (143) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (144) 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) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) if_min(true, z0, z1, z2) -> z2 if_min(false, z0, z1, z2) -> minIter(z0, z1, z2) minIter(add(z0, z1), z2, z3) -> if_min(le(z0, z3), z1, z2, z3) minIter(nil, add(z0, z1), z2) -> minIter(add(z0, z1), add(z0, z1), s(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, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) RM(0, add(0, x2)) -> c21(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c21(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c21(IF_RM(false, s(z0), add(0, x2))) MINITER(add(s(s(z0)), x1), x2, s(0)) -> c12(IF_MIN(false, x1, x2, s(0)), LE(s(s(z0)), s(0))) MINITER(add(s(x0), x1), x2, s(x3)) -> c12(LE(s(x0), s(x3))) MINITER(add(s(0), x1), x2, s(z0)) -> c12(LE(s(0), s(z0))) RM(s(0), add(s(0), x2)) -> c21(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c21(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c21(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(x0), add(s(x1), x2)) -> c21(EQ(s(x0), s(x1))) MINSORT(add(x0, x1), x2) -> c(MIN(add(x0, x1))) MINSORT(add(z0, z1), z2) -> c(EQ(z0, minIter(add(z0, z1), add(z0, z1), 0))) MINITER(add(s(s(s(z0))), x1), x2, s(s(0))) -> c12(IF_MIN(false, x1, x2, s(s(0))), LE(s(s(s(z0))), s(s(0)))) MINITER(add(s(s(s(z0))), x1), x2, s(s(s(z1)))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(s(z1)))), LE(s(s(s(z0))), s(s(s(z1))))) MINITER(add(s(s(x0)), x1), x2, s(s(x3))) -> c12(LE(s(s(x0)), s(s(x3)))) MINITER(add(s(s(0)), x1), x2, s(s(z0))) -> c12(LE(s(s(0)), s(s(z0)))) RM(s(s(0)), add(s(s(0)), x2)) -> c21(IF_RM(true, s(s(0)), add(s(s(0)), x2)), EQ(s(s(0)), s(s(0)))) RM(s(s(0)), add(s(s(s(z0))), x2)) -> c21(IF_RM(false, s(s(0)), add(s(s(s(z0))), x2)), EQ(s(s(0)), s(s(s(z0))))) RM(s(s(s(z0))), add(s(s(0)), x2)) -> c21(IF_RM(false, s(s(s(z0))), add(s(s(0)), x2)), EQ(s(s(s(z0))), s(s(0)))) RM(s(s(s(z0))), add(s(s(s(z1))), x2)) -> c21(IF_RM(eq(z0, z1), s(s(s(z0))), add(s(s(s(z1))), x2)), EQ(s(s(s(z0))), s(s(s(z1))))) RM(s(s(x0)), add(s(s(x1)), x2)) -> c21(EQ(s(s(x0)), s(s(x1)))) IF_MINSORT(true, add(0, add(0, z2)), x1) -> c26(MINSORT(app(rm(0, z2), x1), nil), APP(rm(0, add(0, z2)), x1), RM(0, add(0, z2))) IF_MINSORT(true, add(0, add(0, x0)), x1) -> c26(RM(0, add(0, x0))) IF_MINSORT(true, add(0, add(s(x0), z2)), x2) -> c26(MINSORT(app(add(s(x0), rm(0, z2)), x2), nil), APP(rm(0, add(s(x0), z2)), x2), RM(0, add(s(x0), z2))) IF_MINSORT(true, add(0, add(s(x0), x1)), x2) -> c26(RM(0, add(s(x0), x1))) IF_MINSORT(true, add(s(x0), add(0, z2)), x2) -> c26(MINSORT(app(add(0, rm(s(x0), z2)), x2), nil), APP(rm(s(x0), add(0, z2)), x2), RM(s(x0), add(0, z2))) IF_MINSORT(true, add(s(x0), add(0, x1)), x2) -> c26(RM(s(x0), add(0, x1))) IF_MINSORT(true, add(s(0), add(s(0), x2)), x3) -> c26(MINSORT(app(if_rm(true, s(0), add(s(0), x2)), x3), nil), APP(rm(s(0), add(s(0), x2)), x3), RM(s(0), add(s(0), x2))) IF_MINSORT(true, add(s(0), add(s(s(z0)), x2)), x3) -> c26(MINSORT(app(if_rm(false, s(0), add(s(s(z0)), x2)), x3), nil), APP(rm(s(0), add(s(s(z0)), x2)), x3), RM(s(0), add(s(s(z0)), x2))) IF_MINSORT(true, add(s(s(z0)), add(s(0), x2)), x3) -> c26(MINSORT(app(if_rm(false, s(s(z0)), add(s(0), x2)), x3), nil), APP(rm(s(s(z0)), add(s(0), x2)), x3), RM(s(s(z0)), add(s(0), x2))) IF_MINSORT(true, add(s(s(z0)), add(s(s(z1)), x2)), x3) -> c26(MINSORT(app(if_rm(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), x3), nil), APP(rm(s(s(z0)), add(s(s(z1)), x2)), x3), RM(s(s(z0)), add(s(s(z1)), x2))) IF_MINSORT(true, add(s(x0), add(s(x1), x2)), x3) -> c1(APP(rm(s(x0), add(s(x1), x2)), x3)) IF_MINSORT(true, add(s(x0), add(s(x1), x2)), x3) -> c1(RM(s(x0), add(s(x1), x2))) IF_MINSORT(true, add(z0, add(z1, z2)), x3) -> c26(APP(if_rm(eq(z0, z1), z0, add(z1, z2)), x3)) IF_MINSORT(true, add(z0, nil), x1) -> c26(MINSORT(x1, nil)) MINSORT(add(s(z0), x1), x2) -> c25(IF_MINSORT(eq(s(z0), if_min(false, x1, add(s(z0), x1), 0)), add(s(z0), x1), x2), EQ(s(z0), minIter(add(s(z0), x1), add(s(z0), x1), 0)), MIN(add(s(z0), x1))) MINSORT(add(x0, x1), x2) -> c25(EQ(x0, minIter(add(x0, x1), add(x0, x1), 0)), MIN(add(x0, x1))) MINSORT(add(0, x1), x2) -> c25(IF_MINSORT(eq(0, if_min(true, x1, add(0, x1), 0)), add(0, x1), x2), MIN(add(0, x1))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) MINITER(add(s(s(z0)), x1), x2, s(0)) -> c12(IF_MIN(false, x1, x2, s(0)), LE(s(s(z0)), s(0))) MINITER(add(s(x0), x1), x2, s(x3)) -> c12(LE(s(x0), s(x3))) MINITER(add(s(0), x1), x2, s(z0)) -> c12(LE(s(0), s(z0))) MINITER(add(s(s(s(z0))), x1), x2, s(s(0))) -> c12(IF_MIN(false, x1, x2, s(s(0))), LE(s(s(s(z0))), s(s(0)))) MINITER(add(s(s(s(z0))), x1), x2, s(s(s(z1)))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(s(z1)))), LE(s(s(s(z0))), s(s(s(z1))))) MINITER(add(s(s(x0)), x1), x2, s(s(x3))) -> c12(LE(s(s(x0)), s(s(x3)))) MINITER(add(s(s(0)), x1), x2, s(s(z0))) -> c12(LE(s(s(0)), s(s(z0)))) K tuples: IF_MINSORT(true, add(z0, z1), z2) -> c26(MINSORT(app(rm(z0, z1), z2), nil), APP(rm(z0, z1), z2), RM(z0, z1)) MINSORT(add(z0, z1), z2) -> c25(IF_MINSORT(eq(z0, min(add(z0, z1))), add(z0, z1), z2), EQ(z0, min(add(z0, z1))), MIN(add(z0, z1))) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MIN(add(z0, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) RM(z0, add(z1, z2)) -> c21(IF_RM(eq(z0, z1), z0, add(z1, z2)), EQ(z0, z1)) Defined Rule Symbols: le_2, eq_2, app_2, if_rm_3, rm_2, if_min_4, minIter_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, MIN_1, MINITER_3, IF_MIN_4, IF_RM_3, IF_MINSORT_3, RM_2, MINSORT_2 Compound Symbols: c3_1, c6_1, c8_1, c10_1, c11_1, c14_1, c22_1, c23_1, c27_1, c12_1, c21_1, c12_2, c21_2, c_1, c26_3, c26_1, c1_1, c25_3, c25_2 ---------------------------------------- (145) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (146) 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) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) if_min(true, z0, z1, z2) -> z2 if_min(false, z0, z1, z2) -> minIter(z0, z1, z2) minIter(add(z0, z1), z2, z3) -> if_min(le(z0, z3), z1, z2, z3) minIter(nil, add(z0, z1), z2) -> minIter(add(z0, z1), add(z0, z1), s(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, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) RM(0, add(0, x2)) -> c21(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c21(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c21(IF_RM(false, s(z0), add(0, x2))) MINITER(add(s(s(z0)), x1), x2, s(0)) -> c12(IF_MIN(false, x1, x2, s(0)), LE(s(s(z0)), s(0))) MINITER(add(s(x0), x1), x2, s(x3)) -> c12(LE(s(x0), s(x3))) MINITER(add(s(0), x1), x2, s(z0)) -> c12(LE(s(0), s(z0))) RM(s(0), add(s(0), x2)) -> c21(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c21(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c21(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(x0), add(s(x1), x2)) -> c21(EQ(s(x0), s(x1))) MINSORT(add(x0, x1), x2) -> c(MIN(add(x0, x1))) MINSORT(add(z0, z1), z2) -> c(EQ(z0, minIter(add(z0, z1), add(z0, z1), 0))) MINITER(add(s(s(s(z0))), x1), x2, s(s(0))) -> c12(IF_MIN(false, x1, x2, s(s(0))), LE(s(s(s(z0))), s(s(0)))) MINITER(add(s(s(s(z0))), x1), x2, s(s(s(z1)))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(s(z1)))), LE(s(s(s(z0))), s(s(s(z1))))) MINITER(add(s(s(x0)), x1), x2, s(s(x3))) -> c12(LE(s(s(x0)), s(s(x3)))) MINITER(add(s(s(0)), x1), x2, s(s(z0))) -> c12(LE(s(s(0)), s(s(z0)))) RM(s(s(0)), add(s(s(0)), x2)) -> c21(IF_RM(true, s(s(0)), add(s(s(0)), x2)), EQ(s(s(0)), s(s(0)))) RM(s(s(0)), add(s(s(s(z0))), x2)) -> c21(IF_RM(false, s(s(0)), add(s(s(s(z0))), x2)), EQ(s(s(0)), s(s(s(z0))))) RM(s(s(s(z0))), add(s(s(0)), x2)) -> c21(IF_RM(false, s(s(s(z0))), add(s(s(0)), x2)), EQ(s(s(s(z0))), s(s(0)))) RM(s(s(s(z0))), add(s(s(s(z1))), x2)) -> c21(IF_RM(eq(z0, z1), s(s(s(z0))), add(s(s(s(z1))), x2)), EQ(s(s(s(z0))), s(s(s(z1))))) RM(s(s(x0)), add(s(s(x1)), x2)) -> c21(EQ(s(s(x0)), s(s(x1)))) IF_MINSORT(true, add(0, add(0, z2)), x1) -> c26(MINSORT(app(rm(0, z2), x1), nil), APP(rm(0, add(0, z2)), x1), RM(0, add(0, z2))) IF_MINSORT(true, add(0, add(0, x0)), x1) -> c26(RM(0, add(0, x0))) IF_MINSORT(true, add(0, add(s(x0), z2)), x2) -> c26(MINSORT(app(add(s(x0), rm(0, z2)), x2), nil), APP(rm(0, add(s(x0), z2)), x2), RM(0, add(s(x0), z2))) IF_MINSORT(true, add(0, add(s(x0), x1)), x2) -> c26(RM(0, add(s(x0), x1))) IF_MINSORT(true, add(s(x0), add(0, z2)), x2) -> c26(MINSORT(app(add(0, rm(s(x0), z2)), x2), nil), APP(rm(s(x0), add(0, z2)), x2), RM(s(x0), add(0, z2))) IF_MINSORT(true, add(s(x0), add(0, x1)), x2) -> c26(RM(s(x0), add(0, x1))) IF_MINSORT(true, add(s(0), add(s(0), x2)), x3) -> c26(MINSORT(app(if_rm(true, s(0), add(s(0), x2)), x3), nil), APP(rm(s(0), add(s(0), x2)), x3), RM(s(0), add(s(0), x2))) IF_MINSORT(true, add(s(0), add(s(s(z0)), x2)), x3) -> c26(MINSORT(app(if_rm(false, s(0), add(s(s(z0)), x2)), x3), nil), APP(rm(s(0), add(s(s(z0)), x2)), x3), RM(s(0), add(s(s(z0)), x2))) IF_MINSORT(true, add(s(s(z0)), add(s(0), x2)), x3) -> c26(MINSORT(app(if_rm(false, s(s(z0)), add(s(0), x2)), x3), nil), APP(rm(s(s(z0)), add(s(0), x2)), x3), RM(s(s(z0)), add(s(0), x2))) IF_MINSORT(true, add(s(s(z0)), add(s(s(z1)), x2)), x3) -> c26(MINSORT(app(if_rm(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), x3), nil), APP(rm(s(s(z0)), add(s(s(z1)), x2)), x3), RM(s(s(z0)), add(s(s(z1)), x2))) IF_MINSORT(true, add(s(x0), add(s(x1), x2)), x3) -> c1(APP(rm(s(x0), add(s(x1), x2)), x3)) IF_MINSORT(true, add(s(x0), add(s(x1), x2)), x3) -> c1(RM(s(x0), add(s(x1), x2))) IF_MINSORT(true, add(z0, add(z1, z2)), x3) -> c26(APP(if_rm(eq(z0, z1), z0, add(z1, z2)), x3)) IF_MINSORT(true, add(z0, nil), x1) -> c26(MINSORT(x1, nil)) MINSORT(add(s(z0), x1), x2) -> c25(IF_MINSORT(eq(s(z0), if_min(false, x1, add(s(z0), x1), 0)), add(s(z0), x1), x2), EQ(s(z0), minIter(add(s(z0), x1), add(s(z0), x1), 0)), MIN(add(s(z0), x1))) MINSORT(add(0, x1), x2) -> c25(IF_MINSORT(eq(0, if_min(true, x1, add(0, x1), 0)), add(0, x1), x2), MIN(add(0, x1))) MINSORT(add(x0, x1), x2) -> c2(EQ(x0, minIter(add(x0, x1), add(x0, x1), 0))) MINSORT(add(x0, x1), x2) -> c2(MIN(add(x0, x1))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) MINITER(add(s(s(z0)), x1), x2, s(0)) -> c12(IF_MIN(false, x1, x2, s(0)), LE(s(s(z0)), s(0))) MINITER(add(s(x0), x1), x2, s(x3)) -> c12(LE(s(x0), s(x3))) MINITER(add(s(0), x1), x2, s(z0)) -> c12(LE(s(0), s(z0))) MINITER(add(s(s(s(z0))), x1), x2, s(s(0))) -> c12(IF_MIN(false, x1, x2, s(s(0))), LE(s(s(s(z0))), s(s(0)))) MINITER(add(s(s(s(z0))), x1), x2, s(s(s(z1)))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(s(z1)))), LE(s(s(s(z0))), s(s(s(z1))))) MINITER(add(s(s(x0)), x1), x2, s(s(x3))) -> c12(LE(s(s(x0)), s(s(x3)))) MINITER(add(s(s(0)), x1), x2, s(s(z0))) -> c12(LE(s(s(0)), s(s(z0)))) K tuples: IF_MINSORT(true, add(z0, z1), z2) -> c26(MINSORT(app(rm(z0, z1), z2), nil), APP(rm(z0, z1), z2), RM(z0, z1)) MINSORT(add(z0, z1), z2) -> c25(IF_MINSORT(eq(z0, min(add(z0, z1))), add(z0, z1), z2), EQ(z0, min(add(z0, z1))), MIN(add(z0, z1))) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MIN(add(z0, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) RM(z0, add(z1, z2)) -> c21(IF_RM(eq(z0, z1), z0, add(z1, z2)), EQ(z0, z1)) Defined Rule Symbols: le_2, eq_2, app_2, if_rm_3, rm_2, if_min_4, minIter_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, MIN_1, MINITER_3, IF_MIN_4, IF_RM_3, IF_MINSORT_3, RM_2, MINSORT_2 Compound Symbols: c3_1, c6_1, c8_1, c10_1, c11_1, c14_1, c22_1, c23_1, c27_1, c12_1, c21_1, c12_2, c21_2, c_1, c26_3, c26_1, c1_1, c25_3, c25_2, c2_1 ---------------------------------------- (147) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MINSORT(add(z0, z1), z2) -> c(EQ(z0, minIter(add(z0, z1), add(z0, z1), 0))) by MINSORT(add(z0, z1), x2) -> c(EQ(z0, if_min(le(z0, 0), z1, add(z0, z1), 0))) ---------------------------------------- (148) 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) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) if_min(true, z0, z1, z2) -> z2 if_min(false, z0, z1, z2) -> minIter(z0, z1, z2) minIter(add(z0, z1), z2, z3) -> if_min(le(z0, z3), z1, z2, z3) minIter(nil, add(z0, z1), z2) -> minIter(add(z0, z1), add(z0, z1), s(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, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) RM(0, add(0, x2)) -> c21(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c21(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c21(IF_RM(false, s(z0), add(0, x2))) MINITER(add(s(s(z0)), x1), x2, s(0)) -> c12(IF_MIN(false, x1, x2, s(0)), LE(s(s(z0)), s(0))) MINITER(add(s(x0), x1), x2, s(x3)) -> c12(LE(s(x0), s(x3))) MINITER(add(s(0), x1), x2, s(z0)) -> c12(LE(s(0), s(z0))) RM(s(0), add(s(0), x2)) -> c21(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c21(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c21(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(x0), add(s(x1), x2)) -> c21(EQ(s(x0), s(x1))) MINSORT(add(x0, x1), x2) -> c(MIN(add(x0, x1))) MINITER(add(s(s(s(z0))), x1), x2, s(s(0))) -> c12(IF_MIN(false, x1, x2, s(s(0))), LE(s(s(s(z0))), s(s(0)))) MINITER(add(s(s(s(z0))), x1), x2, s(s(s(z1)))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(s(z1)))), LE(s(s(s(z0))), s(s(s(z1))))) MINITER(add(s(s(x0)), x1), x2, s(s(x3))) -> c12(LE(s(s(x0)), s(s(x3)))) MINITER(add(s(s(0)), x1), x2, s(s(z0))) -> c12(LE(s(s(0)), s(s(z0)))) RM(s(s(0)), add(s(s(0)), x2)) -> c21(IF_RM(true, s(s(0)), add(s(s(0)), x2)), EQ(s(s(0)), s(s(0)))) RM(s(s(0)), add(s(s(s(z0))), x2)) -> c21(IF_RM(false, s(s(0)), add(s(s(s(z0))), x2)), EQ(s(s(0)), s(s(s(z0))))) RM(s(s(s(z0))), add(s(s(0)), x2)) -> c21(IF_RM(false, s(s(s(z0))), add(s(s(0)), x2)), EQ(s(s(s(z0))), s(s(0)))) RM(s(s(s(z0))), add(s(s(s(z1))), x2)) -> c21(IF_RM(eq(z0, z1), s(s(s(z0))), add(s(s(s(z1))), x2)), EQ(s(s(s(z0))), s(s(s(z1))))) RM(s(s(x0)), add(s(s(x1)), x2)) -> c21(EQ(s(s(x0)), s(s(x1)))) IF_MINSORT(true, add(0, add(0, z2)), x1) -> c26(MINSORT(app(rm(0, z2), x1), nil), APP(rm(0, add(0, z2)), x1), RM(0, add(0, z2))) IF_MINSORT(true, add(0, add(0, x0)), x1) -> c26(RM(0, add(0, x0))) IF_MINSORT(true, add(0, add(s(x0), z2)), x2) -> c26(MINSORT(app(add(s(x0), rm(0, z2)), x2), nil), APP(rm(0, add(s(x0), z2)), x2), RM(0, add(s(x0), z2))) IF_MINSORT(true, add(0, add(s(x0), x1)), x2) -> c26(RM(0, add(s(x0), x1))) IF_MINSORT(true, add(s(x0), add(0, z2)), x2) -> c26(MINSORT(app(add(0, rm(s(x0), z2)), x2), nil), APP(rm(s(x0), add(0, z2)), x2), RM(s(x0), add(0, z2))) IF_MINSORT(true, add(s(x0), add(0, x1)), x2) -> c26(RM(s(x0), add(0, x1))) IF_MINSORT(true, add(s(0), add(s(0), x2)), x3) -> c26(MINSORT(app(if_rm(true, s(0), add(s(0), x2)), x3), nil), APP(rm(s(0), add(s(0), x2)), x3), RM(s(0), add(s(0), x2))) IF_MINSORT(true, add(s(0), add(s(s(z0)), x2)), x3) -> c26(MINSORT(app(if_rm(false, s(0), add(s(s(z0)), x2)), x3), nil), APP(rm(s(0), add(s(s(z0)), x2)), x3), RM(s(0), add(s(s(z0)), x2))) IF_MINSORT(true, add(s(s(z0)), add(s(0), x2)), x3) -> c26(MINSORT(app(if_rm(false, s(s(z0)), add(s(0), x2)), x3), nil), APP(rm(s(s(z0)), add(s(0), x2)), x3), RM(s(s(z0)), add(s(0), x2))) IF_MINSORT(true, add(s(s(z0)), add(s(s(z1)), x2)), x3) -> c26(MINSORT(app(if_rm(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), x3), nil), APP(rm(s(s(z0)), add(s(s(z1)), x2)), x3), RM(s(s(z0)), add(s(s(z1)), x2))) IF_MINSORT(true, add(s(x0), add(s(x1), x2)), x3) -> c1(APP(rm(s(x0), add(s(x1), x2)), x3)) IF_MINSORT(true, add(s(x0), add(s(x1), x2)), x3) -> c1(RM(s(x0), add(s(x1), x2))) IF_MINSORT(true, add(z0, add(z1, z2)), x3) -> c26(APP(if_rm(eq(z0, z1), z0, add(z1, z2)), x3)) IF_MINSORT(true, add(z0, nil), x1) -> c26(MINSORT(x1, nil)) MINSORT(add(s(z0), x1), x2) -> c25(IF_MINSORT(eq(s(z0), if_min(false, x1, add(s(z0), x1), 0)), add(s(z0), x1), x2), EQ(s(z0), minIter(add(s(z0), x1), add(s(z0), x1), 0)), MIN(add(s(z0), x1))) MINSORT(add(0, x1), x2) -> c25(IF_MINSORT(eq(0, if_min(true, x1, add(0, x1), 0)), add(0, x1), x2), MIN(add(0, x1))) MINSORT(add(x0, x1), x2) -> c2(EQ(x0, minIter(add(x0, x1), add(x0, x1), 0))) MINSORT(add(x0, x1), x2) -> c2(MIN(add(x0, x1))) MINSORT(add(z0, z1), x2) -> c(EQ(z0, if_min(le(z0, 0), z1, add(z0, z1), 0))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) MINITER(add(s(s(z0)), x1), x2, s(0)) -> c12(IF_MIN(false, x1, x2, s(0)), LE(s(s(z0)), s(0))) MINITER(add(s(x0), x1), x2, s(x3)) -> c12(LE(s(x0), s(x3))) MINITER(add(s(0), x1), x2, s(z0)) -> c12(LE(s(0), s(z0))) MINITER(add(s(s(s(z0))), x1), x2, s(s(0))) -> c12(IF_MIN(false, x1, x2, s(s(0))), LE(s(s(s(z0))), s(s(0)))) MINITER(add(s(s(s(z0))), x1), x2, s(s(s(z1)))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(s(z1)))), LE(s(s(s(z0))), s(s(s(z1))))) MINITER(add(s(s(x0)), x1), x2, s(s(x3))) -> c12(LE(s(s(x0)), s(s(x3)))) MINITER(add(s(s(0)), x1), x2, s(s(z0))) -> c12(LE(s(s(0)), s(s(z0)))) K tuples: IF_MINSORT(true, add(z0, z1), z2) -> c26(MINSORT(app(rm(z0, z1), z2), nil), APP(rm(z0, z1), z2), RM(z0, z1)) MINSORT(add(z0, z1), z2) -> c25(IF_MINSORT(eq(z0, min(add(z0, z1))), add(z0, z1), z2), EQ(z0, min(add(z0, z1))), MIN(add(z0, z1))) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MIN(add(z0, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) RM(z0, add(z1, z2)) -> c21(IF_RM(eq(z0, z1), z0, add(z1, z2)), EQ(z0, z1)) Defined Rule Symbols: le_2, eq_2, app_2, if_rm_3, rm_2, if_min_4, minIter_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, MIN_1, MINITER_3, IF_MIN_4, IF_RM_3, IF_MINSORT_3, RM_2, MINSORT_2 Compound Symbols: c3_1, c6_1, c8_1, c10_1, c11_1, c14_1, c22_1, c23_1, c27_1, c12_1, c21_1, c12_2, c21_2, c_1, c26_3, c26_1, c1_1, c25_3, c25_2, c2_1 ---------------------------------------- (149) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MINITER(add(s(s(s(z0))), x1), x2, s(s(s(z1)))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(s(z1)))), LE(s(s(s(z0))), s(s(s(z1))))) by MINITER(add(s(s(s(0))), x1), x2, s(s(s(z0)))) -> c12(IF_MIN(true, x1, x2, s(s(s(z0)))), LE(s(s(s(0))), s(s(s(z0))))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(0)))) -> c12(IF_MIN(false, x1, x2, s(s(s(0)))), LE(s(s(s(s(z0)))), s(s(s(0))))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(s(z1))))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(s(s(z1))))), LE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MINITER(add(s(s(s(x0))), x1), x2, s(s(s(x3)))) -> c12(LE(s(s(s(x0))), s(s(s(x3))))) ---------------------------------------- (150) 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) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) if_min(true, z0, z1, z2) -> z2 if_min(false, z0, z1, z2) -> minIter(z0, z1, z2) minIter(add(z0, z1), z2, z3) -> if_min(le(z0, z3), z1, z2, z3) minIter(nil, add(z0, z1), z2) -> minIter(add(z0, z1), add(z0, z1), s(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, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) RM(0, add(0, x2)) -> c21(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c21(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c21(IF_RM(false, s(z0), add(0, x2))) MINITER(add(s(s(z0)), x1), x2, s(0)) -> c12(IF_MIN(false, x1, x2, s(0)), LE(s(s(z0)), s(0))) MINITER(add(s(x0), x1), x2, s(x3)) -> c12(LE(s(x0), s(x3))) MINITER(add(s(0), x1), x2, s(z0)) -> c12(LE(s(0), s(z0))) RM(s(0), add(s(0), x2)) -> c21(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c21(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c21(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(x0), add(s(x1), x2)) -> c21(EQ(s(x0), s(x1))) MINSORT(add(x0, x1), x2) -> c(MIN(add(x0, x1))) MINITER(add(s(s(s(z0))), x1), x2, s(s(0))) -> c12(IF_MIN(false, x1, x2, s(s(0))), LE(s(s(s(z0))), s(s(0)))) MINITER(add(s(s(x0)), x1), x2, s(s(x3))) -> c12(LE(s(s(x0)), s(s(x3)))) MINITER(add(s(s(0)), x1), x2, s(s(z0))) -> c12(LE(s(s(0)), s(s(z0)))) RM(s(s(0)), add(s(s(0)), x2)) -> c21(IF_RM(true, s(s(0)), add(s(s(0)), x2)), EQ(s(s(0)), s(s(0)))) RM(s(s(0)), add(s(s(s(z0))), x2)) -> c21(IF_RM(false, s(s(0)), add(s(s(s(z0))), x2)), EQ(s(s(0)), s(s(s(z0))))) RM(s(s(s(z0))), add(s(s(0)), x2)) -> c21(IF_RM(false, s(s(s(z0))), add(s(s(0)), x2)), EQ(s(s(s(z0))), s(s(0)))) RM(s(s(s(z0))), add(s(s(s(z1))), x2)) -> c21(IF_RM(eq(z0, z1), s(s(s(z0))), add(s(s(s(z1))), x2)), EQ(s(s(s(z0))), s(s(s(z1))))) RM(s(s(x0)), add(s(s(x1)), x2)) -> c21(EQ(s(s(x0)), s(s(x1)))) IF_MINSORT(true, add(0, add(0, z2)), x1) -> c26(MINSORT(app(rm(0, z2), x1), nil), APP(rm(0, add(0, z2)), x1), RM(0, add(0, z2))) IF_MINSORT(true, add(0, add(0, x0)), x1) -> c26(RM(0, add(0, x0))) IF_MINSORT(true, add(0, add(s(x0), z2)), x2) -> c26(MINSORT(app(add(s(x0), rm(0, z2)), x2), nil), APP(rm(0, add(s(x0), z2)), x2), RM(0, add(s(x0), z2))) IF_MINSORT(true, add(0, add(s(x0), x1)), x2) -> c26(RM(0, add(s(x0), x1))) IF_MINSORT(true, add(s(x0), add(0, z2)), x2) -> c26(MINSORT(app(add(0, rm(s(x0), z2)), x2), nil), APP(rm(s(x0), add(0, z2)), x2), RM(s(x0), add(0, z2))) IF_MINSORT(true, add(s(x0), add(0, x1)), x2) -> c26(RM(s(x0), add(0, x1))) IF_MINSORT(true, add(s(0), add(s(0), x2)), x3) -> c26(MINSORT(app(if_rm(true, s(0), add(s(0), x2)), x3), nil), APP(rm(s(0), add(s(0), x2)), x3), RM(s(0), add(s(0), x2))) IF_MINSORT(true, add(s(0), add(s(s(z0)), x2)), x3) -> c26(MINSORT(app(if_rm(false, s(0), add(s(s(z0)), x2)), x3), nil), APP(rm(s(0), add(s(s(z0)), x2)), x3), RM(s(0), add(s(s(z0)), x2))) IF_MINSORT(true, add(s(s(z0)), add(s(0), x2)), x3) -> c26(MINSORT(app(if_rm(false, s(s(z0)), add(s(0), x2)), x3), nil), APP(rm(s(s(z0)), add(s(0), x2)), x3), RM(s(s(z0)), add(s(0), x2))) IF_MINSORT(true, add(s(s(z0)), add(s(s(z1)), x2)), x3) -> c26(MINSORT(app(if_rm(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), x3), nil), APP(rm(s(s(z0)), add(s(s(z1)), x2)), x3), RM(s(s(z0)), add(s(s(z1)), x2))) IF_MINSORT(true, add(s(x0), add(s(x1), x2)), x3) -> c1(APP(rm(s(x0), add(s(x1), x2)), x3)) IF_MINSORT(true, add(s(x0), add(s(x1), x2)), x3) -> c1(RM(s(x0), add(s(x1), x2))) IF_MINSORT(true, add(z0, add(z1, z2)), x3) -> c26(APP(if_rm(eq(z0, z1), z0, add(z1, z2)), x3)) IF_MINSORT(true, add(z0, nil), x1) -> c26(MINSORT(x1, nil)) MINSORT(add(s(z0), x1), x2) -> c25(IF_MINSORT(eq(s(z0), if_min(false, x1, add(s(z0), x1), 0)), add(s(z0), x1), x2), EQ(s(z0), minIter(add(s(z0), x1), add(s(z0), x1), 0)), MIN(add(s(z0), x1))) MINSORT(add(0, x1), x2) -> c25(IF_MINSORT(eq(0, if_min(true, x1, add(0, x1), 0)), add(0, x1), x2), MIN(add(0, x1))) MINSORT(add(x0, x1), x2) -> c2(EQ(x0, minIter(add(x0, x1), add(x0, x1), 0))) MINSORT(add(x0, x1), x2) -> c2(MIN(add(x0, x1))) MINSORT(add(z0, z1), x2) -> c(EQ(z0, if_min(le(z0, 0), z1, add(z0, z1), 0))) MINITER(add(s(s(s(0))), x1), x2, s(s(s(z0)))) -> c12(IF_MIN(true, x1, x2, s(s(s(z0)))), LE(s(s(s(0))), s(s(s(z0))))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(0)))) -> c12(IF_MIN(false, x1, x2, s(s(s(0)))), LE(s(s(s(s(z0)))), s(s(s(0))))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(s(z1))))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(s(s(z1))))), LE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MINITER(add(s(s(s(x0))), x1), x2, s(s(s(x3)))) -> c12(LE(s(s(s(x0))), s(s(s(x3))))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) MINITER(add(s(s(z0)), x1), x2, s(0)) -> c12(IF_MIN(false, x1, x2, s(0)), LE(s(s(z0)), s(0))) MINITER(add(s(x0), x1), x2, s(x3)) -> c12(LE(s(x0), s(x3))) MINITER(add(s(0), x1), x2, s(z0)) -> c12(LE(s(0), s(z0))) MINITER(add(s(s(s(z0))), x1), x2, s(s(0))) -> c12(IF_MIN(false, x1, x2, s(s(0))), LE(s(s(s(z0))), s(s(0)))) MINITER(add(s(s(x0)), x1), x2, s(s(x3))) -> c12(LE(s(s(x0)), s(s(x3)))) MINITER(add(s(s(0)), x1), x2, s(s(z0))) -> c12(LE(s(s(0)), s(s(z0)))) MINITER(add(s(s(s(0))), x1), x2, s(s(s(z0)))) -> c12(IF_MIN(true, x1, x2, s(s(s(z0)))), LE(s(s(s(0))), s(s(s(z0))))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(0)))) -> c12(IF_MIN(false, x1, x2, s(s(s(0)))), LE(s(s(s(s(z0)))), s(s(s(0))))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(s(z1))))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(s(s(z1))))), LE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MINITER(add(s(s(s(x0))), x1), x2, s(s(s(x3)))) -> c12(LE(s(s(s(x0))), s(s(s(x3))))) K tuples: IF_MINSORT(true, add(z0, z1), z2) -> c26(MINSORT(app(rm(z0, z1), z2), nil), APP(rm(z0, z1), z2), RM(z0, z1)) MINSORT(add(z0, z1), z2) -> c25(IF_MINSORT(eq(z0, min(add(z0, z1))), add(z0, z1), z2), EQ(z0, min(add(z0, z1))), MIN(add(z0, z1))) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MIN(add(z0, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) RM(z0, add(z1, z2)) -> c21(IF_RM(eq(z0, z1), z0, add(z1, z2)), EQ(z0, z1)) Defined Rule Symbols: le_2, eq_2, app_2, if_rm_3, rm_2, if_min_4, minIter_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, MIN_1, MINITER_3, IF_MIN_4, IF_RM_3, IF_MINSORT_3, RM_2, MINSORT_2 Compound Symbols: c3_1, c6_1, c8_1, c10_1, c11_1, c14_1, c22_1, c23_1, c27_1, c12_1, c21_1, c12_2, c21_2, c_1, c26_3, c26_1, c1_1, c25_3, c25_2, c2_1 ---------------------------------------- (151) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (152) 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) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) if_min(true, z0, z1, z2) -> z2 if_min(false, z0, z1, z2) -> minIter(z0, z1, z2) minIter(add(z0, z1), z2, z3) -> if_min(le(z0, z3), z1, z2, z3) minIter(nil, add(z0, z1), z2) -> minIter(add(z0, z1), add(z0, z1), s(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, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) RM(0, add(0, x2)) -> c21(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c21(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c21(IF_RM(false, s(z0), add(0, x2))) MINITER(add(s(s(z0)), x1), x2, s(0)) -> c12(IF_MIN(false, x1, x2, s(0)), LE(s(s(z0)), s(0))) MINITER(add(s(x0), x1), x2, s(x3)) -> c12(LE(s(x0), s(x3))) MINITER(add(s(0), x1), x2, s(z0)) -> c12(LE(s(0), s(z0))) RM(s(0), add(s(0), x2)) -> c21(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c21(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c21(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(x0), add(s(x1), x2)) -> c21(EQ(s(x0), s(x1))) MINSORT(add(x0, x1), x2) -> c(MIN(add(x0, x1))) MINITER(add(s(s(s(z0))), x1), x2, s(s(0))) -> c12(IF_MIN(false, x1, x2, s(s(0))), LE(s(s(s(z0))), s(s(0)))) MINITER(add(s(s(x0)), x1), x2, s(s(x3))) -> c12(LE(s(s(x0)), s(s(x3)))) MINITER(add(s(s(0)), x1), x2, s(s(z0))) -> c12(LE(s(s(0)), s(s(z0)))) RM(s(s(0)), add(s(s(0)), x2)) -> c21(IF_RM(true, s(s(0)), add(s(s(0)), x2)), EQ(s(s(0)), s(s(0)))) RM(s(s(0)), add(s(s(s(z0))), x2)) -> c21(IF_RM(false, s(s(0)), add(s(s(s(z0))), x2)), EQ(s(s(0)), s(s(s(z0))))) RM(s(s(s(z0))), add(s(s(0)), x2)) -> c21(IF_RM(false, s(s(s(z0))), add(s(s(0)), x2)), EQ(s(s(s(z0))), s(s(0)))) RM(s(s(s(z0))), add(s(s(s(z1))), x2)) -> c21(IF_RM(eq(z0, z1), s(s(s(z0))), add(s(s(s(z1))), x2)), EQ(s(s(s(z0))), s(s(s(z1))))) RM(s(s(x0)), add(s(s(x1)), x2)) -> c21(EQ(s(s(x0)), s(s(x1)))) IF_MINSORT(true, add(0, add(0, z2)), x1) -> c26(MINSORT(app(rm(0, z2), x1), nil), APP(rm(0, add(0, z2)), x1), RM(0, add(0, z2))) IF_MINSORT(true, add(0, add(0, x0)), x1) -> c26(RM(0, add(0, x0))) IF_MINSORT(true, add(0, add(s(x0), z2)), x2) -> c26(MINSORT(app(add(s(x0), rm(0, z2)), x2), nil), APP(rm(0, add(s(x0), z2)), x2), RM(0, add(s(x0), z2))) IF_MINSORT(true, add(0, add(s(x0), x1)), x2) -> c26(RM(0, add(s(x0), x1))) IF_MINSORT(true, add(s(x0), add(0, z2)), x2) -> c26(MINSORT(app(add(0, rm(s(x0), z2)), x2), nil), APP(rm(s(x0), add(0, z2)), x2), RM(s(x0), add(0, z2))) IF_MINSORT(true, add(s(x0), add(0, x1)), x2) -> c26(RM(s(x0), add(0, x1))) IF_MINSORT(true, add(s(0), add(s(0), x2)), x3) -> c26(MINSORT(app(if_rm(true, s(0), add(s(0), x2)), x3), nil), APP(rm(s(0), add(s(0), x2)), x3), RM(s(0), add(s(0), x2))) IF_MINSORT(true, add(s(0), add(s(s(z0)), x2)), x3) -> c26(MINSORT(app(if_rm(false, s(0), add(s(s(z0)), x2)), x3), nil), APP(rm(s(0), add(s(s(z0)), x2)), x3), RM(s(0), add(s(s(z0)), x2))) IF_MINSORT(true, add(s(s(z0)), add(s(0), x2)), x3) -> c26(MINSORT(app(if_rm(false, s(s(z0)), add(s(0), x2)), x3), nil), APP(rm(s(s(z0)), add(s(0), x2)), x3), RM(s(s(z0)), add(s(0), x2))) IF_MINSORT(true, add(s(s(z0)), add(s(s(z1)), x2)), x3) -> c26(MINSORT(app(if_rm(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), x3), nil), APP(rm(s(s(z0)), add(s(s(z1)), x2)), x3), RM(s(s(z0)), add(s(s(z1)), x2))) IF_MINSORT(true, add(s(x0), add(s(x1), x2)), x3) -> c1(APP(rm(s(x0), add(s(x1), x2)), x3)) IF_MINSORT(true, add(s(x0), add(s(x1), x2)), x3) -> c1(RM(s(x0), add(s(x1), x2))) IF_MINSORT(true, add(z0, add(z1, z2)), x3) -> c26(APP(if_rm(eq(z0, z1), z0, add(z1, z2)), x3)) IF_MINSORT(true, add(z0, nil), x1) -> c26(MINSORT(x1, nil)) MINSORT(add(s(z0), x1), x2) -> c25(IF_MINSORT(eq(s(z0), if_min(false, x1, add(s(z0), x1), 0)), add(s(z0), x1), x2), EQ(s(z0), minIter(add(s(z0), x1), add(s(z0), x1), 0)), MIN(add(s(z0), x1))) MINSORT(add(0, x1), x2) -> c25(IF_MINSORT(eq(0, if_min(true, x1, add(0, x1), 0)), add(0, x1), x2), MIN(add(0, x1))) MINSORT(add(x0, x1), x2) -> c2(EQ(x0, minIter(add(x0, x1), add(x0, x1), 0))) MINSORT(add(x0, x1), x2) -> c2(MIN(add(x0, x1))) MINSORT(add(z0, z1), x2) -> c(EQ(z0, if_min(le(z0, 0), z1, add(z0, z1), 0))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(0)))) -> c12(IF_MIN(false, x1, x2, s(s(s(0)))), LE(s(s(s(s(z0)))), s(s(s(0))))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(s(z1))))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(s(s(z1))))), LE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MINITER(add(s(s(s(x0))), x1), x2, s(s(s(x3)))) -> c12(LE(s(s(s(x0))), s(s(s(x3))))) MINITER(add(s(s(s(0))), x1), x2, s(s(s(z0)))) -> c12(LE(s(s(s(0))), s(s(s(z0))))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) MINITER(add(s(s(z0)), x1), x2, s(0)) -> c12(IF_MIN(false, x1, x2, s(0)), LE(s(s(z0)), s(0))) MINITER(add(s(x0), x1), x2, s(x3)) -> c12(LE(s(x0), s(x3))) MINITER(add(s(0), x1), x2, s(z0)) -> c12(LE(s(0), s(z0))) MINITER(add(s(s(s(z0))), x1), x2, s(s(0))) -> c12(IF_MIN(false, x1, x2, s(s(0))), LE(s(s(s(z0))), s(s(0)))) MINITER(add(s(s(x0)), x1), x2, s(s(x3))) -> c12(LE(s(s(x0)), s(s(x3)))) MINITER(add(s(s(0)), x1), x2, s(s(z0))) -> c12(LE(s(s(0)), s(s(z0)))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(0)))) -> c12(IF_MIN(false, x1, x2, s(s(s(0)))), LE(s(s(s(s(z0)))), s(s(s(0))))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(s(z1))))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(s(s(z1))))), LE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MINITER(add(s(s(s(x0))), x1), x2, s(s(s(x3)))) -> c12(LE(s(s(s(x0))), s(s(s(x3))))) MINITER(add(s(s(s(0))), x1), x2, s(s(s(z0)))) -> c12(LE(s(s(s(0))), s(s(s(z0))))) K tuples: IF_MINSORT(true, add(z0, z1), z2) -> c26(MINSORT(app(rm(z0, z1), z2), nil), APP(rm(z0, z1), z2), RM(z0, z1)) MINSORT(add(z0, z1), z2) -> c25(IF_MINSORT(eq(z0, min(add(z0, z1))), add(z0, z1), z2), EQ(z0, min(add(z0, z1))), MIN(add(z0, z1))) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MIN(add(z0, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) RM(z0, add(z1, z2)) -> c21(IF_RM(eq(z0, z1), z0, add(z1, z2)), EQ(z0, z1)) Defined Rule Symbols: le_2, eq_2, app_2, if_rm_3, rm_2, if_min_4, minIter_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, MIN_1, MINITER_3, IF_MIN_4, IF_RM_3, IF_MINSORT_3, RM_2, MINSORT_2 Compound Symbols: c3_1, c6_1, c8_1, c10_1, c11_1, c14_1, c22_1, c23_1, c27_1, c12_1, c21_1, c12_2, c21_2, c_1, c26_3, c26_1, c1_1, c25_3, c25_2, c2_1 ---------------------------------------- (153) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace RM(s(s(s(z0))), add(s(s(s(z1))), x2)) -> c21(IF_RM(eq(z0, z1), s(s(s(z0))), add(s(s(s(z1))), x2)), EQ(s(s(s(z0))), s(s(s(z1))))) by RM(s(s(s(0))), add(s(s(s(0))), x2)) -> c21(IF_RM(true, s(s(s(0))), add(s(s(s(0))), x2)), EQ(s(s(s(0))), s(s(s(0))))) RM(s(s(s(0))), add(s(s(s(s(z0)))), x2)) -> c21(IF_RM(false, s(s(s(0))), add(s(s(s(s(z0)))), x2)), EQ(s(s(s(0))), s(s(s(s(z0)))))) RM(s(s(s(s(z0)))), add(s(s(s(0))), x2)) -> c21(IF_RM(false, s(s(s(s(z0)))), add(s(s(s(0))), x2)), EQ(s(s(s(s(z0)))), s(s(s(0))))) RM(s(s(s(s(z0)))), add(s(s(s(s(z1)))), x2)) -> c21(IF_RM(eq(z0, z1), s(s(s(s(z0)))), add(s(s(s(s(z1)))), x2)), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) RM(s(s(s(x0))), add(s(s(s(x1))), x2)) -> c21(EQ(s(s(s(x0))), s(s(s(x1))))) ---------------------------------------- (154) 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) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) if_min(true, z0, z1, z2) -> z2 if_min(false, z0, z1, z2) -> minIter(z0, z1, z2) minIter(add(z0, z1), z2, z3) -> if_min(le(z0, z3), z1, z2, z3) minIter(nil, add(z0, z1), z2) -> minIter(add(z0, z1), add(z0, z1), s(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, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) RM(0, add(0, x2)) -> c21(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c21(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c21(IF_RM(false, s(z0), add(0, x2))) MINITER(add(s(s(z0)), x1), x2, s(0)) -> c12(IF_MIN(false, x1, x2, s(0)), LE(s(s(z0)), s(0))) MINITER(add(s(x0), x1), x2, s(x3)) -> c12(LE(s(x0), s(x3))) MINITER(add(s(0), x1), x2, s(z0)) -> c12(LE(s(0), s(z0))) RM(s(0), add(s(0), x2)) -> c21(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c21(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c21(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(x0), add(s(x1), x2)) -> c21(EQ(s(x0), s(x1))) MINSORT(add(x0, x1), x2) -> c(MIN(add(x0, x1))) MINITER(add(s(s(s(z0))), x1), x2, s(s(0))) -> c12(IF_MIN(false, x1, x2, s(s(0))), LE(s(s(s(z0))), s(s(0)))) MINITER(add(s(s(x0)), x1), x2, s(s(x3))) -> c12(LE(s(s(x0)), s(s(x3)))) MINITER(add(s(s(0)), x1), x2, s(s(z0))) -> c12(LE(s(s(0)), s(s(z0)))) RM(s(s(0)), add(s(s(0)), x2)) -> c21(IF_RM(true, s(s(0)), add(s(s(0)), x2)), EQ(s(s(0)), s(s(0)))) RM(s(s(0)), add(s(s(s(z0))), x2)) -> c21(IF_RM(false, s(s(0)), add(s(s(s(z0))), x2)), EQ(s(s(0)), s(s(s(z0))))) RM(s(s(s(z0))), add(s(s(0)), x2)) -> c21(IF_RM(false, s(s(s(z0))), add(s(s(0)), x2)), EQ(s(s(s(z0))), s(s(0)))) RM(s(s(x0)), add(s(s(x1)), x2)) -> c21(EQ(s(s(x0)), s(s(x1)))) IF_MINSORT(true, add(0, add(0, z2)), x1) -> c26(MINSORT(app(rm(0, z2), x1), nil), APP(rm(0, add(0, z2)), x1), RM(0, add(0, z2))) IF_MINSORT(true, add(0, add(0, x0)), x1) -> c26(RM(0, add(0, x0))) IF_MINSORT(true, add(0, add(s(x0), z2)), x2) -> c26(MINSORT(app(add(s(x0), rm(0, z2)), x2), nil), APP(rm(0, add(s(x0), z2)), x2), RM(0, add(s(x0), z2))) IF_MINSORT(true, add(0, add(s(x0), x1)), x2) -> c26(RM(0, add(s(x0), x1))) IF_MINSORT(true, add(s(x0), add(0, z2)), x2) -> c26(MINSORT(app(add(0, rm(s(x0), z2)), x2), nil), APP(rm(s(x0), add(0, z2)), x2), RM(s(x0), add(0, z2))) IF_MINSORT(true, add(s(x0), add(0, x1)), x2) -> c26(RM(s(x0), add(0, x1))) IF_MINSORT(true, add(s(0), add(s(0), x2)), x3) -> c26(MINSORT(app(if_rm(true, s(0), add(s(0), x2)), x3), nil), APP(rm(s(0), add(s(0), x2)), x3), RM(s(0), add(s(0), x2))) IF_MINSORT(true, add(s(0), add(s(s(z0)), x2)), x3) -> c26(MINSORT(app(if_rm(false, s(0), add(s(s(z0)), x2)), x3), nil), APP(rm(s(0), add(s(s(z0)), x2)), x3), RM(s(0), add(s(s(z0)), x2))) IF_MINSORT(true, add(s(s(z0)), add(s(0), x2)), x3) -> c26(MINSORT(app(if_rm(false, s(s(z0)), add(s(0), x2)), x3), nil), APP(rm(s(s(z0)), add(s(0), x2)), x3), RM(s(s(z0)), add(s(0), x2))) IF_MINSORT(true, add(s(s(z0)), add(s(s(z1)), x2)), x3) -> c26(MINSORT(app(if_rm(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), x3), nil), APP(rm(s(s(z0)), add(s(s(z1)), x2)), x3), RM(s(s(z0)), add(s(s(z1)), x2))) IF_MINSORT(true, add(s(x0), add(s(x1), x2)), x3) -> c1(APP(rm(s(x0), add(s(x1), x2)), x3)) IF_MINSORT(true, add(s(x0), add(s(x1), x2)), x3) -> c1(RM(s(x0), add(s(x1), x2))) IF_MINSORT(true, add(z0, add(z1, z2)), x3) -> c26(APP(if_rm(eq(z0, z1), z0, add(z1, z2)), x3)) IF_MINSORT(true, add(z0, nil), x1) -> c26(MINSORT(x1, nil)) MINSORT(add(s(z0), x1), x2) -> c25(IF_MINSORT(eq(s(z0), if_min(false, x1, add(s(z0), x1), 0)), add(s(z0), x1), x2), EQ(s(z0), minIter(add(s(z0), x1), add(s(z0), x1), 0)), MIN(add(s(z0), x1))) MINSORT(add(0, x1), x2) -> c25(IF_MINSORT(eq(0, if_min(true, x1, add(0, x1), 0)), add(0, x1), x2), MIN(add(0, x1))) MINSORT(add(x0, x1), x2) -> c2(EQ(x0, minIter(add(x0, x1), add(x0, x1), 0))) MINSORT(add(x0, x1), x2) -> c2(MIN(add(x0, x1))) MINSORT(add(z0, z1), x2) -> c(EQ(z0, if_min(le(z0, 0), z1, add(z0, z1), 0))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(0)))) -> c12(IF_MIN(false, x1, x2, s(s(s(0)))), LE(s(s(s(s(z0)))), s(s(s(0))))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(s(z1))))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(s(s(z1))))), LE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MINITER(add(s(s(s(x0))), x1), x2, s(s(s(x3)))) -> c12(LE(s(s(s(x0))), s(s(s(x3))))) MINITER(add(s(s(s(0))), x1), x2, s(s(s(z0)))) -> c12(LE(s(s(s(0))), s(s(s(z0))))) RM(s(s(s(0))), add(s(s(s(0))), x2)) -> c21(IF_RM(true, s(s(s(0))), add(s(s(s(0))), x2)), EQ(s(s(s(0))), s(s(s(0))))) RM(s(s(s(0))), add(s(s(s(s(z0)))), x2)) -> c21(IF_RM(false, s(s(s(0))), add(s(s(s(s(z0)))), x2)), EQ(s(s(s(0))), s(s(s(s(z0)))))) RM(s(s(s(s(z0)))), add(s(s(s(0))), x2)) -> c21(IF_RM(false, s(s(s(s(z0)))), add(s(s(s(0))), x2)), EQ(s(s(s(s(z0)))), s(s(s(0))))) RM(s(s(s(s(z0)))), add(s(s(s(s(z1)))), x2)) -> c21(IF_RM(eq(z0, z1), s(s(s(s(z0)))), add(s(s(s(s(z1)))), x2)), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) RM(s(s(s(x0))), add(s(s(s(x1))), x2)) -> c21(EQ(s(s(s(x0))), s(s(s(x1))))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) MINITER(add(s(s(z0)), x1), x2, s(0)) -> c12(IF_MIN(false, x1, x2, s(0)), LE(s(s(z0)), s(0))) MINITER(add(s(x0), x1), x2, s(x3)) -> c12(LE(s(x0), s(x3))) MINITER(add(s(0), x1), x2, s(z0)) -> c12(LE(s(0), s(z0))) MINITER(add(s(s(s(z0))), x1), x2, s(s(0))) -> c12(IF_MIN(false, x1, x2, s(s(0))), LE(s(s(s(z0))), s(s(0)))) MINITER(add(s(s(x0)), x1), x2, s(s(x3))) -> c12(LE(s(s(x0)), s(s(x3)))) MINITER(add(s(s(0)), x1), x2, s(s(z0))) -> c12(LE(s(s(0)), s(s(z0)))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(0)))) -> c12(IF_MIN(false, x1, x2, s(s(s(0)))), LE(s(s(s(s(z0)))), s(s(s(0))))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(s(z1))))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(s(s(z1))))), LE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MINITER(add(s(s(s(x0))), x1), x2, s(s(s(x3)))) -> c12(LE(s(s(s(x0))), s(s(s(x3))))) MINITER(add(s(s(s(0))), x1), x2, s(s(s(z0)))) -> c12(LE(s(s(s(0))), s(s(s(z0))))) K tuples: IF_MINSORT(true, add(z0, z1), z2) -> c26(MINSORT(app(rm(z0, z1), z2), nil), APP(rm(z0, z1), z2), RM(z0, z1)) MINSORT(add(z0, z1), z2) -> c25(IF_MINSORT(eq(z0, min(add(z0, z1))), add(z0, z1), z2), EQ(z0, min(add(z0, z1))), MIN(add(z0, z1))) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MIN(add(z0, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) RM(z0, add(z1, z2)) -> c21(IF_RM(eq(z0, z1), z0, add(z1, z2)), EQ(z0, z1)) Defined Rule Symbols: le_2, eq_2, app_2, if_rm_3, rm_2, if_min_4, minIter_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, MIN_1, MINITER_3, IF_MIN_4, IF_RM_3, IF_MINSORT_3, RM_2, MINSORT_2 Compound Symbols: c3_1, c6_1, c8_1, c10_1, c11_1, c14_1, c22_1, c23_1, c27_1, c12_1, c21_1, c12_2, c21_2, c_1, c26_3, c26_1, c1_1, c25_3, c25_2, c2_1 ---------------------------------------- (155) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF_MINSORT(true, add(0, add(0, z2)), x1) -> c26(MINSORT(app(rm(0, z2), x1), nil), APP(rm(0, add(0, z2)), x1), RM(0, add(0, z2))) by IF_MINSORT(true, add(0, add(0, nil)), x1) -> c26(MINSORT(app(nil, x1), nil), APP(rm(0, add(0, nil)), x1), RM(0, add(0, nil))) IF_MINSORT(true, add(0, add(0, add(z1, z2))), x1) -> c26(MINSORT(app(if_rm(eq(0, z1), 0, add(z1, z2)), x1), nil), APP(rm(0, add(0, add(z1, z2))), x1), RM(0, add(0, add(z1, z2)))) IF_MINSORT(true, add(0, add(0, x0)), x1) -> c26(APP(rm(0, add(0, x0)), x1), RM(0, add(0, x0))) ---------------------------------------- (156) 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) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) if_min(true, z0, z1, z2) -> z2 if_min(false, z0, z1, z2) -> minIter(z0, z1, z2) minIter(add(z0, z1), z2, z3) -> if_min(le(z0, z3), z1, z2, z3) minIter(nil, add(z0, z1), z2) -> minIter(add(z0, z1), add(z0, z1), s(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, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) RM(0, add(0, x2)) -> c21(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c21(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c21(IF_RM(false, s(z0), add(0, x2))) MINITER(add(s(s(z0)), x1), x2, s(0)) -> c12(IF_MIN(false, x1, x2, s(0)), LE(s(s(z0)), s(0))) MINITER(add(s(x0), x1), x2, s(x3)) -> c12(LE(s(x0), s(x3))) MINITER(add(s(0), x1), x2, s(z0)) -> c12(LE(s(0), s(z0))) RM(s(0), add(s(0), x2)) -> c21(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c21(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c21(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(x0), add(s(x1), x2)) -> c21(EQ(s(x0), s(x1))) MINSORT(add(x0, x1), x2) -> c(MIN(add(x0, x1))) MINITER(add(s(s(s(z0))), x1), x2, s(s(0))) -> c12(IF_MIN(false, x1, x2, s(s(0))), LE(s(s(s(z0))), s(s(0)))) MINITER(add(s(s(x0)), x1), x2, s(s(x3))) -> c12(LE(s(s(x0)), s(s(x3)))) MINITER(add(s(s(0)), x1), x2, s(s(z0))) -> c12(LE(s(s(0)), s(s(z0)))) RM(s(s(0)), add(s(s(0)), x2)) -> c21(IF_RM(true, s(s(0)), add(s(s(0)), x2)), EQ(s(s(0)), s(s(0)))) RM(s(s(0)), add(s(s(s(z0))), x2)) -> c21(IF_RM(false, s(s(0)), add(s(s(s(z0))), x2)), EQ(s(s(0)), s(s(s(z0))))) RM(s(s(s(z0))), add(s(s(0)), x2)) -> c21(IF_RM(false, s(s(s(z0))), add(s(s(0)), x2)), EQ(s(s(s(z0))), s(s(0)))) RM(s(s(x0)), add(s(s(x1)), x2)) -> c21(EQ(s(s(x0)), s(s(x1)))) IF_MINSORT(true, add(0, add(0, x0)), x1) -> c26(RM(0, add(0, x0))) IF_MINSORT(true, add(0, add(s(x0), z2)), x2) -> c26(MINSORT(app(add(s(x0), rm(0, z2)), x2), nil), APP(rm(0, add(s(x0), z2)), x2), RM(0, add(s(x0), z2))) IF_MINSORT(true, add(0, add(s(x0), x1)), x2) -> c26(RM(0, add(s(x0), x1))) IF_MINSORT(true, add(s(x0), add(0, z2)), x2) -> c26(MINSORT(app(add(0, rm(s(x0), z2)), x2), nil), APP(rm(s(x0), add(0, z2)), x2), RM(s(x0), add(0, z2))) IF_MINSORT(true, add(s(x0), add(0, x1)), x2) -> c26(RM(s(x0), add(0, x1))) IF_MINSORT(true, add(s(0), add(s(0), x2)), x3) -> c26(MINSORT(app(if_rm(true, s(0), add(s(0), x2)), x3), nil), APP(rm(s(0), add(s(0), x2)), x3), RM(s(0), add(s(0), x2))) IF_MINSORT(true, add(s(0), add(s(s(z0)), x2)), x3) -> c26(MINSORT(app(if_rm(false, s(0), add(s(s(z0)), x2)), x3), nil), APP(rm(s(0), add(s(s(z0)), x2)), x3), RM(s(0), add(s(s(z0)), x2))) IF_MINSORT(true, add(s(s(z0)), add(s(0), x2)), x3) -> c26(MINSORT(app(if_rm(false, s(s(z0)), add(s(0), x2)), x3), nil), APP(rm(s(s(z0)), add(s(0), x2)), x3), RM(s(s(z0)), add(s(0), x2))) IF_MINSORT(true, add(s(s(z0)), add(s(s(z1)), x2)), x3) -> c26(MINSORT(app(if_rm(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), x3), nil), APP(rm(s(s(z0)), add(s(s(z1)), x2)), x3), RM(s(s(z0)), add(s(s(z1)), x2))) IF_MINSORT(true, add(s(x0), add(s(x1), x2)), x3) -> c1(APP(rm(s(x0), add(s(x1), x2)), x3)) IF_MINSORT(true, add(s(x0), add(s(x1), x2)), x3) -> c1(RM(s(x0), add(s(x1), x2))) IF_MINSORT(true, add(z0, add(z1, z2)), x3) -> c26(APP(if_rm(eq(z0, z1), z0, add(z1, z2)), x3)) IF_MINSORT(true, add(z0, nil), x1) -> c26(MINSORT(x1, nil)) MINSORT(add(s(z0), x1), x2) -> c25(IF_MINSORT(eq(s(z0), if_min(false, x1, add(s(z0), x1), 0)), add(s(z0), x1), x2), EQ(s(z0), minIter(add(s(z0), x1), add(s(z0), x1), 0)), MIN(add(s(z0), x1))) MINSORT(add(0, x1), x2) -> c25(IF_MINSORT(eq(0, if_min(true, x1, add(0, x1), 0)), add(0, x1), x2), MIN(add(0, x1))) MINSORT(add(x0, x1), x2) -> c2(EQ(x0, minIter(add(x0, x1), add(x0, x1), 0))) MINSORT(add(x0, x1), x2) -> c2(MIN(add(x0, x1))) MINSORT(add(z0, z1), x2) -> c(EQ(z0, if_min(le(z0, 0), z1, add(z0, z1), 0))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(0)))) -> c12(IF_MIN(false, x1, x2, s(s(s(0)))), LE(s(s(s(s(z0)))), s(s(s(0))))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(s(z1))))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(s(s(z1))))), LE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MINITER(add(s(s(s(x0))), x1), x2, s(s(s(x3)))) -> c12(LE(s(s(s(x0))), s(s(s(x3))))) MINITER(add(s(s(s(0))), x1), x2, s(s(s(z0)))) -> c12(LE(s(s(s(0))), s(s(s(z0))))) RM(s(s(s(0))), add(s(s(s(0))), x2)) -> c21(IF_RM(true, s(s(s(0))), add(s(s(s(0))), x2)), EQ(s(s(s(0))), s(s(s(0))))) RM(s(s(s(0))), add(s(s(s(s(z0)))), x2)) -> c21(IF_RM(false, s(s(s(0))), add(s(s(s(s(z0)))), x2)), EQ(s(s(s(0))), s(s(s(s(z0)))))) RM(s(s(s(s(z0)))), add(s(s(s(0))), x2)) -> c21(IF_RM(false, s(s(s(s(z0)))), add(s(s(s(0))), x2)), EQ(s(s(s(s(z0)))), s(s(s(0))))) RM(s(s(s(s(z0)))), add(s(s(s(s(z1)))), x2)) -> c21(IF_RM(eq(z0, z1), s(s(s(s(z0)))), add(s(s(s(s(z1)))), x2)), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) RM(s(s(s(x0))), add(s(s(s(x1))), x2)) -> c21(EQ(s(s(s(x0))), s(s(s(x1))))) IF_MINSORT(true, add(0, add(0, nil)), x1) -> c26(MINSORT(app(nil, x1), nil), APP(rm(0, add(0, nil)), x1), RM(0, add(0, nil))) IF_MINSORT(true, add(0, add(0, add(z1, z2))), x1) -> c26(MINSORT(app(if_rm(eq(0, z1), 0, add(z1, z2)), x1), nil), APP(rm(0, add(0, add(z1, z2))), x1), RM(0, add(0, add(z1, z2)))) IF_MINSORT(true, add(0, add(0, x0)), x1) -> c26(APP(rm(0, add(0, x0)), x1), RM(0, add(0, x0))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) MINITER(add(s(s(z0)), x1), x2, s(0)) -> c12(IF_MIN(false, x1, x2, s(0)), LE(s(s(z0)), s(0))) MINITER(add(s(x0), x1), x2, s(x3)) -> c12(LE(s(x0), s(x3))) MINITER(add(s(0), x1), x2, s(z0)) -> c12(LE(s(0), s(z0))) MINITER(add(s(s(s(z0))), x1), x2, s(s(0))) -> c12(IF_MIN(false, x1, x2, s(s(0))), LE(s(s(s(z0))), s(s(0)))) MINITER(add(s(s(x0)), x1), x2, s(s(x3))) -> c12(LE(s(s(x0)), s(s(x3)))) MINITER(add(s(s(0)), x1), x2, s(s(z0))) -> c12(LE(s(s(0)), s(s(z0)))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(0)))) -> c12(IF_MIN(false, x1, x2, s(s(s(0)))), LE(s(s(s(s(z0)))), s(s(s(0))))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(s(z1))))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(s(s(z1))))), LE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MINITER(add(s(s(s(x0))), x1), x2, s(s(s(x3)))) -> c12(LE(s(s(s(x0))), s(s(s(x3))))) MINITER(add(s(s(s(0))), x1), x2, s(s(s(z0)))) -> c12(LE(s(s(s(0))), s(s(s(z0))))) K tuples: IF_MINSORT(true, add(z0, z1), z2) -> c26(MINSORT(app(rm(z0, z1), z2), nil), APP(rm(z0, z1), z2), RM(z0, z1)) MINSORT(add(z0, z1), z2) -> c25(IF_MINSORT(eq(z0, min(add(z0, z1))), add(z0, z1), z2), EQ(z0, min(add(z0, z1))), MIN(add(z0, z1))) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MIN(add(z0, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) RM(z0, add(z1, z2)) -> c21(IF_RM(eq(z0, z1), z0, add(z1, z2)), EQ(z0, z1)) Defined Rule Symbols: le_2, eq_2, app_2, if_rm_3, rm_2, if_min_4, minIter_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, MIN_1, MINITER_3, IF_MIN_4, IF_RM_3, IF_MINSORT_3, RM_2, MINSORT_2 Compound Symbols: c3_1, c6_1, c8_1, c10_1, c11_1, c14_1, c22_1, c23_1, c27_1, c12_1, c21_1, c12_2, c21_2, c_1, c26_1, c26_3, c1_1, c25_3, c25_2, c2_1, c26_2 ---------------------------------------- (157) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (158) 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) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) if_min(true, z0, z1, z2) -> z2 if_min(false, z0, z1, z2) -> minIter(z0, z1, z2) minIter(add(z0, z1), z2, z3) -> if_min(le(z0, z3), z1, z2, z3) minIter(nil, add(z0, z1), z2) -> minIter(add(z0, z1), add(z0, z1), s(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, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) RM(0, add(0, x2)) -> c21(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c21(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c21(IF_RM(false, s(z0), add(0, x2))) MINITER(add(s(s(z0)), x1), x2, s(0)) -> c12(IF_MIN(false, x1, x2, s(0)), LE(s(s(z0)), s(0))) MINITER(add(s(x0), x1), x2, s(x3)) -> c12(LE(s(x0), s(x3))) MINITER(add(s(0), x1), x2, s(z0)) -> c12(LE(s(0), s(z0))) RM(s(0), add(s(0), x2)) -> c21(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c21(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c21(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(x0), add(s(x1), x2)) -> c21(EQ(s(x0), s(x1))) MINSORT(add(x0, x1), x2) -> c(MIN(add(x0, x1))) MINITER(add(s(s(s(z0))), x1), x2, s(s(0))) -> c12(IF_MIN(false, x1, x2, s(s(0))), LE(s(s(s(z0))), s(s(0)))) MINITER(add(s(s(x0)), x1), x2, s(s(x3))) -> c12(LE(s(s(x0)), s(s(x3)))) MINITER(add(s(s(0)), x1), x2, s(s(z0))) -> c12(LE(s(s(0)), s(s(z0)))) RM(s(s(0)), add(s(s(0)), x2)) -> c21(IF_RM(true, s(s(0)), add(s(s(0)), x2)), EQ(s(s(0)), s(s(0)))) RM(s(s(0)), add(s(s(s(z0))), x2)) -> c21(IF_RM(false, s(s(0)), add(s(s(s(z0))), x2)), EQ(s(s(0)), s(s(s(z0))))) RM(s(s(s(z0))), add(s(s(0)), x2)) -> c21(IF_RM(false, s(s(s(z0))), add(s(s(0)), x2)), EQ(s(s(s(z0))), s(s(0)))) RM(s(s(x0)), add(s(s(x1)), x2)) -> c21(EQ(s(s(x0)), s(s(x1)))) IF_MINSORT(true, add(0, add(0, x0)), x1) -> c26(RM(0, add(0, x0))) IF_MINSORT(true, add(0, add(s(x0), z2)), x2) -> c26(MINSORT(app(add(s(x0), rm(0, z2)), x2), nil), APP(rm(0, add(s(x0), z2)), x2), RM(0, add(s(x0), z2))) IF_MINSORT(true, add(0, add(s(x0), x1)), x2) -> c26(RM(0, add(s(x0), x1))) IF_MINSORT(true, add(s(x0), add(0, z2)), x2) -> c26(MINSORT(app(add(0, rm(s(x0), z2)), x2), nil), APP(rm(s(x0), add(0, z2)), x2), RM(s(x0), add(0, z2))) IF_MINSORT(true, add(s(x0), add(0, x1)), x2) -> c26(RM(s(x0), add(0, x1))) IF_MINSORT(true, add(s(0), add(s(0), x2)), x3) -> c26(MINSORT(app(if_rm(true, s(0), add(s(0), x2)), x3), nil), APP(rm(s(0), add(s(0), x2)), x3), RM(s(0), add(s(0), x2))) IF_MINSORT(true, add(s(0), add(s(s(z0)), x2)), x3) -> c26(MINSORT(app(if_rm(false, s(0), add(s(s(z0)), x2)), x3), nil), APP(rm(s(0), add(s(s(z0)), x2)), x3), RM(s(0), add(s(s(z0)), x2))) IF_MINSORT(true, add(s(s(z0)), add(s(0), x2)), x3) -> c26(MINSORT(app(if_rm(false, s(s(z0)), add(s(0), x2)), x3), nil), APP(rm(s(s(z0)), add(s(0), x2)), x3), RM(s(s(z0)), add(s(0), x2))) IF_MINSORT(true, add(s(s(z0)), add(s(s(z1)), x2)), x3) -> c26(MINSORT(app(if_rm(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), x3), nil), APP(rm(s(s(z0)), add(s(s(z1)), x2)), x3), RM(s(s(z0)), add(s(s(z1)), x2))) IF_MINSORT(true, add(s(x0), add(s(x1), x2)), x3) -> c1(APP(rm(s(x0), add(s(x1), x2)), x3)) IF_MINSORT(true, add(s(x0), add(s(x1), x2)), x3) -> c1(RM(s(x0), add(s(x1), x2))) IF_MINSORT(true, add(z0, add(z1, z2)), x3) -> c26(APP(if_rm(eq(z0, z1), z0, add(z1, z2)), x3)) IF_MINSORT(true, add(z0, nil), x1) -> c26(MINSORT(x1, nil)) MINSORT(add(s(z0), x1), x2) -> c25(IF_MINSORT(eq(s(z0), if_min(false, x1, add(s(z0), x1), 0)), add(s(z0), x1), x2), EQ(s(z0), minIter(add(s(z0), x1), add(s(z0), x1), 0)), MIN(add(s(z0), x1))) MINSORT(add(0, x1), x2) -> c25(IF_MINSORT(eq(0, if_min(true, x1, add(0, x1), 0)), add(0, x1), x2), MIN(add(0, x1))) MINSORT(add(x0, x1), x2) -> c2(EQ(x0, minIter(add(x0, x1), add(x0, x1), 0))) MINSORT(add(x0, x1), x2) -> c2(MIN(add(x0, x1))) MINSORT(add(z0, z1), x2) -> c(EQ(z0, if_min(le(z0, 0), z1, add(z0, z1), 0))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(0)))) -> c12(IF_MIN(false, x1, x2, s(s(s(0)))), LE(s(s(s(s(z0)))), s(s(s(0))))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(s(z1))))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(s(s(z1))))), LE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MINITER(add(s(s(s(x0))), x1), x2, s(s(s(x3)))) -> c12(LE(s(s(s(x0))), s(s(s(x3))))) MINITER(add(s(s(s(0))), x1), x2, s(s(s(z0)))) -> c12(LE(s(s(s(0))), s(s(s(z0))))) RM(s(s(s(0))), add(s(s(s(0))), x2)) -> c21(IF_RM(true, s(s(s(0))), add(s(s(s(0))), x2)), EQ(s(s(s(0))), s(s(s(0))))) RM(s(s(s(0))), add(s(s(s(s(z0)))), x2)) -> c21(IF_RM(false, s(s(s(0))), add(s(s(s(s(z0)))), x2)), EQ(s(s(s(0))), s(s(s(s(z0)))))) RM(s(s(s(s(z0)))), add(s(s(s(0))), x2)) -> c21(IF_RM(false, s(s(s(s(z0)))), add(s(s(s(0))), x2)), EQ(s(s(s(s(z0)))), s(s(s(0))))) RM(s(s(s(s(z0)))), add(s(s(s(s(z1)))), x2)) -> c21(IF_RM(eq(z0, z1), s(s(s(s(z0)))), add(s(s(s(s(z1)))), x2)), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) RM(s(s(s(x0))), add(s(s(s(x1))), x2)) -> c21(EQ(s(s(s(x0))), s(s(s(x1))))) IF_MINSORT(true, add(0, add(0, nil)), x1) -> c26(MINSORT(app(nil, x1), nil), APP(rm(0, add(0, nil)), x1), RM(0, add(0, nil))) IF_MINSORT(true, add(0, add(0, add(z1, z2))), x1) -> c26(MINSORT(app(if_rm(eq(0, z1), 0, add(z1, z2)), x1), nil), APP(rm(0, add(0, add(z1, z2))), x1), RM(0, add(0, add(z1, z2)))) IF_MINSORT(true, add(0, add(0, x0)), x1) -> c4(APP(rm(0, add(0, x0)), x1)) IF_MINSORT(true, add(0, add(0, x0)), x1) -> c4(RM(0, add(0, x0))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) MINITER(add(s(s(z0)), x1), x2, s(0)) -> c12(IF_MIN(false, x1, x2, s(0)), LE(s(s(z0)), s(0))) MINITER(add(s(x0), x1), x2, s(x3)) -> c12(LE(s(x0), s(x3))) MINITER(add(s(0), x1), x2, s(z0)) -> c12(LE(s(0), s(z0))) MINITER(add(s(s(s(z0))), x1), x2, s(s(0))) -> c12(IF_MIN(false, x1, x2, s(s(0))), LE(s(s(s(z0))), s(s(0)))) MINITER(add(s(s(x0)), x1), x2, s(s(x3))) -> c12(LE(s(s(x0)), s(s(x3)))) MINITER(add(s(s(0)), x1), x2, s(s(z0))) -> c12(LE(s(s(0)), s(s(z0)))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(0)))) -> c12(IF_MIN(false, x1, x2, s(s(s(0)))), LE(s(s(s(s(z0)))), s(s(s(0))))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(s(z1))))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(s(s(z1))))), LE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MINITER(add(s(s(s(x0))), x1), x2, s(s(s(x3)))) -> c12(LE(s(s(s(x0))), s(s(s(x3))))) MINITER(add(s(s(s(0))), x1), x2, s(s(s(z0)))) -> c12(LE(s(s(s(0))), s(s(s(z0))))) K tuples: IF_MINSORT(true, add(z0, z1), z2) -> c26(MINSORT(app(rm(z0, z1), z2), nil), APP(rm(z0, z1), z2), RM(z0, z1)) MINSORT(add(z0, z1), z2) -> c25(IF_MINSORT(eq(z0, min(add(z0, z1))), add(z0, z1), z2), EQ(z0, min(add(z0, z1))), MIN(add(z0, z1))) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MIN(add(z0, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) RM(z0, add(z1, z2)) -> c21(IF_RM(eq(z0, z1), z0, add(z1, z2)), EQ(z0, z1)) Defined Rule Symbols: le_2, eq_2, app_2, if_rm_3, rm_2, if_min_4, minIter_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, MIN_1, MINITER_3, IF_MIN_4, IF_RM_3, IF_MINSORT_3, RM_2, MINSORT_2 Compound Symbols: c3_1, c6_1, c8_1, c10_1, c11_1, c14_1, c22_1, c23_1, c27_1, c12_1, c21_1, c12_2, c21_2, c_1, c26_1, c26_3, c1_1, c25_3, c25_2, c2_1, c4_1 ---------------------------------------- (159) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF_MINSORT(true, add(0, add(s(x0), z2)), x2) -> c26(MINSORT(app(add(s(x0), rm(0, z2)), x2), nil), APP(rm(0, add(s(x0), z2)), x2), RM(0, add(s(x0), z2))) by IF_MINSORT(true, add(0, add(s(x0), x1)), z2) -> c26(MINSORT(add(s(x0), app(rm(0, x1), z2)), nil), APP(rm(0, add(s(x0), x1)), z2), RM(0, add(s(x0), x1))) IF_MINSORT(true, add(0, add(s(x0), nil)), x2) -> c26(MINSORT(app(add(s(x0), nil), x2), nil), APP(rm(0, add(s(x0), nil)), x2), RM(0, add(s(x0), nil))) IF_MINSORT(true, add(0, add(s(x0), add(z1, z2))), x2) -> c26(MINSORT(app(add(s(x0), if_rm(eq(0, z1), 0, add(z1, z2))), x2), nil), APP(rm(0, add(s(x0), add(z1, z2))), x2), RM(0, add(s(x0), add(z1, z2)))) IF_MINSORT(true, add(0, add(s(x0), x1)), x2) -> c26(APP(rm(0, add(s(x0), x1)), x2), RM(0, add(s(x0), x1))) ---------------------------------------- (160) 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) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) if_min(true, z0, z1, z2) -> z2 if_min(false, z0, z1, z2) -> minIter(z0, z1, z2) minIter(add(z0, z1), z2, z3) -> if_min(le(z0, z3), z1, z2, z3) minIter(nil, add(z0, z1), z2) -> minIter(add(z0, z1), add(z0, z1), s(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, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) RM(0, add(0, x2)) -> c21(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c21(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c21(IF_RM(false, s(z0), add(0, x2))) MINITER(add(s(s(z0)), x1), x2, s(0)) -> c12(IF_MIN(false, x1, x2, s(0)), LE(s(s(z0)), s(0))) MINITER(add(s(x0), x1), x2, s(x3)) -> c12(LE(s(x0), s(x3))) MINITER(add(s(0), x1), x2, s(z0)) -> c12(LE(s(0), s(z0))) RM(s(0), add(s(0), x2)) -> c21(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c21(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c21(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(x0), add(s(x1), x2)) -> c21(EQ(s(x0), s(x1))) MINSORT(add(x0, x1), x2) -> c(MIN(add(x0, x1))) MINITER(add(s(s(s(z0))), x1), x2, s(s(0))) -> c12(IF_MIN(false, x1, x2, s(s(0))), LE(s(s(s(z0))), s(s(0)))) MINITER(add(s(s(x0)), x1), x2, s(s(x3))) -> c12(LE(s(s(x0)), s(s(x3)))) MINITER(add(s(s(0)), x1), x2, s(s(z0))) -> c12(LE(s(s(0)), s(s(z0)))) RM(s(s(0)), add(s(s(0)), x2)) -> c21(IF_RM(true, s(s(0)), add(s(s(0)), x2)), EQ(s(s(0)), s(s(0)))) RM(s(s(0)), add(s(s(s(z0))), x2)) -> c21(IF_RM(false, s(s(0)), add(s(s(s(z0))), x2)), EQ(s(s(0)), s(s(s(z0))))) RM(s(s(s(z0))), add(s(s(0)), x2)) -> c21(IF_RM(false, s(s(s(z0))), add(s(s(0)), x2)), EQ(s(s(s(z0))), s(s(0)))) RM(s(s(x0)), add(s(s(x1)), x2)) -> c21(EQ(s(s(x0)), s(s(x1)))) IF_MINSORT(true, add(0, add(0, x0)), x1) -> c26(RM(0, add(0, x0))) IF_MINSORT(true, add(0, add(s(x0), x1)), x2) -> c26(RM(0, add(s(x0), x1))) IF_MINSORT(true, add(s(x0), add(0, z2)), x2) -> c26(MINSORT(app(add(0, rm(s(x0), z2)), x2), nil), APP(rm(s(x0), add(0, z2)), x2), RM(s(x0), add(0, z2))) IF_MINSORT(true, add(s(x0), add(0, x1)), x2) -> c26(RM(s(x0), add(0, x1))) IF_MINSORT(true, add(s(0), add(s(0), x2)), x3) -> c26(MINSORT(app(if_rm(true, s(0), add(s(0), x2)), x3), nil), APP(rm(s(0), add(s(0), x2)), x3), RM(s(0), add(s(0), x2))) IF_MINSORT(true, add(s(0), add(s(s(z0)), x2)), x3) -> c26(MINSORT(app(if_rm(false, s(0), add(s(s(z0)), x2)), x3), nil), APP(rm(s(0), add(s(s(z0)), x2)), x3), RM(s(0), add(s(s(z0)), x2))) IF_MINSORT(true, add(s(s(z0)), add(s(0), x2)), x3) -> c26(MINSORT(app(if_rm(false, s(s(z0)), add(s(0), x2)), x3), nil), APP(rm(s(s(z0)), add(s(0), x2)), x3), RM(s(s(z0)), add(s(0), x2))) IF_MINSORT(true, add(s(s(z0)), add(s(s(z1)), x2)), x3) -> c26(MINSORT(app(if_rm(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), x3), nil), APP(rm(s(s(z0)), add(s(s(z1)), x2)), x3), RM(s(s(z0)), add(s(s(z1)), x2))) IF_MINSORT(true, add(s(x0), add(s(x1), x2)), x3) -> c1(APP(rm(s(x0), add(s(x1), x2)), x3)) IF_MINSORT(true, add(s(x0), add(s(x1), x2)), x3) -> c1(RM(s(x0), add(s(x1), x2))) IF_MINSORT(true, add(z0, add(z1, z2)), x3) -> c26(APP(if_rm(eq(z0, z1), z0, add(z1, z2)), x3)) IF_MINSORT(true, add(z0, nil), x1) -> c26(MINSORT(x1, nil)) MINSORT(add(s(z0), x1), x2) -> c25(IF_MINSORT(eq(s(z0), if_min(false, x1, add(s(z0), x1), 0)), add(s(z0), x1), x2), EQ(s(z0), minIter(add(s(z0), x1), add(s(z0), x1), 0)), MIN(add(s(z0), x1))) MINSORT(add(0, x1), x2) -> c25(IF_MINSORT(eq(0, if_min(true, x1, add(0, x1), 0)), add(0, x1), x2), MIN(add(0, x1))) MINSORT(add(x0, x1), x2) -> c2(EQ(x0, minIter(add(x0, x1), add(x0, x1), 0))) MINSORT(add(x0, x1), x2) -> c2(MIN(add(x0, x1))) MINSORT(add(z0, z1), x2) -> c(EQ(z0, if_min(le(z0, 0), z1, add(z0, z1), 0))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(0)))) -> c12(IF_MIN(false, x1, x2, s(s(s(0)))), LE(s(s(s(s(z0)))), s(s(s(0))))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(s(z1))))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(s(s(z1))))), LE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MINITER(add(s(s(s(x0))), x1), x2, s(s(s(x3)))) -> c12(LE(s(s(s(x0))), s(s(s(x3))))) MINITER(add(s(s(s(0))), x1), x2, s(s(s(z0)))) -> c12(LE(s(s(s(0))), s(s(s(z0))))) RM(s(s(s(0))), add(s(s(s(0))), x2)) -> c21(IF_RM(true, s(s(s(0))), add(s(s(s(0))), x2)), EQ(s(s(s(0))), s(s(s(0))))) RM(s(s(s(0))), add(s(s(s(s(z0)))), x2)) -> c21(IF_RM(false, s(s(s(0))), add(s(s(s(s(z0)))), x2)), EQ(s(s(s(0))), s(s(s(s(z0)))))) RM(s(s(s(s(z0)))), add(s(s(s(0))), x2)) -> c21(IF_RM(false, s(s(s(s(z0)))), add(s(s(s(0))), x2)), EQ(s(s(s(s(z0)))), s(s(s(0))))) RM(s(s(s(s(z0)))), add(s(s(s(s(z1)))), x2)) -> c21(IF_RM(eq(z0, z1), s(s(s(s(z0)))), add(s(s(s(s(z1)))), x2)), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) RM(s(s(s(x0))), add(s(s(s(x1))), x2)) -> c21(EQ(s(s(s(x0))), s(s(s(x1))))) IF_MINSORT(true, add(0, add(0, nil)), x1) -> c26(MINSORT(app(nil, x1), nil), APP(rm(0, add(0, nil)), x1), RM(0, add(0, nil))) IF_MINSORT(true, add(0, add(0, add(z1, z2))), x1) -> c26(MINSORT(app(if_rm(eq(0, z1), 0, add(z1, z2)), x1), nil), APP(rm(0, add(0, add(z1, z2))), x1), RM(0, add(0, add(z1, z2)))) IF_MINSORT(true, add(0, add(0, x0)), x1) -> c4(APP(rm(0, add(0, x0)), x1)) IF_MINSORT(true, add(0, add(0, x0)), x1) -> c4(RM(0, add(0, x0))) IF_MINSORT(true, add(0, add(s(x0), x1)), z2) -> c26(MINSORT(add(s(x0), app(rm(0, x1), z2)), nil), APP(rm(0, add(s(x0), x1)), z2), RM(0, add(s(x0), x1))) IF_MINSORT(true, add(0, add(s(x0), nil)), x2) -> c26(MINSORT(app(add(s(x0), nil), x2), nil), APP(rm(0, add(s(x0), nil)), x2), RM(0, add(s(x0), nil))) IF_MINSORT(true, add(0, add(s(x0), add(z1, z2))), x2) -> c26(MINSORT(app(add(s(x0), if_rm(eq(0, z1), 0, add(z1, z2))), x2), nil), APP(rm(0, add(s(x0), add(z1, z2))), x2), RM(0, add(s(x0), add(z1, z2)))) IF_MINSORT(true, add(0, add(s(x0), x1)), x2) -> c26(APP(rm(0, add(s(x0), x1)), x2), RM(0, add(s(x0), x1))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) MINITER(add(s(s(z0)), x1), x2, s(0)) -> c12(IF_MIN(false, x1, x2, s(0)), LE(s(s(z0)), s(0))) MINITER(add(s(x0), x1), x2, s(x3)) -> c12(LE(s(x0), s(x3))) MINITER(add(s(0), x1), x2, s(z0)) -> c12(LE(s(0), s(z0))) MINITER(add(s(s(s(z0))), x1), x2, s(s(0))) -> c12(IF_MIN(false, x1, x2, s(s(0))), LE(s(s(s(z0))), s(s(0)))) MINITER(add(s(s(x0)), x1), x2, s(s(x3))) -> c12(LE(s(s(x0)), s(s(x3)))) MINITER(add(s(s(0)), x1), x2, s(s(z0))) -> c12(LE(s(s(0)), s(s(z0)))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(0)))) -> c12(IF_MIN(false, x1, x2, s(s(s(0)))), LE(s(s(s(s(z0)))), s(s(s(0))))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(s(z1))))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(s(s(z1))))), LE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MINITER(add(s(s(s(x0))), x1), x2, s(s(s(x3)))) -> c12(LE(s(s(s(x0))), s(s(s(x3))))) MINITER(add(s(s(s(0))), x1), x2, s(s(s(z0)))) -> c12(LE(s(s(s(0))), s(s(s(z0))))) K tuples: IF_MINSORT(true, add(z0, z1), z2) -> c26(MINSORT(app(rm(z0, z1), z2), nil), APP(rm(z0, z1), z2), RM(z0, z1)) MINSORT(add(z0, z1), z2) -> c25(IF_MINSORT(eq(z0, min(add(z0, z1))), add(z0, z1), z2), EQ(z0, min(add(z0, z1))), MIN(add(z0, z1))) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MIN(add(z0, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) RM(z0, add(z1, z2)) -> c21(IF_RM(eq(z0, z1), z0, add(z1, z2)), EQ(z0, z1)) Defined Rule Symbols: le_2, eq_2, app_2, if_rm_3, rm_2, if_min_4, minIter_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, MIN_1, MINITER_3, IF_MIN_4, IF_RM_3, IF_MINSORT_3, RM_2, MINSORT_2 Compound Symbols: c3_1, c6_1, c8_1, c10_1, c11_1, c14_1, c22_1, c23_1, c27_1, c12_1, c21_1, c12_2, c21_2, c_1, c26_1, c26_3, c1_1, c25_3, c25_2, c2_1, c4_1, c26_2 ---------------------------------------- (161) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (162) 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) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) if_min(true, z0, z1, z2) -> z2 if_min(false, z0, z1, z2) -> minIter(z0, z1, z2) minIter(add(z0, z1), z2, z3) -> if_min(le(z0, z3), z1, z2, z3) minIter(nil, add(z0, z1), z2) -> minIter(add(z0, z1), add(z0, z1), s(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, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) RM(0, add(0, x2)) -> c21(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c21(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c21(IF_RM(false, s(z0), add(0, x2))) MINITER(add(s(s(z0)), x1), x2, s(0)) -> c12(IF_MIN(false, x1, x2, s(0)), LE(s(s(z0)), s(0))) MINITER(add(s(x0), x1), x2, s(x3)) -> c12(LE(s(x0), s(x3))) MINITER(add(s(0), x1), x2, s(z0)) -> c12(LE(s(0), s(z0))) RM(s(0), add(s(0), x2)) -> c21(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c21(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c21(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(x0), add(s(x1), x2)) -> c21(EQ(s(x0), s(x1))) MINSORT(add(x0, x1), x2) -> c(MIN(add(x0, x1))) MINITER(add(s(s(s(z0))), x1), x2, s(s(0))) -> c12(IF_MIN(false, x1, x2, s(s(0))), LE(s(s(s(z0))), s(s(0)))) MINITER(add(s(s(x0)), x1), x2, s(s(x3))) -> c12(LE(s(s(x0)), s(s(x3)))) MINITER(add(s(s(0)), x1), x2, s(s(z0))) -> c12(LE(s(s(0)), s(s(z0)))) RM(s(s(0)), add(s(s(0)), x2)) -> c21(IF_RM(true, s(s(0)), add(s(s(0)), x2)), EQ(s(s(0)), s(s(0)))) RM(s(s(0)), add(s(s(s(z0))), x2)) -> c21(IF_RM(false, s(s(0)), add(s(s(s(z0))), x2)), EQ(s(s(0)), s(s(s(z0))))) RM(s(s(s(z0))), add(s(s(0)), x2)) -> c21(IF_RM(false, s(s(s(z0))), add(s(s(0)), x2)), EQ(s(s(s(z0))), s(s(0)))) RM(s(s(x0)), add(s(s(x1)), x2)) -> c21(EQ(s(s(x0)), s(s(x1)))) IF_MINSORT(true, add(0, add(0, x0)), x1) -> c26(RM(0, add(0, x0))) IF_MINSORT(true, add(0, add(s(x0), x1)), x2) -> c26(RM(0, add(s(x0), x1))) IF_MINSORT(true, add(s(x0), add(0, z2)), x2) -> c26(MINSORT(app(add(0, rm(s(x0), z2)), x2), nil), APP(rm(s(x0), add(0, z2)), x2), RM(s(x0), add(0, z2))) IF_MINSORT(true, add(s(x0), add(0, x1)), x2) -> c26(RM(s(x0), add(0, x1))) IF_MINSORT(true, add(s(0), add(s(0), x2)), x3) -> c26(MINSORT(app(if_rm(true, s(0), add(s(0), x2)), x3), nil), APP(rm(s(0), add(s(0), x2)), x3), RM(s(0), add(s(0), x2))) IF_MINSORT(true, add(s(0), add(s(s(z0)), x2)), x3) -> c26(MINSORT(app(if_rm(false, s(0), add(s(s(z0)), x2)), x3), nil), APP(rm(s(0), add(s(s(z0)), x2)), x3), RM(s(0), add(s(s(z0)), x2))) IF_MINSORT(true, add(s(s(z0)), add(s(0), x2)), x3) -> c26(MINSORT(app(if_rm(false, s(s(z0)), add(s(0), x2)), x3), nil), APP(rm(s(s(z0)), add(s(0), x2)), x3), RM(s(s(z0)), add(s(0), x2))) IF_MINSORT(true, add(s(s(z0)), add(s(s(z1)), x2)), x3) -> c26(MINSORT(app(if_rm(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), x3), nil), APP(rm(s(s(z0)), add(s(s(z1)), x2)), x3), RM(s(s(z0)), add(s(s(z1)), x2))) IF_MINSORT(true, add(s(x0), add(s(x1), x2)), x3) -> c1(APP(rm(s(x0), add(s(x1), x2)), x3)) IF_MINSORT(true, add(s(x0), add(s(x1), x2)), x3) -> c1(RM(s(x0), add(s(x1), x2))) IF_MINSORT(true, add(z0, add(z1, z2)), x3) -> c26(APP(if_rm(eq(z0, z1), z0, add(z1, z2)), x3)) IF_MINSORT(true, add(z0, nil), x1) -> c26(MINSORT(x1, nil)) MINSORT(add(s(z0), x1), x2) -> c25(IF_MINSORT(eq(s(z0), if_min(false, x1, add(s(z0), x1), 0)), add(s(z0), x1), x2), EQ(s(z0), minIter(add(s(z0), x1), add(s(z0), x1), 0)), MIN(add(s(z0), x1))) MINSORT(add(0, x1), x2) -> c25(IF_MINSORT(eq(0, if_min(true, x1, add(0, x1), 0)), add(0, x1), x2), MIN(add(0, x1))) MINSORT(add(x0, x1), x2) -> c2(EQ(x0, minIter(add(x0, x1), add(x0, x1), 0))) MINSORT(add(x0, x1), x2) -> c2(MIN(add(x0, x1))) MINSORT(add(z0, z1), x2) -> c(EQ(z0, if_min(le(z0, 0), z1, add(z0, z1), 0))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(0)))) -> c12(IF_MIN(false, x1, x2, s(s(s(0)))), LE(s(s(s(s(z0)))), s(s(s(0))))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(s(z1))))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(s(s(z1))))), LE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MINITER(add(s(s(s(x0))), x1), x2, s(s(s(x3)))) -> c12(LE(s(s(s(x0))), s(s(s(x3))))) MINITER(add(s(s(s(0))), x1), x2, s(s(s(z0)))) -> c12(LE(s(s(s(0))), s(s(s(z0))))) RM(s(s(s(0))), add(s(s(s(0))), x2)) -> c21(IF_RM(true, s(s(s(0))), add(s(s(s(0))), x2)), EQ(s(s(s(0))), s(s(s(0))))) RM(s(s(s(0))), add(s(s(s(s(z0)))), x2)) -> c21(IF_RM(false, s(s(s(0))), add(s(s(s(s(z0)))), x2)), EQ(s(s(s(0))), s(s(s(s(z0)))))) RM(s(s(s(s(z0)))), add(s(s(s(0))), x2)) -> c21(IF_RM(false, s(s(s(s(z0)))), add(s(s(s(0))), x2)), EQ(s(s(s(s(z0)))), s(s(s(0))))) RM(s(s(s(s(z0)))), add(s(s(s(s(z1)))), x2)) -> c21(IF_RM(eq(z0, z1), s(s(s(s(z0)))), add(s(s(s(s(z1)))), x2)), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) RM(s(s(s(x0))), add(s(s(s(x1))), x2)) -> c21(EQ(s(s(s(x0))), s(s(s(x1))))) IF_MINSORT(true, add(0, add(0, nil)), x1) -> c26(MINSORT(app(nil, x1), nil), APP(rm(0, add(0, nil)), x1), RM(0, add(0, nil))) IF_MINSORT(true, add(0, add(0, add(z1, z2))), x1) -> c26(MINSORT(app(if_rm(eq(0, z1), 0, add(z1, z2)), x1), nil), APP(rm(0, add(0, add(z1, z2))), x1), RM(0, add(0, add(z1, z2)))) IF_MINSORT(true, add(0, add(0, x0)), x1) -> c4(APP(rm(0, add(0, x0)), x1)) IF_MINSORT(true, add(0, add(0, x0)), x1) -> c4(RM(0, add(0, x0))) IF_MINSORT(true, add(0, add(s(x0), x1)), z2) -> c26(MINSORT(add(s(x0), app(rm(0, x1), z2)), nil), APP(rm(0, add(s(x0), x1)), z2), RM(0, add(s(x0), x1))) IF_MINSORT(true, add(0, add(s(x0), nil)), x2) -> c26(MINSORT(app(add(s(x0), nil), x2), nil), APP(rm(0, add(s(x0), nil)), x2), RM(0, add(s(x0), nil))) IF_MINSORT(true, add(0, add(s(x0), add(z1, z2))), x2) -> c26(MINSORT(app(add(s(x0), if_rm(eq(0, z1), 0, add(z1, z2))), x2), nil), APP(rm(0, add(s(x0), add(z1, z2))), x2), RM(0, add(s(x0), add(z1, z2)))) IF_MINSORT(true, add(0, add(s(x0), x1)), x2) -> c5(APP(rm(0, add(s(x0), x1)), x2)) IF_MINSORT(true, add(0, add(s(x0), x1)), x2) -> c5(RM(0, add(s(x0), x1))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) MINITER(add(s(s(z0)), x1), x2, s(0)) -> c12(IF_MIN(false, x1, x2, s(0)), LE(s(s(z0)), s(0))) MINITER(add(s(x0), x1), x2, s(x3)) -> c12(LE(s(x0), s(x3))) MINITER(add(s(0), x1), x2, s(z0)) -> c12(LE(s(0), s(z0))) MINITER(add(s(s(s(z0))), x1), x2, s(s(0))) -> c12(IF_MIN(false, x1, x2, s(s(0))), LE(s(s(s(z0))), s(s(0)))) MINITER(add(s(s(x0)), x1), x2, s(s(x3))) -> c12(LE(s(s(x0)), s(s(x3)))) MINITER(add(s(s(0)), x1), x2, s(s(z0))) -> c12(LE(s(s(0)), s(s(z0)))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(0)))) -> c12(IF_MIN(false, x1, x2, s(s(s(0)))), LE(s(s(s(s(z0)))), s(s(s(0))))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(s(z1))))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(s(s(z1))))), LE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MINITER(add(s(s(s(x0))), x1), x2, s(s(s(x3)))) -> c12(LE(s(s(s(x0))), s(s(s(x3))))) MINITER(add(s(s(s(0))), x1), x2, s(s(s(z0)))) -> c12(LE(s(s(s(0))), s(s(s(z0))))) K tuples: IF_MINSORT(true, add(z0, z1), z2) -> c26(MINSORT(app(rm(z0, z1), z2), nil), APP(rm(z0, z1), z2), RM(z0, z1)) MINSORT(add(z0, z1), z2) -> c25(IF_MINSORT(eq(z0, min(add(z0, z1))), add(z0, z1), z2), EQ(z0, min(add(z0, z1))), MIN(add(z0, z1))) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MIN(add(z0, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) RM(z0, add(z1, z2)) -> c21(IF_RM(eq(z0, z1), z0, add(z1, z2)), EQ(z0, z1)) Defined Rule Symbols: le_2, eq_2, app_2, if_rm_3, rm_2, if_min_4, minIter_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, MIN_1, MINITER_3, IF_MIN_4, IF_RM_3, IF_MINSORT_3, RM_2, MINSORT_2 Compound Symbols: c3_1, c6_1, c8_1, c10_1, c11_1, c14_1, c22_1, c23_1, c27_1, c12_1, c21_1, c12_2, c21_2, c_1, c26_1, c26_3, c1_1, c25_3, c25_2, c2_1, c4_1, c5_1 ---------------------------------------- (163) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF_MINSORT(true, add(s(x0), add(0, z2)), x2) -> c26(MINSORT(app(add(0, rm(s(x0), z2)), x2), nil), APP(rm(s(x0), add(0, z2)), x2), RM(s(x0), add(0, z2))) by IF_MINSORT(true, add(s(x0), add(0, x1)), z2) -> c26(MINSORT(add(0, app(rm(s(x0), x1), z2)), nil), APP(rm(s(x0), add(0, x1)), z2), RM(s(x0), add(0, x1))) IF_MINSORT(true, add(s(x0), add(0, nil)), x2) -> c26(MINSORT(app(add(0, nil), x2), nil), APP(rm(s(x0), add(0, nil)), x2), RM(s(x0), add(0, nil))) IF_MINSORT(true, add(s(x0), add(0, add(z1, z2))), x2) -> c26(MINSORT(app(add(0, if_rm(eq(s(x0), z1), s(x0), add(z1, z2))), x2), nil), APP(rm(s(x0), add(0, add(z1, z2))), x2), RM(s(x0), add(0, add(z1, z2)))) IF_MINSORT(true, add(s(x0), add(0, x1)), x2) -> c26(APP(rm(s(x0), add(0, x1)), x2), RM(s(x0), add(0, x1))) ---------------------------------------- (164) 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) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) if_min(true, z0, z1, z2) -> z2 if_min(false, z0, z1, z2) -> minIter(z0, z1, z2) minIter(add(z0, z1), z2, z3) -> if_min(le(z0, z3), z1, z2, z3) minIter(nil, add(z0, z1), z2) -> minIter(add(z0, z1), add(z0, z1), s(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, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) RM(0, add(0, x2)) -> c21(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c21(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c21(IF_RM(false, s(z0), add(0, x2))) MINITER(add(s(s(z0)), x1), x2, s(0)) -> c12(IF_MIN(false, x1, x2, s(0)), LE(s(s(z0)), s(0))) MINITER(add(s(x0), x1), x2, s(x3)) -> c12(LE(s(x0), s(x3))) MINITER(add(s(0), x1), x2, s(z0)) -> c12(LE(s(0), s(z0))) RM(s(0), add(s(0), x2)) -> c21(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c21(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c21(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(x0), add(s(x1), x2)) -> c21(EQ(s(x0), s(x1))) MINSORT(add(x0, x1), x2) -> c(MIN(add(x0, x1))) MINITER(add(s(s(s(z0))), x1), x2, s(s(0))) -> c12(IF_MIN(false, x1, x2, s(s(0))), LE(s(s(s(z0))), s(s(0)))) MINITER(add(s(s(x0)), x1), x2, s(s(x3))) -> c12(LE(s(s(x0)), s(s(x3)))) MINITER(add(s(s(0)), x1), x2, s(s(z0))) -> c12(LE(s(s(0)), s(s(z0)))) RM(s(s(0)), add(s(s(0)), x2)) -> c21(IF_RM(true, s(s(0)), add(s(s(0)), x2)), EQ(s(s(0)), s(s(0)))) RM(s(s(0)), add(s(s(s(z0))), x2)) -> c21(IF_RM(false, s(s(0)), add(s(s(s(z0))), x2)), EQ(s(s(0)), s(s(s(z0))))) RM(s(s(s(z0))), add(s(s(0)), x2)) -> c21(IF_RM(false, s(s(s(z0))), add(s(s(0)), x2)), EQ(s(s(s(z0))), s(s(0)))) RM(s(s(x0)), add(s(s(x1)), x2)) -> c21(EQ(s(s(x0)), s(s(x1)))) IF_MINSORT(true, add(0, add(0, x0)), x1) -> c26(RM(0, add(0, x0))) IF_MINSORT(true, add(0, add(s(x0), x1)), x2) -> c26(RM(0, add(s(x0), x1))) IF_MINSORT(true, add(s(x0), add(0, x1)), x2) -> c26(RM(s(x0), add(0, x1))) IF_MINSORT(true, add(s(0), add(s(0), x2)), x3) -> c26(MINSORT(app(if_rm(true, s(0), add(s(0), x2)), x3), nil), APP(rm(s(0), add(s(0), x2)), x3), RM(s(0), add(s(0), x2))) IF_MINSORT(true, add(s(0), add(s(s(z0)), x2)), x3) -> c26(MINSORT(app(if_rm(false, s(0), add(s(s(z0)), x2)), x3), nil), APP(rm(s(0), add(s(s(z0)), x2)), x3), RM(s(0), add(s(s(z0)), x2))) IF_MINSORT(true, add(s(s(z0)), add(s(0), x2)), x3) -> c26(MINSORT(app(if_rm(false, s(s(z0)), add(s(0), x2)), x3), nil), APP(rm(s(s(z0)), add(s(0), x2)), x3), RM(s(s(z0)), add(s(0), x2))) IF_MINSORT(true, add(s(s(z0)), add(s(s(z1)), x2)), x3) -> c26(MINSORT(app(if_rm(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), x3), nil), APP(rm(s(s(z0)), add(s(s(z1)), x2)), x3), RM(s(s(z0)), add(s(s(z1)), x2))) IF_MINSORT(true, add(s(x0), add(s(x1), x2)), x3) -> c1(APP(rm(s(x0), add(s(x1), x2)), x3)) IF_MINSORT(true, add(s(x0), add(s(x1), x2)), x3) -> c1(RM(s(x0), add(s(x1), x2))) IF_MINSORT(true, add(z0, add(z1, z2)), x3) -> c26(APP(if_rm(eq(z0, z1), z0, add(z1, z2)), x3)) IF_MINSORT(true, add(z0, nil), x1) -> c26(MINSORT(x1, nil)) MINSORT(add(s(z0), x1), x2) -> c25(IF_MINSORT(eq(s(z0), if_min(false, x1, add(s(z0), x1), 0)), add(s(z0), x1), x2), EQ(s(z0), minIter(add(s(z0), x1), add(s(z0), x1), 0)), MIN(add(s(z0), x1))) MINSORT(add(0, x1), x2) -> c25(IF_MINSORT(eq(0, if_min(true, x1, add(0, x1), 0)), add(0, x1), x2), MIN(add(0, x1))) MINSORT(add(x0, x1), x2) -> c2(EQ(x0, minIter(add(x0, x1), add(x0, x1), 0))) MINSORT(add(x0, x1), x2) -> c2(MIN(add(x0, x1))) MINSORT(add(z0, z1), x2) -> c(EQ(z0, if_min(le(z0, 0), z1, add(z0, z1), 0))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(0)))) -> c12(IF_MIN(false, x1, x2, s(s(s(0)))), LE(s(s(s(s(z0)))), s(s(s(0))))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(s(z1))))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(s(s(z1))))), LE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MINITER(add(s(s(s(x0))), x1), x2, s(s(s(x3)))) -> c12(LE(s(s(s(x0))), s(s(s(x3))))) MINITER(add(s(s(s(0))), x1), x2, s(s(s(z0)))) -> c12(LE(s(s(s(0))), s(s(s(z0))))) RM(s(s(s(0))), add(s(s(s(0))), x2)) -> c21(IF_RM(true, s(s(s(0))), add(s(s(s(0))), x2)), EQ(s(s(s(0))), s(s(s(0))))) RM(s(s(s(0))), add(s(s(s(s(z0)))), x2)) -> c21(IF_RM(false, s(s(s(0))), add(s(s(s(s(z0)))), x2)), EQ(s(s(s(0))), s(s(s(s(z0)))))) RM(s(s(s(s(z0)))), add(s(s(s(0))), x2)) -> c21(IF_RM(false, s(s(s(s(z0)))), add(s(s(s(0))), x2)), EQ(s(s(s(s(z0)))), s(s(s(0))))) RM(s(s(s(s(z0)))), add(s(s(s(s(z1)))), x2)) -> c21(IF_RM(eq(z0, z1), s(s(s(s(z0)))), add(s(s(s(s(z1)))), x2)), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) RM(s(s(s(x0))), add(s(s(s(x1))), x2)) -> c21(EQ(s(s(s(x0))), s(s(s(x1))))) IF_MINSORT(true, add(0, add(0, nil)), x1) -> c26(MINSORT(app(nil, x1), nil), APP(rm(0, add(0, nil)), x1), RM(0, add(0, nil))) IF_MINSORT(true, add(0, add(0, add(z1, z2))), x1) -> c26(MINSORT(app(if_rm(eq(0, z1), 0, add(z1, z2)), x1), nil), APP(rm(0, add(0, add(z1, z2))), x1), RM(0, add(0, add(z1, z2)))) IF_MINSORT(true, add(0, add(0, x0)), x1) -> c4(APP(rm(0, add(0, x0)), x1)) IF_MINSORT(true, add(0, add(0, x0)), x1) -> c4(RM(0, add(0, x0))) IF_MINSORT(true, add(0, add(s(x0), x1)), z2) -> c26(MINSORT(add(s(x0), app(rm(0, x1), z2)), nil), APP(rm(0, add(s(x0), x1)), z2), RM(0, add(s(x0), x1))) IF_MINSORT(true, add(0, add(s(x0), nil)), x2) -> c26(MINSORT(app(add(s(x0), nil), x2), nil), APP(rm(0, add(s(x0), nil)), x2), RM(0, add(s(x0), nil))) IF_MINSORT(true, add(0, add(s(x0), add(z1, z2))), x2) -> c26(MINSORT(app(add(s(x0), if_rm(eq(0, z1), 0, add(z1, z2))), x2), nil), APP(rm(0, add(s(x0), add(z1, z2))), x2), RM(0, add(s(x0), add(z1, z2)))) IF_MINSORT(true, add(0, add(s(x0), x1)), x2) -> c5(APP(rm(0, add(s(x0), x1)), x2)) IF_MINSORT(true, add(0, add(s(x0), x1)), x2) -> c5(RM(0, add(s(x0), x1))) IF_MINSORT(true, add(s(x0), add(0, x1)), z2) -> c26(MINSORT(add(0, app(rm(s(x0), x1), z2)), nil), APP(rm(s(x0), add(0, x1)), z2), RM(s(x0), add(0, x1))) IF_MINSORT(true, add(s(x0), add(0, nil)), x2) -> c26(MINSORT(app(add(0, nil), x2), nil), APP(rm(s(x0), add(0, nil)), x2), RM(s(x0), add(0, nil))) IF_MINSORT(true, add(s(x0), add(0, add(z1, z2))), x2) -> c26(MINSORT(app(add(0, if_rm(eq(s(x0), z1), s(x0), add(z1, z2))), x2), nil), APP(rm(s(x0), add(0, add(z1, z2))), x2), RM(s(x0), add(0, add(z1, z2)))) IF_MINSORT(true, add(s(x0), add(0, x1)), x2) -> c26(APP(rm(s(x0), add(0, x1)), x2), RM(s(x0), add(0, x1))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) MINITER(add(s(s(z0)), x1), x2, s(0)) -> c12(IF_MIN(false, x1, x2, s(0)), LE(s(s(z0)), s(0))) MINITER(add(s(x0), x1), x2, s(x3)) -> c12(LE(s(x0), s(x3))) MINITER(add(s(0), x1), x2, s(z0)) -> c12(LE(s(0), s(z0))) MINITER(add(s(s(s(z0))), x1), x2, s(s(0))) -> c12(IF_MIN(false, x1, x2, s(s(0))), LE(s(s(s(z0))), s(s(0)))) MINITER(add(s(s(x0)), x1), x2, s(s(x3))) -> c12(LE(s(s(x0)), s(s(x3)))) MINITER(add(s(s(0)), x1), x2, s(s(z0))) -> c12(LE(s(s(0)), s(s(z0)))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(0)))) -> c12(IF_MIN(false, x1, x2, s(s(s(0)))), LE(s(s(s(s(z0)))), s(s(s(0))))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(s(z1))))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(s(s(z1))))), LE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MINITER(add(s(s(s(x0))), x1), x2, s(s(s(x3)))) -> c12(LE(s(s(s(x0))), s(s(s(x3))))) MINITER(add(s(s(s(0))), x1), x2, s(s(s(z0)))) -> c12(LE(s(s(s(0))), s(s(s(z0))))) K tuples: IF_MINSORT(true, add(z0, z1), z2) -> c26(MINSORT(app(rm(z0, z1), z2), nil), APP(rm(z0, z1), z2), RM(z0, z1)) MINSORT(add(z0, z1), z2) -> c25(IF_MINSORT(eq(z0, min(add(z0, z1))), add(z0, z1), z2), EQ(z0, min(add(z0, z1))), MIN(add(z0, z1))) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MIN(add(z0, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) RM(z0, add(z1, z2)) -> c21(IF_RM(eq(z0, z1), z0, add(z1, z2)), EQ(z0, z1)) Defined Rule Symbols: le_2, eq_2, app_2, if_rm_3, rm_2, if_min_4, minIter_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, MIN_1, MINITER_3, IF_MIN_4, IF_RM_3, IF_MINSORT_3, RM_2, MINSORT_2 Compound Symbols: c3_1, c6_1, c8_1, c10_1, c11_1, c14_1, c22_1, c23_1, c27_1, c12_1, c21_1, c12_2, c21_2, c_1, c26_1, c26_3, c1_1, c25_3, c25_2, c2_1, c4_1, c5_1, c26_2 ---------------------------------------- (165) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (166) 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) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) if_min(true, z0, z1, z2) -> z2 if_min(false, z0, z1, z2) -> minIter(z0, z1, z2) minIter(add(z0, z1), z2, z3) -> if_min(le(z0, z3), z1, z2, z3) minIter(nil, add(z0, z1), z2) -> minIter(add(z0, z1), add(z0, z1), s(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, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) RM(0, add(0, x2)) -> c21(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c21(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c21(IF_RM(false, s(z0), add(0, x2))) MINITER(add(s(s(z0)), x1), x2, s(0)) -> c12(IF_MIN(false, x1, x2, s(0)), LE(s(s(z0)), s(0))) MINITER(add(s(x0), x1), x2, s(x3)) -> c12(LE(s(x0), s(x3))) MINITER(add(s(0), x1), x2, s(z0)) -> c12(LE(s(0), s(z0))) RM(s(0), add(s(0), x2)) -> c21(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c21(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c21(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(x0), add(s(x1), x2)) -> c21(EQ(s(x0), s(x1))) MINSORT(add(x0, x1), x2) -> c(MIN(add(x0, x1))) MINITER(add(s(s(s(z0))), x1), x2, s(s(0))) -> c12(IF_MIN(false, x1, x2, s(s(0))), LE(s(s(s(z0))), s(s(0)))) MINITER(add(s(s(x0)), x1), x2, s(s(x3))) -> c12(LE(s(s(x0)), s(s(x3)))) MINITER(add(s(s(0)), x1), x2, s(s(z0))) -> c12(LE(s(s(0)), s(s(z0)))) RM(s(s(0)), add(s(s(0)), x2)) -> c21(IF_RM(true, s(s(0)), add(s(s(0)), x2)), EQ(s(s(0)), s(s(0)))) RM(s(s(0)), add(s(s(s(z0))), x2)) -> c21(IF_RM(false, s(s(0)), add(s(s(s(z0))), x2)), EQ(s(s(0)), s(s(s(z0))))) RM(s(s(s(z0))), add(s(s(0)), x2)) -> c21(IF_RM(false, s(s(s(z0))), add(s(s(0)), x2)), EQ(s(s(s(z0))), s(s(0)))) RM(s(s(x0)), add(s(s(x1)), x2)) -> c21(EQ(s(s(x0)), s(s(x1)))) IF_MINSORT(true, add(0, add(0, x0)), x1) -> c26(RM(0, add(0, x0))) IF_MINSORT(true, add(0, add(s(x0), x1)), x2) -> c26(RM(0, add(s(x0), x1))) IF_MINSORT(true, add(s(x0), add(0, x1)), x2) -> c26(RM(s(x0), add(0, x1))) IF_MINSORT(true, add(s(0), add(s(0), x2)), x3) -> c26(MINSORT(app(if_rm(true, s(0), add(s(0), x2)), x3), nil), APP(rm(s(0), add(s(0), x2)), x3), RM(s(0), add(s(0), x2))) IF_MINSORT(true, add(s(0), add(s(s(z0)), x2)), x3) -> c26(MINSORT(app(if_rm(false, s(0), add(s(s(z0)), x2)), x3), nil), APP(rm(s(0), add(s(s(z0)), x2)), x3), RM(s(0), add(s(s(z0)), x2))) IF_MINSORT(true, add(s(s(z0)), add(s(0), x2)), x3) -> c26(MINSORT(app(if_rm(false, s(s(z0)), add(s(0), x2)), x3), nil), APP(rm(s(s(z0)), add(s(0), x2)), x3), RM(s(s(z0)), add(s(0), x2))) IF_MINSORT(true, add(s(s(z0)), add(s(s(z1)), x2)), x3) -> c26(MINSORT(app(if_rm(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), x3), nil), APP(rm(s(s(z0)), add(s(s(z1)), x2)), x3), RM(s(s(z0)), add(s(s(z1)), x2))) IF_MINSORT(true, add(s(x0), add(s(x1), x2)), x3) -> c1(APP(rm(s(x0), add(s(x1), x2)), x3)) IF_MINSORT(true, add(s(x0), add(s(x1), x2)), x3) -> c1(RM(s(x0), add(s(x1), x2))) IF_MINSORT(true, add(z0, add(z1, z2)), x3) -> c26(APP(if_rm(eq(z0, z1), z0, add(z1, z2)), x3)) IF_MINSORT(true, add(z0, nil), x1) -> c26(MINSORT(x1, nil)) MINSORT(add(s(z0), x1), x2) -> c25(IF_MINSORT(eq(s(z0), if_min(false, x1, add(s(z0), x1), 0)), add(s(z0), x1), x2), EQ(s(z0), minIter(add(s(z0), x1), add(s(z0), x1), 0)), MIN(add(s(z0), x1))) MINSORT(add(0, x1), x2) -> c25(IF_MINSORT(eq(0, if_min(true, x1, add(0, x1), 0)), add(0, x1), x2), MIN(add(0, x1))) MINSORT(add(x0, x1), x2) -> c2(EQ(x0, minIter(add(x0, x1), add(x0, x1), 0))) MINSORT(add(x0, x1), x2) -> c2(MIN(add(x0, x1))) MINSORT(add(z0, z1), x2) -> c(EQ(z0, if_min(le(z0, 0), z1, add(z0, z1), 0))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(0)))) -> c12(IF_MIN(false, x1, x2, s(s(s(0)))), LE(s(s(s(s(z0)))), s(s(s(0))))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(s(z1))))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(s(s(z1))))), LE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MINITER(add(s(s(s(x0))), x1), x2, s(s(s(x3)))) -> c12(LE(s(s(s(x0))), s(s(s(x3))))) MINITER(add(s(s(s(0))), x1), x2, s(s(s(z0)))) -> c12(LE(s(s(s(0))), s(s(s(z0))))) RM(s(s(s(0))), add(s(s(s(0))), x2)) -> c21(IF_RM(true, s(s(s(0))), add(s(s(s(0))), x2)), EQ(s(s(s(0))), s(s(s(0))))) RM(s(s(s(0))), add(s(s(s(s(z0)))), x2)) -> c21(IF_RM(false, s(s(s(0))), add(s(s(s(s(z0)))), x2)), EQ(s(s(s(0))), s(s(s(s(z0)))))) RM(s(s(s(s(z0)))), add(s(s(s(0))), x2)) -> c21(IF_RM(false, s(s(s(s(z0)))), add(s(s(s(0))), x2)), EQ(s(s(s(s(z0)))), s(s(s(0))))) RM(s(s(s(s(z0)))), add(s(s(s(s(z1)))), x2)) -> c21(IF_RM(eq(z0, z1), s(s(s(s(z0)))), add(s(s(s(s(z1)))), x2)), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) RM(s(s(s(x0))), add(s(s(s(x1))), x2)) -> c21(EQ(s(s(s(x0))), s(s(s(x1))))) IF_MINSORT(true, add(0, add(0, nil)), x1) -> c26(MINSORT(app(nil, x1), nil), APP(rm(0, add(0, nil)), x1), RM(0, add(0, nil))) IF_MINSORT(true, add(0, add(0, add(z1, z2))), x1) -> c26(MINSORT(app(if_rm(eq(0, z1), 0, add(z1, z2)), x1), nil), APP(rm(0, add(0, add(z1, z2))), x1), RM(0, add(0, add(z1, z2)))) IF_MINSORT(true, add(0, add(0, x0)), x1) -> c4(APP(rm(0, add(0, x0)), x1)) IF_MINSORT(true, add(0, add(0, x0)), x1) -> c4(RM(0, add(0, x0))) IF_MINSORT(true, add(0, add(s(x0), x1)), z2) -> c26(MINSORT(add(s(x0), app(rm(0, x1), z2)), nil), APP(rm(0, add(s(x0), x1)), z2), RM(0, add(s(x0), x1))) IF_MINSORT(true, add(0, add(s(x0), nil)), x2) -> c26(MINSORT(app(add(s(x0), nil), x2), nil), APP(rm(0, add(s(x0), nil)), x2), RM(0, add(s(x0), nil))) IF_MINSORT(true, add(0, add(s(x0), add(z1, z2))), x2) -> c26(MINSORT(app(add(s(x0), if_rm(eq(0, z1), 0, add(z1, z2))), x2), nil), APP(rm(0, add(s(x0), add(z1, z2))), x2), RM(0, add(s(x0), add(z1, z2)))) IF_MINSORT(true, add(0, add(s(x0), x1)), x2) -> c5(APP(rm(0, add(s(x0), x1)), x2)) IF_MINSORT(true, add(0, add(s(x0), x1)), x2) -> c5(RM(0, add(s(x0), x1))) IF_MINSORT(true, add(s(x0), add(0, x1)), z2) -> c26(MINSORT(add(0, app(rm(s(x0), x1), z2)), nil), APP(rm(s(x0), add(0, x1)), z2), RM(s(x0), add(0, x1))) IF_MINSORT(true, add(s(x0), add(0, nil)), x2) -> c26(MINSORT(app(add(0, nil), x2), nil), APP(rm(s(x0), add(0, nil)), x2), RM(s(x0), add(0, nil))) IF_MINSORT(true, add(s(x0), add(0, add(z1, z2))), x2) -> c26(MINSORT(app(add(0, if_rm(eq(s(x0), z1), s(x0), add(z1, z2))), x2), nil), APP(rm(s(x0), add(0, add(z1, z2))), x2), RM(s(x0), add(0, add(z1, z2)))) IF_MINSORT(true, add(s(x0), add(0, x1)), x2) -> c7(APP(rm(s(x0), add(0, x1)), x2)) IF_MINSORT(true, add(s(x0), add(0, x1)), x2) -> c7(RM(s(x0), add(0, x1))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) MINITER(add(s(s(z0)), x1), x2, s(0)) -> c12(IF_MIN(false, x1, x2, s(0)), LE(s(s(z0)), s(0))) MINITER(add(s(x0), x1), x2, s(x3)) -> c12(LE(s(x0), s(x3))) MINITER(add(s(0), x1), x2, s(z0)) -> c12(LE(s(0), s(z0))) MINITER(add(s(s(s(z0))), x1), x2, s(s(0))) -> c12(IF_MIN(false, x1, x2, s(s(0))), LE(s(s(s(z0))), s(s(0)))) MINITER(add(s(s(x0)), x1), x2, s(s(x3))) -> c12(LE(s(s(x0)), s(s(x3)))) MINITER(add(s(s(0)), x1), x2, s(s(z0))) -> c12(LE(s(s(0)), s(s(z0)))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(0)))) -> c12(IF_MIN(false, x1, x2, s(s(s(0)))), LE(s(s(s(s(z0)))), s(s(s(0))))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(s(z1))))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(s(s(z1))))), LE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MINITER(add(s(s(s(x0))), x1), x2, s(s(s(x3)))) -> c12(LE(s(s(s(x0))), s(s(s(x3))))) MINITER(add(s(s(s(0))), x1), x2, s(s(s(z0)))) -> c12(LE(s(s(s(0))), s(s(s(z0))))) K tuples: IF_MINSORT(true, add(z0, z1), z2) -> c26(MINSORT(app(rm(z0, z1), z2), nil), APP(rm(z0, z1), z2), RM(z0, z1)) MINSORT(add(z0, z1), z2) -> c25(IF_MINSORT(eq(z0, min(add(z0, z1))), add(z0, z1), z2), EQ(z0, min(add(z0, z1))), MIN(add(z0, z1))) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MIN(add(z0, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) RM(z0, add(z1, z2)) -> c21(IF_RM(eq(z0, z1), z0, add(z1, z2)), EQ(z0, z1)) Defined Rule Symbols: le_2, eq_2, app_2, if_rm_3, rm_2, if_min_4, minIter_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, MIN_1, MINITER_3, IF_MIN_4, IF_RM_3, IF_MINSORT_3, RM_2, MINSORT_2 Compound Symbols: c3_1, c6_1, c8_1, c10_1, c11_1, c14_1, c22_1, c23_1, c27_1, c12_1, c21_1, c12_2, c21_2, c_1, c26_1, c26_3, c1_1, c25_3, c25_2, c2_1, c4_1, c5_1, c7_1 ---------------------------------------- (167) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF_MINSORT(true, add(s(0), add(s(0), x2)), x3) -> c26(MINSORT(app(if_rm(true, s(0), add(s(0), x2)), x3), nil), APP(rm(s(0), add(s(0), x2)), x3), RM(s(0), add(s(0), x2))) by IF_MINSORT(true, add(s(0), add(s(0), z2)), x1) -> c26(MINSORT(app(rm(s(0), z2), x1), nil), APP(rm(s(0), add(s(0), z2)), x1), RM(s(0), add(s(0), z2))) IF_MINSORT(true, add(s(0), add(s(0), x0)), x1) -> c26(RM(s(0), add(s(0), x0))) ---------------------------------------- (168) 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) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) if_min(true, z0, z1, z2) -> z2 if_min(false, z0, z1, z2) -> minIter(z0, z1, z2) minIter(add(z0, z1), z2, z3) -> if_min(le(z0, z3), z1, z2, z3) minIter(nil, add(z0, z1), z2) -> minIter(add(z0, z1), add(z0, z1), s(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, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) RM(0, add(0, x2)) -> c21(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c21(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c21(IF_RM(false, s(z0), add(0, x2))) MINITER(add(s(s(z0)), x1), x2, s(0)) -> c12(IF_MIN(false, x1, x2, s(0)), LE(s(s(z0)), s(0))) MINITER(add(s(x0), x1), x2, s(x3)) -> c12(LE(s(x0), s(x3))) MINITER(add(s(0), x1), x2, s(z0)) -> c12(LE(s(0), s(z0))) RM(s(0), add(s(0), x2)) -> c21(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c21(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c21(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(x0), add(s(x1), x2)) -> c21(EQ(s(x0), s(x1))) MINSORT(add(x0, x1), x2) -> c(MIN(add(x0, x1))) MINITER(add(s(s(s(z0))), x1), x2, s(s(0))) -> c12(IF_MIN(false, x1, x2, s(s(0))), LE(s(s(s(z0))), s(s(0)))) MINITER(add(s(s(x0)), x1), x2, s(s(x3))) -> c12(LE(s(s(x0)), s(s(x3)))) MINITER(add(s(s(0)), x1), x2, s(s(z0))) -> c12(LE(s(s(0)), s(s(z0)))) RM(s(s(0)), add(s(s(0)), x2)) -> c21(IF_RM(true, s(s(0)), add(s(s(0)), x2)), EQ(s(s(0)), s(s(0)))) RM(s(s(0)), add(s(s(s(z0))), x2)) -> c21(IF_RM(false, s(s(0)), add(s(s(s(z0))), x2)), EQ(s(s(0)), s(s(s(z0))))) RM(s(s(s(z0))), add(s(s(0)), x2)) -> c21(IF_RM(false, s(s(s(z0))), add(s(s(0)), x2)), EQ(s(s(s(z0))), s(s(0)))) RM(s(s(x0)), add(s(s(x1)), x2)) -> c21(EQ(s(s(x0)), s(s(x1)))) IF_MINSORT(true, add(0, add(0, x0)), x1) -> c26(RM(0, add(0, x0))) IF_MINSORT(true, add(0, add(s(x0), x1)), x2) -> c26(RM(0, add(s(x0), x1))) IF_MINSORT(true, add(s(x0), add(0, x1)), x2) -> c26(RM(s(x0), add(0, x1))) IF_MINSORT(true, add(s(0), add(s(s(z0)), x2)), x3) -> c26(MINSORT(app(if_rm(false, s(0), add(s(s(z0)), x2)), x3), nil), APP(rm(s(0), add(s(s(z0)), x2)), x3), RM(s(0), add(s(s(z0)), x2))) IF_MINSORT(true, add(s(s(z0)), add(s(0), x2)), x3) -> c26(MINSORT(app(if_rm(false, s(s(z0)), add(s(0), x2)), x3), nil), APP(rm(s(s(z0)), add(s(0), x2)), x3), RM(s(s(z0)), add(s(0), x2))) IF_MINSORT(true, add(s(s(z0)), add(s(s(z1)), x2)), x3) -> c26(MINSORT(app(if_rm(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), x3), nil), APP(rm(s(s(z0)), add(s(s(z1)), x2)), x3), RM(s(s(z0)), add(s(s(z1)), x2))) IF_MINSORT(true, add(s(x0), add(s(x1), x2)), x3) -> c1(APP(rm(s(x0), add(s(x1), x2)), x3)) IF_MINSORT(true, add(s(x0), add(s(x1), x2)), x3) -> c1(RM(s(x0), add(s(x1), x2))) IF_MINSORT(true, add(z0, add(z1, z2)), x3) -> c26(APP(if_rm(eq(z0, z1), z0, add(z1, z2)), x3)) IF_MINSORT(true, add(z0, nil), x1) -> c26(MINSORT(x1, nil)) MINSORT(add(s(z0), x1), x2) -> c25(IF_MINSORT(eq(s(z0), if_min(false, x1, add(s(z0), x1), 0)), add(s(z0), x1), x2), EQ(s(z0), minIter(add(s(z0), x1), add(s(z0), x1), 0)), MIN(add(s(z0), x1))) MINSORT(add(0, x1), x2) -> c25(IF_MINSORT(eq(0, if_min(true, x1, add(0, x1), 0)), add(0, x1), x2), MIN(add(0, x1))) MINSORT(add(x0, x1), x2) -> c2(EQ(x0, minIter(add(x0, x1), add(x0, x1), 0))) MINSORT(add(x0, x1), x2) -> c2(MIN(add(x0, x1))) MINSORT(add(z0, z1), x2) -> c(EQ(z0, if_min(le(z0, 0), z1, add(z0, z1), 0))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(0)))) -> c12(IF_MIN(false, x1, x2, s(s(s(0)))), LE(s(s(s(s(z0)))), s(s(s(0))))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(s(z1))))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(s(s(z1))))), LE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MINITER(add(s(s(s(x0))), x1), x2, s(s(s(x3)))) -> c12(LE(s(s(s(x0))), s(s(s(x3))))) MINITER(add(s(s(s(0))), x1), x2, s(s(s(z0)))) -> c12(LE(s(s(s(0))), s(s(s(z0))))) RM(s(s(s(0))), add(s(s(s(0))), x2)) -> c21(IF_RM(true, s(s(s(0))), add(s(s(s(0))), x2)), EQ(s(s(s(0))), s(s(s(0))))) RM(s(s(s(0))), add(s(s(s(s(z0)))), x2)) -> c21(IF_RM(false, s(s(s(0))), add(s(s(s(s(z0)))), x2)), EQ(s(s(s(0))), s(s(s(s(z0)))))) RM(s(s(s(s(z0)))), add(s(s(s(0))), x2)) -> c21(IF_RM(false, s(s(s(s(z0)))), add(s(s(s(0))), x2)), EQ(s(s(s(s(z0)))), s(s(s(0))))) RM(s(s(s(s(z0)))), add(s(s(s(s(z1)))), x2)) -> c21(IF_RM(eq(z0, z1), s(s(s(s(z0)))), add(s(s(s(s(z1)))), x2)), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) RM(s(s(s(x0))), add(s(s(s(x1))), x2)) -> c21(EQ(s(s(s(x0))), s(s(s(x1))))) IF_MINSORT(true, add(0, add(0, nil)), x1) -> c26(MINSORT(app(nil, x1), nil), APP(rm(0, add(0, nil)), x1), RM(0, add(0, nil))) IF_MINSORT(true, add(0, add(0, add(z1, z2))), x1) -> c26(MINSORT(app(if_rm(eq(0, z1), 0, add(z1, z2)), x1), nil), APP(rm(0, add(0, add(z1, z2))), x1), RM(0, add(0, add(z1, z2)))) IF_MINSORT(true, add(0, add(0, x0)), x1) -> c4(APP(rm(0, add(0, x0)), x1)) IF_MINSORT(true, add(0, add(0, x0)), x1) -> c4(RM(0, add(0, x0))) IF_MINSORT(true, add(0, add(s(x0), x1)), z2) -> c26(MINSORT(add(s(x0), app(rm(0, x1), z2)), nil), APP(rm(0, add(s(x0), x1)), z2), RM(0, add(s(x0), x1))) IF_MINSORT(true, add(0, add(s(x0), nil)), x2) -> c26(MINSORT(app(add(s(x0), nil), x2), nil), APP(rm(0, add(s(x0), nil)), x2), RM(0, add(s(x0), nil))) IF_MINSORT(true, add(0, add(s(x0), add(z1, z2))), x2) -> c26(MINSORT(app(add(s(x0), if_rm(eq(0, z1), 0, add(z1, z2))), x2), nil), APP(rm(0, add(s(x0), add(z1, z2))), x2), RM(0, add(s(x0), add(z1, z2)))) IF_MINSORT(true, add(0, add(s(x0), x1)), x2) -> c5(APP(rm(0, add(s(x0), x1)), x2)) IF_MINSORT(true, add(0, add(s(x0), x1)), x2) -> c5(RM(0, add(s(x0), x1))) IF_MINSORT(true, add(s(x0), add(0, x1)), z2) -> c26(MINSORT(add(0, app(rm(s(x0), x1), z2)), nil), APP(rm(s(x0), add(0, x1)), z2), RM(s(x0), add(0, x1))) IF_MINSORT(true, add(s(x0), add(0, nil)), x2) -> c26(MINSORT(app(add(0, nil), x2), nil), APP(rm(s(x0), add(0, nil)), x2), RM(s(x0), add(0, nil))) IF_MINSORT(true, add(s(x0), add(0, add(z1, z2))), x2) -> c26(MINSORT(app(add(0, if_rm(eq(s(x0), z1), s(x0), add(z1, z2))), x2), nil), APP(rm(s(x0), add(0, add(z1, z2))), x2), RM(s(x0), add(0, add(z1, z2)))) IF_MINSORT(true, add(s(x0), add(0, x1)), x2) -> c7(APP(rm(s(x0), add(0, x1)), x2)) IF_MINSORT(true, add(s(x0), add(0, x1)), x2) -> c7(RM(s(x0), add(0, x1))) IF_MINSORT(true, add(s(0), add(s(0), z2)), x1) -> c26(MINSORT(app(rm(s(0), z2), x1), nil), APP(rm(s(0), add(s(0), z2)), x1), RM(s(0), add(s(0), z2))) IF_MINSORT(true, add(s(0), add(s(0), x0)), x1) -> c26(RM(s(0), add(s(0), x0))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) MINITER(add(s(s(z0)), x1), x2, s(0)) -> c12(IF_MIN(false, x1, x2, s(0)), LE(s(s(z0)), s(0))) MINITER(add(s(x0), x1), x2, s(x3)) -> c12(LE(s(x0), s(x3))) MINITER(add(s(0), x1), x2, s(z0)) -> c12(LE(s(0), s(z0))) MINITER(add(s(s(s(z0))), x1), x2, s(s(0))) -> c12(IF_MIN(false, x1, x2, s(s(0))), LE(s(s(s(z0))), s(s(0)))) MINITER(add(s(s(x0)), x1), x2, s(s(x3))) -> c12(LE(s(s(x0)), s(s(x3)))) MINITER(add(s(s(0)), x1), x2, s(s(z0))) -> c12(LE(s(s(0)), s(s(z0)))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(0)))) -> c12(IF_MIN(false, x1, x2, s(s(s(0)))), LE(s(s(s(s(z0)))), s(s(s(0))))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(s(z1))))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(s(s(z1))))), LE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MINITER(add(s(s(s(x0))), x1), x2, s(s(s(x3)))) -> c12(LE(s(s(s(x0))), s(s(s(x3))))) MINITER(add(s(s(s(0))), x1), x2, s(s(s(z0)))) -> c12(LE(s(s(s(0))), s(s(s(z0))))) K tuples: IF_MINSORT(true, add(z0, z1), z2) -> c26(MINSORT(app(rm(z0, z1), z2), nil), APP(rm(z0, z1), z2), RM(z0, z1)) MINSORT(add(z0, z1), z2) -> c25(IF_MINSORT(eq(z0, min(add(z0, z1))), add(z0, z1), z2), EQ(z0, min(add(z0, z1))), MIN(add(z0, z1))) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MIN(add(z0, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) RM(z0, add(z1, z2)) -> c21(IF_RM(eq(z0, z1), z0, add(z1, z2)), EQ(z0, z1)) Defined Rule Symbols: le_2, eq_2, app_2, if_rm_3, rm_2, if_min_4, minIter_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, MIN_1, MINITER_3, IF_MIN_4, IF_RM_3, IF_MINSORT_3, RM_2, MINSORT_2 Compound Symbols: c3_1, c6_1, c8_1, c10_1, c11_1, c14_1, c22_1, c23_1, c27_1, c12_1, c21_1, c12_2, c21_2, c_1, c26_1, c26_3, c1_1, c25_3, c25_2, c2_1, c4_1, c5_1, c7_1 ---------------------------------------- (169) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF_MINSORT(true, add(s(0), add(s(s(z0)), x2)), x3) -> c26(MINSORT(app(if_rm(false, s(0), add(s(s(z0)), x2)), x3), nil), APP(rm(s(0), add(s(s(z0)), x2)), x3), RM(s(0), add(s(s(z0)), x2))) by IF_MINSORT(true, add(s(0), add(s(s(x0)), z2)), x2) -> c26(MINSORT(app(add(s(s(x0)), rm(s(0), z2)), x2), nil), APP(rm(s(0), add(s(s(x0)), z2)), x2), RM(s(0), add(s(s(x0)), z2))) IF_MINSORT(true, add(s(0), add(s(s(x0)), x1)), x2) -> c26(RM(s(0), add(s(s(x0)), x1))) ---------------------------------------- (170) 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) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) if_min(true, z0, z1, z2) -> z2 if_min(false, z0, z1, z2) -> minIter(z0, z1, z2) minIter(add(z0, z1), z2, z3) -> if_min(le(z0, z3), z1, z2, z3) minIter(nil, add(z0, z1), z2) -> minIter(add(z0, z1), add(z0, z1), s(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, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) RM(0, add(0, x2)) -> c21(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c21(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c21(IF_RM(false, s(z0), add(0, x2))) MINITER(add(s(s(z0)), x1), x2, s(0)) -> c12(IF_MIN(false, x1, x2, s(0)), LE(s(s(z0)), s(0))) MINITER(add(s(x0), x1), x2, s(x3)) -> c12(LE(s(x0), s(x3))) MINITER(add(s(0), x1), x2, s(z0)) -> c12(LE(s(0), s(z0))) RM(s(0), add(s(0), x2)) -> c21(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c21(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c21(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(x0), add(s(x1), x2)) -> c21(EQ(s(x0), s(x1))) MINSORT(add(x0, x1), x2) -> c(MIN(add(x0, x1))) MINITER(add(s(s(s(z0))), x1), x2, s(s(0))) -> c12(IF_MIN(false, x1, x2, s(s(0))), LE(s(s(s(z0))), s(s(0)))) MINITER(add(s(s(x0)), x1), x2, s(s(x3))) -> c12(LE(s(s(x0)), s(s(x3)))) MINITER(add(s(s(0)), x1), x2, s(s(z0))) -> c12(LE(s(s(0)), s(s(z0)))) RM(s(s(0)), add(s(s(0)), x2)) -> c21(IF_RM(true, s(s(0)), add(s(s(0)), x2)), EQ(s(s(0)), s(s(0)))) RM(s(s(0)), add(s(s(s(z0))), x2)) -> c21(IF_RM(false, s(s(0)), add(s(s(s(z0))), x2)), EQ(s(s(0)), s(s(s(z0))))) RM(s(s(s(z0))), add(s(s(0)), x2)) -> c21(IF_RM(false, s(s(s(z0))), add(s(s(0)), x2)), EQ(s(s(s(z0))), s(s(0)))) RM(s(s(x0)), add(s(s(x1)), x2)) -> c21(EQ(s(s(x0)), s(s(x1)))) IF_MINSORT(true, add(0, add(0, x0)), x1) -> c26(RM(0, add(0, x0))) IF_MINSORT(true, add(0, add(s(x0), x1)), x2) -> c26(RM(0, add(s(x0), x1))) IF_MINSORT(true, add(s(x0), add(0, x1)), x2) -> c26(RM(s(x0), add(0, x1))) IF_MINSORT(true, add(s(s(z0)), add(s(0), x2)), x3) -> c26(MINSORT(app(if_rm(false, s(s(z0)), add(s(0), x2)), x3), nil), APP(rm(s(s(z0)), add(s(0), x2)), x3), RM(s(s(z0)), add(s(0), x2))) IF_MINSORT(true, add(s(s(z0)), add(s(s(z1)), x2)), x3) -> c26(MINSORT(app(if_rm(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), x3), nil), APP(rm(s(s(z0)), add(s(s(z1)), x2)), x3), RM(s(s(z0)), add(s(s(z1)), x2))) IF_MINSORT(true, add(s(x0), add(s(x1), x2)), x3) -> c1(APP(rm(s(x0), add(s(x1), x2)), x3)) IF_MINSORT(true, add(s(x0), add(s(x1), x2)), x3) -> c1(RM(s(x0), add(s(x1), x2))) IF_MINSORT(true, add(z0, add(z1, z2)), x3) -> c26(APP(if_rm(eq(z0, z1), z0, add(z1, z2)), x3)) IF_MINSORT(true, add(z0, nil), x1) -> c26(MINSORT(x1, nil)) MINSORT(add(s(z0), x1), x2) -> c25(IF_MINSORT(eq(s(z0), if_min(false, x1, add(s(z0), x1), 0)), add(s(z0), x1), x2), EQ(s(z0), minIter(add(s(z0), x1), add(s(z0), x1), 0)), MIN(add(s(z0), x1))) MINSORT(add(0, x1), x2) -> c25(IF_MINSORT(eq(0, if_min(true, x1, add(0, x1), 0)), add(0, x1), x2), MIN(add(0, x1))) MINSORT(add(x0, x1), x2) -> c2(EQ(x0, minIter(add(x0, x1), add(x0, x1), 0))) MINSORT(add(x0, x1), x2) -> c2(MIN(add(x0, x1))) MINSORT(add(z0, z1), x2) -> c(EQ(z0, if_min(le(z0, 0), z1, add(z0, z1), 0))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(0)))) -> c12(IF_MIN(false, x1, x2, s(s(s(0)))), LE(s(s(s(s(z0)))), s(s(s(0))))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(s(z1))))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(s(s(z1))))), LE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MINITER(add(s(s(s(x0))), x1), x2, s(s(s(x3)))) -> c12(LE(s(s(s(x0))), s(s(s(x3))))) MINITER(add(s(s(s(0))), x1), x2, s(s(s(z0)))) -> c12(LE(s(s(s(0))), s(s(s(z0))))) RM(s(s(s(0))), add(s(s(s(0))), x2)) -> c21(IF_RM(true, s(s(s(0))), add(s(s(s(0))), x2)), EQ(s(s(s(0))), s(s(s(0))))) RM(s(s(s(0))), add(s(s(s(s(z0)))), x2)) -> c21(IF_RM(false, s(s(s(0))), add(s(s(s(s(z0)))), x2)), EQ(s(s(s(0))), s(s(s(s(z0)))))) RM(s(s(s(s(z0)))), add(s(s(s(0))), x2)) -> c21(IF_RM(false, s(s(s(s(z0)))), add(s(s(s(0))), x2)), EQ(s(s(s(s(z0)))), s(s(s(0))))) RM(s(s(s(s(z0)))), add(s(s(s(s(z1)))), x2)) -> c21(IF_RM(eq(z0, z1), s(s(s(s(z0)))), add(s(s(s(s(z1)))), x2)), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) RM(s(s(s(x0))), add(s(s(s(x1))), x2)) -> c21(EQ(s(s(s(x0))), s(s(s(x1))))) IF_MINSORT(true, add(0, add(0, nil)), x1) -> c26(MINSORT(app(nil, x1), nil), APP(rm(0, add(0, nil)), x1), RM(0, add(0, nil))) IF_MINSORT(true, add(0, add(0, add(z1, z2))), x1) -> c26(MINSORT(app(if_rm(eq(0, z1), 0, add(z1, z2)), x1), nil), APP(rm(0, add(0, add(z1, z2))), x1), RM(0, add(0, add(z1, z2)))) IF_MINSORT(true, add(0, add(0, x0)), x1) -> c4(APP(rm(0, add(0, x0)), x1)) IF_MINSORT(true, add(0, add(0, x0)), x1) -> c4(RM(0, add(0, x0))) IF_MINSORT(true, add(0, add(s(x0), x1)), z2) -> c26(MINSORT(add(s(x0), app(rm(0, x1), z2)), nil), APP(rm(0, add(s(x0), x1)), z2), RM(0, add(s(x0), x1))) IF_MINSORT(true, add(0, add(s(x0), nil)), x2) -> c26(MINSORT(app(add(s(x0), nil), x2), nil), APP(rm(0, add(s(x0), nil)), x2), RM(0, add(s(x0), nil))) IF_MINSORT(true, add(0, add(s(x0), add(z1, z2))), x2) -> c26(MINSORT(app(add(s(x0), if_rm(eq(0, z1), 0, add(z1, z2))), x2), nil), APP(rm(0, add(s(x0), add(z1, z2))), x2), RM(0, add(s(x0), add(z1, z2)))) IF_MINSORT(true, add(0, add(s(x0), x1)), x2) -> c5(APP(rm(0, add(s(x0), x1)), x2)) IF_MINSORT(true, add(0, add(s(x0), x1)), x2) -> c5(RM(0, add(s(x0), x1))) IF_MINSORT(true, add(s(x0), add(0, x1)), z2) -> c26(MINSORT(add(0, app(rm(s(x0), x1), z2)), nil), APP(rm(s(x0), add(0, x1)), z2), RM(s(x0), add(0, x1))) IF_MINSORT(true, add(s(x0), add(0, nil)), x2) -> c26(MINSORT(app(add(0, nil), x2), nil), APP(rm(s(x0), add(0, nil)), x2), RM(s(x0), add(0, nil))) IF_MINSORT(true, add(s(x0), add(0, add(z1, z2))), x2) -> c26(MINSORT(app(add(0, if_rm(eq(s(x0), z1), s(x0), add(z1, z2))), x2), nil), APP(rm(s(x0), add(0, add(z1, z2))), x2), RM(s(x0), add(0, add(z1, z2)))) IF_MINSORT(true, add(s(x0), add(0, x1)), x2) -> c7(APP(rm(s(x0), add(0, x1)), x2)) IF_MINSORT(true, add(s(x0), add(0, x1)), x2) -> c7(RM(s(x0), add(0, x1))) IF_MINSORT(true, add(s(0), add(s(0), z2)), x1) -> c26(MINSORT(app(rm(s(0), z2), x1), nil), APP(rm(s(0), add(s(0), z2)), x1), RM(s(0), add(s(0), z2))) IF_MINSORT(true, add(s(0), add(s(0), x0)), x1) -> c26(RM(s(0), add(s(0), x0))) IF_MINSORT(true, add(s(0), add(s(s(x0)), z2)), x2) -> c26(MINSORT(app(add(s(s(x0)), rm(s(0), z2)), x2), nil), APP(rm(s(0), add(s(s(x0)), z2)), x2), RM(s(0), add(s(s(x0)), z2))) IF_MINSORT(true, add(s(0), add(s(s(x0)), x1)), x2) -> c26(RM(s(0), add(s(s(x0)), x1))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) MINITER(add(s(s(z0)), x1), x2, s(0)) -> c12(IF_MIN(false, x1, x2, s(0)), LE(s(s(z0)), s(0))) MINITER(add(s(x0), x1), x2, s(x3)) -> c12(LE(s(x0), s(x3))) MINITER(add(s(0), x1), x2, s(z0)) -> c12(LE(s(0), s(z0))) MINITER(add(s(s(s(z0))), x1), x2, s(s(0))) -> c12(IF_MIN(false, x1, x2, s(s(0))), LE(s(s(s(z0))), s(s(0)))) MINITER(add(s(s(x0)), x1), x2, s(s(x3))) -> c12(LE(s(s(x0)), s(s(x3)))) MINITER(add(s(s(0)), x1), x2, s(s(z0))) -> c12(LE(s(s(0)), s(s(z0)))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(0)))) -> c12(IF_MIN(false, x1, x2, s(s(s(0)))), LE(s(s(s(s(z0)))), s(s(s(0))))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(s(z1))))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(s(s(z1))))), LE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MINITER(add(s(s(s(x0))), x1), x2, s(s(s(x3)))) -> c12(LE(s(s(s(x0))), s(s(s(x3))))) MINITER(add(s(s(s(0))), x1), x2, s(s(s(z0)))) -> c12(LE(s(s(s(0))), s(s(s(z0))))) K tuples: IF_MINSORT(true, add(z0, z1), z2) -> c26(MINSORT(app(rm(z0, z1), z2), nil), APP(rm(z0, z1), z2), RM(z0, z1)) MINSORT(add(z0, z1), z2) -> c25(IF_MINSORT(eq(z0, min(add(z0, z1))), add(z0, z1), z2), EQ(z0, min(add(z0, z1))), MIN(add(z0, z1))) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MIN(add(z0, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) RM(z0, add(z1, z2)) -> c21(IF_RM(eq(z0, z1), z0, add(z1, z2)), EQ(z0, z1)) Defined Rule Symbols: le_2, eq_2, app_2, if_rm_3, rm_2, if_min_4, minIter_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, MIN_1, MINITER_3, IF_MIN_4, IF_RM_3, IF_MINSORT_3, RM_2, MINSORT_2 Compound Symbols: c3_1, c6_1, c8_1, c10_1, c11_1, c14_1, c22_1, c23_1, c27_1, c12_1, c21_1, c12_2, c21_2, c_1, c26_1, c26_3, c1_1, c25_3, c25_2, c2_1, c4_1, c5_1, c7_1 ---------------------------------------- (171) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF_MINSORT(true, add(s(s(z0)), add(s(0), x2)), x3) -> c26(MINSORT(app(if_rm(false, s(s(z0)), add(s(0), x2)), x3), nil), APP(rm(s(s(z0)), add(s(0), x2)), x3), RM(s(s(z0)), add(s(0), x2))) by IF_MINSORT(true, add(s(s(x0)), add(s(0), z2)), x2) -> c26(MINSORT(app(add(s(0), rm(s(s(x0)), z2)), x2), nil), APP(rm(s(s(x0)), add(s(0), z2)), x2), RM(s(s(x0)), add(s(0), z2))) IF_MINSORT(true, add(s(s(x0)), add(s(0), x1)), x2) -> c26(RM(s(s(x0)), add(s(0), x1))) ---------------------------------------- (172) 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) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) if_min(true, z0, z1, z2) -> z2 if_min(false, z0, z1, z2) -> minIter(z0, z1, z2) minIter(add(z0, z1), z2, z3) -> if_min(le(z0, z3), z1, z2, z3) minIter(nil, add(z0, z1), z2) -> minIter(add(z0, z1), add(z0, z1), s(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, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) RM(0, add(0, x2)) -> c21(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c21(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c21(IF_RM(false, s(z0), add(0, x2))) MINITER(add(s(s(z0)), x1), x2, s(0)) -> c12(IF_MIN(false, x1, x2, s(0)), LE(s(s(z0)), s(0))) MINITER(add(s(x0), x1), x2, s(x3)) -> c12(LE(s(x0), s(x3))) MINITER(add(s(0), x1), x2, s(z0)) -> c12(LE(s(0), s(z0))) RM(s(0), add(s(0), x2)) -> c21(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c21(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c21(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(x0), add(s(x1), x2)) -> c21(EQ(s(x0), s(x1))) MINSORT(add(x0, x1), x2) -> c(MIN(add(x0, x1))) MINITER(add(s(s(s(z0))), x1), x2, s(s(0))) -> c12(IF_MIN(false, x1, x2, s(s(0))), LE(s(s(s(z0))), s(s(0)))) MINITER(add(s(s(x0)), x1), x2, s(s(x3))) -> c12(LE(s(s(x0)), s(s(x3)))) MINITER(add(s(s(0)), x1), x2, s(s(z0))) -> c12(LE(s(s(0)), s(s(z0)))) RM(s(s(0)), add(s(s(0)), x2)) -> c21(IF_RM(true, s(s(0)), add(s(s(0)), x2)), EQ(s(s(0)), s(s(0)))) RM(s(s(0)), add(s(s(s(z0))), x2)) -> c21(IF_RM(false, s(s(0)), add(s(s(s(z0))), x2)), EQ(s(s(0)), s(s(s(z0))))) RM(s(s(s(z0))), add(s(s(0)), x2)) -> c21(IF_RM(false, s(s(s(z0))), add(s(s(0)), x2)), EQ(s(s(s(z0))), s(s(0)))) RM(s(s(x0)), add(s(s(x1)), x2)) -> c21(EQ(s(s(x0)), s(s(x1)))) IF_MINSORT(true, add(0, add(0, x0)), x1) -> c26(RM(0, add(0, x0))) IF_MINSORT(true, add(0, add(s(x0), x1)), x2) -> c26(RM(0, add(s(x0), x1))) IF_MINSORT(true, add(s(x0), add(0, x1)), x2) -> c26(RM(s(x0), add(0, x1))) IF_MINSORT(true, add(s(s(z0)), add(s(s(z1)), x2)), x3) -> c26(MINSORT(app(if_rm(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), x3), nil), APP(rm(s(s(z0)), add(s(s(z1)), x2)), x3), RM(s(s(z0)), add(s(s(z1)), x2))) IF_MINSORT(true, add(s(x0), add(s(x1), x2)), x3) -> c1(APP(rm(s(x0), add(s(x1), x2)), x3)) IF_MINSORT(true, add(s(x0), add(s(x1), x2)), x3) -> c1(RM(s(x0), add(s(x1), x2))) IF_MINSORT(true, add(z0, add(z1, z2)), x3) -> c26(APP(if_rm(eq(z0, z1), z0, add(z1, z2)), x3)) IF_MINSORT(true, add(z0, nil), x1) -> c26(MINSORT(x1, nil)) MINSORT(add(s(z0), x1), x2) -> c25(IF_MINSORT(eq(s(z0), if_min(false, x1, add(s(z0), x1), 0)), add(s(z0), x1), x2), EQ(s(z0), minIter(add(s(z0), x1), add(s(z0), x1), 0)), MIN(add(s(z0), x1))) MINSORT(add(0, x1), x2) -> c25(IF_MINSORT(eq(0, if_min(true, x1, add(0, x1), 0)), add(0, x1), x2), MIN(add(0, x1))) MINSORT(add(x0, x1), x2) -> c2(EQ(x0, minIter(add(x0, x1), add(x0, x1), 0))) MINSORT(add(x0, x1), x2) -> c2(MIN(add(x0, x1))) MINSORT(add(z0, z1), x2) -> c(EQ(z0, if_min(le(z0, 0), z1, add(z0, z1), 0))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(0)))) -> c12(IF_MIN(false, x1, x2, s(s(s(0)))), LE(s(s(s(s(z0)))), s(s(s(0))))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(s(z1))))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(s(s(z1))))), LE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MINITER(add(s(s(s(x0))), x1), x2, s(s(s(x3)))) -> c12(LE(s(s(s(x0))), s(s(s(x3))))) MINITER(add(s(s(s(0))), x1), x2, s(s(s(z0)))) -> c12(LE(s(s(s(0))), s(s(s(z0))))) RM(s(s(s(0))), add(s(s(s(0))), x2)) -> c21(IF_RM(true, s(s(s(0))), add(s(s(s(0))), x2)), EQ(s(s(s(0))), s(s(s(0))))) RM(s(s(s(0))), add(s(s(s(s(z0)))), x2)) -> c21(IF_RM(false, s(s(s(0))), add(s(s(s(s(z0)))), x2)), EQ(s(s(s(0))), s(s(s(s(z0)))))) RM(s(s(s(s(z0)))), add(s(s(s(0))), x2)) -> c21(IF_RM(false, s(s(s(s(z0)))), add(s(s(s(0))), x2)), EQ(s(s(s(s(z0)))), s(s(s(0))))) RM(s(s(s(s(z0)))), add(s(s(s(s(z1)))), x2)) -> c21(IF_RM(eq(z0, z1), s(s(s(s(z0)))), add(s(s(s(s(z1)))), x2)), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) RM(s(s(s(x0))), add(s(s(s(x1))), x2)) -> c21(EQ(s(s(s(x0))), s(s(s(x1))))) IF_MINSORT(true, add(0, add(0, nil)), x1) -> c26(MINSORT(app(nil, x1), nil), APP(rm(0, add(0, nil)), x1), RM(0, add(0, nil))) IF_MINSORT(true, add(0, add(0, add(z1, z2))), x1) -> c26(MINSORT(app(if_rm(eq(0, z1), 0, add(z1, z2)), x1), nil), APP(rm(0, add(0, add(z1, z2))), x1), RM(0, add(0, add(z1, z2)))) IF_MINSORT(true, add(0, add(0, x0)), x1) -> c4(APP(rm(0, add(0, x0)), x1)) IF_MINSORT(true, add(0, add(0, x0)), x1) -> c4(RM(0, add(0, x0))) IF_MINSORT(true, add(0, add(s(x0), x1)), z2) -> c26(MINSORT(add(s(x0), app(rm(0, x1), z2)), nil), APP(rm(0, add(s(x0), x1)), z2), RM(0, add(s(x0), x1))) IF_MINSORT(true, add(0, add(s(x0), nil)), x2) -> c26(MINSORT(app(add(s(x0), nil), x2), nil), APP(rm(0, add(s(x0), nil)), x2), RM(0, add(s(x0), nil))) IF_MINSORT(true, add(0, add(s(x0), add(z1, z2))), x2) -> c26(MINSORT(app(add(s(x0), if_rm(eq(0, z1), 0, add(z1, z2))), x2), nil), APP(rm(0, add(s(x0), add(z1, z2))), x2), RM(0, add(s(x0), add(z1, z2)))) IF_MINSORT(true, add(0, add(s(x0), x1)), x2) -> c5(APP(rm(0, add(s(x0), x1)), x2)) IF_MINSORT(true, add(0, add(s(x0), x1)), x2) -> c5(RM(0, add(s(x0), x1))) IF_MINSORT(true, add(s(x0), add(0, x1)), z2) -> c26(MINSORT(add(0, app(rm(s(x0), x1), z2)), nil), APP(rm(s(x0), add(0, x1)), z2), RM(s(x0), add(0, x1))) IF_MINSORT(true, add(s(x0), add(0, nil)), x2) -> c26(MINSORT(app(add(0, nil), x2), nil), APP(rm(s(x0), add(0, nil)), x2), RM(s(x0), add(0, nil))) IF_MINSORT(true, add(s(x0), add(0, add(z1, z2))), x2) -> c26(MINSORT(app(add(0, if_rm(eq(s(x0), z1), s(x0), add(z1, z2))), x2), nil), APP(rm(s(x0), add(0, add(z1, z2))), x2), RM(s(x0), add(0, add(z1, z2)))) IF_MINSORT(true, add(s(x0), add(0, x1)), x2) -> c7(APP(rm(s(x0), add(0, x1)), x2)) IF_MINSORT(true, add(s(x0), add(0, x1)), x2) -> c7(RM(s(x0), add(0, x1))) IF_MINSORT(true, add(s(0), add(s(0), z2)), x1) -> c26(MINSORT(app(rm(s(0), z2), x1), nil), APP(rm(s(0), add(s(0), z2)), x1), RM(s(0), add(s(0), z2))) IF_MINSORT(true, add(s(0), add(s(0), x0)), x1) -> c26(RM(s(0), add(s(0), x0))) IF_MINSORT(true, add(s(0), add(s(s(x0)), z2)), x2) -> c26(MINSORT(app(add(s(s(x0)), rm(s(0), z2)), x2), nil), APP(rm(s(0), add(s(s(x0)), z2)), x2), RM(s(0), add(s(s(x0)), z2))) IF_MINSORT(true, add(s(0), add(s(s(x0)), x1)), x2) -> c26(RM(s(0), add(s(s(x0)), x1))) IF_MINSORT(true, add(s(s(x0)), add(s(0), z2)), x2) -> c26(MINSORT(app(add(s(0), rm(s(s(x0)), z2)), x2), nil), APP(rm(s(s(x0)), add(s(0), z2)), x2), RM(s(s(x0)), add(s(0), z2))) IF_MINSORT(true, add(s(s(x0)), add(s(0), x1)), x2) -> c26(RM(s(s(x0)), add(s(0), x1))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) MINITER(add(s(s(z0)), x1), x2, s(0)) -> c12(IF_MIN(false, x1, x2, s(0)), LE(s(s(z0)), s(0))) MINITER(add(s(x0), x1), x2, s(x3)) -> c12(LE(s(x0), s(x3))) MINITER(add(s(0), x1), x2, s(z0)) -> c12(LE(s(0), s(z0))) MINITER(add(s(s(s(z0))), x1), x2, s(s(0))) -> c12(IF_MIN(false, x1, x2, s(s(0))), LE(s(s(s(z0))), s(s(0)))) MINITER(add(s(s(x0)), x1), x2, s(s(x3))) -> c12(LE(s(s(x0)), s(s(x3)))) MINITER(add(s(s(0)), x1), x2, s(s(z0))) -> c12(LE(s(s(0)), s(s(z0)))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(0)))) -> c12(IF_MIN(false, x1, x2, s(s(s(0)))), LE(s(s(s(s(z0)))), s(s(s(0))))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(s(z1))))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(s(s(z1))))), LE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MINITER(add(s(s(s(x0))), x1), x2, s(s(s(x3)))) -> c12(LE(s(s(s(x0))), s(s(s(x3))))) MINITER(add(s(s(s(0))), x1), x2, s(s(s(z0)))) -> c12(LE(s(s(s(0))), s(s(s(z0))))) K tuples: IF_MINSORT(true, add(z0, z1), z2) -> c26(MINSORT(app(rm(z0, z1), z2), nil), APP(rm(z0, z1), z2), RM(z0, z1)) MINSORT(add(z0, z1), z2) -> c25(IF_MINSORT(eq(z0, min(add(z0, z1))), add(z0, z1), z2), EQ(z0, min(add(z0, z1))), MIN(add(z0, z1))) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MIN(add(z0, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) RM(z0, add(z1, z2)) -> c21(IF_RM(eq(z0, z1), z0, add(z1, z2)), EQ(z0, z1)) Defined Rule Symbols: le_2, eq_2, app_2, if_rm_3, rm_2, if_min_4, minIter_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, MIN_1, MINITER_3, IF_MIN_4, IF_RM_3, IF_MINSORT_3, RM_2, MINSORT_2 Compound Symbols: c3_1, c6_1, c8_1, c10_1, c11_1, c14_1, c22_1, c23_1, c27_1, c12_1, c21_1, c12_2, c21_2, c_1, c26_1, c26_3, c1_1, c25_3, c25_2, c2_1, c4_1, c5_1, c7_1 ---------------------------------------- (173) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF_MINSORT(true, add(s(s(z0)), add(s(s(z1)), x2)), x3) -> c26(MINSORT(app(if_rm(eq(z0, z1), s(s(z0)), add(s(s(z1)), x2)), x3), nil), APP(rm(s(s(z0)), add(s(s(z1)), x2)), x3), RM(s(s(z0)), add(s(s(z1)), x2))) by IF_MINSORT(true, add(s(s(0)), add(s(s(0)), x2)), x3) -> c26(MINSORT(app(if_rm(true, s(s(0)), add(s(s(0)), x2)), x3), nil), APP(rm(s(s(0)), add(s(s(0)), x2)), x3), RM(s(s(0)), add(s(s(0)), x2))) IF_MINSORT(true, add(s(s(0)), add(s(s(s(z0))), x2)), x3) -> c26(MINSORT(app(if_rm(false, s(s(0)), add(s(s(s(z0))), x2)), x3), nil), APP(rm(s(s(0)), add(s(s(s(z0))), x2)), x3), RM(s(s(0)), add(s(s(s(z0))), x2))) IF_MINSORT(true, add(s(s(s(z0))), add(s(s(0)), x2)), x3) -> c26(MINSORT(app(if_rm(false, s(s(s(z0))), add(s(s(0)), x2)), x3), nil), APP(rm(s(s(s(z0))), add(s(s(0)), x2)), x3), RM(s(s(s(z0))), add(s(s(0)), x2))) IF_MINSORT(true, add(s(s(s(z0))), add(s(s(s(z1))), x2)), x3) -> c26(MINSORT(app(if_rm(eq(z0, z1), s(s(s(z0))), add(s(s(s(z1))), x2)), x3), nil), APP(rm(s(s(s(z0))), add(s(s(s(z1))), x2)), x3), RM(s(s(s(z0))), add(s(s(s(z1))), x2))) IF_MINSORT(true, add(s(s(x0)), add(s(s(x1)), x2)), x3) -> c26(APP(rm(s(s(x0)), add(s(s(x1)), x2)), x3), RM(s(s(x0)), add(s(s(x1)), x2))) ---------------------------------------- (174) 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) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) if_min(true, z0, z1, z2) -> z2 if_min(false, z0, z1, z2) -> minIter(z0, z1, z2) minIter(add(z0, z1), z2, z3) -> if_min(le(z0, z3), z1, z2, z3) minIter(nil, add(z0, z1), z2) -> minIter(add(z0, z1), add(z0, z1), s(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, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) RM(0, add(0, x2)) -> c21(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c21(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c21(IF_RM(false, s(z0), add(0, x2))) MINITER(add(s(s(z0)), x1), x2, s(0)) -> c12(IF_MIN(false, x1, x2, s(0)), LE(s(s(z0)), s(0))) MINITER(add(s(x0), x1), x2, s(x3)) -> c12(LE(s(x0), s(x3))) MINITER(add(s(0), x1), x2, s(z0)) -> c12(LE(s(0), s(z0))) RM(s(0), add(s(0), x2)) -> c21(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c21(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c21(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(x0), add(s(x1), x2)) -> c21(EQ(s(x0), s(x1))) MINSORT(add(x0, x1), x2) -> c(MIN(add(x0, x1))) MINITER(add(s(s(s(z0))), x1), x2, s(s(0))) -> c12(IF_MIN(false, x1, x2, s(s(0))), LE(s(s(s(z0))), s(s(0)))) MINITER(add(s(s(x0)), x1), x2, s(s(x3))) -> c12(LE(s(s(x0)), s(s(x3)))) MINITER(add(s(s(0)), x1), x2, s(s(z0))) -> c12(LE(s(s(0)), s(s(z0)))) RM(s(s(0)), add(s(s(0)), x2)) -> c21(IF_RM(true, s(s(0)), add(s(s(0)), x2)), EQ(s(s(0)), s(s(0)))) RM(s(s(0)), add(s(s(s(z0))), x2)) -> c21(IF_RM(false, s(s(0)), add(s(s(s(z0))), x2)), EQ(s(s(0)), s(s(s(z0))))) RM(s(s(s(z0))), add(s(s(0)), x2)) -> c21(IF_RM(false, s(s(s(z0))), add(s(s(0)), x2)), EQ(s(s(s(z0))), s(s(0)))) RM(s(s(x0)), add(s(s(x1)), x2)) -> c21(EQ(s(s(x0)), s(s(x1)))) IF_MINSORT(true, add(0, add(0, x0)), x1) -> c26(RM(0, add(0, x0))) IF_MINSORT(true, add(0, add(s(x0), x1)), x2) -> c26(RM(0, add(s(x0), x1))) IF_MINSORT(true, add(s(x0), add(0, x1)), x2) -> c26(RM(s(x0), add(0, x1))) IF_MINSORT(true, add(s(x0), add(s(x1), x2)), x3) -> c1(APP(rm(s(x0), add(s(x1), x2)), x3)) IF_MINSORT(true, add(s(x0), add(s(x1), x2)), x3) -> c1(RM(s(x0), add(s(x1), x2))) IF_MINSORT(true, add(z0, add(z1, z2)), x3) -> c26(APP(if_rm(eq(z0, z1), z0, add(z1, z2)), x3)) IF_MINSORT(true, add(z0, nil), x1) -> c26(MINSORT(x1, nil)) MINSORT(add(s(z0), x1), x2) -> c25(IF_MINSORT(eq(s(z0), if_min(false, x1, add(s(z0), x1), 0)), add(s(z0), x1), x2), EQ(s(z0), minIter(add(s(z0), x1), add(s(z0), x1), 0)), MIN(add(s(z0), x1))) MINSORT(add(0, x1), x2) -> c25(IF_MINSORT(eq(0, if_min(true, x1, add(0, x1), 0)), add(0, x1), x2), MIN(add(0, x1))) MINSORT(add(x0, x1), x2) -> c2(EQ(x0, minIter(add(x0, x1), add(x0, x1), 0))) MINSORT(add(x0, x1), x2) -> c2(MIN(add(x0, x1))) MINSORT(add(z0, z1), x2) -> c(EQ(z0, if_min(le(z0, 0), z1, add(z0, z1), 0))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(0)))) -> c12(IF_MIN(false, x1, x2, s(s(s(0)))), LE(s(s(s(s(z0)))), s(s(s(0))))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(s(z1))))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(s(s(z1))))), LE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MINITER(add(s(s(s(x0))), x1), x2, s(s(s(x3)))) -> c12(LE(s(s(s(x0))), s(s(s(x3))))) MINITER(add(s(s(s(0))), x1), x2, s(s(s(z0)))) -> c12(LE(s(s(s(0))), s(s(s(z0))))) RM(s(s(s(0))), add(s(s(s(0))), x2)) -> c21(IF_RM(true, s(s(s(0))), add(s(s(s(0))), x2)), EQ(s(s(s(0))), s(s(s(0))))) RM(s(s(s(0))), add(s(s(s(s(z0)))), x2)) -> c21(IF_RM(false, s(s(s(0))), add(s(s(s(s(z0)))), x2)), EQ(s(s(s(0))), s(s(s(s(z0)))))) RM(s(s(s(s(z0)))), add(s(s(s(0))), x2)) -> c21(IF_RM(false, s(s(s(s(z0)))), add(s(s(s(0))), x2)), EQ(s(s(s(s(z0)))), s(s(s(0))))) RM(s(s(s(s(z0)))), add(s(s(s(s(z1)))), x2)) -> c21(IF_RM(eq(z0, z1), s(s(s(s(z0)))), add(s(s(s(s(z1)))), x2)), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) RM(s(s(s(x0))), add(s(s(s(x1))), x2)) -> c21(EQ(s(s(s(x0))), s(s(s(x1))))) IF_MINSORT(true, add(0, add(0, nil)), x1) -> c26(MINSORT(app(nil, x1), nil), APP(rm(0, add(0, nil)), x1), RM(0, add(0, nil))) IF_MINSORT(true, add(0, add(0, add(z1, z2))), x1) -> c26(MINSORT(app(if_rm(eq(0, z1), 0, add(z1, z2)), x1), nil), APP(rm(0, add(0, add(z1, z2))), x1), RM(0, add(0, add(z1, z2)))) IF_MINSORT(true, add(0, add(0, x0)), x1) -> c4(APP(rm(0, add(0, x0)), x1)) IF_MINSORT(true, add(0, add(0, x0)), x1) -> c4(RM(0, add(0, x0))) IF_MINSORT(true, add(0, add(s(x0), x1)), z2) -> c26(MINSORT(add(s(x0), app(rm(0, x1), z2)), nil), APP(rm(0, add(s(x0), x1)), z2), RM(0, add(s(x0), x1))) IF_MINSORT(true, add(0, add(s(x0), nil)), x2) -> c26(MINSORT(app(add(s(x0), nil), x2), nil), APP(rm(0, add(s(x0), nil)), x2), RM(0, add(s(x0), nil))) IF_MINSORT(true, add(0, add(s(x0), add(z1, z2))), x2) -> c26(MINSORT(app(add(s(x0), if_rm(eq(0, z1), 0, add(z1, z2))), x2), nil), APP(rm(0, add(s(x0), add(z1, z2))), x2), RM(0, add(s(x0), add(z1, z2)))) IF_MINSORT(true, add(0, add(s(x0), x1)), x2) -> c5(APP(rm(0, add(s(x0), x1)), x2)) IF_MINSORT(true, add(0, add(s(x0), x1)), x2) -> c5(RM(0, add(s(x0), x1))) IF_MINSORT(true, add(s(x0), add(0, x1)), z2) -> c26(MINSORT(add(0, app(rm(s(x0), x1), z2)), nil), APP(rm(s(x0), add(0, x1)), z2), RM(s(x0), add(0, x1))) IF_MINSORT(true, add(s(x0), add(0, nil)), x2) -> c26(MINSORT(app(add(0, nil), x2), nil), APP(rm(s(x0), add(0, nil)), x2), RM(s(x0), add(0, nil))) IF_MINSORT(true, add(s(x0), add(0, add(z1, z2))), x2) -> c26(MINSORT(app(add(0, if_rm(eq(s(x0), z1), s(x0), add(z1, z2))), x2), nil), APP(rm(s(x0), add(0, add(z1, z2))), x2), RM(s(x0), add(0, add(z1, z2)))) IF_MINSORT(true, add(s(x0), add(0, x1)), x2) -> c7(APP(rm(s(x0), add(0, x1)), x2)) IF_MINSORT(true, add(s(x0), add(0, x1)), x2) -> c7(RM(s(x0), add(0, x1))) IF_MINSORT(true, add(s(0), add(s(0), z2)), x1) -> c26(MINSORT(app(rm(s(0), z2), x1), nil), APP(rm(s(0), add(s(0), z2)), x1), RM(s(0), add(s(0), z2))) IF_MINSORT(true, add(s(0), add(s(0), x0)), x1) -> c26(RM(s(0), add(s(0), x0))) IF_MINSORT(true, add(s(0), add(s(s(x0)), z2)), x2) -> c26(MINSORT(app(add(s(s(x0)), rm(s(0), z2)), x2), nil), APP(rm(s(0), add(s(s(x0)), z2)), x2), RM(s(0), add(s(s(x0)), z2))) IF_MINSORT(true, add(s(0), add(s(s(x0)), x1)), x2) -> c26(RM(s(0), add(s(s(x0)), x1))) IF_MINSORT(true, add(s(s(x0)), add(s(0), z2)), x2) -> c26(MINSORT(app(add(s(0), rm(s(s(x0)), z2)), x2), nil), APP(rm(s(s(x0)), add(s(0), z2)), x2), RM(s(s(x0)), add(s(0), z2))) IF_MINSORT(true, add(s(s(x0)), add(s(0), x1)), x2) -> c26(RM(s(s(x0)), add(s(0), x1))) IF_MINSORT(true, add(s(s(0)), add(s(s(0)), x2)), x3) -> c26(MINSORT(app(if_rm(true, s(s(0)), add(s(s(0)), x2)), x3), nil), APP(rm(s(s(0)), add(s(s(0)), x2)), x3), RM(s(s(0)), add(s(s(0)), x2))) IF_MINSORT(true, add(s(s(0)), add(s(s(s(z0))), x2)), x3) -> c26(MINSORT(app(if_rm(false, s(s(0)), add(s(s(s(z0))), x2)), x3), nil), APP(rm(s(s(0)), add(s(s(s(z0))), x2)), x3), RM(s(s(0)), add(s(s(s(z0))), x2))) IF_MINSORT(true, add(s(s(s(z0))), add(s(s(0)), x2)), x3) -> c26(MINSORT(app(if_rm(false, s(s(s(z0))), add(s(s(0)), x2)), x3), nil), APP(rm(s(s(s(z0))), add(s(s(0)), x2)), x3), RM(s(s(s(z0))), add(s(s(0)), x2))) IF_MINSORT(true, add(s(s(s(z0))), add(s(s(s(z1))), x2)), x3) -> c26(MINSORT(app(if_rm(eq(z0, z1), s(s(s(z0))), add(s(s(s(z1))), x2)), x3), nil), APP(rm(s(s(s(z0))), add(s(s(s(z1))), x2)), x3), RM(s(s(s(z0))), add(s(s(s(z1))), x2))) IF_MINSORT(true, add(s(s(x0)), add(s(s(x1)), x2)), x3) -> c26(APP(rm(s(s(x0)), add(s(s(x1)), x2)), x3), RM(s(s(x0)), add(s(s(x1)), x2))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) MINITER(add(s(s(z0)), x1), x2, s(0)) -> c12(IF_MIN(false, x1, x2, s(0)), LE(s(s(z0)), s(0))) MINITER(add(s(x0), x1), x2, s(x3)) -> c12(LE(s(x0), s(x3))) MINITER(add(s(0), x1), x2, s(z0)) -> c12(LE(s(0), s(z0))) MINITER(add(s(s(s(z0))), x1), x2, s(s(0))) -> c12(IF_MIN(false, x1, x2, s(s(0))), LE(s(s(s(z0))), s(s(0)))) MINITER(add(s(s(x0)), x1), x2, s(s(x3))) -> c12(LE(s(s(x0)), s(s(x3)))) MINITER(add(s(s(0)), x1), x2, s(s(z0))) -> c12(LE(s(s(0)), s(s(z0)))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(0)))) -> c12(IF_MIN(false, x1, x2, s(s(s(0)))), LE(s(s(s(s(z0)))), s(s(s(0))))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(s(z1))))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(s(s(z1))))), LE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MINITER(add(s(s(s(x0))), x1), x2, s(s(s(x3)))) -> c12(LE(s(s(s(x0))), s(s(s(x3))))) MINITER(add(s(s(s(0))), x1), x2, s(s(s(z0)))) -> c12(LE(s(s(s(0))), s(s(s(z0))))) K tuples: IF_MINSORT(true, add(z0, z1), z2) -> c26(MINSORT(app(rm(z0, z1), z2), nil), APP(rm(z0, z1), z2), RM(z0, z1)) MINSORT(add(z0, z1), z2) -> c25(IF_MINSORT(eq(z0, min(add(z0, z1))), add(z0, z1), z2), EQ(z0, min(add(z0, z1))), MIN(add(z0, z1))) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MIN(add(z0, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) RM(z0, add(z1, z2)) -> c21(IF_RM(eq(z0, z1), z0, add(z1, z2)), EQ(z0, z1)) Defined Rule Symbols: le_2, eq_2, app_2, if_rm_3, rm_2, if_min_4, minIter_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, MIN_1, MINITER_3, IF_MIN_4, IF_RM_3, IF_MINSORT_3, RM_2, MINSORT_2 Compound Symbols: c3_1, c6_1, c8_1, c10_1, c11_1, c14_1, c22_1, c23_1, c27_1, c12_1, c21_1, c12_2, c21_2, c_1, c26_1, c1_1, c25_3, c25_2, c2_1, c26_3, c4_1, c5_1, c7_1, c26_2 ---------------------------------------- (175) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (176) 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) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) if_min(true, z0, z1, z2) -> z2 if_min(false, z0, z1, z2) -> minIter(z0, z1, z2) minIter(add(z0, z1), z2, z3) -> if_min(le(z0, z3), z1, z2, z3) minIter(nil, add(z0, z1), z2) -> minIter(add(z0, z1), add(z0, z1), s(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, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) RM(0, add(0, x2)) -> c21(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c21(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c21(IF_RM(false, s(z0), add(0, x2))) MINITER(add(s(s(z0)), x1), x2, s(0)) -> c12(IF_MIN(false, x1, x2, s(0)), LE(s(s(z0)), s(0))) MINITER(add(s(x0), x1), x2, s(x3)) -> c12(LE(s(x0), s(x3))) MINITER(add(s(0), x1), x2, s(z0)) -> c12(LE(s(0), s(z0))) RM(s(0), add(s(0), x2)) -> c21(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c21(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c21(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(x0), add(s(x1), x2)) -> c21(EQ(s(x0), s(x1))) MINSORT(add(x0, x1), x2) -> c(MIN(add(x0, x1))) MINITER(add(s(s(s(z0))), x1), x2, s(s(0))) -> c12(IF_MIN(false, x1, x2, s(s(0))), LE(s(s(s(z0))), s(s(0)))) MINITER(add(s(s(x0)), x1), x2, s(s(x3))) -> c12(LE(s(s(x0)), s(s(x3)))) MINITER(add(s(s(0)), x1), x2, s(s(z0))) -> c12(LE(s(s(0)), s(s(z0)))) RM(s(s(0)), add(s(s(0)), x2)) -> c21(IF_RM(true, s(s(0)), add(s(s(0)), x2)), EQ(s(s(0)), s(s(0)))) RM(s(s(0)), add(s(s(s(z0))), x2)) -> c21(IF_RM(false, s(s(0)), add(s(s(s(z0))), x2)), EQ(s(s(0)), s(s(s(z0))))) RM(s(s(s(z0))), add(s(s(0)), x2)) -> c21(IF_RM(false, s(s(s(z0))), add(s(s(0)), x2)), EQ(s(s(s(z0))), s(s(0)))) RM(s(s(x0)), add(s(s(x1)), x2)) -> c21(EQ(s(s(x0)), s(s(x1)))) IF_MINSORT(true, add(0, add(0, x0)), x1) -> c26(RM(0, add(0, x0))) IF_MINSORT(true, add(0, add(s(x0), x1)), x2) -> c26(RM(0, add(s(x0), x1))) IF_MINSORT(true, add(s(x0), add(0, x1)), x2) -> c26(RM(s(x0), add(0, x1))) IF_MINSORT(true, add(s(x0), add(s(x1), x2)), x3) -> c1(APP(rm(s(x0), add(s(x1), x2)), x3)) IF_MINSORT(true, add(s(x0), add(s(x1), x2)), x3) -> c1(RM(s(x0), add(s(x1), x2))) IF_MINSORT(true, add(z0, add(z1, z2)), x3) -> c26(APP(if_rm(eq(z0, z1), z0, add(z1, z2)), x3)) IF_MINSORT(true, add(z0, nil), x1) -> c26(MINSORT(x1, nil)) MINSORT(add(s(z0), x1), x2) -> c25(IF_MINSORT(eq(s(z0), if_min(false, x1, add(s(z0), x1), 0)), add(s(z0), x1), x2), EQ(s(z0), minIter(add(s(z0), x1), add(s(z0), x1), 0)), MIN(add(s(z0), x1))) MINSORT(add(0, x1), x2) -> c25(IF_MINSORT(eq(0, if_min(true, x1, add(0, x1), 0)), add(0, x1), x2), MIN(add(0, x1))) MINSORT(add(x0, x1), x2) -> c2(EQ(x0, minIter(add(x0, x1), add(x0, x1), 0))) MINSORT(add(x0, x1), x2) -> c2(MIN(add(x0, x1))) MINSORT(add(z0, z1), x2) -> c(EQ(z0, if_min(le(z0, 0), z1, add(z0, z1), 0))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(0)))) -> c12(IF_MIN(false, x1, x2, s(s(s(0)))), LE(s(s(s(s(z0)))), s(s(s(0))))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(s(z1))))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(s(s(z1))))), LE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MINITER(add(s(s(s(x0))), x1), x2, s(s(s(x3)))) -> c12(LE(s(s(s(x0))), s(s(s(x3))))) MINITER(add(s(s(s(0))), x1), x2, s(s(s(z0)))) -> c12(LE(s(s(s(0))), s(s(s(z0))))) RM(s(s(s(0))), add(s(s(s(0))), x2)) -> c21(IF_RM(true, s(s(s(0))), add(s(s(s(0))), x2)), EQ(s(s(s(0))), s(s(s(0))))) RM(s(s(s(0))), add(s(s(s(s(z0)))), x2)) -> c21(IF_RM(false, s(s(s(0))), add(s(s(s(s(z0)))), x2)), EQ(s(s(s(0))), s(s(s(s(z0)))))) RM(s(s(s(s(z0)))), add(s(s(s(0))), x2)) -> c21(IF_RM(false, s(s(s(s(z0)))), add(s(s(s(0))), x2)), EQ(s(s(s(s(z0)))), s(s(s(0))))) RM(s(s(s(s(z0)))), add(s(s(s(s(z1)))), x2)) -> c21(IF_RM(eq(z0, z1), s(s(s(s(z0)))), add(s(s(s(s(z1)))), x2)), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) RM(s(s(s(x0))), add(s(s(s(x1))), x2)) -> c21(EQ(s(s(s(x0))), s(s(s(x1))))) IF_MINSORT(true, add(0, add(0, nil)), x1) -> c26(MINSORT(app(nil, x1), nil), APP(rm(0, add(0, nil)), x1), RM(0, add(0, nil))) IF_MINSORT(true, add(0, add(0, add(z1, z2))), x1) -> c26(MINSORT(app(if_rm(eq(0, z1), 0, add(z1, z2)), x1), nil), APP(rm(0, add(0, add(z1, z2))), x1), RM(0, add(0, add(z1, z2)))) IF_MINSORT(true, add(0, add(0, x0)), x1) -> c4(APP(rm(0, add(0, x0)), x1)) IF_MINSORT(true, add(0, add(0, x0)), x1) -> c4(RM(0, add(0, x0))) IF_MINSORT(true, add(0, add(s(x0), x1)), z2) -> c26(MINSORT(add(s(x0), app(rm(0, x1), z2)), nil), APP(rm(0, add(s(x0), x1)), z2), RM(0, add(s(x0), x1))) IF_MINSORT(true, add(0, add(s(x0), nil)), x2) -> c26(MINSORT(app(add(s(x0), nil), x2), nil), APP(rm(0, add(s(x0), nil)), x2), RM(0, add(s(x0), nil))) IF_MINSORT(true, add(0, add(s(x0), add(z1, z2))), x2) -> c26(MINSORT(app(add(s(x0), if_rm(eq(0, z1), 0, add(z1, z2))), x2), nil), APP(rm(0, add(s(x0), add(z1, z2))), x2), RM(0, add(s(x0), add(z1, z2)))) IF_MINSORT(true, add(0, add(s(x0), x1)), x2) -> c5(APP(rm(0, add(s(x0), x1)), x2)) IF_MINSORT(true, add(0, add(s(x0), x1)), x2) -> c5(RM(0, add(s(x0), x1))) IF_MINSORT(true, add(s(x0), add(0, x1)), z2) -> c26(MINSORT(add(0, app(rm(s(x0), x1), z2)), nil), APP(rm(s(x0), add(0, x1)), z2), RM(s(x0), add(0, x1))) IF_MINSORT(true, add(s(x0), add(0, nil)), x2) -> c26(MINSORT(app(add(0, nil), x2), nil), APP(rm(s(x0), add(0, nil)), x2), RM(s(x0), add(0, nil))) IF_MINSORT(true, add(s(x0), add(0, add(z1, z2))), x2) -> c26(MINSORT(app(add(0, if_rm(eq(s(x0), z1), s(x0), add(z1, z2))), x2), nil), APP(rm(s(x0), add(0, add(z1, z2))), x2), RM(s(x0), add(0, add(z1, z2)))) IF_MINSORT(true, add(s(x0), add(0, x1)), x2) -> c7(APP(rm(s(x0), add(0, x1)), x2)) IF_MINSORT(true, add(s(x0), add(0, x1)), x2) -> c7(RM(s(x0), add(0, x1))) IF_MINSORT(true, add(s(0), add(s(0), z2)), x1) -> c26(MINSORT(app(rm(s(0), z2), x1), nil), APP(rm(s(0), add(s(0), z2)), x1), RM(s(0), add(s(0), z2))) IF_MINSORT(true, add(s(0), add(s(0), x0)), x1) -> c26(RM(s(0), add(s(0), x0))) IF_MINSORT(true, add(s(0), add(s(s(x0)), z2)), x2) -> c26(MINSORT(app(add(s(s(x0)), rm(s(0), z2)), x2), nil), APP(rm(s(0), add(s(s(x0)), z2)), x2), RM(s(0), add(s(s(x0)), z2))) IF_MINSORT(true, add(s(0), add(s(s(x0)), x1)), x2) -> c26(RM(s(0), add(s(s(x0)), x1))) IF_MINSORT(true, add(s(s(x0)), add(s(0), z2)), x2) -> c26(MINSORT(app(add(s(0), rm(s(s(x0)), z2)), x2), nil), APP(rm(s(s(x0)), add(s(0), z2)), x2), RM(s(s(x0)), add(s(0), z2))) IF_MINSORT(true, add(s(s(x0)), add(s(0), x1)), x2) -> c26(RM(s(s(x0)), add(s(0), x1))) IF_MINSORT(true, add(s(s(0)), add(s(s(0)), x2)), x3) -> c26(MINSORT(app(if_rm(true, s(s(0)), add(s(s(0)), x2)), x3), nil), APP(rm(s(s(0)), add(s(s(0)), x2)), x3), RM(s(s(0)), add(s(s(0)), x2))) IF_MINSORT(true, add(s(s(0)), add(s(s(s(z0))), x2)), x3) -> c26(MINSORT(app(if_rm(false, s(s(0)), add(s(s(s(z0))), x2)), x3), nil), APP(rm(s(s(0)), add(s(s(s(z0))), x2)), x3), RM(s(s(0)), add(s(s(s(z0))), x2))) IF_MINSORT(true, add(s(s(s(z0))), add(s(s(0)), x2)), x3) -> c26(MINSORT(app(if_rm(false, s(s(s(z0))), add(s(s(0)), x2)), x3), nil), APP(rm(s(s(s(z0))), add(s(s(0)), x2)), x3), RM(s(s(s(z0))), add(s(s(0)), x2))) IF_MINSORT(true, add(s(s(s(z0))), add(s(s(s(z1))), x2)), x3) -> c26(MINSORT(app(if_rm(eq(z0, z1), s(s(s(z0))), add(s(s(s(z1))), x2)), x3), nil), APP(rm(s(s(s(z0))), add(s(s(s(z1))), x2)), x3), RM(s(s(s(z0))), add(s(s(s(z1))), x2))) IF_MINSORT(true, add(s(s(x0)), add(s(s(x1)), x2)), x3) -> c9(APP(rm(s(s(x0)), add(s(s(x1)), x2)), x3)) IF_MINSORT(true, add(s(s(x0)), add(s(s(x1)), x2)), x3) -> c9(RM(s(s(x0)), add(s(s(x1)), x2))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) MINITER(add(s(s(z0)), x1), x2, s(0)) -> c12(IF_MIN(false, x1, x2, s(0)), LE(s(s(z0)), s(0))) MINITER(add(s(x0), x1), x2, s(x3)) -> c12(LE(s(x0), s(x3))) MINITER(add(s(0), x1), x2, s(z0)) -> c12(LE(s(0), s(z0))) MINITER(add(s(s(s(z0))), x1), x2, s(s(0))) -> c12(IF_MIN(false, x1, x2, s(s(0))), LE(s(s(s(z0))), s(s(0)))) MINITER(add(s(s(x0)), x1), x2, s(s(x3))) -> c12(LE(s(s(x0)), s(s(x3)))) MINITER(add(s(s(0)), x1), x2, s(s(z0))) -> c12(LE(s(s(0)), s(s(z0)))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(0)))) -> c12(IF_MIN(false, x1, x2, s(s(s(0)))), LE(s(s(s(s(z0)))), s(s(s(0))))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(s(z1))))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(s(s(z1))))), LE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MINITER(add(s(s(s(x0))), x1), x2, s(s(s(x3)))) -> c12(LE(s(s(s(x0))), s(s(s(x3))))) MINITER(add(s(s(s(0))), x1), x2, s(s(s(z0)))) -> c12(LE(s(s(s(0))), s(s(s(z0))))) K tuples: IF_MINSORT(true, add(z0, z1), z2) -> c26(MINSORT(app(rm(z0, z1), z2), nil), APP(rm(z0, z1), z2), RM(z0, z1)) MINSORT(add(z0, z1), z2) -> c25(IF_MINSORT(eq(z0, min(add(z0, z1))), add(z0, z1), z2), EQ(z0, min(add(z0, z1))), MIN(add(z0, z1))) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MIN(add(z0, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) RM(z0, add(z1, z2)) -> c21(IF_RM(eq(z0, z1), z0, add(z1, z2)), EQ(z0, z1)) Defined Rule Symbols: le_2, eq_2, app_2, if_rm_3, rm_2, if_min_4, minIter_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, MIN_1, MINITER_3, IF_MIN_4, IF_RM_3, IF_MINSORT_3, RM_2, MINSORT_2 Compound Symbols: c3_1, c6_1, c8_1, c10_1, c11_1, c14_1, c22_1, c23_1, c27_1, c12_1, c21_1, c12_2, c21_2, c_1, c26_1, c1_1, c25_3, c25_2, c2_1, c26_3, c4_1, c5_1, c7_1, c9_1 ---------------------------------------- (177) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF_MINSORT(true, add(s(x0), add(s(x1), x2)), x3) -> c1(APP(rm(s(x0), add(s(x1), x2)), x3)) by IF_MINSORT(true, add(s(x0), add(s(x1), z2)), x3) -> c1(APP(if_rm(eq(s(x0), s(x1)), s(x0), add(s(x1), z2)), x3)) ---------------------------------------- (178) 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) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) if_min(true, z0, z1, z2) -> z2 if_min(false, z0, z1, z2) -> minIter(z0, z1, z2) minIter(add(z0, z1), z2, z3) -> if_min(le(z0, z3), z1, z2, z3) minIter(nil, add(z0, z1), z2) -> minIter(add(z0, z1), add(z0, z1), s(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, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) RM(0, add(0, x2)) -> c21(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c21(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c21(IF_RM(false, s(z0), add(0, x2))) MINITER(add(s(s(z0)), x1), x2, s(0)) -> c12(IF_MIN(false, x1, x2, s(0)), LE(s(s(z0)), s(0))) MINITER(add(s(x0), x1), x2, s(x3)) -> c12(LE(s(x0), s(x3))) MINITER(add(s(0), x1), x2, s(z0)) -> c12(LE(s(0), s(z0))) RM(s(0), add(s(0), x2)) -> c21(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c21(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c21(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(x0), add(s(x1), x2)) -> c21(EQ(s(x0), s(x1))) MINSORT(add(x0, x1), x2) -> c(MIN(add(x0, x1))) MINITER(add(s(s(s(z0))), x1), x2, s(s(0))) -> c12(IF_MIN(false, x1, x2, s(s(0))), LE(s(s(s(z0))), s(s(0)))) MINITER(add(s(s(x0)), x1), x2, s(s(x3))) -> c12(LE(s(s(x0)), s(s(x3)))) MINITER(add(s(s(0)), x1), x2, s(s(z0))) -> c12(LE(s(s(0)), s(s(z0)))) RM(s(s(0)), add(s(s(0)), x2)) -> c21(IF_RM(true, s(s(0)), add(s(s(0)), x2)), EQ(s(s(0)), s(s(0)))) RM(s(s(0)), add(s(s(s(z0))), x2)) -> c21(IF_RM(false, s(s(0)), add(s(s(s(z0))), x2)), EQ(s(s(0)), s(s(s(z0))))) RM(s(s(s(z0))), add(s(s(0)), x2)) -> c21(IF_RM(false, s(s(s(z0))), add(s(s(0)), x2)), EQ(s(s(s(z0))), s(s(0)))) RM(s(s(x0)), add(s(s(x1)), x2)) -> c21(EQ(s(s(x0)), s(s(x1)))) IF_MINSORT(true, add(0, add(0, x0)), x1) -> c26(RM(0, add(0, x0))) IF_MINSORT(true, add(0, add(s(x0), x1)), x2) -> c26(RM(0, add(s(x0), x1))) IF_MINSORT(true, add(s(x0), add(0, x1)), x2) -> c26(RM(s(x0), add(0, x1))) IF_MINSORT(true, add(s(x0), add(s(x1), x2)), x3) -> c1(RM(s(x0), add(s(x1), x2))) IF_MINSORT(true, add(z0, add(z1, z2)), x3) -> c26(APP(if_rm(eq(z0, z1), z0, add(z1, z2)), x3)) IF_MINSORT(true, add(z0, nil), x1) -> c26(MINSORT(x1, nil)) MINSORT(add(s(z0), x1), x2) -> c25(IF_MINSORT(eq(s(z0), if_min(false, x1, add(s(z0), x1), 0)), add(s(z0), x1), x2), EQ(s(z0), minIter(add(s(z0), x1), add(s(z0), x1), 0)), MIN(add(s(z0), x1))) MINSORT(add(0, x1), x2) -> c25(IF_MINSORT(eq(0, if_min(true, x1, add(0, x1), 0)), add(0, x1), x2), MIN(add(0, x1))) MINSORT(add(x0, x1), x2) -> c2(EQ(x0, minIter(add(x0, x1), add(x0, x1), 0))) MINSORT(add(x0, x1), x2) -> c2(MIN(add(x0, x1))) MINSORT(add(z0, z1), x2) -> c(EQ(z0, if_min(le(z0, 0), z1, add(z0, z1), 0))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(0)))) -> c12(IF_MIN(false, x1, x2, s(s(s(0)))), LE(s(s(s(s(z0)))), s(s(s(0))))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(s(z1))))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(s(s(z1))))), LE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MINITER(add(s(s(s(x0))), x1), x2, s(s(s(x3)))) -> c12(LE(s(s(s(x0))), s(s(s(x3))))) MINITER(add(s(s(s(0))), x1), x2, s(s(s(z0)))) -> c12(LE(s(s(s(0))), s(s(s(z0))))) RM(s(s(s(0))), add(s(s(s(0))), x2)) -> c21(IF_RM(true, s(s(s(0))), add(s(s(s(0))), x2)), EQ(s(s(s(0))), s(s(s(0))))) RM(s(s(s(0))), add(s(s(s(s(z0)))), x2)) -> c21(IF_RM(false, s(s(s(0))), add(s(s(s(s(z0)))), x2)), EQ(s(s(s(0))), s(s(s(s(z0)))))) RM(s(s(s(s(z0)))), add(s(s(s(0))), x2)) -> c21(IF_RM(false, s(s(s(s(z0)))), add(s(s(s(0))), x2)), EQ(s(s(s(s(z0)))), s(s(s(0))))) RM(s(s(s(s(z0)))), add(s(s(s(s(z1)))), x2)) -> c21(IF_RM(eq(z0, z1), s(s(s(s(z0)))), add(s(s(s(s(z1)))), x2)), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) RM(s(s(s(x0))), add(s(s(s(x1))), x2)) -> c21(EQ(s(s(s(x0))), s(s(s(x1))))) IF_MINSORT(true, add(0, add(0, nil)), x1) -> c26(MINSORT(app(nil, x1), nil), APP(rm(0, add(0, nil)), x1), RM(0, add(0, nil))) IF_MINSORT(true, add(0, add(0, add(z1, z2))), x1) -> c26(MINSORT(app(if_rm(eq(0, z1), 0, add(z1, z2)), x1), nil), APP(rm(0, add(0, add(z1, z2))), x1), RM(0, add(0, add(z1, z2)))) IF_MINSORT(true, add(0, add(0, x0)), x1) -> c4(APP(rm(0, add(0, x0)), x1)) IF_MINSORT(true, add(0, add(0, x0)), x1) -> c4(RM(0, add(0, x0))) IF_MINSORT(true, add(0, add(s(x0), x1)), z2) -> c26(MINSORT(add(s(x0), app(rm(0, x1), z2)), nil), APP(rm(0, add(s(x0), x1)), z2), RM(0, add(s(x0), x1))) IF_MINSORT(true, add(0, add(s(x0), nil)), x2) -> c26(MINSORT(app(add(s(x0), nil), x2), nil), APP(rm(0, add(s(x0), nil)), x2), RM(0, add(s(x0), nil))) IF_MINSORT(true, add(0, add(s(x0), add(z1, z2))), x2) -> c26(MINSORT(app(add(s(x0), if_rm(eq(0, z1), 0, add(z1, z2))), x2), nil), APP(rm(0, add(s(x0), add(z1, z2))), x2), RM(0, add(s(x0), add(z1, z2)))) IF_MINSORT(true, add(0, add(s(x0), x1)), x2) -> c5(APP(rm(0, add(s(x0), x1)), x2)) IF_MINSORT(true, add(0, add(s(x0), x1)), x2) -> c5(RM(0, add(s(x0), x1))) IF_MINSORT(true, add(s(x0), add(0, x1)), z2) -> c26(MINSORT(add(0, app(rm(s(x0), x1), z2)), nil), APP(rm(s(x0), add(0, x1)), z2), RM(s(x0), add(0, x1))) IF_MINSORT(true, add(s(x0), add(0, nil)), x2) -> c26(MINSORT(app(add(0, nil), x2), nil), APP(rm(s(x0), add(0, nil)), x2), RM(s(x0), add(0, nil))) IF_MINSORT(true, add(s(x0), add(0, add(z1, z2))), x2) -> c26(MINSORT(app(add(0, if_rm(eq(s(x0), z1), s(x0), add(z1, z2))), x2), nil), APP(rm(s(x0), add(0, add(z1, z2))), x2), RM(s(x0), add(0, add(z1, z2)))) IF_MINSORT(true, add(s(x0), add(0, x1)), x2) -> c7(APP(rm(s(x0), add(0, x1)), x2)) IF_MINSORT(true, add(s(x0), add(0, x1)), x2) -> c7(RM(s(x0), add(0, x1))) IF_MINSORT(true, add(s(0), add(s(0), z2)), x1) -> c26(MINSORT(app(rm(s(0), z2), x1), nil), APP(rm(s(0), add(s(0), z2)), x1), RM(s(0), add(s(0), z2))) IF_MINSORT(true, add(s(0), add(s(0), x0)), x1) -> c26(RM(s(0), add(s(0), x0))) IF_MINSORT(true, add(s(0), add(s(s(x0)), z2)), x2) -> c26(MINSORT(app(add(s(s(x0)), rm(s(0), z2)), x2), nil), APP(rm(s(0), add(s(s(x0)), z2)), x2), RM(s(0), add(s(s(x0)), z2))) IF_MINSORT(true, add(s(0), add(s(s(x0)), x1)), x2) -> c26(RM(s(0), add(s(s(x0)), x1))) IF_MINSORT(true, add(s(s(x0)), add(s(0), z2)), x2) -> c26(MINSORT(app(add(s(0), rm(s(s(x0)), z2)), x2), nil), APP(rm(s(s(x0)), add(s(0), z2)), x2), RM(s(s(x0)), add(s(0), z2))) IF_MINSORT(true, add(s(s(x0)), add(s(0), x1)), x2) -> c26(RM(s(s(x0)), add(s(0), x1))) IF_MINSORT(true, add(s(s(0)), add(s(s(0)), x2)), x3) -> c26(MINSORT(app(if_rm(true, s(s(0)), add(s(s(0)), x2)), x3), nil), APP(rm(s(s(0)), add(s(s(0)), x2)), x3), RM(s(s(0)), add(s(s(0)), x2))) IF_MINSORT(true, add(s(s(0)), add(s(s(s(z0))), x2)), x3) -> c26(MINSORT(app(if_rm(false, s(s(0)), add(s(s(s(z0))), x2)), x3), nil), APP(rm(s(s(0)), add(s(s(s(z0))), x2)), x3), RM(s(s(0)), add(s(s(s(z0))), x2))) IF_MINSORT(true, add(s(s(s(z0))), add(s(s(0)), x2)), x3) -> c26(MINSORT(app(if_rm(false, s(s(s(z0))), add(s(s(0)), x2)), x3), nil), APP(rm(s(s(s(z0))), add(s(s(0)), x2)), x3), RM(s(s(s(z0))), add(s(s(0)), x2))) IF_MINSORT(true, add(s(s(s(z0))), add(s(s(s(z1))), x2)), x3) -> c26(MINSORT(app(if_rm(eq(z0, z1), s(s(s(z0))), add(s(s(s(z1))), x2)), x3), nil), APP(rm(s(s(s(z0))), add(s(s(s(z1))), x2)), x3), RM(s(s(s(z0))), add(s(s(s(z1))), x2))) IF_MINSORT(true, add(s(s(x0)), add(s(s(x1)), x2)), x3) -> c9(APP(rm(s(s(x0)), add(s(s(x1)), x2)), x3)) IF_MINSORT(true, add(s(s(x0)), add(s(s(x1)), x2)), x3) -> c9(RM(s(s(x0)), add(s(s(x1)), x2))) IF_MINSORT(true, add(s(x0), add(s(x1), z2)), x3) -> c1(APP(if_rm(eq(s(x0), s(x1)), s(x0), add(s(x1), z2)), x3)) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) MINITER(add(s(s(z0)), x1), x2, s(0)) -> c12(IF_MIN(false, x1, x2, s(0)), LE(s(s(z0)), s(0))) MINITER(add(s(x0), x1), x2, s(x3)) -> c12(LE(s(x0), s(x3))) MINITER(add(s(0), x1), x2, s(z0)) -> c12(LE(s(0), s(z0))) MINITER(add(s(s(s(z0))), x1), x2, s(s(0))) -> c12(IF_MIN(false, x1, x2, s(s(0))), LE(s(s(s(z0))), s(s(0)))) MINITER(add(s(s(x0)), x1), x2, s(s(x3))) -> c12(LE(s(s(x0)), s(s(x3)))) MINITER(add(s(s(0)), x1), x2, s(s(z0))) -> c12(LE(s(s(0)), s(s(z0)))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(0)))) -> c12(IF_MIN(false, x1, x2, s(s(s(0)))), LE(s(s(s(s(z0)))), s(s(s(0))))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(s(z1))))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(s(s(z1))))), LE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MINITER(add(s(s(s(x0))), x1), x2, s(s(s(x3)))) -> c12(LE(s(s(s(x0))), s(s(s(x3))))) MINITER(add(s(s(s(0))), x1), x2, s(s(s(z0)))) -> c12(LE(s(s(s(0))), s(s(s(z0))))) K tuples: IF_MINSORT(true, add(z0, z1), z2) -> c26(MINSORT(app(rm(z0, z1), z2), nil), APP(rm(z0, z1), z2), RM(z0, z1)) MINSORT(add(z0, z1), z2) -> c25(IF_MINSORT(eq(z0, min(add(z0, z1))), add(z0, z1), z2), EQ(z0, min(add(z0, z1))), MIN(add(z0, z1))) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MIN(add(z0, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) RM(z0, add(z1, z2)) -> c21(IF_RM(eq(z0, z1), z0, add(z1, z2)), EQ(z0, z1)) Defined Rule Symbols: le_2, eq_2, app_2, if_rm_3, rm_2, if_min_4, minIter_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, MIN_1, MINITER_3, IF_MIN_4, IF_RM_3, IF_MINSORT_3, RM_2, MINSORT_2 Compound Symbols: c3_1, c6_1, c8_1, c10_1, c11_1, c14_1, c22_1, c23_1, c27_1, c12_1, c21_1, c12_2, c21_2, c_1, c26_1, c1_1, c25_3, c25_2, c2_1, c26_3, c4_1, c5_1, c7_1, c9_1 ---------------------------------------- (179) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF_MINSORT(true, add(z0, add(z1, z2)), x3) -> c26(APP(if_rm(eq(z0, z1), z0, add(z1, z2)), x3)) by IF_MINSORT(true, add(0, add(0, x2)), x3) -> c26(APP(if_rm(true, 0, add(0, x2)), x3)) IF_MINSORT(true, add(0, add(s(z0), x2)), x3) -> c26(APP(if_rm(false, 0, add(s(z0), x2)), x3)) IF_MINSORT(true, add(s(z0), add(0, x2)), x3) -> c26(APP(if_rm(false, s(z0), add(0, x2)), x3)) IF_MINSORT(true, add(s(z0), add(s(z1), x2)), x3) -> c26(APP(if_rm(eq(z0, z1), s(z0), add(s(z1), x2)), x3)) ---------------------------------------- (180) 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) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) if_min(true, z0, z1, z2) -> z2 if_min(false, z0, z1, z2) -> minIter(z0, z1, z2) minIter(add(z0, z1), z2, z3) -> if_min(le(z0, z3), z1, z2, z3) minIter(nil, add(z0, z1), z2) -> minIter(add(z0, z1), add(z0, z1), s(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, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) RM(0, add(0, x2)) -> c21(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c21(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c21(IF_RM(false, s(z0), add(0, x2))) MINITER(add(s(s(z0)), x1), x2, s(0)) -> c12(IF_MIN(false, x1, x2, s(0)), LE(s(s(z0)), s(0))) MINITER(add(s(x0), x1), x2, s(x3)) -> c12(LE(s(x0), s(x3))) MINITER(add(s(0), x1), x2, s(z0)) -> c12(LE(s(0), s(z0))) RM(s(0), add(s(0), x2)) -> c21(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c21(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c21(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(x0), add(s(x1), x2)) -> c21(EQ(s(x0), s(x1))) MINSORT(add(x0, x1), x2) -> c(MIN(add(x0, x1))) MINITER(add(s(s(s(z0))), x1), x2, s(s(0))) -> c12(IF_MIN(false, x1, x2, s(s(0))), LE(s(s(s(z0))), s(s(0)))) MINITER(add(s(s(x0)), x1), x2, s(s(x3))) -> c12(LE(s(s(x0)), s(s(x3)))) MINITER(add(s(s(0)), x1), x2, s(s(z0))) -> c12(LE(s(s(0)), s(s(z0)))) RM(s(s(0)), add(s(s(0)), x2)) -> c21(IF_RM(true, s(s(0)), add(s(s(0)), x2)), EQ(s(s(0)), s(s(0)))) RM(s(s(0)), add(s(s(s(z0))), x2)) -> c21(IF_RM(false, s(s(0)), add(s(s(s(z0))), x2)), EQ(s(s(0)), s(s(s(z0))))) RM(s(s(s(z0))), add(s(s(0)), x2)) -> c21(IF_RM(false, s(s(s(z0))), add(s(s(0)), x2)), EQ(s(s(s(z0))), s(s(0)))) RM(s(s(x0)), add(s(s(x1)), x2)) -> c21(EQ(s(s(x0)), s(s(x1)))) IF_MINSORT(true, add(0, add(0, x0)), x1) -> c26(RM(0, add(0, x0))) IF_MINSORT(true, add(0, add(s(x0), x1)), x2) -> c26(RM(0, add(s(x0), x1))) IF_MINSORT(true, add(s(x0), add(0, x1)), x2) -> c26(RM(s(x0), add(0, x1))) IF_MINSORT(true, add(s(x0), add(s(x1), x2)), x3) -> c1(RM(s(x0), add(s(x1), x2))) IF_MINSORT(true, add(z0, nil), x1) -> c26(MINSORT(x1, nil)) MINSORT(add(s(z0), x1), x2) -> c25(IF_MINSORT(eq(s(z0), if_min(false, x1, add(s(z0), x1), 0)), add(s(z0), x1), x2), EQ(s(z0), minIter(add(s(z0), x1), add(s(z0), x1), 0)), MIN(add(s(z0), x1))) MINSORT(add(0, x1), x2) -> c25(IF_MINSORT(eq(0, if_min(true, x1, add(0, x1), 0)), add(0, x1), x2), MIN(add(0, x1))) MINSORT(add(x0, x1), x2) -> c2(EQ(x0, minIter(add(x0, x1), add(x0, x1), 0))) MINSORT(add(x0, x1), x2) -> c2(MIN(add(x0, x1))) MINSORT(add(z0, z1), x2) -> c(EQ(z0, if_min(le(z0, 0), z1, add(z0, z1), 0))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(0)))) -> c12(IF_MIN(false, x1, x2, s(s(s(0)))), LE(s(s(s(s(z0)))), s(s(s(0))))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(s(z1))))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(s(s(z1))))), LE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MINITER(add(s(s(s(x0))), x1), x2, s(s(s(x3)))) -> c12(LE(s(s(s(x0))), s(s(s(x3))))) MINITER(add(s(s(s(0))), x1), x2, s(s(s(z0)))) -> c12(LE(s(s(s(0))), s(s(s(z0))))) RM(s(s(s(0))), add(s(s(s(0))), x2)) -> c21(IF_RM(true, s(s(s(0))), add(s(s(s(0))), x2)), EQ(s(s(s(0))), s(s(s(0))))) RM(s(s(s(0))), add(s(s(s(s(z0)))), x2)) -> c21(IF_RM(false, s(s(s(0))), add(s(s(s(s(z0)))), x2)), EQ(s(s(s(0))), s(s(s(s(z0)))))) RM(s(s(s(s(z0)))), add(s(s(s(0))), x2)) -> c21(IF_RM(false, s(s(s(s(z0)))), add(s(s(s(0))), x2)), EQ(s(s(s(s(z0)))), s(s(s(0))))) RM(s(s(s(s(z0)))), add(s(s(s(s(z1)))), x2)) -> c21(IF_RM(eq(z0, z1), s(s(s(s(z0)))), add(s(s(s(s(z1)))), x2)), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) RM(s(s(s(x0))), add(s(s(s(x1))), x2)) -> c21(EQ(s(s(s(x0))), s(s(s(x1))))) IF_MINSORT(true, add(0, add(0, nil)), x1) -> c26(MINSORT(app(nil, x1), nil), APP(rm(0, add(0, nil)), x1), RM(0, add(0, nil))) IF_MINSORT(true, add(0, add(0, add(z1, z2))), x1) -> c26(MINSORT(app(if_rm(eq(0, z1), 0, add(z1, z2)), x1), nil), APP(rm(0, add(0, add(z1, z2))), x1), RM(0, add(0, add(z1, z2)))) IF_MINSORT(true, add(0, add(0, x0)), x1) -> c4(APP(rm(0, add(0, x0)), x1)) IF_MINSORT(true, add(0, add(0, x0)), x1) -> c4(RM(0, add(0, x0))) IF_MINSORT(true, add(0, add(s(x0), x1)), z2) -> c26(MINSORT(add(s(x0), app(rm(0, x1), z2)), nil), APP(rm(0, add(s(x0), x1)), z2), RM(0, add(s(x0), x1))) IF_MINSORT(true, add(0, add(s(x0), nil)), x2) -> c26(MINSORT(app(add(s(x0), nil), x2), nil), APP(rm(0, add(s(x0), nil)), x2), RM(0, add(s(x0), nil))) IF_MINSORT(true, add(0, add(s(x0), add(z1, z2))), x2) -> c26(MINSORT(app(add(s(x0), if_rm(eq(0, z1), 0, add(z1, z2))), x2), nil), APP(rm(0, add(s(x0), add(z1, z2))), x2), RM(0, add(s(x0), add(z1, z2)))) IF_MINSORT(true, add(0, add(s(x0), x1)), x2) -> c5(APP(rm(0, add(s(x0), x1)), x2)) IF_MINSORT(true, add(0, add(s(x0), x1)), x2) -> c5(RM(0, add(s(x0), x1))) IF_MINSORT(true, add(s(x0), add(0, x1)), z2) -> c26(MINSORT(add(0, app(rm(s(x0), x1), z2)), nil), APP(rm(s(x0), add(0, x1)), z2), RM(s(x0), add(0, x1))) IF_MINSORT(true, add(s(x0), add(0, nil)), x2) -> c26(MINSORT(app(add(0, nil), x2), nil), APP(rm(s(x0), add(0, nil)), x2), RM(s(x0), add(0, nil))) IF_MINSORT(true, add(s(x0), add(0, add(z1, z2))), x2) -> c26(MINSORT(app(add(0, if_rm(eq(s(x0), z1), s(x0), add(z1, z2))), x2), nil), APP(rm(s(x0), add(0, add(z1, z2))), x2), RM(s(x0), add(0, add(z1, z2)))) IF_MINSORT(true, add(s(x0), add(0, x1)), x2) -> c7(APP(rm(s(x0), add(0, x1)), x2)) IF_MINSORT(true, add(s(x0), add(0, x1)), x2) -> c7(RM(s(x0), add(0, x1))) IF_MINSORT(true, add(s(0), add(s(0), z2)), x1) -> c26(MINSORT(app(rm(s(0), z2), x1), nil), APP(rm(s(0), add(s(0), z2)), x1), RM(s(0), add(s(0), z2))) IF_MINSORT(true, add(s(0), add(s(0), x0)), x1) -> c26(RM(s(0), add(s(0), x0))) IF_MINSORT(true, add(s(0), add(s(s(x0)), z2)), x2) -> c26(MINSORT(app(add(s(s(x0)), rm(s(0), z2)), x2), nil), APP(rm(s(0), add(s(s(x0)), z2)), x2), RM(s(0), add(s(s(x0)), z2))) IF_MINSORT(true, add(s(0), add(s(s(x0)), x1)), x2) -> c26(RM(s(0), add(s(s(x0)), x1))) IF_MINSORT(true, add(s(s(x0)), add(s(0), z2)), x2) -> c26(MINSORT(app(add(s(0), rm(s(s(x0)), z2)), x2), nil), APP(rm(s(s(x0)), add(s(0), z2)), x2), RM(s(s(x0)), add(s(0), z2))) IF_MINSORT(true, add(s(s(x0)), add(s(0), x1)), x2) -> c26(RM(s(s(x0)), add(s(0), x1))) IF_MINSORT(true, add(s(s(0)), add(s(s(0)), x2)), x3) -> c26(MINSORT(app(if_rm(true, s(s(0)), add(s(s(0)), x2)), x3), nil), APP(rm(s(s(0)), add(s(s(0)), x2)), x3), RM(s(s(0)), add(s(s(0)), x2))) IF_MINSORT(true, add(s(s(0)), add(s(s(s(z0))), x2)), x3) -> c26(MINSORT(app(if_rm(false, s(s(0)), add(s(s(s(z0))), x2)), x3), nil), APP(rm(s(s(0)), add(s(s(s(z0))), x2)), x3), RM(s(s(0)), add(s(s(s(z0))), x2))) IF_MINSORT(true, add(s(s(s(z0))), add(s(s(0)), x2)), x3) -> c26(MINSORT(app(if_rm(false, s(s(s(z0))), add(s(s(0)), x2)), x3), nil), APP(rm(s(s(s(z0))), add(s(s(0)), x2)), x3), RM(s(s(s(z0))), add(s(s(0)), x2))) IF_MINSORT(true, add(s(s(s(z0))), add(s(s(s(z1))), x2)), x3) -> c26(MINSORT(app(if_rm(eq(z0, z1), s(s(s(z0))), add(s(s(s(z1))), x2)), x3), nil), APP(rm(s(s(s(z0))), add(s(s(s(z1))), x2)), x3), RM(s(s(s(z0))), add(s(s(s(z1))), x2))) IF_MINSORT(true, add(s(s(x0)), add(s(s(x1)), x2)), x3) -> c9(APP(rm(s(s(x0)), add(s(s(x1)), x2)), x3)) IF_MINSORT(true, add(s(s(x0)), add(s(s(x1)), x2)), x3) -> c9(RM(s(s(x0)), add(s(s(x1)), x2))) IF_MINSORT(true, add(s(x0), add(s(x1), z2)), x3) -> c1(APP(if_rm(eq(s(x0), s(x1)), s(x0), add(s(x1), z2)), x3)) IF_MINSORT(true, add(0, add(0, x2)), x3) -> c26(APP(if_rm(true, 0, add(0, x2)), x3)) IF_MINSORT(true, add(0, add(s(z0), x2)), x3) -> c26(APP(if_rm(false, 0, add(s(z0), x2)), x3)) IF_MINSORT(true, add(s(z0), add(0, x2)), x3) -> c26(APP(if_rm(false, s(z0), add(0, x2)), x3)) IF_MINSORT(true, add(s(z0), add(s(z1), x2)), x3) -> c26(APP(if_rm(eq(z0, z1), s(z0), add(s(z1), x2)), x3)) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) MINITER(add(s(s(z0)), x1), x2, s(0)) -> c12(IF_MIN(false, x1, x2, s(0)), LE(s(s(z0)), s(0))) MINITER(add(s(x0), x1), x2, s(x3)) -> c12(LE(s(x0), s(x3))) MINITER(add(s(0), x1), x2, s(z0)) -> c12(LE(s(0), s(z0))) MINITER(add(s(s(s(z0))), x1), x2, s(s(0))) -> c12(IF_MIN(false, x1, x2, s(s(0))), LE(s(s(s(z0))), s(s(0)))) MINITER(add(s(s(x0)), x1), x2, s(s(x3))) -> c12(LE(s(s(x0)), s(s(x3)))) MINITER(add(s(s(0)), x1), x2, s(s(z0))) -> c12(LE(s(s(0)), s(s(z0)))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(0)))) -> c12(IF_MIN(false, x1, x2, s(s(s(0)))), LE(s(s(s(s(z0)))), s(s(s(0))))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(s(z1))))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(s(s(z1))))), LE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MINITER(add(s(s(s(x0))), x1), x2, s(s(s(x3)))) -> c12(LE(s(s(s(x0))), s(s(s(x3))))) MINITER(add(s(s(s(0))), x1), x2, s(s(s(z0)))) -> c12(LE(s(s(s(0))), s(s(s(z0))))) K tuples: IF_MINSORT(true, add(z0, z1), z2) -> c26(MINSORT(app(rm(z0, z1), z2), nil), APP(rm(z0, z1), z2), RM(z0, z1)) MINSORT(add(z0, z1), z2) -> c25(IF_MINSORT(eq(z0, min(add(z0, z1))), add(z0, z1), z2), EQ(z0, min(add(z0, z1))), MIN(add(z0, z1))) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MIN(add(z0, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) RM(z0, add(z1, z2)) -> c21(IF_RM(eq(z0, z1), z0, add(z1, z2)), EQ(z0, z1)) Defined Rule Symbols: le_2, eq_2, app_2, if_rm_3, rm_2, if_min_4, minIter_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, MIN_1, MINITER_3, IF_MIN_4, IF_RM_3, IF_MINSORT_3, RM_2, MINSORT_2 Compound Symbols: c3_1, c6_1, c8_1, c10_1, c11_1, c14_1, c22_1, c23_1, c27_1, c12_1, c21_1, c12_2, c21_2, c_1, c26_1, c1_1, c25_3, c25_2, c2_1, c26_3, c4_1, c5_1, c7_1, c9_1 ---------------------------------------- (181) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MINSORT(add(s(z0), x1), x2) -> c25(IF_MINSORT(eq(s(z0), if_min(false, x1, add(s(z0), x1), 0)), add(s(z0), x1), x2), EQ(s(z0), minIter(add(s(z0), x1), add(s(z0), x1), 0)), MIN(add(s(z0), x1))) by MINSORT(add(s(x0), z0), x2) -> c25(IF_MINSORT(eq(s(x0), minIter(z0, add(s(x0), z0), 0)), add(s(x0), z0), x2), EQ(s(x0), minIter(add(s(x0), z0), add(s(x0), z0), 0)), MIN(add(s(x0), z0))) MINSORT(add(s(x0), x1), x2) -> c25(EQ(s(x0), minIter(add(s(x0), x1), add(s(x0), x1), 0)), MIN(add(s(x0), x1))) ---------------------------------------- (182) 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) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) if_min(true, z0, z1, z2) -> z2 if_min(false, z0, z1, z2) -> minIter(z0, z1, z2) minIter(add(z0, z1), z2, z3) -> if_min(le(z0, z3), z1, z2, z3) minIter(nil, add(z0, z1), z2) -> minIter(add(z0, z1), add(z0, z1), s(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, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) RM(0, add(0, x2)) -> c21(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c21(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c21(IF_RM(false, s(z0), add(0, x2))) MINITER(add(s(s(z0)), x1), x2, s(0)) -> c12(IF_MIN(false, x1, x2, s(0)), LE(s(s(z0)), s(0))) MINITER(add(s(x0), x1), x2, s(x3)) -> c12(LE(s(x0), s(x3))) MINITER(add(s(0), x1), x2, s(z0)) -> c12(LE(s(0), s(z0))) RM(s(0), add(s(0), x2)) -> c21(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c21(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c21(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(x0), add(s(x1), x2)) -> c21(EQ(s(x0), s(x1))) MINSORT(add(x0, x1), x2) -> c(MIN(add(x0, x1))) MINITER(add(s(s(s(z0))), x1), x2, s(s(0))) -> c12(IF_MIN(false, x1, x2, s(s(0))), LE(s(s(s(z0))), s(s(0)))) MINITER(add(s(s(x0)), x1), x2, s(s(x3))) -> c12(LE(s(s(x0)), s(s(x3)))) MINITER(add(s(s(0)), x1), x2, s(s(z0))) -> c12(LE(s(s(0)), s(s(z0)))) RM(s(s(0)), add(s(s(0)), x2)) -> c21(IF_RM(true, s(s(0)), add(s(s(0)), x2)), EQ(s(s(0)), s(s(0)))) RM(s(s(0)), add(s(s(s(z0))), x2)) -> c21(IF_RM(false, s(s(0)), add(s(s(s(z0))), x2)), EQ(s(s(0)), s(s(s(z0))))) RM(s(s(s(z0))), add(s(s(0)), x2)) -> c21(IF_RM(false, s(s(s(z0))), add(s(s(0)), x2)), EQ(s(s(s(z0))), s(s(0)))) RM(s(s(x0)), add(s(s(x1)), x2)) -> c21(EQ(s(s(x0)), s(s(x1)))) IF_MINSORT(true, add(0, add(0, x0)), x1) -> c26(RM(0, add(0, x0))) IF_MINSORT(true, add(0, add(s(x0), x1)), x2) -> c26(RM(0, add(s(x0), x1))) IF_MINSORT(true, add(s(x0), add(0, x1)), x2) -> c26(RM(s(x0), add(0, x1))) IF_MINSORT(true, add(s(x0), add(s(x1), x2)), x3) -> c1(RM(s(x0), add(s(x1), x2))) IF_MINSORT(true, add(z0, nil), x1) -> c26(MINSORT(x1, nil)) MINSORT(add(0, x1), x2) -> c25(IF_MINSORT(eq(0, if_min(true, x1, add(0, x1), 0)), add(0, x1), x2), MIN(add(0, x1))) MINSORT(add(x0, x1), x2) -> c2(EQ(x0, minIter(add(x0, x1), add(x0, x1), 0))) MINSORT(add(x0, x1), x2) -> c2(MIN(add(x0, x1))) MINSORT(add(z0, z1), x2) -> c(EQ(z0, if_min(le(z0, 0), z1, add(z0, z1), 0))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(0)))) -> c12(IF_MIN(false, x1, x2, s(s(s(0)))), LE(s(s(s(s(z0)))), s(s(s(0))))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(s(z1))))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(s(s(z1))))), LE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MINITER(add(s(s(s(x0))), x1), x2, s(s(s(x3)))) -> c12(LE(s(s(s(x0))), s(s(s(x3))))) MINITER(add(s(s(s(0))), x1), x2, s(s(s(z0)))) -> c12(LE(s(s(s(0))), s(s(s(z0))))) RM(s(s(s(0))), add(s(s(s(0))), x2)) -> c21(IF_RM(true, s(s(s(0))), add(s(s(s(0))), x2)), EQ(s(s(s(0))), s(s(s(0))))) RM(s(s(s(0))), add(s(s(s(s(z0)))), x2)) -> c21(IF_RM(false, s(s(s(0))), add(s(s(s(s(z0)))), x2)), EQ(s(s(s(0))), s(s(s(s(z0)))))) RM(s(s(s(s(z0)))), add(s(s(s(0))), x2)) -> c21(IF_RM(false, s(s(s(s(z0)))), add(s(s(s(0))), x2)), EQ(s(s(s(s(z0)))), s(s(s(0))))) RM(s(s(s(s(z0)))), add(s(s(s(s(z1)))), x2)) -> c21(IF_RM(eq(z0, z1), s(s(s(s(z0)))), add(s(s(s(s(z1)))), x2)), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) RM(s(s(s(x0))), add(s(s(s(x1))), x2)) -> c21(EQ(s(s(s(x0))), s(s(s(x1))))) IF_MINSORT(true, add(0, add(0, nil)), x1) -> c26(MINSORT(app(nil, x1), nil), APP(rm(0, add(0, nil)), x1), RM(0, add(0, nil))) IF_MINSORT(true, add(0, add(0, add(z1, z2))), x1) -> c26(MINSORT(app(if_rm(eq(0, z1), 0, add(z1, z2)), x1), nil), APP(rm(0, add(0, add(z1, z2))), x1), RM(0, add(0, add(z1, z2)))) IF_MINSORT(true, add(0, add(0, x0)), x1) -> c4(APP(rm(0, add(0, x0)), x1)) IF_MINSORT(true, add(0, add(0, x0)), x1) -> c4(RM(0, add(0, x0))) IF_MINSORT(true, add(0, add(s(x0), x1)), z2) -> c26(MINSORT(add(s(x0), app(rm(0, x1), z2)), nil), APP(rm(0, add(s(x0), x1)), z2), RM(0, add(s(x0), x1))) IF_MINSORT(true, add(0, add(s(x0), nil)), x2) -> c26(MINSORT(app(add(s(x0), nil), x2), nil), APP(rm(0, add(s(x0), nil)), x2), RM(0, add(s(x0), nil))) IF_MINSORT(true, add(0, add(s(x0), add(z1, z2))), x2) -> c26(MINSORT(app(add(s(x0), if_rm(eq(0, z1), 0, add(z1, z2))), x2), nil), APP(rm(0, add(s(x0), add(z1, z2))), x2), RM(0, add(s(x0), add(z1, z2)))) IF_MINSORT(true, add(0, add(s(x0), x1)), x2) -> c5(APP(rm(0, add(s(x0), x1)), x2)) IF_MINSORT(true, add(0, add(s(x0), x1)), x2) -> c5(RM(0, add(s(x0), x1))) IF_MINSORT(true, add(s(x0), add(0, x1)), z2) -> c26(MINSORT(add(0, app(rm(s(x0), x1), z2)), nil), APP(rm(s(x0), add(0, x1)), z2), RM(s(x0), add(0, x1))) IF_MINSORT(true, add(s(x0), add(0, nil)), x2) -> c26(MINSORT(app(add(0, nil), x2), nil), APP(rm(s(x0), add(0, nil)), x2), RM(s(x0), add(0, nil))) IF_MINSORT(true, add(s(x0), add(0, add(z1, z2))), x2) -> c26(MINSORT(app(add(0, if_rm(eq(s(x0), z1), s(x0), add(z1, z2))), x2), nil), APP(rm(s(x0), add(0, add(z1, z2))), x2), RM(s(x0), add(0, add(z1, z2)))) IF_MINSORT(true, add(s(x0), add(0, x1)), x2) -> c7(APP(rm(s(x0), add(0, x1)), x2)) IF_MINSORT(true, add(s(x0), add(0, x1)), x2) -> c7(RM(s(x0), add(0, x1))) IF_MINSORT(true, add(s(0), add(s(0), z2)), x1) -> c26(MINSORT(app(rm(s(0), z2), x1), nil), APP(rm(s(0), add(s(0), z2)), x1), RM(s(0), add(s(0), z2))) IF_MINSORT(true, add(s(0), add(s(0), x0)), x1) -> c26(RM(s(0), add(s(0), x0))) IF_MINSORT(true, add(s(0), add(s(s(x0)), z2)), x2) -> c26(MINSORT(app(add(s(s(x0)), rm(s(0), z2)), x2), nil), APP(rm(s(0), add(s(s(x0)), z2)), x2), RM(s(0), add(s(s(x0)), z2))) IF_MINSORT(true, add(s(0), add(s(s(x0)), x1)), x2) -> c26(RM(s(0), add(s(s(x0)), x1))) IF_MINSORT(true, add(s(s(x0)), add(s(0), z2)), x2) -> c26(MINSORT(app(add(s(0), rm(s(s(x0)), z2)), x2), nil), APP(rm(s(s(x0)), add(s(0), z2)), x2), RM(s(s(x0)), add(s(0), z2))) IF_MINSORT(true, add(s(s(x0)), add(s(0), x1)), x2) -> c26(RM(s(s(x0)), add(s(0), x1))) IF_MINSORT(true, add(s(s(0)), add(s(s(0)), x2)), x3) -> c26(MINSORT(app(if_rm(true, s(s(0)), add(s(s(0)), x2)), x3), nil), APP(rm(s(s(0)), add(s(s(0)), x2)), x3), RM(s(s(0)), add(s(s(0)), x2))) IF_MINSORT(true, add(s(s(0)), add(s(s(s(z0))), x2)), x3) -> c26(MINSORT(app(if_rm(false, s(s(0)), add(s(s(s(z0))), x2)), x3), nil), APP(rm(s(s(0)), add(s(s(s(z0))), x2)), x3), RM(s(s(0)), add(s(s(s(z0))), x2))) IF_MINSORT(true, add(s(s(s(z0))), add(s(s(0)), x2)), x3) -> c26(MINSORT(app(if_rm(false, s(s(s(z0))), add(s(s(0)), x2)), x3), nil), APP(rm(s(s(s(z0))), add(s(s(0)), x2)), x3), RM(s(s(s(z0))), add(s(s(0)), x2))) IF_MINSORT(true, add(s(s(s(z0))), add(s(s(s(z1))), x2)), x3) -> c26(MINSORT(app(if_rm(eq(z0, z1), s(s(s(z0))), add(s(s(s(z1))), x2)), x3), nil), APP(rm(s(s(s(z0))), add(s(s(s(z1))), x2)), x3), RM(s(s(s(z0))), add(s(s(s(z1))), x2))) IF_MINSORT(true, add(s(s(x0)), add(s(s(x1)), x2)), x3) -> c9(APP(rm(s(s(x0)), add(s(s(x1)), x2)), x3)) IF_MINSORT(true, add(s(s(x0)), add(s(s(x1)), x2)), x3) -> c9(RM(s(s(x0)), add(s(s(x1)), x2))) IF_MINSORT(true, add(s(x0), add(s(x1), z2)), x3) -> c1(APP(if_rm(eq(s(x0), s(x1)), s(x0), add(s(x1), z2)), x3)) IF_MINSORT(true, add(0, add(0, x2)), x3) -> c26(APP(if_rm(true, 0, add(0, x2)), x3)) IF_MINSORT(true, add(0, add(s(z0), x2)), x3) -> c26(APP(if_rm(false, 0, add(s(z0), x2)), x3)) IF_MINSORT(true, add(s(z0), add(0, x2)), x3) -> c26(APP(if_rm(false, s(z0), add(0, x2)), x3)) IF_MINSORT(true, add(s(z0), add(s(z1), x2)), x3) -> c26(APP(if_rm(eq(z0, z1), s(z0), add(s(z1), x2)), x3)) MINSORT(add(s(x0), z0), x2) -> c25(IF_MINSORT(eq(s(x0), minIter(z0, add(s(x0), z0), 0)), add(s(x0), z0), x2), EQ(s(x0), minIter(add(s(x0), z0), add(s(x0), z0), 0)), MIN(add(s(x0), z0))) MINSORT(add(s(x0), x1), x2) -> c25(EQ(s(x0), minIter(add(s(x0), x1), add(s(x0), x1), 0)), MIN(add(s(x0), x1))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) MINITER(add(s(s(z0)), x1), x2, s(0)) -> c12(IF_MIN(false, x1, x2, s(0)), LE(s(s(z0)), s(0))) MINITER(add(s(x0), x1), x2, s(x3)) -> c12(LE(s(x0), s(x3))) MINITER(add(s(0), x1), x2, s(z0)) -> c12(LE(s(0), s(z0))) MINITER(add(s(s(s(z0))), x1), x2, s(s(0))) -> c12(IF_MIN(false, x1, x2, s(s(0))), LE(s(s(s(z0))), s(s(0)))) MINITER(add(s(s(x0)), x1), x2, s(s(x3))) -> c12(LE(s(s(x0)), s(s(x3)))) MINITER(add(s(s(0)), x1), x2, s(s(z0))) -> c12(LE(s(s(0)), s(s(z0)))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(0)))) -> c12(IF_MIN(false, x1, x2, s(s(s(0)))), LE(s(s(s(s(z0)))), s(s(s(0))))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(s(z1))))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(s(s(z1))))), LE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MINITER(add(s(s(s(x0))), x1), x2, s(s(s(x3)))) -> c12(LE(s(s(s(x0))), s(s(s(x3))))) MINITER(add(s(s(s(0))), x1), x2, s(s(s(z0)))) -> c12(LE(s(s(s(0))), s(s(s(z0))))) K tuples: IF_MINSORT(true, add(z0, z1), z2) -> c26(MINSORT(app(rm(z0, z1), z2), nil), APP(rm(z0, z1), z2), RM(z0, z1)) MINSORT(add(z0, z1), z2) -> c25(IF_MINSORT(eq(z0, min(add(z0, z1))), add(z0, z1), z2), EQ(z0, min(add(z0, z1))), MIN(add(z0, z1))) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MIN(add(z0, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) RM(z0, add(z1, z2)) -> c21(IF_RM(eq(z0, z1), z0, add(z1, z2)), EQ(z0, z1)) Defined Rule Symbols: le_2, eq_2, app_2, if_rm_3, rm_2, if_min_4, minIter_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, MIN_1, MINITER_3, IF_MIN_4, IF_RM_3, IF_MINSORT_3, RM_2, MINSORT_2 Compound Symbols: c3_1, c6_1, c8_1, c10_1, c11_1, c14_1, c22_1, c23_1, c27_1, c12_1, c21_1, c12_2, c21_2, c_1, c26_1, c1_1, c25_2, c2_1, c26_3, c4_1, c5_1, c7_1, c9_1, c25_3 ---------------------------------------- (183) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (184) 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) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) if_min(true, z0, z1, z2) -> z2 if_min(false, z0, z1, z2) -> minIter(z0, z1, z2) minIter(add(z0, z1), z2, z3) -> if_min(le(z0, z3), z1, z2, z3) minIter(nil, add(z0, z1), z2) -> minIter(add(z0, z1), add(z0, z1), s(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, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) RM(0, add(0, x2)) -> c21(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c21(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c21(IF_RM(false, s(z0), add(0, x2))) MINITER(add(s(s(z0)), x1), x2, s(0)) -> c12(IF_MIN(false, x1, x2, s(0)), LE(s(s(z0)), s(0))) MINITER(add(s(x0), x1), x2, s(x3)) -> c12(LE(s(x0), s(x3))) MINITER(add(s(0), x1), x2, s(z0)) -> c12(LE(s(0), s(z0))) RM(s(0), add(s(0), x2)) -> c21(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c21(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c21(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(x0), add(s(x1), x2)) -> c21(EQ(s(x0), s(x1))) MINSORT(add(x0, x1), x2) -> c(MIN(add(x0, x1))) MINITER(add(s(s(s(z0))), x1), x2, s(s(0))) -> c12(IF_MIN(false, x1, x2, s(s(0))), LE(s(s(s(z0))), s(s(0)))) MINITER(add(s(s(x0)), x1), x2, s(s(x3))) -> c12(LE(s(s(x0)), s(s(x3)))) MINITER(add(s(s(0)), x1), x2, s(s(z0))) -> c12(LE(s(s(0)), s(s(z0)))) RM(s(s(0)), add(s(s(0)), x2)) -> c21(IF_RM(true, s(s(0)), add(s(s(0)), x2)), EQ(s(s(0)), s(s(0)))) RM(s(s(0)), add(s(s(s(z0))), x2)) -> c21(IF_RM(false, s(s(0)), add(s(s(s(z0))), x2)), EQ(s(s(0)), s(s(s(z0))))) RM(s(s(s(z0))), add(s(s(0)), x2)) -> c21(IF_RM(false, s(s(s(z0))), add(s(s(0)), x2)), EQ(s(s(s(z0))), s(s(0)))) RM(s(s(x0)), add(s(s(x1)), x2)) -> c21(EQ(s(s(x0)), s(s(x1)))) IF_MINSORT(true, add(0, add(0, x0)), x1) -> c26(RM(0, add(0, x0))) IF_MINSORT(true, add(0, add(s(x0), x1)), x2) -> c26(RM(0, add(s(x0), x1))) IF_MINSORT(true, add(s(x0), add(0, x1)), x2) -> c26(RM(s(x0), add(0, x1))) IF_MINSORT(true, add(s(x0), add(s(x1), x2)), x3) -> c1(RM(s(x0), add(s(x1), x2))) IF_MINSORT(true, add(z0, nil), x1) -> c26(MINSORT(x1, nil)) MINSORT(add(0, x1), x2) -> c25(IF_MINSORT(eq(0, if_min(true, x1, add(0, x1), 0)), add(0, x1), x2), MIN(add(0, x1))) MINSORT(add(x0, x1), x2) -> c2(EQ(x0, minIter(add(x0, x1), add(x0, x1), 0))) MINSORT(add(x0, x1), x2) -> c2(MIN(add(x0, x1))) MINSORT(add(z0, z1), x2) -> c(EQ(z0, if_min(le(z0, 0), z1, add(z0, z1), 0))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(0)))) -> c12(IF_MIN(false, x1, x2, s(s(s(0)))), LE(s(s(s(s(z0)))), s(s(s(0))))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(s(z1))))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(s(s(z1))))), LE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MINITER(add(s(s(s(x0))), x1), x2, s(s(s(x3)))) -> c12(LE(s(s(s(x0))), s(s(s(x3))))) MINITER(add(s(s(s(0))), x1), x2, s(s(s(z0)))) -> c12(LE(s(s(s(0))), s(s(s(z0))))) RM(s(s(s(0))), add(s(s(s(0))), x2)) -> c21(IF_RM(true, s(s(s(0))), add(s(s(s(0))), x2)), EQ(s(s(s(0))), s(s(s(0))))) RM(s(s(s(0))), add(s(s(s(s(z0)))), x2)) -> c21(IF_RM(false, s(s(s(0))), add(s(s(s(s(z0)))), x2)), EQ(s(s(s(0))), s(s(s(s(z0)))))) RM(s(s(s(s(z0)))), add(s(s(s(0))), x2)) -> c21(IF_RM(false, s(s(s(s(z0)))), add(s(s(s(0))), x2)), EQ(s(s(s(s(z0)))), s(s(s(0))))) RM(s(s(s(s(z0)))), add(s(s(s(s(z1)))), x2)) -> c21(IF_RM(eq(z0, z1), s(s(s(s(z0)))), add(s(s(s(s(z1)))), x2)), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) RM(s(s(s(x0))), add(s(s(s(x1))), x2)) -> c21(EQ(s(s(s(x0))), s(s(s(x1))))) IF_MINSORT(true, add(0, add(0, nil)), x1) -> c26(MINSORT(app(nil, x1), nil), APP(rm(0, add(0, nil)), x1), RM(0, add(0, nil))) IF_MINSORT(true, add(0, add(0, add(z1, z2))), x1) -> c26(MINSORT(app(if_rm(eq(0, z1), 0, add(z1, z2)), x1), nil), APP(rm(0, add(0, add(z1, z2))), x1), RM(0, add(0, add(z1, z2)))) IF_MINSORT(true, add(0, add(0, x0)), x1) -> c4(APP(rm(0, add(0, x0)), x1)) IF_MINSORT(true, add(0, add(0, x0)), x1) -> c4(RM(0, add(0, x0))) IF_MINSORT(true, add(0, add(s(x0), x1)), z2) -> c26(MINSORT(add(s(x0), app(rm(0, x1), z2)), nil), APP(rm(0, add(s(x0), x1)), z2), RM(0, add(s(x0), x1))) IF_MINSORT(true, add(0, add(s(x0), nil)), x2) -> c26(MINSORT(app(add(s(x0), nil), x2), nil), APP(rm(0, add(s(x0), nil)), x2), RM(0, add(s(x0), nil))) IF_MINSORT(true, add(0, add(s(x0), add(z1, z2))), x2) -> c26(MINSORT(app(add(s(x0), if_rm(eq(0, z1), 0, add(z1, z2))), x2), nil), APP(rm(0, add(s(x0), add(z1, z2))), x2), RM(0, add(s(x0), add(z1, z2)))) IF_MINSORT(true, add(0, add(s(x0), x1)), x2) -> c5(APP(rm(0, add(s(x0), x1)), x2)) IF_MINSORT(true, add(0, add(s(x0), x1)), x2) -> c5(RM(0, add(s(x0), x1))) IF_MINSORT(true, add(s(x0), add(0, x1)), z2) -> c26(MINSORT(add(0, app(rm(s(x0), x1), z2)), nil), APP(rm(s(x0), add(0, x1)), z2), RM(s(x0), add(0, x1))) IF_MINSORT(true, add(s(x0), add(0, nil)), x2) -> c26(MINSORT(app(add(0, nil), x2), nil), APP(rm(s(x0), add(0, nil)), x2), RM(s(x0), add(0, nil))) IF_MINSORT(true, add(s(x0), add(0, add(z1, z2))), x2) -> c26(MINSORT(app(add(0, if_rm(eq(s(x0), z1), s(x0), add(z1, z2))), x2), nil), APP(rm(s(x0), add(0, add(z1, z2))), x2), RM(s(x0), add(0, add(z1, z2)))) IF_MINSORT(true, add(s(x0), add(0, x1)), x2) -> c7(APP(rm(s(x0), add(0, x1)), x2)) IF_MINSORT(true, add(s(x0), add(0, x1)), x2) -> c7(RM(s(x0), add(0, x1))) IF_MINSORT(true, add(s(0), add(s(0), z2)), x1) -> c26(MINSORT(app(rm(s(0), z2), x1), nil), APP(rm(s(0), add(s(0), z2)), x1), RM(s(0), add(s(0), z2))) IF_MINSORT(true, add(s(0), add(s(0), x0)), x1) -> c26(RM(s(0), add(s(0), x0))) IF_MINSORT(true, add(s(0), add(s(s(x0)), z2)), x2) -> c26(MINSORT(app(add(s(s(x0)), rm(s(0), z2)), x2), nil), APP(rm(s(0), add(s(s(x0)), z2)), x2), RM(s(0), add(s(s(x0)), z2))) IF_MINSORT(true, add(s(0), add(s(s(x0)), x1)), x2) -> c26(RM(s(0), add(s(s(x0)), x1))) IF_MINSORT(true, add(s(s(x0)), add(s(0), z2)), x2) -> c26(MINSORT(app(add(s(0), rm(s(s(x0)), z2)), x2), nil), APP(rm(s(s(x0)), add(s(0), z2)), x2), RM(s(s(x0)), add(s(0), z2))) IF_MINSORT(true, add(s(s(x0)), add(s(0), x1)), x2) -> c26(RM(s(s(x0)), add(s(0), x1))) IF_MINSORT(true, add(s(s(0)), add(s(s(0)), x2)), x3) -> c26(MINSORT(app(if_rm(true, s(s(0)), add(s(s(0)), x2)), x3), nil), APP(rm(s(s(0)), add(s(s(0)), x2)), x3), RM(s(s(0)), add(s(s(0)), x2))) IF_MINSORT(true, add(s(s(0)), add(s(s(s(z0))), x2)), x3) -> c26(MINSORT(app(if_rm(false, s(s(0)), add(s(s(s(z0))), x2)), x3), nil), APP(rm(s(s(0)), add(s(s(s(z0))), x2)), x3), RM(s(s(0)), add(s(s(s(z0))), x2))) IF_MINSORT(true, add(s(s(s(z0))), add(s(s(0)), x2)), x3) -> c26(MINSORT(app(if_rm(false, s(s(s(z0))), add(s(s(0)), x2)), x3), nil), APP(rm(s(s(s(z0))), add(s(s(0)), x2)), x3), RM(s(s(s(z0))), add(s(s(0)), x2))) IF_MINSORT(true, add(s(s(s(z0))), add(s(s(s(z1))), x2)), x3) -> c26(MINSORT(app(if_rm(eq(z0, z1), s(s(s(z0))), add(s(s(s(z1))), x2)), x3), nil), APP(rm(s(s(s(z0))), add(s(s(s(z1))), x2)), x3), RM(s(s(s(z0))), add(s(s(s(z1))), x2))) IF_MINSORT(true, add(s(s(x0)), add(s(s(x1)), x2)), x3) -> c9(APP(rm(s(s(x0)), add(s(s(x1)), x2)), x3)) IF_MINSORT(true, add(s(s(x0)), add(s(s(x1)), x2)), x3) -> c9(RM(s(s(x0)), add(s(s(x1)), x2))) IF_MINSORT(true, add(s(x0), add(s(x1), z2)), x3) -> c1(APP(if_rm(eq(s(x0), s(x1)), s(x0), add(s(x1), z2)), x3)) IF_MINSORT(true, add(0, add(0, x2)), x3) -> c26(APP(if_rm(true, 0, add(0, x2)), x3)) IF_MINSORT(true, add(0, add(s(z0), x2)), x3) -> c26(APP(if_rm(false, 0, add(s(z0), x2)), x3)) IF_MINSORT(true, add(s(z0), add(0, x2)), x3) -> c26(APP(if_rm(false, s(z0), add(0, x2)), x3)) IF_MINSORT(true, add(s(z0), add(s(z1), x2)), x3) -> c26(APP(if_rm(eq(z0, z1), s(z0), add(s(z1), x2)), x3)) MINSORT(add(s(x0), z0), x2) -> c25(IF_MINSORT(eq(s(x0), minIter(z0, add(s(x0), z0), 0)), add(s(x0), z0), x2), EQ(s(x0), minIter(add(s(x0), z0), add(s(x0), z0), 0)), MIN(add(s(x0), z0))) MINSORT(add(s(x0), x1), x2) -> c13(EQ(s(x0), minIter(add(s(x0), x1), add(s(x0), x1), 0))) MINSORT(add(s(x0), x1), x2) -> c13(MIN(add(s(x0), x1))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) MINITER(add(s(s(z0)), x1), x2, s(0)) -> c12(IF_MIN(false, x1, x2, s(0)), LE(s(s(z0)), s(0))) MINITER(add(s(x0), x1), x2, s(x3)) -> c12(LE(s(x0), s(x3))) MINITER(add(s(0), x1), x2, s(z0)) -> c12(LE(s(0), s(z0))) MINITER(add(s(s(s(z0))), x1), x2, s(s(0))) -> c12(IF_MIN(false, x1, x2, s(s(0))), LE(s(s(s(z0))), s(s(0)))) MINITER(add(s(s(x0)), x1), x2, s(s(x3))) -> c12(LE(s(s(x0)), s(s(x3)))) MINITER(add(s(s(0)), x1), x2, s(s(z0))) -> c12(LE(s(s(0)), s(s(z0)))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(0)))) -> c12(IF_MIN(false, x1, x2, s(s(s(0)))), LE(s(s(s(s(z0)))), s(s(s(0))))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(s(z1))))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(s(s(z1))))), LE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MINITER(add(s(s(s(x0))), x1), x2, s(s(s(x3)))) -> c12(LE(s(s(s(x0))), s(s(s(x3))))) MINITER(add(s(s(s(0))), x1), x2, s(s(s(z0)))) -> c12(LE(s(s(s(0))), s(s(s(z0))))) K tuples: IF_MINSORT(true, add(z0, z1), z2) -> c26(MINSORT(app(rm(z0, z1), z2), nil), APP(rm(z0, z1), z2), RM(z0, z1)) MINSORT(add(z0, z1), z2) -> c25(IF_MINSORT(eq(z0, min(add(z0, z1))), add(z0, z1), z2), EQ(z0, min(add(z0, z1))), MIN(add(z0, z1))) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MIN(add(z0, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) RM(z0, add(z1, z2)) -> c21(IF_RM(eq(z0, z1), z0, add(z1, z2)), EQ(z0, z1)) Defined Rule Symbols: le_2, eq_2, app_2, if_rm_3, rm_2, if_min_4, minIter_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, MIN_1, MINITER_3, IF_MIN_4, IF_RM_3, IF_MINSORT_3, RM_2, MINSORT_2 Compound Symbols: c3_1, c6_1, c8_1, c10_1, c11_1, c14_1, c22_1, c23_1, c27_1, c12_1, c21_1, c12_2, c21_2, c_1, c26_1, c1_1, c25_2, c2_1, c26_3, c4_1, c5_1, c7_1, c9_1, c25_3, c13_1 ---------------------------------------- (185) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MINSORT(add(0, x1), x2) -> c25(IF_MINSORT(eq(0, if_min(true, x1, add(0, x1), 0)), add(0, x1), x2), MIN(add(0, x1))) by MINSORT(add(0, z0), x1) -> c25(IF_MINSORT(eq(0, 0), add(0, z0), x1), MIN(add(0, z0))) MINSORT(add(0, x0), x1) -> c25(MIN(add(0, x0))) ---------------------------------------- (186) 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) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) if_min(true, z0, z1, z2) -> z2 if_min(false, z0, z1, z2) -> minIter(z0, z1, z2) minIter(add(z0, z1), z2, z3) -> if_min(le(z0, z3), z1, z2, z3) minIter(nil, add(z0, z1), z2) -> minIter(add(z0, z1), add(z0, z1), s(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, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) RM(0, add(0, x2)) -> c21(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c21(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c21(IF_RM(false, s(z0), add(0, x2))) MINITER(add(s(s(z0)), x1), x2, s(0)) -> c12(IF_MIN(false, x1, x2, s(0)), LE(s(s(z0)), s(0))) MINITER(add(s(x0), x1), x2, s(x3)) -> c12(LE(s(x0), s(x3))) MINITER(add(s(0), x1), x2, s(z0)) -> c12(LE(s(0), s(z0))) RM(s(0), add(s(0), x2)) -> c21(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c21(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c21(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(x0), add(s(x1), x2)) -> c21(EQ(s(x0), s(x1))) MINSORT(add(x0, x1), x2) -> c(MIN(add(x0, x1))) MINITER(add(s(s(s(z0))), x1), x2, s(s(0))) -> c12(IF_MIN(false, x1, x2, s(s(0))), LE(s(s(s(z0))), s(s(0)))) MINITER(add(s(s(x0)), x1), x2, s(s(x3))) -> c12(LE(s(s(x0)), s(s(x3)))) MINITER(add(s(s(0)), x1), x2, s(s(z0))) -> c12(LE(s(s(0)), s(s(z0)))) RM(s(s(0)), add(s(s(0)), x2)) -> c21(IF_RM(true, s(s(0)), add(s(s(0)), x2)), EQ(s(s(0)), s(s(0)))) RM(s(s(0)), add(s(s(s(z0))), x2)) -> c21(IF_RM(false, s(s(0)), add(s(s(s(z0))), x2)), EQ(s(s(0)), s(s(s(z0))))) RM(s(s(s(z0))), add(s(s(0)), x2)) -> c21(IF_RM(false, s(s(s(z0))), add(s(s(0)), x2)), EQ(s(s(s(z0))), s(s(0)))) RM(s(s(x0)), add(s(s(x1)), x2)) -> c21(EQ(s(s(x0)), s(s(x1)))) IF_MINSORT(true, add(0, add(0, x0)), x1) -> c26(RM(0, add(0, x0))) IF_MINSORT(true, add(0, add(s(x0), x1)), x2) -> c26(RM(0, add(s(x0), x1))) IF_MINSORT(true, add(s(x0), add(0, x1)), x2) -> c26(RM(s(x0), add(0, x1))) IF_MINSORT(true, add(s(x0), add(s(x1), x2)), x3) -> c1(RM(s(x0), add(s(x1), x2))) IF_MINSORT(true, add(z0, nil), x1) -> c26(MINSORT(x1, nil)) MINSORT(add(x0, x1), x2) -> c2(EQ(x0, minIter(add(x0, x1), add(x0, x1), 0))) MINSORT(add(x0, x1), x2) -> c2(MIN(add(x0, x1))) MINSORT(add(z0, z1), x2) -> c(EQ(z0, if_min(le(z0, 0), z1, add(z0, z1), 0))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(0)))) -> c12(IF_MIN(false, x1, x2, s(s(s(0)))), LE(s(s(s(s(z0)))), s(s(s(0))))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(s(z1))))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(s(s(z1))))), LE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MINITER(add(s(s(s(x0))), x1), x2, s(s(s(x3)))) -> c12(LE(s(s(s(x0))), s(s(s(x3))))) MINITER(add(s(s(s(0))), x1), x2, s(s(s(z0)))) -> c12(LE(s(s(s(0))), s(s(s(z0))))) RM(s(s(s(0))), add(s(s(s(0))), x2)) -> c21(IF_RM(true, s(s(s(0))), add(s(s(s(0))), x2)), EQ(s(s(s(0))), s(s(s(0))))) RM(s(s(s(0))), add(s(s(s(s(z0)))), x2)) -> c21(IF_RM(false, s(s(s(0))), add(s(s(s(s(z0)))), x2)), EQ(s(s(s(0))), s(s(s(s(z0)))))) RM(s(s(s(s(z0)))), add(s(s(s(0))), x2)) -> c21(IF_RM(false, s(s(s(s(z0)))), add(s(s(s(0))), x2)), EQ(s(s(s(s(z0)))), s(s(s(0))))) RM(s(s(s(s(z0)))), add(s(s(s(s(z1)))), x2)) -> c21(IF_RM(eq(z0, z1), s(s(s(s(z0)))), add(s(s(s(s(z1)))), x2)), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) RM(s(s(s(x0))), add(s(s(s(x1))), x2)) -> c21(EQ(s(s(s(x0))), s(s(s(x1))))) IF_MINSORT(true, add(0, add(0, nil)), x1) -> c26(MINSORT(app(nil, x1), nil), APP(rm(0, add(0, nil)), x1), RM(0, add(0, nil))) IF_MINSORT(true, add(0, add(0, add(z1, z2))), x1) -> c26(MINSORT(app(if_rm(eq(0, z1), 0, add(z1, z2)), x1), nil), APP(rm(0, add(0, add(z1, z2))), x1), RM(0, add(0, add(z1, z2)))) IF_MINSORT(true, add(0, add(0, x0)), x1) -> c4(APP(rm(0, add(0, x0)), x1)) IF_MINSORT(true, add(0, add(0, x0)), x1) -> c4(RM(0, add(0, x0))) IF_MINSORT(true, add(0, add(s(x0), x1)), z2) -> c26(MINSORT(add(s(x0), app(rm(0, x1), z2)), nil), APP(rm(0, add(s(x0), x1)), z2), RM(0, add(s(x0), x1))) IF_MINSORT(true, add(0, add(s(x0), nil)), x2) -> c26(MINSORT(app(add(s(x0), nil), x2), nil), APP(rm(0, add(s(x0), nil)), x2), RM(0, add(s(x0), nil))) IF_MINSORT(true, add(0, add(s(x0), add(z1, z2))), x2) -> c26(MINSORT(app(add(s(x0), if_rm(eq(0, z1), 0, add(z1, z2))), x2), nil), APP(rm(0, add(s(x0), add(z1, z2))), x2), RM(0, add(s(x0), add(z1, z2)))) IF_MINSORT(true, add(0, add(s(x0), x1)), x2) -> c5(APP(rm(0, add(s(x0), x1)), x2)) IF_MINSORT(true, add(0, add(s(x0), x1)), x2) -> c5(RM(0, add(s(x0), x1))) IF_MINSORT(true, add(s(x0), add(0, x1)), z2) -> c26(MINSORT(add(0, app(rm(s(x0), x1), z2)), nil), APP(rm(s(x0), add(0, x1)), z2), RM(s(x0), add(0, x1))) IF_MINSORT(true, add(s(x0), add(0, nil)), x2) -> c26(MINSORT(app(add(0, nil), x2), nil), APP(rm(s(x0), add(0, nil)), x2), RM(s(x0), add(0, nil))) IF_MINSORT(true, add(s(x0), add(0, add(z1, z2))), x2) -> c26(MINSORT(app(add(0, if_rm(eq(s(x0), z1), s(x0), add(z1, z2))), x2), nil), APP(rm(s(x0), add(0, add(z1, z2))), x2), RM(s(x0), add(0, add(z1, z2)))) IF_MINSORT(true, add(s(x0), add(0, x1)), x2) -> c7(APP(rm(s(x0), add(0, x1)), x2)) IF_MINSORT(true, add(s(x0), add(0, x1)), x2) -> c7(RM(s(x0), add(0, x1))) IF_MINSORT(true, add(s(0), add(s(0), z2)), x1) -> c26(MINSORT(app(rm(s(0), z2), x1), nil), APP(rm(s(0), add(s(0), z2)), x1), RM(s(0), add(s(0), z2))) IF_MINSORT(true, add(s(0), add(s(0), x0)), x1) -> c26(RM(s(0), add(s(0), x0))) IF_MINSORT(true, add(s(0), add(s(s(x0)), z2)), x2) -> c26(MINSORT(app(add(s(s(x0)), rm(s(0), z2)), x2), nil), APP(rm(s(0), add(s(s(x0)), z2)), x2), RM(s(0), add(s(s(x0)), z2))) IF_MINSORT(true, add(s(0), add(s(s(x0)), x1)), x2) -> c26(RM(s(0), add(s(s(x0)), x1))) IF_MINSORT(true, add(s(s(x0)), add(s(0), z2)), x2) -> c26(MINSORT(app(add(s(0), rm(s(s(x0)), z2)), x2), nil), APP(rm(s(s(x0)), add(s(0), z2)), x2), RM(s(s(x0)), add(s(0), z2))) IF_MINSORT(true, add(s(s(x0)), add(s(0), x1)), x2) -> c26(RM(s(s(x0)), add(s(0), x1))) IF_MINSORT(true, add(s(s(0)), add(s(s(0)), x2)), x3) -> c26(MINSORT(app(if_rm(true, s(s(0)), add(s(s(0)), x2)), x3), nil), APP(rm(s(s(0)), add(s(s(0)), x2)), x3), RM(s(s(0)), add(s(s(0)), x2))) IF_MINSORT(true, add(s(s(0)), add(s(s(s(z0))), x2)), x3) -> c26(MINSORT(app(if_rm(false, s(s(0)), add(s(s(s(z0))), x2)), x3), nil), APP(rm(s(s(0)), add(s(s(s(z0))), x2)), x3), RM(s(s(0)), add(s(s(s(z0))), x2))) IF_MINSORT(true, add(s(s(s(z0))), add(s(s(0)), x2)), x3) -> c26(MINSORT(app(if_rm(false, s(s(s(z0))), add(s(s(0)), x2)), x3), nil), APP(rm(s(s(s(z0))), add(s(s(0)), x2)), x3), RM(s(s(s(z0))), add(s(s(0)), x2))) IF_MINSORT(true, add(s(s(s(z0))), add(s(s(s(z1))), x2)), x3) -> c26(MINSORT(app(if_rm(eq(z0, z1), s(s(s(z0))), add(s(s(s(z1))), x2)), x3), nil), APP(rm(s(s(s(z0))), add(s(s(s(z1))), x2)), x3), RM(s(s(s(z0))), add(s(s(s(z1))), x2))) IF_MINSORT(true, add(s(s(x0)), add(s(s(x1)), x2)), x3) -> c9(APP(rm(s(s(x0)), add(s(s(x1)), x2)), x3)) IF_MINSORT(true, add(s(s(x0)), add(s(s(x1)), x2)), x3) -> c9(RM(s(s(x0)), add(s(s(x1)), x2))) IF_MINSORT(true, add(s(x0), add(s(x1), z2)), x3) -> c1(APP(if_rm(eq(s(x0), s(x1)), s(x0), add(s(x1), z2)), x3)) IF_MINSORT(true, add(0, add(0, x2)), x3) -> c26(APP(if_rm(true, 0, add(0, x2)), x3)) IF_MINSORT(true, add(0, add(s(z0), x2)), x3) -> c26(APP(if_rm(false, 0, add(s(z0), x2)), x3)) IF_MINSORT(true, add(s(z0), add(0, x2)), x3) -> c26(APP(if_rm(false, s(z0), add(0, x2)), x3)) IF_MINSORT(true, add(s(z0), add(s(z1), x2)), x3) -> c26(APP(if_rm(eq(z0, z1), s(z0), add(s(z1), x2)), x3)) MINSORT(add(s(x0), z0), x2) -> c25(IF_MINSORT(eq(s(x0), minIter(z0, add(s(x0), z0), 0)), add(s(x0), z0), x2), EQ(s(x0), minIter(add(s(x0), z0), add(s(x0), z0), 0)), MIN(add(s(x0), z0))) MINSORT(add(s(x0), x1), x2) -> c13(EQ(s(x0), minIter(add(s(x0), x1), add(s(x0), x1), 0))) MINSORT(add(s(x0), x1), x2) -> c13(MIN(add(s(x0), x1))) MINSORT(add(0, z0), x1) -> c25(IF_MINSORT(eq(0, 0), add(0, z0), x1), MIN(add(0, z0))) MINSORT(add(0, x0), x1) -> c25(MIN(add(0, x0))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) MINITER(add(s(s(z0)), x1), x2, s(0)) -> c12(IF_MIN(false, x1, x2, s(0)), LE(s(s(z0)), s(0))) MINITER(add(s(x0), x1), x2, s(x3)) -> c12(LE(s(x0), s(x3))) MINITER(add(s(0), x1), x2, s(z0)) -> c12(LE(s(0), s(z0))) MINITER(add(s(s(s(z0))), x1), x2, s(s(0))) -> c12(IF_MIN(false, x1, x2, s(s(0))), LE(s(s(s(z0))), s(s(0)))) MINITER(add(s(s(x0)), x1), x2, s(s(x3))) -> c12(LE(s(s(x0)), s(s(x3)))) MINITER(add(s(s(0)), x1), x2, s(s(z0))) -> c12(LE(s(s(0)), s(s(z0)))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(0)))) -> c12(IF_MIN(false, x1, x2, s(s(s(0)))), LE(s(s(s(s(z0)))), s(s(s(0))))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(s(z1))))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(s(s(z1))))), LE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MINITER(add(s(s(s(x0))), x1), x2, s(s(s(x3)))) -> c12(LE(s(s(s(x0))), s(s(s(x3))))) MINITER(add(s(s(s(0))), x1), x2, s(s(s(z0)))) -> c12(LE(s(s(s(0))), s(s(s(z0))))) K tuples: IF_MINSORT(true, add(z0, z1), z2) -> c26(MINSORT(app(rm(z0, z1), z2), nil), APP(rm(z0, z1), z2), RM(z0, z1)) MINSORT(add(z0, z1), z2) -> c25(IF_MINSORT(eq(z0, min(add(z0, z1))), add(z0, z1), z2), EQ(z0, min(add(z0, z1))), MIN(add(z0, z1))) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MIN(add(z0, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) RM(z0, add(z1, z2)) -> c21(IF_RM(eq(z0, z1), z0, add(z1, z2)), EQ(z0, z1)) Defined Rule Symbols: le_2, eq_2, app_2, if_rm_3, rm_2, if_min_4, minIter_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, MIN_1, MINITER_3, IF_MIN_4, IF_RM_3, IF_MINSORT_3, RM_2, MINSORT_2 Compound Symbols: c3_1, c6_1, c8_1, c10_1, c11_1, c14_1, c22_1, c23_1, c27_1, c12_1, c21_1, c12_2, c21_2, c_1, c26_1, c1_1, c2_1, c26_3, c4_1, c5_1, c7_1, c9_1, c25_3, c13_1, c25_2, c25_1 ---------------------------------------- (187) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MINSORT(add(x0, x1), x2) -> c2(EQ(x0, minIter(add(x0, x1), add(x0, x1), 0))) by MINSORT(add(z0, z1), x2) -> c2(EQ(z0, if_min(le(z0, 0), z1, add(z0, z1), 0))) ---------------------------------------- (188) 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) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) if_min(true, z0, z1, z2) -> z2 if_min(false, z0, z1, z2) -> minIter(z0, z1, z2) minIter(add(z0, z1), z2, z3) -> if_min(le(z0, z3), z1, z2, z3) minIter(nil, add(z0, z1), z2) -> minIter(add(z0, z1), add(z0, z1), s(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, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) RM(0, add(0, x2)) -> c21(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c21(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c21(IF_RM(false, s(z0), add(0, x2))) MINITER(add(s(s(z0)), x1), x2, s(0)) -> c12(IF_MIN(false, x1, x2, s(0)), LE(s(s(z0)), s(0))) MINITER(add(s(x0), x1), x2, s(x3)) -> c12(LE(s(x0), s(x3))) MINITER(add(s(0), x1), x2, s(z0)) -> c12(LE(s(0), s(z0))) RM(s(0), add(s(0), x2)) -> c21(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c21(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c21(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(x0), add(s(x1), x2)) -> c21(EQ(s(x0), s(x1))) MINSORT(add(x0, x1), x2) -> c(MIN(add(x0, x1))) MINITER(add(s(s(s(z0))), x1), x2, s(s(0))) -> c12(IF_MIN(false, x1, x2, s(s(0))), LE(s(s(s(z0))), s(s(0)))) MINITER(add(s(s(x0)), x1), x2, s(s(x3))) -> c12(LE(s(s(x0)), s(s(x3)))) MINITER(add(s(s(0)), x1), x2, s(s(z0))) -> c12(LE(s(s(0)), s(s(z0)))) RM(s(s(0)), add(s(s(0)), x2)) -> c21(IF_RM(true, s(s(0)), add(s(s(0)), x2)), EQ(s(s(0)), s(s(0)))) RM(s(s(0)), add(s(s(s(z0))), x2)) -> c21(IF_RM(false, s(s(0)), add(s(s(s(z0))), x2)), EQ(s(s(0)), s(s(s(z0))))) RM(s(s(s(z0))), add(s(s(0)), x2)) -> c21(IF_RM(false, s(s(s(z0))), add(s(s(0)), x2)), EQ(s(s(s(z0))), s(s(0)))) RM(s(s(x0)), add(s(s(x1)), x2)) -> c21(EQ(s(s(x0)), s(s(x1)))) IF_MINSORT(true, add(0, add(0, x0)), x1) -> c26(RM(0, add(0, x0))) IF_MINSORT(true, add(0, add(s(x0), x1)), x2) -> c26(RM(0, add(s(x0), x1))) IF_MINSORT(true, add(s(x0), add(0, x1)), x2) -> c26(RM(s(x0), add(0, x1))) IF_MINSORT(true, add(s(x0), add(s(x1), x2)), x3) -> c1(RM(s(x0), add(s(x1), x2))) IF_MINSORT(true, add(z0, nil), x1) -> c26(MINSORT(x1, nil)) MINSORT(add(x0, x1), x2) -> c2(MIN(add(x0, x1))) MINSORT(add(z0, z1), x2) -> c(EQ(z0, if_min(le(z0, 0), z1, add(z0, z1), 0))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(0)))) -> c12(IF_MIN(false, x1, x2, s(s(s(0)))), LE(s(s(s(s(z0)))), s(s(s(0))))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(s(z1))))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(s(s(z1))))), LE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MINITER(add(s(s(s(x0))), x1), x2, s(s(s(x3)))) -> c12(LE(s(s(s(x0))), s(s(s(x3))))) MINITER(add(s(s(s(0))), x1), x2, s(s(s(z0)))) -> c12(LE(s(s(s(0))), s(s(s(z0))))) RM(s(s(s(0))), add(s(s(s(0))), x2)) -> c21(IF_RM(true, s(s(s(0))), add(s(s(s(0))), x2)), EQ(s(s(s(0))), s(s(s(0))))) RM(s(s(s(0))), add(s(s(s(s(z0)))), x2)) -> c21(IF_RM(false, s(s(s(0))), add(s(s(s(s(z0)))), x2)), EQ(s(s(s(0))), s(s(s(s(z0)))))) RM(s(s(s(s(z0)))), add(s(s(s(0))), x2)) -> c21(IF_RM(false, s(s(s(s(z0)))), add(s(s(s(0))), x2)), EQ(s(s(s(s(z0)))), s(s(s(0))))) RM(s(s(s(s(z0)))), add(s(s(s(s(z1)))), x2)) -> c21(IF_RM(eq(z0, z1), s(s(s(s(z0)))), add(s(s(s(s(z1)))), x2)), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) RM(s(s(s(x0))), add(s(s(s(x1))), x2)) -> c21(EQ(s(s(s(x0))), s(s(s(x1))))) IF_MINSORT(true, add(0, add(0, nil)), x1) -> c26(MINSORT(app(nil, x1), nil), APP(rm(0, add(0, nil)), x1), RM(0, add(0, nil))) IF_MINSORT(true, add(0, add(0, add(z1, z2))), x1) -> c26(MINSORT(app(if_rm(eq(0, z1), 0, add(z1, z2)), x1), nil), APP(rm(0, add(0, add(z1, z2))), x1), RM(0, add(0, add(z1, z2)))) IF_MINSORT(true, add(0, add(0, x0)), x1) -> c4(APP(rm(0, add(0, x0)), x1)) IF_MINSORT(true, add(0, add(0, x0)), x1) -> c4(RM(0, add(0, x0))) IF_MINSORT(true, add(0, add(s(x0), x1)), z2) -> c26(MINSORT(add(s(x0), app(rm(0, x1), z2)), nil), APP(rm(0, add(s(x0), x1)), z2), RM(0, add(s(x0), x1))) IF_MINSORT(true, add(0, add(s(x0), nil)), x2) -> c26(MINSORT(app(add(s(x0), nil), x2), nil), APP(rm(0, add(s(x0), nil)), x2), RM(0, add(s(x0), nil))) IF_MINSORT(true, add(0, add(s(x0), add(z1, z2))), x2) -> c26(MINSORT(app(add(s(x0), if_rm(eq(0, z1), 0, add(z1, z2))), x2), nil), APP(rm(0, add(s(x0), add(z1, z2))), x2), RM(0, add(s(x0), add(z1, z2)))) IF_MINSORT(true, add(0, add(s(x0), x1)), x2) -> c5(APP(rm(0, add(s(x0), x1)), x2)) IF_MINSORT(true, add(0, add(s(x0), x1)), x2) -> c5(RM(0, add(s(x0), x1))) IF_MINSORT(true, add(s(x0), add(0, x1)), z2) -> c26(MINSORT(add(0, app(rm(s(x0), x1), z2)), nil), APP(rm(s(x0), add(0, x1)), z2), RM(s(x0), add(0, x1))) IF_MINSORT(true, add(s(x0), add(0, nil)), x2) -> c26(MINSORT(app(add(0, nil), x2), nil), APP(rm(s(x0), add(0, nil)), x2), RM(s(x0), add(0, nil))) IF_MINSORT(true, add(s(x0), add(0, add(z1, z2))), x2) -> c26(MINSORT(app(add(0, if_rm(eq(s(x0), z1), s(x0), add(z1, z2))), x2), nil), APP(rm(s(x0), add(0, add(z1, z2))), x2), RM(s(x0), add(0, add(z1, z2)))) IF_MINSORT(true, add(s(x0), add(0, x1)), x2) -> c7(APP(rm(s(x0), add(0, x1)), x2)) IF_MINSORT(true, add(s(x0), add(0, x1)), x2) -> c7(RM(s(x0), add(0, x1))) IF_MINSORT(true, add(s(0), add(s(0), z2)), x1) -> c26(MINSORT(app(rm(s(0), z2), x1), nil), APP(rm(s(0), add(s(0), z2)), x1), RM(s(0), add(s(0), z2))) IF_MINSORT(true, add(s(0), add(s(0), x0)), x1) -> c26(RM(s(0), add(s(0), x0))) IF_MINSORT(true, add(s(0), add(s(s(x0)), z2)), x2) -> c26(MINSORT(app(add(s(s(x0)), rm(s(0), z2)), x2), nil), APP(rm(s(0), add(s(s(x0)), z2)), x2), RM(s(0), add(s(s(x0)), z2))) IF_MINSORT(true, add(s(0), add(s(s(x0)), x1)), x2) -> c26(RM(s(0), add(s(s(x0)), x1))) IF_MINSORT(true, add(s(s(x0)), add(s(0), z2)), x2) -> c26(MINSORT(app(add(s(0), rm(s(s(x0)), z2)), x2), nil), APP(rm(s(s(x0)), add(s(0), z2)), x2), RM(s(s(x0)), add(s(0), z2))) IF_MINSORT(true, add(s(s(x0)), add(s(0), x1)), x2) -> c26(RM(s(s(x0)), add(s(0), x1))) IF_MINSORT(true, add(s(s(0)), add(s(s(0)), x2)), x3) -> c26(MINSORT(app(if_rm(true, s(s(0)), add(s(s(0)), x2)), x3), nil), APP(rm(s(s(0)), add(s(s(0)), x2)), x3), RM(s(s(0)), add(s(s(0)), x2))) IF_MINSORT(true, add(s(s(0)), add(s(s(s(z0))), x2)), x3) -> c26(MINSORT(app(if_rm(false, s(s(0)), add(s(s(s(z0))), x2)), x3), nil), APP(rm(s(s(0)), add(s(s(s(z0))), x2)), x3), RM(s(s(0)), add(s(s(s(z0))), x2))) IF_MINSORT(true, add(s(s(s(z0))), add(s(s(0)), x2)), x3) -> c26(MINSORT(app(if_rm(false, s(s(s(z0))), add(s(s(0)), x2)), x3), nil), APP(rm(s(s(s(z0))), add(s(s(0)), x2)), x3), RM(s(s(s(z0))), add(s(s(0)), x2))) IF_MINSORT(true, add(s(s(s(z0))), add(s(s(s(z1))), x2)), x3) -> c26(MINSORT(app(if_rm(eq(z0, z1), s(s(s(z0))), add(s(s(s(z1))), x2)), x3), nil), APP(rm(s(s(s(z0))), add(s(s(s(z1))), x2)), x3), RM(s(s(s(z0))), add(s(s(s(z1))), x2))) IF_MINSORT(true, add(s(s(x0)), add(s(s(x1)), x2)), x3) -> c9(APP(rm(s(s(x0)), add(s(s(x1)), x2)), x3)) IF_MINSORT(true, add(s(s(x0)), add(s(s(x1)), x2)), x3) -> c9(RM(s(s(x0)), add(s(s(x1)), x2))) IF_MINSORT(true, add(s(x0), add(s(x1), z2)), x3) -> c1(APP(if_rm(eq(s(x0), s(x1)), s(x0), add(s(x1), z2)), x3)) IF_MINSORT(true, add(0, add(0, x2)), x3) -> c26(APP(if_rm(true, 0, add(0, x2)), x3)) IF_MINSORT(true, add(0, add(s(z0), x2)), x3) -> c26(APP(if_rm(false, 0, add(s(z0), x2)), x3)) IF_MINSORT(true, add(s(z0), add(0, x2)), x3) -> c26(APP(if_rm(false, s(z0), add(0, x2)), x3)) IF_MINSORT(true, add(s(z0), add(s(z1), x2)), x3) -> c26(APP(if_rm(eq(z0, z1), s(z0), add(s(z1), x2)), x3)) MINSORT(add(s(x0), z0), x2) -> c25(IF_MINSORT(eq(s(x0), minIter(z0, add(s(x0), z0), 0)), add(s(x0), z0), x2), EQ(s(x0), minIter(add(s(x0), z0), add(s(x0), z0), 0)), MIN(add(s(x0), z0))) MINSORT(add(s(x0), x1), x2) -> c13(EQ(s(x0), minIter(add(s(x0), x1), add(s(x0), x1), 0))) MINSORT(add(s(x0), x1), x2) -> c13(MIN(add(s(x0), x1))) MINSORT(add(0, z0), x1) -> c25(IF_MINSORT(eq(0, 0), add(0, z0), x1), MIN(add(0, z0))) MINSORT(add(0, x0), x1) -> c25(MIN(add(0, x0))) MINSORT(add(z0, z1), x2) -> c2(EQ(z0, if_min(le(z0, 0), z1, add(z0, z1), 0))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) MINITER(add(s(s(z0)), x1), x2, s(0)) -> c12(IF_MIN(false, x1, x2, s(0)), LE(s(s(z0)), s(0))) MINITER(add(s(x0), x1), x2, s(x3)) -> c12(LE(s(x0), s(x3))) MINITER(add(s(0), x1), x2, s(z0)) -> c12(LE(s(0), s(z0))) MINITER(add(s(s(s(z0))), x1), x2, s(s(0))) -> c12(IF_MIN(false, x1, x2, s(s(0))), LE(s(s(s(z0))), s(s(0)))) MINITER(add(s(s(x0)), x1), x2, s(s(x3))) -> c12(LE(s(s(x0)), s(s(x3)))) MINITER(add(s(s(0)), x1), x2, s(s(z0))) -> c12(LE(s(s(0)), s(s(z0)))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(0)))) -> c12(IF_MIN(false, x1, x2, s(s(s(0)))), LE(s(s(s(s(z0)))), s(s(s(0))))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(s(z1))))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(s(s(z1))))), LE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MINITER(add(s(s(s(x0))), x1), x2, s(s(s(x3)))) -> c12(LE(s(s(s(x0))), s(s(s(x3))))) MINITER(add(s(s(s(0))), x1), x2, s(s(s(z0)))) -> c12(LE(s(s(s(0))), s(s(s(z0))))) K tuples: IF_MINSORT(true, add(z0, z1), z2) -> c26(MINSORT(app(rm(z0, z1), z2), nil), APP(rm(z0, z1), z2), RM(z0, z1)) MINSORT(add(z0, z1), z2) -> c25(IF_MINSORT(eq(z0, min(add(z0, z1))), add(z0, z1), z2), EQ(z0, min(add(z0, z1))), MIN(add(z0, z1))) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MIN(add(z0, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) RM(z0, add(z1, z2)) -> c21(IF_RM(eq(z0, z1), z0, add(z1, z2)), EQ(z0, z1)) Defined Rule Symbols: le_2, eq_2, app_2, if_rm_3, rm_2, if_min_4, minIter_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, MIN_1, MINITER_3, IF_MIN_4, IF_RM_3, IF_MINSORT_3, RM_2, MINSORT_2 Compound Symbols: c3_1, c6_1, c8_1, c10_1, c11_1, c14_1, c22_1, c23_1, c27_1, c12_1, c21_1, c12_2, c21_2, c_1, c26_1, c1_1, c2_1, c26_3, c4_1, c5_1, c7_1, c9_1, c25_3, c13_1, c25_2, c25_1 ---------------------------------------- (189) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MINSORT(add(z0, z1), x2) -> c(EQ(z0, if_min(le(z0, 0), z1, add(z0, z1), 0))) by MINSORT(add(0, x1), x2) -> c(EQ(0, if_min(true, x1, add(0, x1), 0))) MINSORT(add(s(z0), x1), x2) -> c(EQ(s(z0), if_min(false, x1, add(s(z0), x1), 0))) ---------------------------------------- (190) 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) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) if_min(true, z0, z1, z2) -> z2 if_min(false, z0, z1, z2) -> minIter(z0, z1, z2) minIter(add(z0, z1), z2, z3) -> if_min(le(z0, z3), z1, z2, z3) minIter(nil, add(z0, z1), z2) -> minIter(add(z0, z1), add(z0, z1), s(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, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) RM(0, add(0, x2)) -> c21(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c21(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c21(IF_RM(false, s(z0), add(0, x2))) MINITER(add(s(s(z0)), x1), x2, s(0)) -> c12(IF_MIN(false, x1, x2, s(0)), LE(s(s(z0)), s(0))) MINITER(add(s(x0), x1), x2, s(x3)) -> c12(LE(s(x0), s(x3))) MINITER(add(s(0), x1), x2, s(z0)) -> c12(LE(s(0), s(z0))) RM(s(0), add(s(0), x2)) -> c21(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c21(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c21(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(x0), add(s(x1), x2)) -> c21(EQ(s(x0), s(x1))) MINSORT(add(x0, x1), x2) -> c(MIN(add(x0, x1))) MINITER(add(s(s(s(z0))), x1), x2, s(s(0))) -> c12(IF_MIN(false, x1, x2, s(s(0))), LE(s(s(s(z0))), s(s(0)))) MINITER(add(s(s(x0)), x1), x2, s(s(x3))) -> c12(LE(s(s(x0)), s(s(x3)))) MINITER(add(s(s(0)), x1), x2, s(s(z0))) -> c12(LE(s(s(0)), s(s(z0)))) RM(s(s(0)), add(s(s(0)), x2)) -> c21(IF_RM(true, s(s(0)), add(s(s(0)), x2)), EQ(s(s(0)), s(s(0)))) RM(s(s(0)), add(s(s(s(z0))), x2)) -> c21(IF_RM(false, s(s(0)), add(s(s(s(z0))), x2)), EQ(s(s(0)), s(s(s(z0))))) RM(s(s(s(z0))), add(s(s(0)), x2)) -> c21(IF_RM(false, s(s(s(z0))), add(s(s(0)), x2)), EQ(s(s(s(z0))), s(s(0)))) RM(s(s(x0)), add(s(s(x1)), x2)) -> c21(EQ(s(s(x0)), s(s(x1)))) IF_MINSORT(true, add(0, add(0, x0)), x1) -> c26(RM(0, add(0, x0))) IF_MINSORT(true, add(0, add(s(x0), x1)), x2) -> c26(RM(0, add(s(x0), x1))) IF_MINSORT(true, add(s(x0), add(0, x1)), x2) -> c26(RM(s(x0), add(0, x1))) IF_MINSORT(true, add(s(x0), add(s(x1), x2)), x3) -> c1(RM(s(x0), add(s(x1), x2))) IF_MINSORT(true, add(z0, nil), x1) -> c26(MINSORT(x1, nil)) MINSORT(add(x0, x1), x2) -> c2(MIN(add(x0, x1))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(0)))) -> c12(IF_MIN(false, x1, x2, s(s(s(0)))), LE(s(s(s(s(z0)))), s(s(s(0))))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(s(z1))))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(s(s(z1))))), LE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MINITER(add(s(s(s(x0))), x1), x2, s(s(s(x3)))) -> c12(LE(s(s(s(x0))), s(s(s(x3))))) MINITER(add(s(s(s(0))), x1), x2, s(s(s(z0)))) -> c12(LE(s(s(s(0))), s(s(s(z0))))) RM(s(s(s(0))), add(s(s(s(0))), x2)) -> c21(IF_RM(true, s(s(s(0))), add(s(s(s(0))), x2)), EQ(s(s(s(0))), s(s(s(0))))) RM(s(s(s(0))), add(s(s(s(s(z0)))), x2)) -> c21(IF_RM(false, s(s(s(0))), add(s(s(s(s(z0)))), x2)), EQ(s(s(s(0))), s(s(s(s(z0)))))) RM(s(s(s(s(z0)))), add(s(s(s(0))), x2)) -> c21(IF_RM(false, s(s(s(s(z0)))), add(s(s(s(0))), x2)), EQ(s(s(s(s(z0)))), s(s(s(0))))) RM(s(s(s(s(z0)))), add(s(s(s(s(z1)))), x2)) -> c21(IF_RM(eq(z0, z1), s(s(s(s(z0)))), add(s(s(s(s(z1)))), x2)), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) RM(s(s(s(x0))), add(s(s(s(x1))), x2)) -> c21(EQ(s(s(s(x0))), s(s(s(x1))))) IF_MINSORT(true, add(0, add(0, nil)), x1) -> c26(MINSORT(app(nil, x1), nil), APP(rm(0, add(0, nil)), x1), RM(0, add(0, nil))) IF_MINSORT(true, add(0, add(0, add(z1, z2))), x1) -> c26(MINSORT(app(if_rm(eq(0, z1), 0, add(z1, z2)), x1), nil), APP(rm(0, add(0, add(z1, z2))), x1), RM(0, add(0, add(z1, z2)))) IF_MINSORT(true, add(0, add(0, x0)), x1) -> c4(APP(rm(0, add(0, x0)), x1)) IF_MINSORT(true, add(0, add(0, x0)), x1) -> c4(RM(0, add(0, x0))) IF_MINSORT(true, add(0, add(s(x0), x1)), z2) -> c26(MINSORT(add(s(x0), app(rm(0, x1), z2)), nil), APP(rm(0, add(s(x0), x1)), z2), RM(0, add(s(x0), x1))) IF_MINSORT(true, add(0, add(s(x0), nil)), x2) -> c26(MINSORT(app(add(s(x0), nil), x2), nil), APP(rm(0, add(s(x0), nil)), x2), RM(0, add(s(x0), nil))) IF_MINSORT(true, add(0, add(s(x0), add(z1, z2))), x2) -> c26(MINSORT(app(add(s(x0), if_rm(eq(0, z1), 0, add(z1, z2))), x2), nil), APP(rm(0, add(s(x0), add(z1, z2))), x2), RM(0, add(s(x0), add(z1, z2)))) IF_MINSORT(true, add(0, add(s(x0), x1)), x2) -> c5(APP(rm(0, add(s(x0), x1)), x2)) IF_MINSORT(true, add(0, add(s(x0), x1)), x2) -> c5(RM(0, add(s(x0), x1))) IF_MINSORT(true, add(s(x0), add(0, x1)), z2) -> c26(MINSORT(add(0, app(rm(s(x0), x1), z2)), nil), APP(rm(s(x0), add(0, x1)), z2), RM(s(x0), add(0, x1))) IF_MINSORT(true, add(s(x0), add(0, nil)), x2) -> c26(MINSORT(app(add(0, nil), x2), nil), APP(rm(s(x0), add(0, nil)), x2), RM(s(x0), add(0, nil))) IF_MINSORT(true, add(s(x0), add(0, add(z1, z2))), x2) -> c26(MINSORT(app(add(0, if_rm(eq(s(x0), z1), s(x0), add(z1, z2))), x2), nil), APP(rm(s(x0), add(0, add(z1, z2))), x2), RM(s(x0), add(0, add(z1, z2)))) IF_MINSORT(true, add(s(x0), add(0, x1)), x2) -> c7(APP(rm(s(x0), add(0, x1)), x2)) IF_MINSORT(true, add(s(x0), add(0, x1)), x2) -> c7(RM(s(x0), add(0, x1))) IF_MINSORT(true, add(s(0), add(s(0), z2)), x1) -> c26(MINSORT(app(rm(s(0), z2), x1), nil), APP(rm(s(0), add(s(0), z2)), x1), RM(s(0), add(s(0), z2))) IF_MINSORT(true, add(s(0), add(s(0), x0)), x1) -> c26(RM(s(0), add(s(0), x0))) IF_MINSORT(true, add(s(0), add(s(s(x0)), z2)), x2) -> c26(MINSORT(app(add(s(s(x0)), rm(s(0), z2)), x2), nil), APP(rm(s(0), add(s(s(x0)), z2)), x2), RM(s(0), add(s(s(x0)), z2))) IF_MINSORT(true, add(s(0), add(s(s(x0)), x1)), x2) -> c26(RM(s(0), add(s(s(x0)), x1))) IF_MINSORT(true, add(s(s(x0)), add(s(0), z2)), x2) -> c26(MINSORT(app(add(s(0), rm(s(s(x0)), z2)), x2), nil), APP(rm(s(s(x0)), add(s(0), z2)), x2), RM(s(s(x0)), add(s(0), z2))) IF_MINSORT(true, add(s(s(x0)), add(s(0), x1)), x2) -> c26(RM(s(s(x0)), add(s(0), x1))) IF_MINSORT(true, add(s(s(0)), add(s(s(0)), x2)), x3) -> c26(MINSORT(app(if_rm(true, s(s(0)), add(s(s(0)), x2)), x3), nil), APP(rm(s(s(0)), add(s(s(0)), x2)), x3), RM(s(s(0)), add(s(s(0)), x2))) IF_MINSORT(true, add(s(s(0)), add(s(s(s(z0))), x2)), x3) -> c26(MINSORT(app(if_rm(false, s(s(0)), add(s(s(s(z0))), x2)), x3), nil), APP(rm(s(s(0)), add(s(s(s(z0))), x2)), x3), RM(s(s(0)), add(s(s(s(z0))), x2))) IF_MINSORT(true, add(s(s(s(z0))), add(s(s(0)), x2)), x3) -> c26(MINSORT(app(if_rm(false, s(s(s(z0))), add(s(s(0)), x2)), x3), nil), APP(rm(s(s(s(z0))), add(s(s(0)), x2)), x3), RM(s(s(s(z0))), add(s(s(0)), x2))) IF_MINSORT(true, add(s(s(s(z0))), add(s(s(s(z1))), x2)), x3) -> c26(MINSORT(app(if_rm(eq(z0, z1), s(s(s(z0))), add(s(s(s(z1))), x2)), x3), nil), APP(rm(s(s(s(z0))), add(s(s(s(z1))), x2)), x3), RM(s(s(s(z0))), add(s(s(s(z1))), x2))) IF_MINSORT(true, add(s(s(x0)), add(s(s(x1)), x2)), x3) -> c9(APP(rm(s(s(x0)), add(s(s(x1)), x2)), x3)) IF_MINSORT(true, add(s(s(x0)), add(s(s(x1)), x2)), x3) -> c9(RM(s(s(x0)), add(s(s(x1)), x2))) IF_MINSORT(true, add(s(x0), add(s(x1), z2)), x3) -> c1(APP(if_rm(eq(s(x0), s(x1)), s(x0), add(s(x1), z2)), x3)) IF_MINSORT(true, add(0, add(0, x2)), x3) -> c26(APP(if_rm(true, 0, add(0, x2)), x3)) IF_MINSORT(true, add(0, add(s(z0), x2)), x3) -> c26(APP(if_rm(false, 0, add(s(z0), x2)), x3)) IF_MINSORT(true, add(s(z0), add(0, x2)), x3) -> c26(APP(if_rm(false, s(z0), add(0, x2)), x3)) IF_MINSORT(true, add(s(z0), add(s(z1), x2)), x3) -> c26(APP(if_rm(eq(z0, z1), s(z0), add(s(z1), x2)), x3)) MINSORT(add(s(x0), z0), x2) -> c25(IF_MINSORT(eq(s(x0), minIter(z0, add(s(x0), z0), 0)), add(s(x0), z0), x2), EQ(s(x0), minIter(add(s(x0), z0), add(s(x0), z0), 0)), MIN(add(s(x0), z0))) MINSORT(add(s(x0), x1), x2) -> c13(EQ(s(x0), minIter(add(s(x0), x1), add(s(x0), x1), 0))) MINSORT(add(s(x0), x1), x2) -> c13(MIN(add(s(x0), x1))) MINSORT(add(0, z0), x1) -> c25(IF_MINSORT(eq(0, 0), add(0, z0), x1), MIN(add(0, z0))) MINSORT(add(0, x0), x1) -> c25(MIN(add(0, x0))) MINSORT(add(z0, z1), x2) -> c2(EQ(z0, if_min(le(z0, 0), z1, add(z0, z1), 0))) MINSORT(add(0, x1), x2) -> c(EQ(0, if_min(true, x1, add(0, x1), 0))) MINSORT(add(s(z0), x1), x2) -> c(EQ(s(z0), if_min(false, x1, add(s(z0), x1), 0))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) MINITER(add(s(s(z0)), x1), x2, s(0)) -> c12(IF_MIN(false, x1, x2, s(0)), LE(s(s(z0)), s(0))) MINITER(add(s(x0), x1), x2, s(x3)) -> c12(LE(s(x0), s(x3))) MINITER(add(s(0), x1), x2, s(z0)) -> c12(LE(s(0), s(z0))) MINITER(add(s(s(s(z0))), x1), x2, s(s(0))) -> c12(IF_MIN(false, x1, x2, s(s(0))), LE(s(s(s(z0))), s(s(0)))) MINITER(add(s(s(x0)), x1), x2, s(s(x3))) -> c12(LE(s(s(x0)), s(s(x3)))) MINITER(add(s(s(0)), x1), x2, s(s(z0))) -> c12(LE(s(s(0)), s(s(z0)))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(0)))) -> c12(IF_MIN(false, x1, x2, s(s(s(0)))), LE(s(s(s(s(z0)))), s(s(s(0))))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(s(z1))))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(s(s(z1))))), LE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MINITER(add(s(s(s(x0))), x1), x2, s(s(s(x3)))) -> c12(LE(s(s(s(x0))), s(s(s(x3))))) MINITER(add(s(s(s(0))), x1), x2, s(s(s(z0)))) -> c12(LE(s(s(s(0))), s(s(s(z0))))) K tuples: IF_MINSORT(true, add(z0, z1), z2) -> c26(MINSORT(app(rm(z0, z1), z2), nil), APP(rm(z0, z1), z2), RM(z0, z1)) MINSORT(add(z0, z1), z2) -> c25(IF_MINSORT(eq(z0, min(add(z0, z1))), add(z0, z1), z2), EQ(z0, min(add(z0, z1))), MIN(add(z0, z1))) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MIN(add(z0, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) RM(z0, add(z1, z2)) -> c21(IF_RM(eq(z0, z1), z0, add(z1, z2)), EQ(z0, z1)) Defined Rule Symbols: le_2, eq_2, app_2, if_rm_3, rm_2, if_min_4, minIter_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, MIN_1, MINITER_3, IF_MIN_4, IF_RM_3, IF_MINSORT_3, RM_2, MINSORT_2 Compound Symbols: c3_1, c6_1, c8_1, c10_1, c11_1, c14_1, c22_1, c23_1, c27_1, c12_1, c21_1, c12_2, c21_2, c_1, c26_1, c1_1, c2_1, c26_3, c4_1, c5_1, c7_1, c9_1, c25_3, c13_1, c25_2, c25_1 ---------------------------------------- (191) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MINSORT(add(0, x1), x2) -> c(EQ(0, if_min(true, x1, add(0, x1), 0))) ---------------------------------------- (192) 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) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) if_min(true, z0, z1, z2) -> z2 if_min(false, z0, z1, z2) -> minIter(z0, z1, z2) minIter(add(z0, z1), z2, z3) -> if_min(le(z0, z3), z1, z2, z3) minIter(nil, add(z0, z1), z2) -> minIter(add(z0, z1), add(z0, z1), s(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, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) RM(0, add(0, x2)) -> c21(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c21(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c21(IF_RM(false, s(z0), add(0, x2))) MINITER(add(s(s(z0)), x1), x2, s(0)) -> c12(IF_MIN(false, x1, x2, s(0)), LE(s(s(z0)), s(0))) MINITER(add(s(x0), x1), x2, s(x3)) -> c12(LE(s(x0), s(x3))) MINITER(add(s(0), x1), x2, s(z0)) -> c12(LE(s(0), s(z0))) RM(s(0), add(s(0), x2)) -> c21(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c21(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c21(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(x0), add(s(x1), x2)) -> c21(EQ(s(x0), s(x1))) MINSORT(add(x0, x1), x2) -> c(MIN(add(x0, x1))) MINITER(add(s(s(s(z0))), x1), x2, s(s(0))) -> c12(IF_MIN(false, x1, x2, s(s(0))), LE(s(s(s(z0))), s(s(0)))) MINITER(add(s(s(x0)), x1), x2, s(s(x3))) -> c12(LE(s(s(x0)), s(s(x3)))) MINITER(add(s(s(0)), x1), x2, s(s(z0))) -> c12(LE(s(s(0)), s(s(z0)))) RM(s(s(0)), add(s(s(0)), x2)) -> c21(IF_RM(true, s(s(0)), add(s(s(0)), x2)), EQ(s(s(0)), s(s(0)))) RM(s(s(0)), add(s(s(s(z0))), x2)) -> c21(IF_RM(false, s(s(0)), add(s(s(s(z0))), x2)), EQ(s(s(0)), s(s(s(z0))))) RM(s(s(s(z0))), add(s(s(0)), x2)) -> c21(IF_RM(false, s(s(s(z0))), add(s(s(0)), x2)), EQ(s(s(s(z0))), s(s(0)))) RM(s(s(x0)), add(s(s(x1)), x2)) -> c21(EQ(s(s(x0)), s(s(x1)))) IF_MINSORT(true, add(0, add(0, x0)), x1) -> c26(RM(0, add(0, x0))) IF_MINSORT(true, add(0, add(s(x0), x1)), x2) -> c26(RM(0, add(s(x0), x1))) IF_MINSORT(true, add(s(x0), add(0, x1)), x2) -> c26(RM(s(x0), add(0, x1))) IF_MINSORT(true, add(s(x0), add(s(x1), x2)), x3) -> c1(RM(s(x0), add(s(x1), x2))) IF_MINSORT(true, add(z0, nil), x1) -> c26(MINSORT(x1, nil)) MINSORT(add(x0, x1), x2) -> c2(MIN(add(x0, x1))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(0)))) -> c12(IF_MIN(false, x1, x2, s(s(s(0)))), LE(s(s(s(s(z0)))), s(s(s(0))))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(s(z1))))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(s(s(z1))))), LE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MINITER(add(s(s(s(x0))), x1), x2, s(s(s(x3)))) -> c12(LE(s(s(s(x0))), s(s(s(x3))))) MINITER(add(s(s(s(0))), x1), x2, s(s(s(z0)))) -> c12(LE(s(s(s(0))), s(s(s(z0))))) RM(s(s(s(0))), add(s(s(s(0))), x2)) -> c21(IF_RM(true, s(s(s(0))), add(s(s(s(0))), x2)), EQ(s(s(s(0))), s(s(s(0))))) RM(s(s(s(0))), add(s(s(s(s(z0)))), x2)) -> c21(IF_RM(false, s(s(s(0))), add(s(s(s(s(z0)))), x2)), EQ(s(s(s(0))), s(s(s(s(z0)))))) RM(s(s(s(s(z0)))), add(s(s(s(0))), x2)) -> c21(IF_RM(false, s(s(s(s(z0)))), add(s(s(s(0))), x2)), EQ(s(s(s(s(z0)))), s(s(s(0))))) RM(s(s(s(s(z0)))), add(s(s(s(s(z1)))), x2)) -> c21(IF_RM(eq(z0, z1), s(s(s(s(z0)))), add(s(s(s(s(z1)))), x2)), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) RM(s(s(s(x0))), add(s(s(s(x1))), x2)) -> c21(EQ(s(s(s(x0))), s(s(s(x1))))) IF_MINSORT(true, add(0, add(0, nil)), x1) -> c26(MINSORT(app(nil, x1), nil), APP(rm(0, add(0, nil)), x1), RM(0, add(0, nil))) IF_MINSORT(true, add(0, add(0, add(z1, z2))), x1) -> c26(MINSORT(app(if_rm(eq(0, z1), 0, add(z1, z2)), x1), nil), APP(rm(0, add(0, add(z1, z2))), x1), RM(0, add(0, add(z1, z2)))) IF_MINSORT(true, add(0, add(0, x0)), x1) -> c4(APP(rm(0, add(0, x0)), x1)) IF_MINSORT(true, add(0, add(0, x0)), x1) -> c4(RM(0, add(0, x0))) IF_MINSORT(true, add(0, add(s(x0), x1)), z2) -> c26(MINSORT(add(s(x0), app(rm(0, x1), z2)), nil), APP(rm(0, add(s(x0), x1)), z2), RM(0, add(s(x0), x1))) IF_MINSORT(true, add(0, add(s(x0), nil)), x2) -> c26(MINSORT(app(add(s(x0), nil), x2), nil), APP(rm(0, add(s(x0), nil)), x2), RM(0, add(s(x0), nil))) IF_MINSORT(true, add(0, add(s(x0), add(z1, z2))), x2) -> c26(MINSORT(app(add(s(x0), if_rm(eq(0, z1), 0, add(z1, z2))), x2), nil), APP(rm(0, add(s(x0), add(z1, z2))), x2), RM(0, add(s(x0), add(z1, z2)))) IF_MINSORT(true, add(0, add(s(x0), x1)), x2) -> c5(APP(rm(0, add(s(x0), x1)), x2)) IF_MINSORT(true, add(0, add(s(x0), x1)), x2) -> c5(RM(0, add(s(x0), x1))) IF_MINSORT(true, add(s(x0), add(0, x1)), z2) -> c26(MINSORT(add(0, app(rm(s(x0), x1), z2)), nil), APP(rm(s(x0), add(0, x1)), z2), RM(s(x0), add(0, x1))) IF_MINSORT(true, add(s(x0), add(0, nil)), x2) -> c26(MINSORT(app(add(0, nil), x2), nil), APP(rm(s(x0), add(0, nil)), x2), RM(s(x0), add(0, nil))) IF_MINSORT(true, add(s(x0), add(0, add(z1, z2))), x2) -> c26(MINSORT(app(add(0, if_rm(eq(s(x0), z1), s(x0), add(z1, z2))), x2), nil), APP(rm(s(x0), add(0, add(z1, z2))), x2), RM(s(x0), add(0, add(z1, z2)))) IF_MINSORT(true, add(s(x0), add(0, x1)), x2) -> c7(APP(rm(s(x0), add(0, x1)), x2)) IF_MINSORT(true, add(s(x0), add(0, x1)), x2) -> c7(RM(s(x0), add(0, x1))) IF_MINSORT(true, add(s(0), add(s(0), z2)), x1) -> c26(MINSORT(app(rm(s(0), z2), x1), nil), APP(rm(s(0), add(s(0), z2)), x1), RM(s(0), add(s(0), z2))) IF_MINSORT(true, add(s(0), add(s(0), x0)), x1) -> c26(RM(s(0), add(s(0), x0))) IF_MINSORT(true, add(s(0), add(s(s(x0)), z2)), x2) -> c26(MINSORT(app(add(s(s(x0)), rm(s(0), z2)), x2), nil), APP(rm(s(0), add(s(s(x0)), z2)), x2), RM(s(0), add(s(s(x0)), z2))) IF_MINSORT(true, add(s(0), add(s(s(x0)), x1)), x2) -> c26(RM(s(0), add(s(s(x0)), x1))) IF_MINSORT(true, add(s(s(x0)), add(s(0), z2)), x2) -> c26(MINSORT(app(add(s(0), rm(s(s(x0)), z2)), x2), nil), APP(rm(s(s(x0)), add(s(0), z2)), x2), RM(s(s(x0)), add(s(0), z2))) IF_MINSORT(true, add(s(s(x0)), add(s(0), x1)), x2) -> c26(RM(s(s(x0)), add(s(0), x1))) IF_MINSORT(true, add(s(s(0)), add(s(s(0)), x2)), x3) -> c26(MINSORT(app(if_rm(true, s(s(0)), add(s(s(0)), x2)), x3), nil), APP(rm(s(s(0)), add(s(s(0)), x2)), x3), RM(s(s(0)), add(s(s(0)), x2))) IF_MINSORT(true, add(s(s(0)), add(s(s(s(z0))), x2)), x3) -> c26(MINSORT(app(if_rm(false, s(s(0)), add(s(s(s(z0))), x2)), x3), nil), APP(rm(s(s(0)), add(s(s(s(z0))), x2)), x3), RM(s(s(0)), add(s(s(s(z0))), x2))) IF_MINSORT(true, add(s(s(s(z0))), add(s(s(0)), x2)), x3) -> c26(MINSORT(app(if_rm(false, s(s(s(z0))), add(s(s(0)), x2)), x3), nil), APP(rm(s(s(s(z0))), add(s(s(0)), x2)), x3), RM(s(s(s(z0))), add(s(s(0)), x2))) IF_MINSORT(true, add(s(s(s(z0))), add(s(s(s(z1))), x2)), x3) -> c26(MINSORT(app(if_rm(eq(z0, z1), s(s(s(z0))), add(s(s(s(z1))), x2)), x3), nil), APP(rm(s(s(s(z0))), add(s(s(s(z1))), x2)), x3), RM(s(s(s(z0))), add(s(s(s(z1))), x2))) IF_MINSORT(true, add(s(s(x0)), add(s(s(x1)), x2)), x3) -> c9(APP(rm(s(s(x0)), add(s(s(x1)), x2)), x3)) IF_MINSORT(true, add(s(s(x0)), add(s(s(x1)), x2)), x3) -> c9(RM(s(s(x0)), add(s(s(x1)), x2))) IF_MINSORT(true, add(s(x0), add(s(x1), z2)), x3) -> c1(APP(if_rm(eq(s(x0), s(x1)), s(x0), add(s(x1), z2)), x3)) IF_MINSORT(true, add(0, add(0, x2)), x3) -> c26(APP(if_rm(true, 0, add(0, x2)), x3)) IF_MINSORT(true, add(0, add(s(z0), x2)), x3) -> c26(APP(if_rm(false, 0, add(s(z0), x2)), x3)) IF_MINSORT(true, add(s(z0), add(0, x2)), x3) -> c26(APP(if_rm(false, s(z0), add(0, x2)), x3)) IF_MINSORT(true, add(s(z0), add(s(z1), x2)), x3) -> c26(APP(if_rm(eq(z0, z1), s(z0), add(s(z1), x2)), x3)) MINSORT(add(s(x0), z0), x2) -> c25(IF_MINSORT(eq(s(x0), minIter(z0, add(s(x0), z0), 0)), add(s(x0), z0), x2), EQ(s(x0), minIter(add(s(x0), z0), add(s(x0), z0), 0)), MIN(add(s(x0), z0))) MINSORT(add(s(x0), x1), x2) -> c13(EQ(s(x0), minIter(add(s(x0), x1), add(s(x0), x1), 0))) MINSORT(add(s(x0), x1), x2) -> c13(MIN(add(s(x0), x1))) MINSORT(add(0, z0), x1) -> c25(IF_MINSORT(eq(0, 0), add(0, z0), x1), MIN(add(0, z0))) MINSORT(add(0, x0), x1) -> c25(MIN(add(0, x0))) MINSORT(add(z0, z1), x2) -> c2(EQ(z0, if_min(le(z0, 0), z1, add(z0, z1), 0))) MINSORT(add(s(z0), x1), x2) -> c(EQ(s(z0), if_min(false, x1, add(s(z0), x1), 0))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) MINITER(add(s(s(z0)), x1), x2, s(0)) -> c12(IF_MIN(false, x1, x2, s(0)), LE(s(s(z0)), s(0))) MINITER(add(s(x0), x1), x2, s(x3)) -> c12(LE(s(x0), s(x3))) MINITER(add(s(0), x1), x2, s(z0)) -> c12(LE(s(0), s(z0))) MINITER(add(s(s(s(z0))), x1), x2, s(s(0))) -> c12(IF_MIN(false, x1, x2, s(s(0))), LE(s(s(s(z0))), s(s(0)))) MINITER(add(s(s(x0)), x1), x2, s(s(x3))) -> c12(LE(s(s(x0)), s(s(x3)))) MINITER(add(s(s(0)), x1), x2, s(s(z0))) -> c12(LE(s(s(0)), s(s(z0)))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(0)))) -> c12(IF_MIN(false, x1, x2, s(s(s(0)))), LE(s(s(s(s(z0)))), s(s(s(0))))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(s(z1))))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(s(s(z1))))), LE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MINITER(add(s(s(s(x0))), x1), x2, s(s(s(x3)))) -> c12(LE(s(s(s(x0))), s(s(s(x3))))) MINITER(add(s(s(s(0))), x1), x2, s(s(s(z0)))) -> c12(LE(s(s(s(0))), s(s(s(z0))))) K tuples: IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MIN(add(z0, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) Defined Rule Symbols: le_2, eq_2, app_2, if_rm_3, rm_2, if_min_4, minIter_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, MIN_1, MINITER_3, IF_MIN_4, IF_RM_3, IF_MINSORT_3, RM_2, MINSORT_2 Compound Symbols: c3_1, c6_1, c8_1, c10_1, c11_1, c14_1, c22_1, c23_1, c27_1, c12_1, c21_1, c12_2, c21_2, c_1, c26_1, c1_1, c2_1, c26_3, c4_1, c5_1, c7_1, c9_1, c25_3, c13_1, c25_2, c25_1 ---------------------------------------- (193) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace IF_MINSORT(true, add(0, add(0, nil)), x1) -> c26(MINSORT(app(nil, x1), nil), APP(rm(0, add(0, nil)), x1), RM(0, add(0, nil))) by IF_MINSORT(true, add(0, add(0, nil)), z0) -> c26(MINSORT(app(nil, z0), nil), APP(if_rm(eq(0, 0), 0, add(0, nil)), z0), RM(0, add(0, nil))) ---------------------------------------- (194) 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) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) if_min(true, z0, z1, z2) -> z2 if_min(false, z0, z1, z2) -> minIter(z0, z1, z2) minIter(add(z0, z1), z2, z3) -> if_min(le(z0, z3), z1, z2, z3) minIter(nil, add(z0, z1), z2) -> minIter(add(z0, z1), add(z0, z1), s(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, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) RM(0, add(0, x2)) -> c21(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c21(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c21(IF_RM(false, s(z0), add(0, x2))) MINITER(add(s(s(z0)), x1), x2, s(0)) -> c12(IF_MIN(false, x1, x2, s(0)), LE(s(s(z0)), s(0))) MINITER(add(s(x0), x1), x2, s(x3)) -> c12(LE(s(x0), s(x3))) MINITER(add(s(0), x1), x2, s(z0)) -> c12(LE(s(0), s(z0))) RM(s(0), add(s(0), x2)) -> c21(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c21(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c21(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(x0), add(s(x1), x2)) -> c21(EQ(s(x0), s(x1))) MINSORT(add(x0, x1), x2) -> c(MIN(add(x0, x1))) MINITER(add(s(s(s(z0))), x1), x2, s(s(0))) -> c12(IF_MIN(false, x1, x2, s(s(0))), LE(s(s(s(z0))), s(s(0)))) MINITER(add(s(s(x0)), x1), x2, s(s(x3))) -> c12(LE(s(s(x0)), s(s(x3)))) MINITER(add(s(s(0)), x1), x2, s(s(z0))) -> c12(LE(s(s(0)), s(s(z0)))) RM(s(s(0)), add(s(s(0)), x2)) -> c21(IF_RM(true, s(s(0)), add(s(s(0)), x2)), EQ(s(s(0)), s(s(0)))) RM(s(s(0)), add(s(s(s(z0))), x2)) -> c21(IF_RM(false, s(s(0)), add(s(s(s(z0))), x2)), EQ(s(s(0)), s(s(s(z0))))) RM(s(s(s(z0))), add(s(s(0)), x2)) -> c21(IF_RM(false, s(s(s(z0))), add(s(s(0)), x2)), EQ(s(s(s(z0))), s(s(0)))) RM(s(s(x0)), add(s(s(x1)), x2)) -> c21(EQ(s(s(x0)), s(s(x1)))) IF_MINSORT(true, add(0, add(0, x0)), x1) -> c26(RM(0, add(0, x0))) IF_MINSORT(true, add(0, add(s(x0), x1)), x2) -> c26(RM(0, add(s(x0), x1))) IF_MINSORT(true, add(s(x0), add(0, x1)), x2) -> c26(RM(s(x0), add(0, x1))) IF_MINSORT(true, add(s(x0), add(s(x1), x2)), x3) -> c1(RM(s(x0), add(s(x1), x2))) IF_MINSORT(true, add(z0, nil), x1) -> c26(MINSORT(x1, nil)) MINSORT(add(x0, x1), x2) -> c2(MIN(add(x0, x1))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(0)))) -> c12(IF_MIN(false, x1, x2, s(s(s(0)))), LE(s(s(s(s(z0)))), s(s(s(0))))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(s(z1))))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(s(s(z1))))), LE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MINITER(add(s(s(s(x0))), x1), x2, s(s(s(x3)))) -> c12(LE(s(s(s(x0))), s(s(s(x3))))) MINITER(add(s(s(s(0))), x1), x2, s(s(s(z0)))) -> c12(LE(s(s(s(0))), s(s(s(z0))))) RM(s(s(s(0))), add(s(s(s(0))), x2)) -> c21(IF_RM(true, s(s(s(0))), add(s(s(s(0))), x2)), EQ(s(s(s(0))), s(s(s(0))))) RM(s(s(s(0))), add(s(s(s(s(z0)))), x2)) -> c21(IF_RM(false, s(s(s(0))), add(s(s(s(s(z0)))), x2)), EQ(s(s(s(0))), s(s(s(s(z0)))))) RM(s(s(s(s(z0)))), add(s(s(s(0))), x2)) -> c21(IF_RM(false, s(s(s(s(z0)))), add(s(s(s(0))), x2)), EQ(s(s(s(s(z0)))), s(s(s(0))))) RM(s(s(s(s(z0)))), add(s(s(s(s(z1)))), x2)) -> c21(IF_RM(eq(z0, z1), s(s(s(s(z0)))), add(s(s(s(s(z1)))), x2)), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) RM(s(s(s(x0))), add(s(s(s(x1))), x2)) -> c21(EQ(s(s(s(x0))), s(s(s(x1))))) IF_MINSORT(true, add(0, add(0, add(z1, z2))), x1) -> c26(MINSORT(app(if_rm(eq(0, z1), 0, add(z1, z2)), x1), nil), APP(rm(0, add(0, add(z1, z2))), x1), RM(0, add(0, add(z1, z2)))) IF_MINSORT(true, add(0, add(0, x0)), x1) -> c4(APP(rm(0, add(0, x0)), x1)) IF_MINSORT(true, add(0, add(0, x0)), x1) -> c4(RM(0, add(0, x0))) IF_MINSORT(true, add(0, add(s(x0), x1)), z2) -> c26(MINSORT(add(s(x0), app(rm(0, x1), z2)), nil), APP(rm(0, add(s(x0), x1)), z2), RM(0, add(s(x0), x1))) IF_MINSORT(true, add(0, add(s(x0), nil)), x2) -> c26(MINSORT(app(add(s(x0), nil), x2), nil), APP(rm(0, add(s(x0), nil)), x2), RM(0, add(s(x0), nil))) IF_MINSORT(true, add(0, add(s(x0), add(z1, z2))), x2) -> c26(MINSORT(app(add(s(x0), if_rm(eq(0, z1), 0, add(z1, z2))), x2), nil), APP(rm(0, add(s(x0), add(z1, z2))), x2), RM(0, add(s(x0), add(z1, z2)))) IF_MINSORT(true, add(0, add(s(x0), x1)), x2) -> c5(APP(rm(0, add(s(x0), x1)), x2)) IF_MINSORT(true, add(0, add(s(x0), x1)), x2) -> c5(RM(0, add(s(x0), x1))) IF_MINSORT(true, add(s(x0), add(0, x1)), z2) -> c26(MINSORT(add(0, app(rm(s(x0), x1), z2)), nil), APP(rm(s(x0), add(0, x1)), z2), RM(s(x0), add(0, x1))) IF_MINSORT(true, add(s(x0), add(0, nil)), x2) -> c26(MINSORT(app(add(0, nil), x2), nil), APP(rm(s(x0), add(0, nil)), x2), RM(s(x0), add(0, nil))) IF_MINSORT(true, add(s(x0), add(0, add(z1, z2))), x2) -> c26(MINSORT(app(add(0, if_rm(eq(s(x0), z1), s(x0), add(z1, z2))), x2), nil), APP(rm(s(x0), add(0, add(z1, z2))), x2), RM(s(x0), add(0, add(z1, z2)))) IF_MINSORT(true, add(s(x0), add(0, x1)), x2) -> c7(APP(rm(s(x0), add(0, x1)), x2)) IF_MINSORT(true, add(s(x0), add(0, x1)), x2) -> c7(RM(s(x0), add(0, x1))) IF_MINSORT(true, add(s(0), add(s(0), z2)), x1) -> c26(MINSORT(app(rm(s(0), z2), x1), nil), APP(rm(s(0), add(s(0), z2)), x1), RM(s(0), add(s(0), z2))) IF_MINSORT(true, add(s(0), add(s(0), x0)), x1) -> c26(RM(s(0), add(s(0), x0))) IF_MINSORT(true, add(s(0), add(s(s(x0)), z2)), x2) -> c26(MINSORT(app(add(s(s(x0)), rm(s(0), z2)), x2), nil), APP(rm(s(0), add(s(s(x0)), z2)), x2), RM(s(0), add(s(s(x0)), z2))) IF_MINSORT(true, add(s(0), add(s(s(x0)), x1)), x2) -> c26(RM(s(0), add(s(s(x0)), x1))) IF_MINSORT(true, add(s(s(x0)), add(s(0), z2)), x2) -> c26(MINSORT(app(add(s(0), rm(s(s(x0)), z2)), x2), nil), APP(rm(s(s(x0)), add(s(0), z2)), x2), RM(s(s(x0)), add(s(0), z2))) IF_MINSORT(true, add(s(s(x0)), add(s(0), x1)), x2) -> c26(RM(s(s(x0)), add(s(0), x1))) IF_MINSORT(true, add(s(s(0)), add(s(s(0)), x2)), x3) -> c26(MINSORT(app(if_rm(true, s(s(0)), add(s(s(0)), x2)), x3), nil), APP(rm(s(s(0)), add(s(s(0)), x2)), x3), RM(s(s(0)), add(s(s(0)), x2))) IF_MINSORT(true, add(s(s(0)), add(s(s(s(z0))), x2)), x3) -> c26(MINSORT(app(if_rm(false, s(s(0)), add(s(s(s(z0))), x2)), x3), nil), APP(rm(s(s(0)), add(s(s(s(z0))), x2)), x3), RM(s(s(0)), add(s(s(s(z0))), x2))) IF_MINSORT(true, add(s(s(s(z0))), add(s(s(0)), x2)), x3) -> c26(MINSORT(app(if_rm(false, s(s(s(z0))), add(s(s(0)), x2)), x3), nil), APP(rm(s(s(s(z0))), add(s(s(0)), x2)), x3), RM(s(s(s(z0))), add(s(s(0)), x2))) IF_MINSORT(true, add(s(s(s(z0))), add(s(s(s(z1))), x2)), x3) -> c26(MINSORT(app(if_rm(eq(z0, z1), s(s(s(z0))), add(s(s(s(z1))), x2)), x3), nil), APP(rm(s(s(s(z0))), add(s(s(s(z1))), x2)), x3), RM(s(s(s(z0))), add(s(s(s(z1))), x2))) IF_MINSORT(true, add(s(s(x0)), add(s(s(x1)), x2)), x3) -> c9(APP(rm(s(s(x0)), add(s(s(x1)), x2)), x3)) IF_MINSORT(true, add(s(s(x0)), add(s(s(x1)), x2)), x3) -> c9(RM(s(s(x0)), add(s(s(x1)), x2))) IF_MINSORT(true, add(s(x0), add(s(x1), z2)), x3) -> c1(APP(if_rm(eq(s(x0), s(x1)), s(x0), add(s(x1), z2)), x3)) IF_MINSORT(true, add(0, add(0, x2)), x3) -> c26(APP(if_rm(true, 0, add(0, x2)), x3)) IF_MINSORT(true, add(0, add(s(z0), x2)), x3) -> c26(APP(if_rm(false, 0, add(s(z0), x2)), x3)) IF_MINSORT(true, add(s(z0), add(0, x2)), x3) -> c26(APP(if_rm(false, s(z0), add(0, x2)), x3)) IF_MINSORT(true, add(s(z0), add(s(z1), x2)), x3) -> c26(APP(if_rm(eq(z0, z1), s(z0), add(s(z1), x2)), x3)) MINSORT(add(s(x0), z0), x2) -> c25(IF_MINSORT(eq(s(x0), minIter(z0, add(s(x0), z0), 0)), add(s(x0), z0), x2), EQ(s(x0), minIter(add(s(x0), z0), add(s(x0), z0), 0)), MIN(add(s(x0), z0))) MINSORT(add(s(x0), x1), x2) -> c13(EQ(s(x0), minIter(add(s(x0), x1), add(s(x0), x1), 0))) MINSORT(add(s(x0), x1), x2) -> c13(MIN(add(s(x0), x1))) MINSORT(add(0, z0), x1) -> c25(IF_MINSORT(eq(0, 0), add(0, z0), x1), MIN(add(0, z0))) MINSORT(add(0, x0), x1) -> c25(MIN(add(0, x0))) MINSORT(add(z0, z1), x2) -> c2(EQ(z0, if_min(le(z0, 0), z1, add(z0, z1), 0))) MINSORT(add(s(z0), x1), x2) -> c(EQ(s(z0), if_min(false, x1, add(s(z0), x1), 0))) IF_MINSORT(true, add(0, add(0, nil)), z0) -> c26(MINSORT(app(nil, z0), nil), APP(if_rm(eq(0, 0), 0, add(0, nil)), z0), RM(0, add(0, nil))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) MINITER(add(s(s(z0)), x1), x2, s(0)) -> c12(IF_MIN(false, x1, x2, s(0)), LE(s(s(z0)), s(0))) MINITER(add(s(x0), x1), x2, s(x3)) -> c12(LE(s(x0), s(x3))) MINITER(add(s(0), x1), x2, s(z0)) -> c12(LE(s(0), s(z0))) MINITER(add(s(s(s(z0))), x1), x2, s(s(0))) -> c12(IF_MIN(false, x1, x2, s(s(0))), LE(s(s(s(z0))), s(s(0)))) MINITER(add(s(s(x0)), x1), x2, s(s(x3))) -> c12(LE(s(s(x0)), s(s(x3)))) MINITER(add(s(s(0)), x1), x2, s(s(z0))) -> c12(LE(s(s(0)), s(s(z0)))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(0)))) -> c12(IF_MIN(false, x1, x2, s(s(s(0)))), LE(s(s(s(s(z0)))), s(s(s(0))))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(s(z1))))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(s(s(z1))))), LE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MINITER(add(s(s(s(x0))), x1), x2, s(s(s(x3)))) -> c12(LE(s(s(s(x0))), s(s(s(x3))))) MINITER(add(s(s(s(0))), x1), x2, s(s(s(z0)))) -> c12(LE(s(s(s(0))), s(s(s(z0))))) K tuples: IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MIN(add(z0, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) Defined Rule Symbols: le_2, eq_2, app_2, if_rm_3, rm_2, if_min_4, minIter_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, MIN_1, MINITER_3, IF_MIN_4, IF_RM_3, IF_MINSORT_3, RM_2, MINSORT_2 Compound Symbols: c3_1, c6_1, c8_1, c10_1, c11_1, c14_1, c22_1, c23_1, c27_1, c12_1, c21_1, c12_2, c21_2, c_1, c26_1, c1_1, c2_1, c26_3, c4_1, c5_1, c7_1, c9_1, c25_3, c13_1, c25_2, c25_1 ---------------------------------------- (195) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IF_MIN(false, z0, z1, z2) -> c14(MINITER(z0, z1, z2)) by IF_MIN(false, x1, x2, 0) -> c14(MINITER(x1, x2, 0)) IF_MIN(false, x1, x2, s(0)) -> c14(MINITER(x1, x2, s(0))) IF_MIN(false, x1, x2, s(s(0))) -> c14(MINITER(x1, x2, s(s(0)))) IF_MIN(false, x1, x2, s(s(s(0)))) -> c14(MINITER(x1, x2, s(s(s(0))))) IF_MIN(false, x1, x2, s(s(s(s(x3))))) -> c14(MINITER(x1, x2, s(s(s(s(x3)))))) ---------------------------------------- (196) 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) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) if_min(true, z0, z1, z2) -> z2 if_min(false, z0, z1, z2) -> minIter(z0, z1, z2) minIter(add(z0, z1), z2, z3) -> if_min(le(z0, z3), z1, z2, z3) minIter(nil, add(z0, z1), z2) -> minIter(add(z0, z1), add(z0, z1), s(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, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) RM(0, add(0, x2)) -> c21(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c21(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c21(IF_RM(false, s(z0), add(0, x2))) MINITER(add(s(s(z0)), x1), x2, s(0)) -> c12(IF_MIN(false, x1, x2, s(0)), LE(s(s(z0)), s(0))) MINITER(add(s(x0), x1), x2, s(x3)) -> c12(LE(s(x0), s(x3))) MINITER(add(s(0), x1), x2, s(z0)) -> c12(LE(s(0), s(z0))) RM(s(0), add(s(0), x2)) -> c21(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c21(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c21(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(x0), add(s(x1), x2)) -> c21(EQ(s(x0), s(x1))) MINSORT(add(x0, x1), x2) -> c(MIN(add(x0, x1))) MINITER(add(s(s(s(z0))), x1), x2, s(s(0))) -> c12(IF_MIN(false, x1, x2, s(s(0))), LE(s(s(s(z0))), s(s(0)))) MINITER(add(s(s(x0)), x1), x2, s(s(x3))) -> c12(LE(s(s(x0)), s(s(x3)))) MINITER(add(s(s(0)), x1), x2, s(s(z0))) -> c12(LE(s(s(0)), s(s(z0)))) RM(s(s(0)), add(s(s(0)), x2)) -> c21(IF_RM(true, s(s(0)), add(s(s(0)), x2)), EQ(s(s(0)), s(s(0)))) RM(s(s(0)), add(s(s(s(z0))), x2)) -> c21(IF_RM(false, s(s(0)), add(s(s(s(z0))), x2)), EQ(s(s(0)), s(s(s(z0))))) RM(s(s(s(z0))), add(s(s(0)), x2)) -> c21(IF_RM(false, s(s(s(z0))), add(s(s(0)), x2)), EQ(s(s(s(z0))), s(s(0)))) RM(s(s(x0)), add(s(s(x1)), x2)) -> c21(EQ(s(s(x0)), s(s(x1)))) IF_MINSORT(true, add(0, add(0, x0)), x1) -> c26(RM(0, add(0, x0))) IF_MINSORT(true, add(0, add(s(x0), x1)), x2) -> c26(RM(0, add(s(x0), x1))) IF_MINSORT(true, add(s(x0), add(0, x1)), x2) -> c26(RM(s(x0), add(0, x1))) IF_MINSORT(true, add(s(x0), add(s(x1), x2)), x3) -> c1(RM(s(x0), add(s(x1), x2))) IF_MINSORT(true, add(z0, nil), x1) -> c26(MINSORT(x1, nil)) MINSORT(add(x0, x1), x2) -> c2(MIN(add(x0, x1))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(0)))) -> c12(IF_MIN(false, x1, x2, s(s(s(0)))), LE(s(s(s(s(z0)))), s(s(s(0))))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(s(z1))))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(s(s(z1))))), LE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MINITER(add(s(s(s(x0))), x1), x2, s(s(s(x3)))) -> c12(LE(s(s(s(x0))), s(s(s(x3))))) MINITER(add(s(s(s(0))), x1), x2, s(s(s(z0)))) -> c12(LE(s(s(s(0))), s(s(s(z0))))) RM(s(s(s(0))), add(s(s(s(0))), x2)) -> c21(IF_RM(true, s(s(s(0))), add(s(s(s(0))), x2)), EQ(s(s(s(0))), s(s(s(0))))) RM(s(s(s(0))), add(s(s(s(s(z0)))), x2)) -> c21(IF_RM(false, s(s(s(0))), add(s(s(s(s(z0)))), x2)), EQ(s(s(s(0))), s(s(s(s(z0)))))) RM(s(s(s(s(z0)))), add(s(s(s(0))), x2)) -> c21(IF_RM(false, s(s(s(s(z0)))), add(s(s(s(0))), x2)), EQ(s(s(s(s(z0)))), s(s(s(0))))) RM(s(s(s(s(z0)))), add(s(s(s(s(z1)))), x2)) -> c21(IF_RM(eq(z0, z1), s(s(s(s(z0)))), add(s(s(s(s(z1)))), x2)), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) RM(s(s(s(x0))), add(s(s(s(x1))), x2)) -> c21(EQ(s(s(s(x0))), s(s(s(x1))))) IF_MINSORT(true, add(0, add(0, add(z1, z2))), x1) -> c26(MINSORT(app(if_rm(eq(0, z1), 0, add(z1, z2)), x1), nil), APP(rm(0, add(0, add(z1, z2))), x1), RM(0, add(0, add(z1, z2)))) IF_MINSORT(true, add(0, add(0, x0)), x1) -> c4(APP(rm(0, add(0, x0)), x1)) IF_MINSORT(true, add(0, add(0, x0)), x1) -> c4(RM(0, add(0, x0))) IF_MINSORT(true, add(0, add(s(x0), x1)), z2) -> c26(MINSORT(add(s(x0), app(rm(0, x1), z2)), nil), APP(rm(0, add(s(x0), x1)), z2), RM(0, add(s(x0), x1))) IF_MINSORT(true, add(0, add(s(x0), nil)), x2) -> c26(MINSORT(app(add(s(x0), nil), x2), nil), APP(rm(0, add(s(x0), nil)), x2), RM(0, add(s(x0), nil))) IF_MINSORT(true, add(0, add(s(x0), add(z1, z2))), x2) -> c26(MINSORT(app(add(s(x0), if_rm(eq(0, z1), 0, add(z1, z2))), x2), nil), APP(rm(0, add(s(x0), add(z1, z2))), x2), RM(0, add(s(x0), add(z1, z2)))) IF_MINSORT(true, add(0, add(s(x0), x1)), x2) -> c5(APP(rm(0, add(s(x0), x1)), x2)) IF_MINSORT(true, add(0, add(s(x0), x1)), x2) -> c5(RM(0, add(s(x0), x1))) IF_MINSORT(true, add(s(x0), add(0, x1)), z2) -> c26(MINSORT(add(0, app(rm(s(x0), x1), z2)), nil), APP(rm(s(x0), add(0, x1)), z2), RM(s(x0), add(0, x1))) IF_MINSORT(true, add(s(x0), add(0, nil)), x2) -> c26(MINSORT(app(add(0, nil), x2), nil), APP(rm(s(x0), add(0, nil)), x2), RM(s(x0), add(0, nil))) IF_MINSORT(true, add(s(x0), add(0, add(z1, z2))), x2) -> c26(MINSORT(app(add(0, if_rm(eq(s(x0), z1), s(x0), add(z1, z2))), x2), nil), APP(rm(s(x0), add(0, add(z1, z2))), x2), RM(s(x0), add(0, add(z1, z2)))) IF_MINSORT(true, add(s(x0), add(0, x1)), x2) -> c7(APP(rm(s(x0), add(0, x1)), x2)) IF_MINSORT(true, add(s(x0), add(0, x1)), x2) -> c7(RM(s(x0), add(0, x1))) IF_MINSORT(true, add(s(0), add(s(0), z2)), x1) -> c26(MINSORT(app(rm(s(0), z2), x1), nil), APP(rm(s(0), add(s(0), z2)), x1), RM(s(0), add(s(0), z2))) IF_MINSORT(true, add(s(0), add(s(0), x0)), x1) -> c26(RM(s(0), add(s(0), x0))) IF_MINSORT(true, add(s(0), add(s(s(x0)), z2)), x2) -> c26(MINSORT(app(add(s(s(x0)), rm(s(0), z2)), x2), nil), APP(rm(s(0), add(s(s(x0)), z2)), x2), RM(s(0), add(s(s(x0)), z2))) IF_MINSORT(true, add(s(0), add(s(s(x0)), x1)), x2) -> c26(RM(s(0), add(s(s(x0)), x1))) IF_MINSORT(true, add(s(s(x0)), add(s(0), z2)), x2) -> c26(MINSORT(app(add(s(0), rm(s(s(x0)), z2)), x2), nil), APP(rm(s(s(x0)), add(s(0), z2)), x2), RM(s(s(x0)), add(s(0), z2))) IF_MINSORT(true, add(s(s(x0)), add(s(0), x1)), x2) -> c26(RM(s(s(x0)), add(s(0), x1))) IF_MINSORT(true, add(s(s(0)), add(s(s(0)), x2)), x3) -> c26(MINSORT(app(if_rm(true, s(s(0)), add(s(s(0)), x2)), x3), nil), APP(rm(s(s(0)), add(s(s(0)), x2)), x3), RM(s(s(0)), add(s(s(0)), x2))) IF_MINSORT(true, add(s(s(0)), add(s(s(s(z0))), x2)), x3) -> c26(MINSORT(app(if_rm(false, s(s(0)), add(s(s(s(z0))), x2)), x3), nil), APP(rm(s(s(0)), add(s(s(s(z0))), x2)), x3), RM(s(s(0)), add(s(s(s(z0))), x2))) IF_MINSORT(true, add(s(s(s(z0))), add(s(s(0)), x2)), x3) -> c26(MINSORT(app(if_rm(false, s(s(s(z0))), add(s(s(0)), x2)), x3), nil), APP(rm(s(s(s(z0))), add(s(s(0)), x2)), x3), RM(s(s(s(z0))), add(s(s(0)), x2))) IF_MINSORT(true, add(s(s(s(z0))), add(s(s(s(z1))), x2)), x3) -> c26(MINSORT(app(if_rm(eq(z0, z1), s(s(s(z0))), add(s(s(s(z1))), x2)), x3), nil), APP(rm(s(s(s(z0))), add(s(s(s(z1))), x2)), x3), RM(s(s(s(z0))), add(s(s(s(z1))), x2))) IF_MINSORT(true, add(s(s(x0)), add(s(s(x1)), x2)), x3) -> c9(APP(rm(s(s(x0)), add(s(s(x1)), x2)), x3)) IF_MINSORT(true, add(s(s(x0)), add(s(s(x1)), x2)), x3) -> c9(RM(s(s(x0)), add(s(s(x1)), x2))) IF_MINSORT(true, add(s(x0), add(s(x1), z2)), x3) -> c1(APP(if_rm(eq(s(x0), s(x1)), s(x0), add(s(x1), z2)), x3)) IF_MINSORT(true, add(0, add(0, x2)), x3) -> c26(APP(if_rm(true, 0, add(0, x2)), x3)) IF_MINSORT(true, add(0, add(s(z0), x2)), x3) -> c26(APP(if_rm(false, 0, add(s(z0), x2)), x3)) IF_MINSORT(true, add(s(z0), add(0, x2)), x3) -> c26(APP(if_rm(false, s(z0), add(0, x2)), x3)) IF_MINSORT(true, add(s(z0), add(s(z1), x2)), x3) -> c26(APP(if_rm(eq(z0, z1), s(z0), add(s(z1), x2)), x3)) MINSORT(add(s(x0), z0), x2) -> c25(IF_MINSORT(eq(s(x0), minIter(z0, add(s(x0), z0), 0)), add(s(x0), z0), x2), EQ(s(x0), minIter(add(s(x0), z0), add(s(x0), z0), 0)), MIN(add(s(x0), z0))) MINSORT(add(s(x0), x1), x2) -> c13(EQ(s(x0), minIter(add(s(x0), x1), add(s(x0), x1), 0))) MINSORT(add(s(x0), x1), x2) -> c13(MIN(add(s(x0), x1))) MINSORT(add(0, z0), x1) -> c25(IF_MINSORT(eq(0, 0), add(0, z0), x1), MIN(add(0, z0))) MINSORT(add(0, x0), x1) -> c25(MIN(add(0, x0))) MINSORT(add(z0, z1), x2) -> c2(EQ(z0, if_min(le(z0, 0), z1, add(z0, z1), 0))) MINSORT(add(s(z0), x1), x2) -> c(EQ(s(z0), if_min(false, x1, add(s(z0), x1), 0))) IF_MINSORT(true, add(0, add(0, nil)), z0) -> c26(MINSORT(app(nil, z0), nil), APP(if_rm(eq(0, 0), 0, add(0, nil)), z0), RM(0, add(0, nil))) IF_MIN(false, x1, x2, 0) -> c14(MINITER(x1, x2, 0)) IF_MIN(false, x1, x2, s(0)) -> c14(MINITER(x1, x2, s(0))) IF_MIN(false, x1, x2, s(s(0))) -> c14(MINITER(x1, x2, s(s(0)))) IF_MIN(false, x1, x2, s(s(s(0)))) -> c14(MINITER(x1, x2, s(s(s(0))))) IF_MIN(false, x1, x2, s(s(s(s(x3))))) -> c14(MINITER(x1, x2, s(s(s(s(x3)))))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) MINITER(add(s(s(z0)), x1), x2, s(0)) -> c12(IF_MIN(false, x1, x2, s(0)), LE(s(s(z0)), s(0))) MINITER(add(s(x0), x1), x2, s(x3)) -> c12(LE(s(x0), s(x3))) MINITER(add(s(0), x1), x2, s(z0)) -> c12(LE(s(0), s(z0))) MINITER(add(s(s(s(z0))), x1), x2, s(s(0))) -> c12(IF_MIN(false, x1, x2, s(s(0))), LE(s(s(s(z0))), s(s(0)))) MINITER(add(s(s(x0)), x1), x2, s(s(x3))) -> c12(LE(s(s(x0)), s(s(x3)))) MINITER(add(s(s(0)), x1), x2, s(s(z0))) -> c12(LE(s(s(0)), s(s(z0)))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(0)))) -> c12(IF_MIN(false, x1, x2, s(s(s(0)))), LE(s(s(s(s(z0)))), s(s(s(0))))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(s(z1))))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(s(s(z1))))), LE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MINITER(add(s(s(s(x0))), x1), x2, s(s(s(x3)))) -> c12(LE(s(s(s(x0))), s(s(s(x3))))) MINITER(add(s(s(s(0))), x1), x2, s(s(s(z0)))) -> c12(LE(s(s(s(0))), s(s(s(z0))))) IF_MIN(false, x1, x2, 0) -> c14(MINITER(x1, x2, 0)) IF_MIN(false, x1, x2, s(0)) -> c14(MINITER(x1, x2, s(0))) IF_MIN(false, x1, x2, s(s(0))) -> c14(MINITER(x1, x2, s(s(0)))) IF_MIN(false, x1, x2, s(s(s(0)))) -> c14(MINITER(x1, x2, s(s(s(0))))) IF_MIN(false, x1, x2, s(s(s(s(x3))))) -> c14(MINITER(x1, x2, s(s(s(s(x3)))))) K tuples: IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MIN(add(z0, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) Defined Rule Symbols: le_2, eq_2, app_2, if_rm_3, rm_2, if_min_4, minIter_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, MIN_1, MINITER_3, IF_RM_3, IF_MINSORT_3, RM_2, MINSORT_2, IF_MIN_4 Compound Symbols: c3_1, c6_1, c8_1, c10_1, c11_1, c22_1, c23_1, c27_1, c12_1, c21_1, c12_2, c21_2, c_1, c26_1, c1_1, c2_1, c26_3, c4_1, c5_1, c7_1, c9_1, c25_3, c13_1, c25_2, c25_1, c14_1 ---------------------------------------- (197) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace MINITER(nil, add(z0, z1), z2) -> c11(MINITER(add(z0, z1), add(z0, z1), s(z2))) by MINITER(nil, add(z0, z1), 0) -> c11(MINITER(add(z0, z1), add(z0, z1), s(0))) MINITER(nil, add(z0, z1), s(0)) -> c11(MINITER(add(z0, z1), add(z0, z1), s(s(0)))) MINITER(nil, add(z0, z1), s(s(0))) -> c11(MINITER(add(z0, z1), add(z0, z1), s(s(s(0))))) MINITER(nil, add(z0, z1), s(s(s(0)))) -> c11(MINITER(add(z0, z1), add(z0, z1), s(s(s(s(0)))))) MINITER(nil, add(z0, z1), s(s(s(s(x2))))) -> c11(MINITER(add(z0, z1), add(z0, z1), s(s(s(s(s(x2))))))) ---------------------------------------- (198) 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) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) if_min(true, z0, z1, z2) -> z2 if_min(false, z0, z1, z2) -> minIter(z0, z1, z2) minIter(add(z0, z1), z2, z3) -> if_min(le(z0, z3), z1, z2, z3) minIter(nil, add(z0, z1), z2) -> minIter(add(z0, z1), add(z0, z1), s(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, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) RM(0, add(0, x2)) -> c21(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c21(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c21(IF_RM(false, s(z0), add(0, x2))) MINITER(add(s(s(z0)), x1), x2, s(0)) -> c12(IF_MIN(false, x1, x2, s(0)), LE(s(s(z0)), s(0))) MINITER(add(s(x0), x1), x2, s(x3)) -> c12(LE(s(x0), s(x3))) MINITER(add(s(0), x1), x2, s(z0)) -> c12(LE(s(0), s(z0))) RM(s(0), add(s(0), x2)) -> c21(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c21(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c21(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(x0), add(s(x1), x2)) -> c21(EQ(s(x0), s(x1))) MINSORT(add(x0, x1), x2) -> c(MIN(add(x0, x1))) MINITER(add(s(s(s(z0))), x1), x2, s(s(0))) -> c12(IF_MIN(false, x1, x2, s(s(0))), LE(s(s(s(z0))), s(s(0)))) MINITER(add(s(s(x0)), x1), x2, s(s(x3))) -> c12(LE(s(s(x0)), s(s(x3)))) MINITER(add(s(s(0)), x1), x2, s(s(z0))) -> c12(LE(s(s(0)), s(s(z0)))) RM(s(s(0)), add(s(s(0)), x2)) -> c21(IF_RM(true, s(s(0)), add(s(s(0)), x2)), EQ(s(s(0)), s(s(0)))) RM(s(s(0)), add(s(s(s(z0))), x2)) -> c21(IF_RM(false, s(s(0)), add(s(s(s(z0))), x2)), EQ(s(s(0)), s(s(s(z0))))) RM(s(s(s(z0))), add(s(s(0)), x2)) -> c21(IF_RM(false, s(s(s(z0))), add(s(s(0)), x2)), EQ(s(s(s(z0))), s(s(0)))) RM(s(s(x0)), add(s(s(x1)), x2)) -> c21(EQ(s(s(x0)), s(s(x1)))) IF_MINSORT(true, add(0, add(0, x0)), x1) -> c26(RM(0, add(0, x0))) IF_MINSORT(true, add(0, add(s(x0), x1)), x2) -> c26(RM(0, add(s(x0), x1))) IF_MINSORT(true, add(s(x0), add(0, x1)), x2) -> c26(RM(s(x0), add(0, x1))) IF_MINSORT(true, add(s(x0), add(s(x1), x2)), x3) -> c1(RM(s(x0), add(s(x1), x2))) IF_MINSORT(true, add(z0, nil), x1) -> c26(MINSORT(x1, nil)) MINSORT(add(x0, x1), x2) -> c2(MIN(add(x0, x1))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(0)))) -> c12(IF_MIN(false, x1, x2, s(s(s(0)))), LE(s(s(s(s(z0)))), s(s(s(0))))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(s(z1))))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(s(s(z1))))), LE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MINITER(add(s(s(s(x0))), x1), x2, s(s(s(x3)))) -> c12(LE(s(s(s(x0))), s(s(s(x3))))) MINITER(add(s(s(s(0))), x1), x2, s(s(s(z0)))) -> c12(LE(s(s(s(0))), s(s(s(z0))))) RM(s(s(s(0))), add(s(s(s(0))), x2)) -> c21(IF_RM(true, s(s(s(0))), add(s(s(s(0))), x2)), EQ(s(s(s(0))), s(s(s(0))))) RM(s(s(s(0))), add(s(s(s(s(z0)))), x2)) -> c21(IF_RM(false, s(s(s(0))), add(s(s(s(s(z0)))), x2)), EQ(s(s(s(0))), s(s(s(s(z0)))))) RM(s(s(s(s(z0)))), add(s(s(s(0))), x2)) -> c21(IF_RM(false, s(s(s(s(z0)))), add(s(s(s(0))), x2)), EQ(s(s(s(s(z0)))), s(s(s(0))))) RM(s(s(s(s(z0)))), add(s(s(s(s(z1)))), x2)) -> c21(IF_RM(eq(z0, z1), s(s(s(s(z0)))), add(s(s(s(s(z1)))), x2)), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) RM(s(s(s(x0))), add(s(s(s(x1))), x2)) -> c21(EQ(s(s(s(x0))), s(s(s(x1))))) IF_MINSORT(true, add(0, add(0, add(z1, z2))), x1) -> c26(MINSORT(app(if_rm(eq(0, z1), 0, add(z1, z2)), x1), nil), APP(rm(0, add(0, add(z1, z2))), x1), RM(0, add(0, add(z1, z2)))) IF_MINSORT(true, add(0, add(0, x0)), x1) -> c4(APP(rm(0, add(0, x0)), x1)) IF_MINSORT(true, add(0, add(0, x0)), x1) -> c4(RM(0, add(0, x0))) IF_MINSORT(true, add(0, add(s(x0), x1)), z2) -> c26(MINSORT(add(s(x0), app(rm(0, x1), z2)), nil), APP(rm(0, add(s(x0), x1)), z2), RM(0, add(s(x0), x1))) IF_MINSORT(true, add(0, add(s(x0), nil)), x2) -> c26(MINSORT(app(add(s(x0), nil), x2), nil), APP(rm(0, add(s(x0), nil)), x2), RM(0, add(s(x0), nil))) IF_MINSORT(true, add(0, add(s(x0), add(z1, z2))), x2) -> c26(MINSORT(app(add(s(x0), if_rm(eq(0, z1), 0, add(z1, z2))), x2), nil), APP(rm(0, add(s(x0), add(z1, z2))), x2), RM(0, add(s(x0), add(z1, z2)))) IF_MINSORT(true, add(0, add(s(x0), x1)), x2) -> c5(APP(rm(0, add(s(x0), x1)), x2)) IF_MINSORT(true, add(0, add(s(x0), x1)), x2) -> c5(RM(0, add(s(x0), x1))) IF_MINSORT(true, add(s(x0), add(0, x1)), z2) -> c26(MINSORT(add(0, app(rm(s(x0), x1), z2)), nil), APP(rm(s(x0), add(0, x1)), z2), RM(s(x0), add(0, x1))) IF_MINSORT(true, add(s(x0), add(0, nil)), x2) -> c26(MINSORT(app(add(0, nil), x2), nil), APP(rm(s(x0), add(0, nil)), x2), RM(s(x0), add(0, nil))) IF_MINSORT(true, add(s(x0), add(0, add(z1, z2))), x2) -> c26(MINSORT(app(add(0, if_rm(eq(s(x0), z1), s(x0), add(z1, z2))), x2), nil), APP(rm(s(x0), add(0, add(z1, z2))), x2), RM(s(x0), add(0, add(z1, z2)))) IF_MINSORT(true, add(s(x0), add(0, x1)), x2) -> c7(APP(rm(s(x0), add(0, x1)), x2)) IF_MINSORT(true, add(s(x0), add(0, x1)), x2) -> c7(RM(s(x0), add(0, x1))) IF_MINSORT(true, add(s(0), add(s(0), z2)), x1) -> c26(MINSORT(app(rm(s(0), z2), x1), nil), APP(rm(s(0), add(s(0), z2)), x1), RM(s(0), add(s(0), z2))) IF_MINSORT(true, add(s(0), add(s(0), x0)), x1) -> c26(RM(s(0), add(s(0), x0))) IF_MINSORT(true, add(s(0), add(s(s(x0)), z2)), x2) -> c26(MINSORT(app(add(s(s(x0)), rm(s(0), z2)), x2), nil), APP(rm(s(0), add(s(s(x0)), z2)), x2), RM(s(0), add(s(s(x0)), z2))) IF_MINSORT(true, add(s(0), add(s(s(x0)), x1)), x2) -> c26(RM(s(0), add(s(s(x0)), x1))) IF_MINSORT(true, add(s(s(x0)), add(s(0), z2)), x2) -> c26(MINSORT(app(add(s(0), rm(s(s(x0)), z2)), x2), nil), APP(rm(s(s(x0)), add(s(0), z2)), x2), RM(s(s(x0)), add(s(0), z2))) IF_MINSORT(true, add(s(s(x0)), add(s(0), x1)), x2) -> c26(RM(s(s(x0)), add(s(0), x1))) IF_MINSORT(true, add(s(s(0)), add(s(s(0)), x2)), x3) -> c26(MINSORT(app(if_rm(true, s(s(0)), add(s(s(0)), x2)), x3), nil), APP(rm(s(s(0)), add(s(s(0)), x2)), x3), RM(s(s(0)), add(s(s(0)), x2))) IF_MINSORT(true, add(s(s(0)), add(s(s(s(z0))), x2)), x3) -> c26(MINSORT(app(if_rm(false, s(s(0)), add(s(s(s(z0))), x2)), x3), nil), APP(rm(s(s(0)), add(s(s(s(z0))), x2)), x3), RM(s(s(0)), add(s(s(s(z0))), x2))) IF_MINSORT(true, add(s(s(s(z0))), add(s(s(0)), x2)), x3) -> c26(MINSORT(app(if_rm(false, s(s(s(z0))), add(s(s(0)), x2)), x3), nil), APP(rm(s(s(s(z0))), add(s(s(0)), x2)), x3), RM(s(s(s(z0))), add(s(s(0)), x2))) IF_MINSORT(true, add(s(s(s(z0))), add(s(s(s(z1))), x2)), x3) -> c26(MINSORT(app(if_rm(eq(z0, z1), s(s(s(z0))), add(s(s(s(z1))), x2)), x3), nil), APP(rm(s(s(s(z0))), add(s(s(s(z1))), x2)), x3), RM(s(s(s(z0))), add(s(s(s(z1))), x2))) IF_MINSORT(true, add(s(s(x0)), add(s(s(x1)), x2)), x3) -> c9(APP(rm(s(s(x0)), add(s(s(x1)), x2)), x3)) IF_MINSORT(true, add(s(s(x0)), add(s(s(x1)), x2)), x3) -> c9(RM(s(s(x0)), add(s(s(x1)), x2))) IF_MINSORT(true, add(s(x0), add(s(x1), z2)), x3) -> c1(APP(if_rm(eq(s(x0), s(x1)), s(x0), add(s(x1), z2)), x3)) IF_MINSORT(true, add(0, add(0, x2)), x3) -> c26(APP(if_rm(true, 0, add(0, x2)), x3)) IF_MINSORT(true, add(0, add(s(z0), x2)), x3) -> c26(APP(if_rm(false, 0, add(s(z0), x2)), x3)) IF_MINSORT(true, add(s(z0), add(0, x2)), x3) -> c26(APP(if_rm(false, s(z0), add(0, x2)), x3)) IF_MINSORT(true, add(s(z0), add(s(z1), x2)), x3) -> c26(APP(if_rm(eq(z0, z1), s(z0), add(s(z1), x2)), x3)) MINSORT(add(s(x0), z0), x2) -> c25(IF_MINSORT(eq(s(x0), minIter(z0, add(s(x0), z0), 0)), add(s(x0), z0), x2), EQ(s(x0), minIter(add(s(x0), z0), add(s(x0), z0), 0)), MIN(add(s(x0), z0))) MINSORT(add(s(x0), x1), x2) -> c13(EQ(s(x0), minIter(add(s(x0), x1), add(s(x0), x1), 0))) MINSORT(add(s(x0), x1), x2) -> c13(MIN(add(s(x0), x1))) MINSORT(add(0, z0), x1) -> c25(IF_MINSORT(eq(0, 0), add(0, z0), x1), MIN(add(0, z0))) MINSORT(add(0, x0), x1) -> c25(MIN(add(0, x0))) MINSORT(add(z0, z1), x2) -> c2(EQ(z0, if_min(le(z0, 0), z1, add(z0, z1), 0))) MINSORT(add(s(z0), x1), x2) -> c(EQ(s(z0), if_min(false, x1, add(s(z0), x1), 0))) IF_MINSORT(true, add(0, add(0, nil)), z0) -> c26(MINSORT(app(nil, z0), nil), APP(if_rm(eq(0, 0), 0, add(0, nil)), z0), RM(0, add(0, nil))) IF_MIN(false, x1, x2, 0) -> c14(MINITER(x1, x2, 0)) IF_MIN(false, x1, x2, s(0)) -> c14(MINITER(x1, x2, s(0))) IF_MIN(false, x1, x2, s(s(0))) -> c14(MINITER(x1, x2, s(s(0)))) IF_MIN(false, x1, x2, s(s(s(0)))) -> c14(MINITER(x1, x2, s(s(s(0))))) IF_MIN(false, x1, x2, s(s(s(s(x3))))) -> c14(MINITER(x1, x2, s(s(s(s(x3)))))) MINITER(nil, add(z0, z1), 0) -> c11(MINITER(add(z0, z1), add(z0, z1), s(0))) MINITER(nil, add(z0, z1), s(0)) -> c11(MINITER(add(z0, z1), add(z0, z1), s(s(0)))) MINITER(nil, add(z0, z1), s(s(0))) -> c11(MINITER(add(z0, z1), add(z0, z1), s(s(s(0))))) MINITER(nil, add(z0, z1), s(s(s(0)))) -> c11(MINITER(add(z0, z1), add(z0, z1), s(s(s(s(0)))))) MINITER(nil, add(z0, z1), s(s(s(s(x2))))) -> c11(MINITER(add(z0, z1), add(z0, z1), s(s(s(s(s(x2))))))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) MINITER(add(s(s(z0)), x1), x2, s(0)) -> c12(IF_MIN(false, x1, x2, s(0)), LE(s(s(z0)), s(0))) MINITER(add(s(x0), x1), x2, s(x3)) -> c12(LE(s(x0), s(x3))) MINITER(add(s(0), x1), x2, s(z0)) -> c12(LE(s(0), s(z0))) MINITER(add(s(s(s(z0))), x1), x2, s(s(0))) -> c12(IF_MIN(false, x1, x2, s(s(0))), LE(s(s(s(z0))), s(s(0)))) MINITER(add(s(s(x0)), x1), x2, s(s(x3))) -> c12(LE(s(s(x0)), s(s(x3)))) MINITER(add(s(s(0)), x1), x2, s(s(z0))) -> c12(LE(s(s(0)), s(s(z0)))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(0)))) -> c12(IF_MIN(false, x1, x2, s(s(s(0)))), LE(s(s(s(s(z0)))), s(s(s(0))))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(s(z1))))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(s(s(z1))))), LE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MINITER(add(s(s(s(x0))), x1), x2, s(s(s(x3)))) -> c12(LE(s(s(s(x0))), s(s(s(x3))))) MINITER(add(s(s(s(0))), x1), x2, s(s(s(z0)))) -> c12(LE(s(s(s(0))), s(s(s(z0))))) IF_MIN(false, x1, x2, 0) -> c14(MINITER(x1, x2, 0)) IF_MIN(false, x1, x2, s(0)) -> c14(MINITER(x1, x2, s(0))) IF_MIN(false, x1, x2, s(s(0))) -> c14(MINITER(x1, x2, s(s(0)))) IF_MIN(false, x1, x2, s(s(s(0)))) -> c14(MINITER(x1, x2, s(s(s(0))))) IF_MIN(false, x1, x2, s(s(s(s(x3))))) -> c14(MINITER(x1, x2, s(s(s(s(x3)))))) MINITER(nil, add(z0, z1), 0) -> c11(MINITER(add(z0, z1), add(z0, z1), s(0))) MINITER(nil, add(z0, z1), s(0)) -> c11(MINITER(add(z0, z1), add(z0, z1), s(s(0)))) MINITER(nil, add(z0, z1), s(s(0))) -> c11(MINITER(add(z0, z1), add(z0, z1), s(s(s(0))))) MINITER(nil, add(z0, z1), s(s(s(0)))) -> c11(MINITER(add(z0, z1), add(z0, z1), s(s(s(s(0)))))) MINITER(nil, add(z0, z1), s(s(s(s(x2))))) -> c11(MINITER(add(z0, z1), add(z0, z1), s(s(s(s(s(x2))))))) K tuples: IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MIN(add(z0, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) Defined Rule Symbols: le_2, eq_2, app_2, if_rm_3, rm_2, if_min_4, minIter_3 Defined Pair Symbols: EQ_2, LE_2, APP_2, MIN_1, IF_RM_3, IF_MINSORT_3, MINITER_3, RM_2, MINSORT_2, IF_MIN_4 Compound Symbols: c3_1, c6_1, c8_1, c10_1, c22_1, c23_1, c27_1, c12_1, c21_1, c12_2, c21_2, c_1, c26_1, c1_1, c2_1, c26_3, c4_1, c5_1, c7_1, c9_1, c25_3, c13_1, c25_2, c25_1, c14_1, c11_1 ---------------------------------------- (199) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) by EQ(s(s(y0)), s(s(y1))) -> c3(EQ(s(y0), s(y1))) ---------------------------------------- (200) 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) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) if_min(true, z0, z1, z2) -> z2 if_min(false, z0, z1, z2) -> minIter(z0, z1, z2) minIter(add(z0, z1), z2, z3) -> if_min(le(z0, z3), z1, z2, z3) minIter(nil, add(z0, z1), z2) -> minIter(add(z0, z1), add(z0, z1), s(z2)) Tuples: LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) MIN(add(z0, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) RM(0, add(0, x2)) -> c21(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c21(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c21(IF_RM(false, s(z0), add(0, x2))) MINITER(add(s(s(z0)), x1), x2, s(0)) -> c12(IF_MIN(false, x1, x2, s(0)), LE(s(s(z0)), s(0))) MINITER(add(s(x0), x1), x2, s(x3)) -> c12(LE(s(x0), s(x3))) MINITER(add(s(0), x1), x2, s(z0)) -> c12(LE(s(0), s(z0))) RM(s(0), add(s(0), x2)) -> c21(IF_RM(true, s(0), add(s(0), x2)), EQ(s(0), s(0))) RM(s(0), add(s(s(z0)), x2)) -> c21(IF_RM(false, s(0), add(s(s(z0)), x2)), EQ(s(0), s(s(z0)))) RM(s(s(z0)), add(s(0), x2)) -> c21(IF_RM(false, s(s(z0)), add(s(0), x2)), EQ(s(s(z0)), s(0))) RM(s(x0), add(s(x1), x2)) -> c21(EQ(s(x0), s(x1))) MINSORT(add(x0, x1), x2) -> c(MIN(add(x0, x1))) MINITER(add(s(s(s(z0))), x1), x2, s(s(0))) -> c12(IF_MIN(false, x1, x2, s(s(0))), LE(s(s(s(z0))), s(s(0)))) MINITER(add(s(s(x0)), x1), x2, s(s(x3))) -> c12(LE(s(s(x0)), s(s(x3)))) MINITER(add(s(s(0)), x1), x2, s(s(z0))) -> c12(LE(s(s(0)), s(s(z0)))) RM(s(s(0)), add(s(s(0)), x2)) -> c21(IF_RM(true, s(s(0)), add(s(s(0)), x2)), EQ(s(s(0)), s(s(0)))) RM(s(s(0)), add(s(s(s(z0))), x2)) -> c21(IF_RM(false, s(s(0)), add(s(s(s(z0))), x2)), EQ(s(s(0)), s(s(s(z0))))) RM(s(s(s(z0))), add(s(s(0)), x2)) -> c21(IF_RM(false, s(s(s(z0))), add(s(s(0)), x2)), EQ(s(s(s(z0))), s(s(0)))) RM(s(s(x0)), add(s(s(x1)), x2)) -> c21(EQ(s(s(x0)), s(s(x1)))) IF_MINSORT(true, add(0, add(0, x0)), x1) -> c26(RM(0, add(0, x0))) IF_MINSORT(true, add(0, add(s(x0), x1)), x2) -> c26(RM(0, add(s(x0), x1))) IF_MINSORT(true, add(s(x0), add(0, x1)), x2) -> c26(RM(s(x0), add(0, x1))) IF_MINSORT(true, add(s(x0), add(s(x1), x2)), x3) -> c1(RM(s(x0), add(s(x1), x2))) IF_MINSORT(true, add(z0, nil), x1) -> c26(MINSORT(x1, nil)) MINSORT(add(x0, x1), x2) -> c2(MIN(add(x0, x1))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(0)))) -> c12(IF_MIN(false, x1, x2, s(s(s(0)))), LE(s(s(s(s(z0)))), s(s(s(0))))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(s(z1))))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(s(s(z1))))), LE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MINITER(add(s(s(s(x0))), x1), x2, s(s(s(x3)))) -> c12(LE(s(s(s(x0))), s(s(s(x3))))) MINITER(add(s(s(s(0))), x1), x2, s(s(s(z0)))) -> c12(LE(s(s(s(0))), s(s(s(z0))))) RM(s(s(s(0))), add(s(s(s(0))), x2)) -> c21(IF_RM(true, s(s(s(0))), add(s(s(s(0))), x2)), EQ(s(s(s(0))), s(s(s(0))))) RM(s(s(s(0))), add(s(s(s(s(z0)))), x2)) -> c21(IF_RM(false, s(s(s(0))), add(s(s(s(s(z0)))), x2)), EQ(s(s(s(0))), s(s(s(s(z0)))))) RM(s(s(s(s(z0)))), add(s(s(s(0))), x2)) -> c21(IF_RM(false, s(s(s(s(z0)))), add(s(s(s(0))), x2)), EQ(s(s(s(s(z0)))), s(s(s(0))))) RM(s(s(s(s(z0)))), add(s(s(s(s(z1)))), x2)) -> c21(IF_RM(eq(z0, z1), s(s(s(s(z0)))), add(s(s(s(s(z1)))), x2)), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) RM(s(s(s(x0))), add(s(s(s(x1))), x2)) -> c21(EQ(s(s(s(x0))), s(s(s(x1))))) IF_MINSORT(true, add(0, add(0, add(z1, z2))), x1) -> c26(MINSORT(app(if_rm(eq(0, z1), 0, add(z1, z2)), x1), nil), APP(rm(0, add(0, add(z1, z2))), x1), RM(0, add(0, add(z1, z2)))) IF_MINSORT(true, add(0, add(0, x0)), x1) -> c4(APP(rm(0, add(0, x0)), x1)) IF_MINSORT(true, add(0, add(0, x0)), x1) -> c4(RM(0, add(0, x0))) IF_MINSORT(true, add(0, add(s(x0), x1)), z2) -> c26(MINSORT(add(s(x0), app(rm(0, x1), z2)), nil), APP(rm(0, add(s(x0), x1)), z2), RM(0, add(s(x0), x1))) IF_MINSORT(true, add(0, add(s(x0), nil)), x2) -> c26(MINSORT(app(add(s(x0), nil), x2), nil), APP(rm(0, add(s(x0), nil)), x2), RM(0, add(s(x0), nil))) IF_MINSORT(true, add(0, add(s(x0), add(z1, z2))), x2) -> c26(MINSORT(app(add(s(x0), if_rm(eq(0, z1), 0, add(z1, z2))), x2), nil), APP(rm(0, add(s(x0), add(z1, z2))), x2), RM(0, add(s(x0), add(z1, z2)))) IF_MINSORT(true, add(0, add(s(x0), x1)), x2) -> c5(APP(rm(0, add(s(x0), x1)), x2)) IF_MINSORT(true, add(0, add(s(x0), x1)), x2) -> c5(RM(0, add(s(x0), x1))) IF_MINSORT(true, add(s(x0), add(0, x1)), z2) -> c26(MINSORT(add(0, app(rm(s(x0), x1), z2)), nil), APP(rm(s(x0), add(0, x1)), z2), RM(s(x0), add(0, x1))) IF_MINSORT(true, add(s(x0), add(0, nil)), x2) -> c26(MINSORT(app(add(0, nil), x2), nil), APP(rm(s(x0), add(0, nil)), x2), RM(s(x0), add(0, nil))) IF_MINSORT(true, add(s(x0), add(0, add(z1, z2))), x2) -> c26(MINSORT(app(add(0, if_rm(eq(s(x0), z1), s(x0), add(z1, z2))), x2), nil), APP(rm(s(x0), add(0, add(z1, z2))), x2), RM(s(x0), add(0, add(z1, z2)))) IF_MINSORT(true, add(s(x0), add(0, x1)), x2) -> c7(APP(rm(s(x0), add(0, x1)), x2)) IF_MINSORT(true, add(s(x0), add(0, x1)), x2) -> c7(RM(s(x0), add(0, x1))) IF_MINSORT(true, add(s(0), add(s(0), z2)), x1) -> c26(MINSORT(app(rm(s(0), z2), x1), nil), APP(rm(s(0), add(s(0), z2)), x1), RM(s(0), add(s(0), z2))) IF_MINSORT(true, add(s(0), add(s(0), x0)), x1) -> c26(RM(s(0), add(s(0), x0))) IF_MINSORT(true, add(s(0), add(s(s(x0)), z2)), x2) -> c26(MINSORT(app(add(s(s(x0)), rm(s(0), z2)), x2), nil), APP(rm(s(0), add(s(s(x0)), z2)), x2), RM(s(0), add(s(s(x0)), z2))) IF_MINSORT(true, add(s(0), add(s(s(x0)), x1)), x2) -> c26(RM(s(0), add(s(s(x0)), x1))) IF_MINSORT(true, add(s(s(x0)), add(s(0), z2)), x2) -> c26(MINSORT(app(add(s(0), rm(s(s(x0)), z2)), x2), nil), APP(rm(s(s(x0)), add(s(0), z2)), x2), RM(s(s(x0)), add(s(0), z2))) IF_MINSORT(true, add(s(s(x0)), add(s(0), x1)), x2) -> c26(RM(s(s(x0)), add(s(0), x1))) IF_MINSORT(true, add(s(s(0)), add(s(s(0)), x2)), x3) -> c26(MINSORT(app(if_rm(true, s(s(0)), add(s(s(0)), x2)), x3), nil), APP(rm(s(s(0)), add(s(s(0)), x2)), x3), RM(s(s(0)), add(s(s(0)), x2))) IF_MINSORT(true, add(s(s(0)), add(s(s(s(z0))), x2)), x3) -> c26(MINSORT(app(if_rm(false, s(s(0)), add(s(s(s(z0))), x2)), x3), nil), APP(rm(s(s(0)), add(s(s(s(z0))), x2)), x3), RM(s(s(0)), add(s(s(s(z0))), x2))) IF_MINSORT(true, add(s(s(s(z0))), add(s(s(0)), x2)), x3) -> c26(MINSORT(app(if_rm(false, s(s(s(z0))), add(s(s(0)), x2)), x3), nil), APP(rm(s(s(s(z0))), add(s(s(0)), x2)), x3), RM(s(s(s(z0))), add(s(s(0)), x2))) IF_MINSORT(true, add(s(s(s(z0))), add(s(s(s(z1))), x2)), x3) -> c26(MINSORT(app(if_rm(eq(z0, z1), s(s(s(z0))), add(s(s(s(z1))), x2)), x3), nil), APP(rm(s(s(s(z0))), add(s(s(s(z1))), x2)), x3), RM(s(s(s(z0))), add(s(s(s(z1))), x2))) IF_MINSORT(true, add(s(s(x0)), add(s(s(x1)), x2)), x3) -> c9(APP(rm(s(s(x0)), add(s(s(x1)), x2)), x3)) IF_MINSORT(true, add(s(s(x0)), add(s(s(x1)), x2)), x3) -> c9(RM(s(s(x0)), add(s(s(x1)), x2))) IF_MINSORT(true, add(s(x0), add(s(x1), z2)), x3) -> c1(APP(if_rm(eq(s(x0), s(x1)), s(x0), add(s(x1), z2)), x3)) IF_MINSORT(true, add(0, add(0, x2)), x3) -> c26(APP(if_rm(true, 0, add(0, x2)), x3)) IF_MINSORT(true, add(0, add(s(z0), x2)), x3) -> c26(APP(if_rm(false, 0, add(s(z0), x2)), x3)) IF_MINSORT(true, add(s(z0), add(0, x2)), x3) -> c26(APP(if_rm(false, s(z0), add(0, x2)), x3)) IF_MINSORT(true, add(s(z0), add(s(z1), x2)), x3) -> c26(APP(if_rm(eq(z0, z1), s(z0), add(s(z1), x2)), x3)) MINSORT(add(s(x0), z0), x2) -> c25(IF_MINSORT(eq(s(x0), minIter(z0, add(s(x0), z0), 0)), add(s(x0), z0), x2), EQ(s(x0), minIter(add(s(x0), z0), add(s(x0), z0), 0)), MIN(add(s(x0), z0))) MINSORT(add(s(x0), x1), x2) -> c13(EQ(s(x0), minIter(add(s(x0), x1), add(s(x0), x1), 0))) MINSORT(add(s(x0), x1), x2) -> c13(MIN(add(s(x0), x1))) MINSORT(add(0, z0), x1) -> c25(IF_MINSORT(eq(0, 0), add(0, z0), x1), MIN(add(0, z0))) MINSORT(add(0, x0), x1) -> c25(MIN(add(0, x0))) MINSORT(add(z0, z1), x2) -> c2(EQ(z0, if_min(le(z0, 0), z1, add(z0, z1), 0))) MINSORT(add(s(z0), x1), x2) -> c(EQ(s(z0), if_min(false, x1, add(s(z0), x1), 0))) IF_MINSORT(true, add(0, add(0, nil)), z0) -> c26(MINSORT(app(nil, z0), nil), APP(if_rm(eq(0, 0), 0, add(0, nil)), z0), RM(0, add(0, nil))) IF_MIN(false, x1, x2, 0) -> c14(MINITER(x1, x2, 0)) IF_MIN(false, x1, x2, s(0)) -> c14(MINITER(x1, x2, s(0))) IF_MIN(false, x1, x2, s(s(0))) -> c14(MINITER(x1, x2, s(s(0)))) IF_MIN(false, x1, x2, s(s(s(0)))) -> c14(MINITER(x1, x2, s(s(s(0))))) IF_MIN(false, x1, x2, s(s(s(s(x3))))) -> c14(MINITER(x1, x2, s(s(s(s(x3)))))) MINITER(nil, add(z0, z1), 0) -> c11(MINITER(add(z0, z1), add(z0, z1), s(0))) MINITER(nil, add(z0, z1), s(0)) -> c11(MINITER(add(z0, z1), add(z0, z1), s(s(0)))) MINITER(nil, add(z0, z1), s(s(0))) -> c11(MINITER(add(z0, z1), add(z0, z1), s(s(s(0))))) MINITER(nil, add(z0, z1), s(s(s(0)))) -> c11(MINITER(add(z0, z1), add(z0, z1), s(s(s(s(0)))))) MINITER(nil, add(z0, z1), s(s(s(s(x2))))) -> c11(MINITER(add(z0, z1), add(z0, z1), s(s(s(s(s(x2))))))) EQ(s(s(y0)), s(s(y1))) -> c3(EQ(s(y0), s(y1))) S tuples: LE(s(z0), s(z1)) -> c6(LE(z0, z1)) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) MINITER(add(s(s(z0)), x1), x2, s(0)) -> c12(IF_MIN(false, x1, x2, s(0)), LE(s(s(z0)), s(0))) MINITER(add(s(x0), x1), x2, s(x3)) -> c12(LE(s(x0), s(x3))) MINITER(add(s(0), x1), x2, s(z0)) -> c12(LE(s(0), s(z0))) MINITER(add(s(s(s(z0))), x1), x2, s(s(0))) -> c12(IF_MIN(false, x1, x2, s(s(0))), LE(s(s(s(z0))), s(s(0)))) MINITER(add(s(s(x0)), x1), x2, s(s(x3))) -> c12(LE(s(s(x0)), s(s(x3)))) MINITER(add(s(s(0)), x1), x2, s(s(z0))) -> c12(LE(s(s(0)), s(s(z0)))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(0)))) -> c12(IF_MIN(false, x1, x2, s(s(s(0)))), LE(s(s(s(s(z0)))), s(s(s(0))))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(s(z1))))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(s(s(z1))))), LE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MINITER(add(s(s(s(x0))), x1), x2, s(s(s(x3)))) -> c12(LE(s(s(s(x0))), s(s(s(x3))))) MINITER(add(s(s(s(0))), x1), x2, s(s(s(z0)))) -> c12(LE(s(s(s(0))), s(s(s(z0))))) IF_MIN(false, x1, x2, 0) -> c14(MINITER(x1, x2, 0)) IF_MIN(false, x1, x2, s(0)) -> c14(MINITER(x1, x2, s(0))) IF_MIN(false, x1, x2, s(s(0))) -> c14(MINITER(x1, x2, s(s(0)))) IF_MIN(false, x1, x2, s(s(s(0)))) -> c14(MINITER(x1, x2, s(s(s(0))))) IF_MIN(false, x1, x2, s(s(s(s(x3))))) -> c14(MINITER(x1, x2, s(s(s(s(x3)))))) MINITER(nil, add(z0, z1), 0) -> c11(MINITER(add(z0, z1), add(z0, z1), s(0))) MINITER(nil, add(z0, z1), s(0)) -> c11(MINITER(add(z0, z1), add(z0, z1), s(s(0)))) MINITER(nil, add(z0, z1), s(s(0))) -> c11(MINITER(add(z0, z1), add(z0, z1), s(s(s(0))))) MINITER(nil, add(z0, z1), s(s(s(0)))) -> c11(MINITER(add(z0, z1), add(z0, z1), s(s(s(s(0)))))) MINITER(nil, add(z0, z1), s(s(s(s(x2))))) -> c11(MINITER(add(z0, z1), add(z0, z1), s(s(s(s(s(x2))))))) EQ(s(s(y0)), s(s(y1))) -> c3(EQ(s(y0), s(y1))) K tuples: IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MIN(add(z0, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) Defined Rule Symbols: le_2, eq_2, app_2, if_rm_3, rm_2, if_min_4, minIter_3 Defined Pair Symbols: LE_2, APP_2, MIN_1, IF_RM_3, IF_MINSORT_3, MINITER_3, RM_2, MINSORT_2, IF_MIN_4, EQ_2 Compound Symbols: c6_1, c8_1, c10_1, c22_1, c23_1, c27_1, c12_1, c21_1, c12_2, c21_2, c_1, c26_1, c1_1, c2_1, c26_3, c4_1, c5_1, c7_1, c9_1, c25_3, c13_1, c25_2, c25_1, c14_1, c11_1, c3_1 ---------------------------------------- (201) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (202) 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) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) if_min(true, z0, z1, z2) -> z2 if_min(false, z0, z1, z2) -> minIter(z0, z1, z2) minIter(add(z0, z1), z2, z3) -> if_min(le(z0, z3), z1, z2, z3) minIter(nil, add(z0, z1), z2) -> minIter(add(z0, z1), add(z0, z1), s(z2)) Tuples: LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) MIN(add(z0, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) RM(0, add(0, x2)) -> c21(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c21(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c21(IF_RM(false, s(z0), add(0, x2))) MINITER(add(s(s(z0)), x1), x2, s(0)) -> c12(IF_MIN(false, x1, x2, s(0)), LE(s(s(z0)), s(0))) MINITER(add(s(x0), x1), x2, s(x3)) -> c12(LE(s(x0), s(x3))) MINITER(add(s(0), x1), x2, s(z0)) -> c12(LE(s(0), s(z0))) RM(s(x0), add(s(x1), x2)) -> c21(EQ(s(x0), s(x1))) MINSORT(add(x0, x1), x2) -> c(MIN(add(x0, x1))) MINITER(add(s(s(s(z0))), x1), x2, s(s(0))) -> c12(IF_MIN(false, x1, x2, s(s(0))), LE(s(s(s(z0))), s(s(0)))) MINITER(add(s(s(x0)), x1), x2, s(s(x3))) -> c12(LE(s(s(x0)), s(s(x3)))) MINITER(add(s(s(0)), x1), x2, s(s(z0))) -> c12(LE(s(s(0)), s(s(z0)))) RM(s(s(0)), add(s(s(0)), x2)) -> c21(IF_RM(true, s(s(0)), add(s(s(0)), x2)), EQ(s(s(0)), s(s(0)))) RM(s(s(0)), add(s(s(s(z0))), x2)) -> c21(IF_RM(false, s(s(0)), add(s(s(s(z0))), x2)), EQ(s(s(0)), s(s(s(z0))))) RM(s(s(s(z0))), add(s(s(0)), x2)) -> c21(IF_RM(false, s(s(s(z0))), add(s(s(0)), x2)), EQ(s(s(s(z0))), s(s(0)))) RM(s(s(x0)), add(s(s(x1)), x2)) -> c21(EQ(s(s(x0)), s(s(x1)))) IF_MINSORT(true, add(0, add(0, x0)), x1) -> c26(RM(0, add(0, x0))) IF_MINSORT(true, add(0, add(s(x0), x1)), x2) -> c26(RM(0, add(s(x0), x1))) IF_MINSORT(true, add(s(x0), add(0, x1)), x2) -> c26(RM(s(x0), add(0, x1))) IF_MINSORT(true, add(s(x0), add(s(x1), x2)), x3) -> c1(RM(s(x0), add(s(x1), x2))) IF_MINSORT(true, add(z0, nil), x1) -> c26(MINSORT(x1, nil)) MINSORT(add(x0, x1), x2) -> c2(MIN(add(x0, x1))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(0)))) -> c12(IF_MIN(false, x1, x2, s(s(s(0)))), LE(s(s(s(s(z0)))), s(s(s(0))))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(s(z1))))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(s(s(z1))))), LE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MINITER(add(s(s(s(x0))), x1), x2, s(s(s(x3)))) -> c12(LE(s(s(s(x0))), s(s(s(x3))))) MINITER(add(s(s(s(0))), x1), x2, s(s(s(z0)))) -> c12(LE(s(s(s(0))), s(s(s(z0))))) RM(s(s(s(0))), add(s(s(s(0))), x2)) -> c21(IF_RM(true, s(s(s(0))), add(s(s(s(0))), x2)), EQ(s(s(s(0))), s(s(s(0))))) RM(s(s(s(0))), add(s(s(s(s(z0)))), x2)) -> c21(IF_RM(false, s(s(s(0))), add(s(s(s(s(z0)))), x2)), EQ(s(s(s(0))), s(s(s(s(z0)))))) RM(s(s(s(s(z0)))), add(s(s(s(0))), x2)) -> c21(IF_RM(false, s(s(s(s(z0)))), add(s(s(s(0))), x2)), EQ(s(s(s(s(z0)))), s(s(s(0))))) RM(s(s(s(s(z0)))), add(s(s(s(s(z1)))), x2)) -> c21(IF_RM(eq(z0, z1), s(s(s(s(z0)))), add(s(s(s(s(z1)))), x2)), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) RM(s(s(s(x0))), add(s(s(s(x1))), x2)) -> c21(EQ(s(s(s(x0))), s(s(s(x1))))) IF_MINSORT(true, add(0, add(0, add(z1, z2))), x1) -> c26(MINSORT(app(if_rm(eq(0, z1), 0, add(z1, z2)), x1), nil), APP(rm(0, add(0, add(z1, z2))), x1), RM(0, add(0, add(z1, z2)))) IF_MINSORT(true, add(0, add(0, x0)), x1) -> c4(APP(rm(0, add(0, x0)), x1)) IF_MINSORT(true, add(0, add(0, x0)), x1) -> c4(RM(0, add(0, x0))) IF_MINSORT(true, add(0, add(s(x0), x1)), z2) -> c26(MINSORT(add(s(x0), app(rm(0, x1), z2)), nil), APP(rm(0, add(s(x0), x1)), z2), RM(0, add(s(x0), x1))) IF_MINSORT(true, add(0, add(s(x0), nil)), x2) -> c26(MINSORT(app(add(s(x0), nil), x2), nil), APP(rm(0, add(s(x0), nil)), x2), RM(0, add(s(x0), nil))) IF_MINSORT(true, add(0, add(s(x0), add(z1, z2))), x2) -> c26(MINSORT(app(add(s(x0), if_rm(eq(0, z1), 0, add(z1, z2))), x2), nil), APP(rm(0, add(s(x0), add(z1, z2))), x2), RM(0, add(s(x0), add(z1, z2)))) IF_MINSORT(true, add(0, add(s(x0), x1)), x2) -> c5(APP(rm(0, add(s(x0), x1)), x2)) IF_MINSORT(true, add(0, add(s(x0), x1)), x2) -> c5(RM(0, add(s(x0), x1))) IF_MINSORT(true, add(s(x0), add(0, x1)), z2) -> c26(MINSORT(add(0, app(rm(s(x0), x1), z2)), nil), APP(rm(s(x0), add(0, x1)), z2), RM(s(x0), add(0, x1))) IF_MINSORT(true, add(s(x0), add(0, nil)), x2) -> c26(MINSORT(app(add(0, nil), x2), nil), APP(rm(s(x0), add(0, nil)), x2), RM(s(x0), add(0, nil))) IF_MINSORT(true, add(s(x0), add(0, add(z1, z2))), x2) -> c26(MINSORT(app(add(0, if_rm(eq(s(x0), z1), s(x0), add(z1, z2))), x2), nil), APP(rm(s(x0), add(0, add(z1, z2))), x2), RM(s(x0), add(0, add(z1, z2)))) IF_MINSORT(true, add(s(x0), add(0, x1)), x2) -> c7(APP(rm(s(x0), add(0, x1)), x2)) IF_MINSORT(true, add(s(x0), add(0, x1)), x2) -> c7(RM(s(x0), add(0, x1))) IF_MINSORT(true, add(s(0), add(s(0), z2)), x1) -> c26(MINSORT(app(rm(s(0), z2), x1), nil), APP(rm(s(0), add(s(0), z2)), x1), RM(s(0), add(s(0), z2))) IF_MINSORT(true, add(s(0), add(s(0), x0)), x1) -> c26(RM(s(0), add(s(0), x0))) IF_MINSORT(true, add(s(0), add(s(s(x0)), z2)), x2) -> c26(MINSORT(app(add(s(s(x0)), rm(s(0), z2)), x2), nil), APP(rm(s(0), add(s(s(x0)), z2)), x2), RM(s(0), add(s(s(x0)), z2))) IF_MINSORT(true, add(s(0), add(s(s(x0)), x1)), x2) -> c26(RM(s(0), add(s(s(x0)), x1))) IF_MINSORT(true, add(s(s(x0)), add(s(0), z2)), x2) -> c26(MINSORT(app(add(s(0), rm(s(s(x0)), z2)), x2), nil), APP(rm(s(s(x0)), add(s(0), z2)), x2), RM(s(s(x0)), add(s(0), z2))) IF_MINSORT(true, add(s(s(x0)), add(s(0), x1)), x2) -> c26(RM(s(s(x0)), add(s(0), x1))) IF_MINSORT(true, add(s(s(0)), add(s(s(0)), x2)), x3) -> c26(MINSORT(app(if_rm(true, s(s(0)), add(s(s(0)), x2)), x3), nil), APP(rm(s(s(0)), add(s(s(0)), x2)), x3), RM(s(s(0)), add(s(s(0)), x2))) IF_MINSORT(true, add(s(s(0)), add(s(s(s(z0))), x2)), x3) -> c26(MINSORT(app(if_rm(false, s(s(0)), add(s(s(s(z0))), x2)), x3), nil), APP(rm(s(s(0)), add(s(s(s(z0))), x2)), x3), RM(s(s(0)), add(s(s(s(z0))), x2))) IF_MINSORT(true, add(s(s(s(z0))), add(s(s(0)), x2)), x3) -> c26(MINSORT(app(if_rm(false, s(s(s(z0))), add(s(s(0)), x2)), x3), nil), APP(rm(s(s(s(z0))), add(s(s(0)), x2)), x3), RM(s(s(s(z0))), add(s(s(0)), x2))) IF_MINSORT(true, add(s(s(s(z0))), add(s(s(s(z1))), x2)), x3) -> c26(MINSORT(app(if_rm(eq(z0, z1), s(s(s(z0))), add(s(s(s(z1))), x2)), x3), nil), APP(rm(s(s(s(z0))), add(s(s(s(z1))), x2)), x3), RM(s(s(s(z0))), add(s(s(s(z1))), x2))) IF_MINSORT(true, add(s(s(x0)), add(s(s(x1)), x2)), x3) -> c9(APP(rm(s(s(x0)), add(s(s(x1)), x2)), x3)) IF_MINSORT(true, add(s(s(x0)), add(s(s(x1)), x2)), x3) -> c9(RM(s(s(x0)), add(s(s(x1)), x2))) IF_MINSORT(true, add(s(x0), add(s(x1), z2)), x3) -> c1(APP(if_rm(eq(s(x0), s(x1)), s(x0), add(s(x1), z2)), x3)) IF_MINSORT(true, add(0, add(0, x2)), x3) -> c26(APP(if_rm(true, 0, add(0, x2)), x3)) IF_MINSORT(true, add(0, add(s(z0), x2)), x3) -> c26(APP(if_rm(false, 0, add(s(z0), x2)), x3)) IF_MINSORT(true, add(s(z0), add(0, x2)), x3) -> c26(APP(if_rm(false, s(z0), add(0, x2)), x3)) IF_MINSORT(true, add(s(z0), add(s(z1), x2)), x3) -> c26(APP(if_rm(eq(z0, z1), s(z0), add(s(z1), x2)), x3)) MINSORT(add(s(x0), z0), x2) -> c25(IF_MINSORT(eq(s(x0), minIter(z0, add(s(x0), z0), 0)), add(s(x0), z0), x2), EQ(s(x0), minIter(add(s(x0), z0), add(s(x0), z0), 0)), MIN(add(s(x0), z0))) MINSORT(add(s(x0), x1), x2) -> c13(EQ(s(x0), minIter(add(s(x0), x1), add(s(x0), x1), 0))) MINSORT(add(s(x0), x1), x2) -> c13(MIN(add(s(x0), x1))) MINSORT(add(0, z0), x1) -> c25(IF_MINSORT(eq(0, 0), add(0, z0), x1), MIN(add(0, z0))) MINSORT(add(0, x0), x1) -> c25(MIN(add(0, x0))) MINSORT(add(z0, z1), x2) -> c2(EQ(z0, if_min(le(z0, 0), z1, add(z0, z1), 0))) MINSORT(add(s(z0), x1), x2) -> c(EQ(s(z0), if_min(false, x1, add(s(z0), x1), 0))) IF_MINSORT(true, add(0, add(0, nil)), z0) -> c26(MINSORT(app(nil, z0), nil), APP(if_rm(eq(0, 0), 0, add(0, nil)), z0), RM(0, add(0, nil))) IF_MIN(false, x1, x2, 0) -> c14(MINITER(x1, x2, 0)) IF_MIN(false, x1, x2, s(0)) -> c14(MINITER(x1, x2, s(0))) IF_MIN(false, x1, x2, s(s(0))) -> c14(MINITER(x1, x2, s(s(0)))) IF_MIN(false, x1, x2, s(s(s(0)))) -> c14(MINITER(x1, x2, s(s(s(0))))) IF_MIN(false, x1, x2, s(s(s(s(x3))))) -> c14(MINITER(x1, x2, s(s(s(s(x3)))))) MINITER(nil, add(z0, z1), 0) -> c11(MINITER(add(z0, z1), add(z0, z1), s(0))) MINITER(nil, add(z0, z1), s(0)) -> c11(MINITER(add(z0, z1), add(z0, z1), s(s(0)))) MINITER(nil, add(z0, z1), s(s(0))) -> c11(MINITER(add(z0, z1), add(z0, z1), s(s(s(0))))) MINITER(nil, add(z0, z1), s(s(s(0)))) -> c11(MINITER(add(z0, z1), add(z0, z1), s(s(s(s(0)))))) MINITER(nil, add(z0, z1), s(s(s(s(x2))))) -> c11(MINITER(add(z0, z1), add(z0, z1), s(s(s(s(s(x2))))))) EQ(s(s(y0)), s(s(y1))) -> c3(EQ(s(y0), s(y1))) RM(s(0), add(s(0), x2)) -> c21(IF_RM(true, s(0), add(s(0), x2))) RM(s(0), add(s(s(z0)), x2)) -> c21(IF_RM(false, s(0), add(s(s(z0)), x2))) RM(s(s(z0)), add(s(0), x2)) -> c21(IF_RM(false, s(s(z0)), add(s(0), x2))) S tuples: LE(s(z0), s(z1)) -> c6(LE(z0, z1)) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) MINITER(add(s(s(z0)), x1), x2, s(0)) -> c12(IF_MIN(false, x1, x2, s(0)), LE(s(s(z0)), s(0))) MINITER(add(s(x0), x1), x2, s(x3)) -> c12(LE(s(x0), s(x3))) MINITER(add(s(0), x1), x2, s(z0)) -> c12(LE(s(0), s(z0))) MINITER(add(s(s(s(z0))), x1), x2, s(s(0))) -> c12(IF_MIN(false, x1, x2, s(s(0))), LE(s(s(s(z0))), s(s(0)))) MINITER(add(s(s(x0)), x1), x2, s(s(x3))) -> c12(LE(s(s(x0)), s(s(x3)))) MINITER(add(s(s(0)), x1), x2, s(s(z0))) -> c12(LE(s(s(0)), s(s(z0)))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(0)))) -> c12(IF_MIN(false, x1, x2, s(s(s(0)))), LE(s(s(s(s(z0)))), s(s(s(0))))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(s(z1))))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(s(s(z1))))), LE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MINITER(add(s(s(s(x0))), x1), x2, s(s(s(x3)))) -> c12(LE(s(s(s(x0))), s(s(s(x3))))) MINITER(add(s(s(s(0))), x1), x2, s(s(s(z0)))) -> c12(LE(s(s(s(0))), s(s(s(z0))))) IF_MIN(false, x1, x2, 0) -> c14(MINITER(x1, x2, 0)) IF_MIN(false, x1, x2, s(0)) -> c14(MINITER(x1, x2, s(0))) IF_MIN(false, x1, x2, s(s(0))) -> c14(MINITER(x1, x2, s(s(0)))) IF_MIN(false, x1, x2, s(s(s(0)))) -> c14(MINITER(x1, x2, s(s(s(0))))) IF_MIN(false, x1, x2, s(s(s(s(x3))))) -> c14(MINITER(x1, x2, s(s(s(s(x3)))))) MINITER(nil, add(z0, z1), 0) -> c11(MINITER(add(z0, z1), add(z0, z1), s(0))) MINITER(nil, add(z0, z1), s(0)) -> c11(MINITER(add(z0, z1), add(z0, z1), s(s(0)))) MINITER(nil, add(z0, z1), s(s(0))) -> c11(MINITER(add(z0, z1), add(z0, z1), s(s(s(0))))) MINITER(nil, add(z0, z1), s(s(s(0)))) -> c11(MINITER(add(z0, z1), add(z0, z1), s(s(s(s(0)))))) MINITER(nil, add(z0, z1), s(s(s(s(x2))))) -> c11(MINITER(add(z0, z1), add(z0, z1), s(s(s(s(s(x2))))))) EQ(s(s(y0)), s(s(y1))) -> c3(EQ(s(y0), s(y1))) K tuples: IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MIN(add(z0, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) Defined Rule Symbols: le_2, eq_2, app_2, if_rm_3, rm_2, if_min_4, minIter_3 Defined Pair Symbols: LE_2, APP_2, MIN_1, IF_RM_3, IF_MINSORT_3, MINITER_3, RM_2, MINSORT_2, IF_MIN_4, EQ_2 Compound Symbols: c6_1, c8_1, c10_1, c22_1, c23_1, c27_1, c12_1, c21_1, c12_2, c_1, c21_2, c26_1, c1_1, c2_1, c26_3, c4_1, c5_1, c7_1, c9_1, c25_3, c13_1, c25_2, c25_1, c14_1, c11_1, c3_1 ---------------------------------------- (203) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IF_RM(true, z0, add(z1, z2)) -> c22(RM(z0, z2)) by IF_RM(true, 0, add(0, x0)) -> c22(RM(0, x0)) IF_RM(true, s(s(0)), add(s(s(0)), x0)) -> c22(RM(s(s(0)), x0)) IF_RM(true, s(s(s(0))), add(s(s(s(0))), x0)) -> c22(RM(s(s(s(0))), x0)) IF_RM(true, s(s(s(s(x0)))), add(s(s(s(s(x1)))), x2)) -> c22(RM(s(s(s(s(x0)))), x2)) IF_RM(true, s(0), add(s(0), x0)) -> c22(RM(s(0), x0)) ---------------------------------------- (204) 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) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) if_min(true, z0, z1, z2) -> z2 if_min(false, z0, z1, z2) -> minIter(z0, z1, z2) minIter(add(z0, z1), z2, z3) -> if_min(le(z0, z3), z1, z2, z3) minIter(nil, add(z0, z1), z2) -> minIter(add(z0, z1), add(z0, z1), s(z2)) Tuples: LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) MIN(add(z0, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) RM(0, add(0, x2)) -> c21(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c21(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c21(IF_RM(false, s(z0), add(0, x2))) MINITER(add(s(s(z0)), x1), x2, s(0)) -> c12(IF_MIN(false, x1, x2, s(0)), LE(s(s(z0)), s(0))) MINITER(add(s(x0), x1), x2, s(x3)) -> c12(LE(s(x0), s(x3))) MINITER(add(s(0), x1), x2, s(z0)) -> c12(LE(s(0), s(z0))) RM(s(x0), add(s(x1), x2)) -> c21(EQ(s(x0), s(x1))) MINSORT(add(x0, x1), x2) -> c(MIN(add(x0, x1))) MINITER(add(s(s(s(z0))), x1), x2, s(s(0))) -> c12(IF_MIN(false, x1, x2, s(s(0))), LE(s(s(s(z0))), s(s(0)))) MINITER(add(s(s(x0)), x1), x2, s(s(x3))) -> c12(LE(s(s(x0)), s(s(x3)))) MINITER(add(s(s(0)), x1), x2, s(s(z0))) -> c12(LE(s(s(0)), s(s(z0)))) RM(s(s(0)), add(s(s(0)), x2)) -> c21(IF_RM(true, s(s(0)), add(s(s(0)), x2)), EQ(s(s(0)), s(s(0)))) RM(s(s(0)), add(s(s(s(z0))), x2)) -> c21(IF_RM(false, s(s(0)), add(s(s(s(z0))), x2)), EQ(s(s(0)), s(s(s(z0))))) RM(s(s(s(z0))), add(s(s(0)), x2)) -> c21(IF_RM(false, s(s(s(z0))), add(s(s(0)), x2)), EQ(s(s(s(z0))), s(s(0)))) RM(s(s(x0)), add(s(s(x1)), x2)) -> c21(EQ(s(s(x0)), s(s(x1)))) IF_MINSORT(true, add(0, add(0, x0)), x1) -> c26(RM(0, add(0, x0))) IF_MINSORT(true, add(0, add(s(x0), x1)), x2) -> c26(RM(0, add(s(x0), x1))) IF_MINSORT(true, add(s(x0), add(0, x1)), x2) -> c26(RM(s(x0), add(0, x1))) IF_MINSORT(true, add(s(x0), add(s(x1), x2)), x3) -> c1(RM(s(x0), add(s(x1), x2))) IF_MINSORT(true, add(z0, nil), x1) -> c26(MINSORT(x1, nil)) MINSORT(add(x0, x1), x2) -> c2(MIN(add(x0, x1))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(0)))) -> c12(IF_MIN(false, x1, x2, s(s(s(0)))), LE(s(s(s(s(z0)))), s(s(s(0))))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(s(z1))))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(s(s(z1))))), LE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MINITER(add(s(s(s(x0))), x1), x2, s(s(s(x3)))) -> c12(LE(s(s(s(x0))), s(s(s(x3))))) MINITER(add(s(s(s(0))), x1), x2, s(s(s(z0)))) -> c12(LE(s(s(s(0))), s(s(s(z0))))) RM(s(s(s(0))), add(s(s(s(0))), x2)) -> c21(IF_RM(true, s(s(s(0))), add(s(s(s(0))), x2)), EQ(s(s(s(0))), s(s(s(0))))) RM(s(s(s(0))), add(s(s(s(s(z0)))), x2)) -> c21(IF_RM(false, s(s(s(0))), add(s(s(s(s(z0)))), x2)), EQ(s(s(s(0))), s(s(s(s(z0)))))) RM(s(s(s(s(z0)))), add(s(s(s(0))), x2)) -> c21(IF_RM(false, s(s(s(s(z0)))), add(s(s(s(0))), x2)), EQ(s(s(s(s(z0)))), s(s(s(0))))) RM(s(s(s(s(z0)))), add(s(s(s(s(z1)))), x2)) -> c21(IF_RM(eq(z0, z1), s(s(s(s(z0)))), add(s(s(s(s(z1)))), x2)), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) RM(s(s(s(x0))), add(s(s(s(x1))), x2)) -> c21(EQ(s(s(s(x0))), s(s(s(x1))))) IF_MINSORT(true, add(0, add(0, add(z1, z2))), x1) -> c26(MINSORT(app(if_rm(eq(0, z1), 0, add(z1, z2)), x1), nil), APP(rm(0, add(0, add(z1, z2))), x1), RM(0, add(0, add(z1, z2)))) IF_MINSORT(true, add(0, add(0, x0)), x1) -> c4(APP(rm(0, add(0, x0)), x1)) IF_MINSORT(true, add(0, add(0, x0)), x1) -> c4(RM(0, add(0, x0))) IF_MINSORT(true, add(0, add(s(x0), x1)), z2) -> c26(MINSORT(add(s(x0), app(rm(0, x1), z2)), nil), APP(rm(0, add(s(x0), x1)), z2), RM(0, add(s(x0), x1))) IF_MINSORT(true, add(0, add(s(x0), nil)), x2) -> c26(MINSORT(app(add(s(x0), nil), x2), nil), APP(rm(0, add(s(x0), nil)), x2), RM(0, add(s(x0), nil))) IF_MINSORT(true, add(0, add(s(x0), add(z1, z2))), x2) -> c26(MINSORT(app(add(s(x0), if_rm(eq(0, z1), 0, add(z1, z2))), x2), nil), APP(rm(0, add(s(x0), add(z1, z2))), x2), RM(0, add(s(x0), add(z1, z2)))) IF_MINSORT(true, add(0, add(s(x0), x1)), x2) -> c5(APP(rm(0, add(s(x0), x1)), x2)) IF_MINSORT(true, add(0, add(s(x0), x1)), x2) -> c5(RM(0, add(s(x0), x1))) IF_MINSORT(true, add(s(x0), add(0, x1)), z2) -> c26(MINSORT(add(0, app(rm(s(x0), x1), z2)), nil), APP(rm(s(x0), add(0, x1)), z2), RM(s(x0), add(0, x1))) IF_MINSORT(true, add(s(x0), add(0, nil)), x2) -> c26(MINSORT(app(add(0, nil), x2), nil), APP(rm(s(x0), add(0, nil)), x2), RM(s(x0), add(0, nil))) IF_MINSORT(true, add(s(x0), add(0, add(z1, z2))), x2) -> c26(MINSORT(app(add(0, if_rm(eq(s(x0), z1), s(x0), add(z1, z2))), x2), nil), APP(rm(s(x0), add(0, add(z1, z2))), x2), RM(s(x0), add(0, add(z1, z2)))) IF_MINSORT(true, add(s(x0), add(0, x1)), x2) -> c7(APP(rm(s(x0), add(0, x1)), x2)) IF_MINSORT(true, add(s(x0), add(0, x1)), x2) -> c7(RM(s(x0), add(0, x1))) IF_MINSORT(true, add(s(0), add(s(0), z2)), x1) -> c26(MINSORT(app(rm(s(0), z2), x1), nil), APP(rm(s(0), add(s(0), z2)), x1), RM(s(0), add(s(0), z2))) IF_MINSORT(true, add(s(0), add(s(0), x0)), x1) -> c26(RM(s(0), add(s(0), x0))) IF_MINSORT(true, add(s(0), add(s(s(x0)), z2)), x2) -> c26(MINSORT(app(add(s(s(x0)), rm(s(0), z2)), x2), nil), APP(rm(s(0), add(s(s(x0)), z2)), x2), RM(s(0), add(s(s(x0)), z2))) IF_MINSORT(true, add(s(0), add(s(s(x0)), x1)), x2) -> c26(RM(s(0), add(s(s(x0)), x1))) IF_MINSORT(true, add(s(s(x0)), add(s(0), z2)), x2) -> c26(MINSORT(app(add(s(0), rm(s(s(x0)), z2)), x2), nil), APP(rm(s(s(x0)), add(s(0), z2)), x2), RM(s(s(x0)), add(s(0), z2))) IF_MINSORT(true, add(s(s(x0)), add(s(0), x1)), x2) -> c26(RM(s(s(x0)), add(s(0), x1))) IF_MINSORT(true, add(s(s(0)), add(s(s(0)), x2)), x3) -> c26(MINSORT(app(if_rm(true, s(s(0)), add(s(s(0)), x2)), x3), nil), APP(rm(s(s(0)), add(s(s(0)), x2)), x3), RM(s(s(0)), add(s(s(0)), x2))) IF_MINSORT(true, add(s(s(0)), add(s(s(s(z0))), x2)), x3) -> c26(MINSORT(app(if_rm(false, s(s(0)), add(s(s(s(z0))), x2)), x3), nil), APP(rm(s(s(0)), add(s(s(s(z0))), x2)), x3), RM(s(s(0)), add(s(s(s(z0))), x2))) IF_MINSORT(true, add(s(s(s(z0))), add(s(s(0)), x2)), x3) -> c26(MINSORT(app(if_rm(false, s(s(s(z0))), add(s(s(0)), x2)), x3), nil), APP(rm(s(s(s(z0))), add(s(s(0)), x2)), x3), RM(s(s(s(z0))), add(s(s(0)), x2))) IF_MINSORT(true, add(s(s(s(z0))), add(s(s(s(z1))), x2)), x3) -> c26(MINSORT(app(if_rm(eq(z0, z1), s(s(s(z0))), add(s(s(s(z1))), x2)), x3), nil), APP(rm(s(s(s(z0))), add(s(s(s(z1))), x2)), x3), RM(s(s(s(z0))), add(s(s(s(z1))), x2))) IF_MINSORT(true, add(s(s(x0)), add(s(s(x1)), x2)), x3) -> c9(APP(rm(s(s(x0)), add(s(s(x1)), x2)), x3)) IF_MINSORT(true, add(s(s(x0)), add(s(s(x1)), x2)), x3) -> c9(RM(s(s(x0)), add(s(s(x1)), x2))) IF_MINSORT(true, add(s(x0), add(s(x1), z2)), x3) -> c1(APP(if_rm(eq(s(x0), s(x1)), s(x0), add(s(x1), z2)), x3)) IF_MINSORT(true, add(0, add(0, x2)), x3) -> c26(APP(if_rm(true, 0, add(0, x2)), x3)) IF_MINSORT(true, add(0, add(s(z0), x2)), x3) -> c26(APP(if_rm(false, 0, add(s(z0), x2)), x3)) IF_MINSORT(true, add(s(z0), add(0, x2)), x3) -> c26(APP(if_rm(false, s(z0), add(0, x2)), x3)) IF_MINSORT(true, add(s(z0), add(s(z1), x2)), x3) -> c26(APP(if_rm(eq(z0, z1), s(z0), add(s(z1), x2)), x3)) MINSORT(add(s(x0), z0), x2) -> c25(IF_MINSORT(eq(s(x0), minIter(z0, add(s(x0), z0), 0)), add(s(x0), z0), x2), EQ(s(x0), minIter(add(s(x0), z0), add(s(x0), z0), 0)), MIN(add(s(x0), z0))) MINSORT(add(s(x0), x1), x2) -> c13(EQ(s(x0), minIter(add(s(x0), x1), add(s(x0), x1), 0))) MINSORT(add(s(x0), x1), x2) -> c13(MIN(add(s(x0), x1))) MINSORT(add(0, z0), x1) -> c25(IF_MINSORT(eq(0, 0), add(0, z0), x1), MIN(add(0, z0))) MINSORT(add(0, x0), x1) -> c25(MIN(add(0, x0))) MINSORT(add(z0, z1), x2) -> c2(EQ(z0, if_min(le(z0, 0), z1, add(z0, z1), 0))) MINSORT(add(s(z0), x1), x2) -> c(EQ(s(z0), if_min(false, x1, add(s(z0), x1), 0))) IF_MINSORT(true, add(0, add(0, nil)), z0) -> c26(MINSORT(app(nil, z0), nil), APP(if_rm(eq(0, 0), 0, add(0, nil)), z0), RM(0, add(0, nil))) IF_MIN(false, x1, x2, 0) -> c14(MINITER(x1, x2, 0)) IF_MIN(false, x1, x2, s(0)) -> c14(MINITER(x1, x2, s(0))) IF_MIN(false, x1, x2, s(s(0))) -> c14(MINITER(x1, x2, s(s(0)))) IF_MIN(false, x1, x2, s(s(s(0)))) -> c14(MINITER(x1, x2, s(s(s(0))))) IF_MIN(false, x1, x2, s(s(s(s(x3))))) -> c14(MINITER(x1, x2, s(s(s(s(x3)))))) MINITER(nil, add(z0, z1), 0) -> c11(MINITER(add(z0, z1), add(z0, z1), s(0))) MINITER(nil, add(z0, z1), s(0)) -> c11(MINITER(add(z0, z1), add(z0, z1), s(s(0)))) MINITER(nil, add(z0, z1), s(s(0))) -> c11(MINITER(add(z0, z1), add(z0, z1), s(s(s(0))))) MINITER(nil, add(z0, z1), s(s(s(0)))) -> c11(MINITER(add(z0, z1), add(z0, z1), s(s(s(s(0)))))) MINITER(nil, add(z0, z1), s(s(s(s(x2))))) -> c11(MINITER(add(z0, z1), add(z0, z1), s(s(s(s(s(x2))))))) EQ(s(s(y0)), s(s(y1))) -> c3(EQ(s(y0), s(y1))) RM(s(0), add(s(0), x2)) -> c21(IF_RM(true, s(0), add(s(0), x2))) RM(s(0), add(s(s(z0)), x2)) -> c21(IF_RM(false, s(0), add(s(s(z0)), x2))) RM(s(s(z0)), add(s(0), x2)) -> c21(IF_RM(false, s(s(z0)), add(s(0), x2))) IF_RM(true, 0, add(0, x0)) -> c22(RM(0, x0)) IF_RM(true, s(s(0)), add(s(s(0)), x0)) -> c22(RM(s(s(0)), x0)) IF_RM(true, s(s(s(0))), add(s(s(s(0))), x0)) -> c22(RM(s(s(s(0))), x0)) IF_RM(true, s(s(s(s(x0)))), add(s(s(s(s(x1)))), x2)) -> c22(RM(s(s(s(s(x0)))), x2)) IF_RM(true, s(0), add(s(0), x0)) -> c22(RM(s(0), x0)) S tuples: LE(s(z0), s(z1)) -> c6(LE(z0, z1)) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) MINITER(add(s(s(z0)), x1), x2, s(0)) -> c12(IF_MIN(false, x1, x2, s(0)), LE(s(s(z0)), s(0))) MINITER(add(s(x0), x1), x2, s(x3)) -> c12(LE(s(x0), s(x3))) MINITER(add(s(0), x1), x2, s(z0)) -> c12(LE(s(0), s(z0))) MINITER(add(s(s(s(z0))), x1), x2, s(s(0))) -> c12(IF_MIN(false, x1, x2, s(s(0))), LE(s(s(s(z0))), s(s(0)))) MINITER(add(s(s(x0)), x1), x2, s(s(x3))) -> c12(LE(s(s(x0)), s(s(x3)))) MINITER(add(s(s(0)), x1), x2, s(s(z0))) -> c12(LE(s(s(0)), s(s(z0)))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(0)))) -> c12(IF_MIN(false, x1, x2, s(s(s(0)))), LE(s(s(s(s(z0)))), s(s(s(0))))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(s(z1))))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(s(s(z1))))), LE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MINITER(add(s(s(s(x0))), x1), x2, s(s(s(x3)))) -> c12(LE(s(s(s(x0))), s(s(s(x3))))) MINITER(add(s(s(s(0))), x1), x2, s(s(s(z0)))) -> c12(LE(s(s(s(0))), s(s(s(z0))))) IF_MIN(false, x1, x2, 0) -> c14(MINITER(x1, x2, 0)) IF_MIN(false, x1, x2, s(0)) -> c14(MINITER(x1, x2, s(0))) IF_MIN(false, x1, x2, s(s(0))) -> c14(MINITER(x1, x2, s(s(0)))) IF_MIN(false, x1, x2, s(s(s(0)))) -> c14(MINITER(x1, x2, s(s(s(0))))) IF_MIN(false, x1, x2, s(s(s(s(x3))))) -> c14(MINITER(x1, x2, s(s(s(s(x3)))))) MINITER(nil, add(z0, z1), 0) -> c11(MINITER(add(z0, z1), add(z0, z1), s(0))) MINITER(nil, add(z0, z1), s(0)) -> c11(MINITER(add(z0, z1), add(z0, z1), s(s(0)))) MINITER(nil, add(z0, z1), s(s(0))) -> c11(MINITER(add(z0, z1), add(z0, z1), s(s(s(0))))) MINITER(nil, add(z0, z1), s(s(s(0)))) -> c11(MINITER(add(z0, z1), add(z0, z1), s(s(s(s(0)))))) MINITER(nil, add(z0, z1), s(s(s(s(x2))))) -> c11(MINITER(add(z0, z1), add(z0, z1), s(s(s(s(s(x2))))))) EQ(s(s(y0)), s(s(y1))) -> c3(EQ(s(y0), s(y1))) K tuples: IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MIN(add(z0, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) IF_RM(true, 0, add(0, x0)) -> c22(RM(0, x0)) IF_RM(true, s(s(0)), add(s(s(0)), x0)) -> c22(RM(s(s(0)), x0)) IF_RM(true, s(s(s(0))), add(s(s(s(0))), x0)) -> c22(RM(s(s(s(0))), x0)) IF_RM(true, s(s(s(s(x0)))), add(s(s(s(s(x1)))), x2)) -> c22(RM(s(s(s(s(x0)))), x2)) IF_RM(true, s(0), add(s(0), x0)) -> c22(RM(s(0), x0)) Defined Rule Symbols: le_2, eq_2, app_2, if_rm_3, rm_2, if_min_4, minIter_3 Defined Pair Symbols: LE_2, APP_2, MIN_1, IF_RM_3, IF_MINSORT_3, MINITER_3, RM_2, MINSORT_2, IF_MIN_4, EQ_2 Compound Symbols: c6_1, c8_1, c10_1, c23_1, c27_1, c12_1, c21_1, c12_2, c_1, c21_2, c26_1, c1_1, c2_1, c26_3, c4_1, c5_1, c7_1, c9_1, c25_3, c13_1, c25_2, c25_1, c14_1, c11_1, c3_1, c22_1 ---------------------------------------- (205) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IF_RM(false, z0, add(z1, z2)) -> c23(RM(z0, z2)) by IF_RM(false, 0, add(s(x0), x1)) -> c23(RM(0, x1)) IF_RM(false, s(x0), add(0, x1)) -> c23(RM(s(x0), x1)) IF_RM(false, s(s(0)), add(s(s(s(x0))), x1)) -> c23(RM(s(s(0)), x1)) IF_RM(false, s(s(s(x0))), add(s(s(0)), x1)) -> c23(RM(s(s(s(x0))), x1)) IF_RM(false, s(s(s(0))), add(s(s(s(s(x0)))), x1)) -> c23(RM(s(s(s(0))), x1)) IF_RM(false, s(s(s(s(x0)))), add(s(s(s(0))), x1)) -> c23(RM(s(s(s(s(x0)))), x1)) IF_RM(false, s(s(s(s(x0)))), add(s(s(s(s(x1)))), x2)) -> c23(RM(s(s(s(s(x0)))), x2)) IF_RM(false, s(0), add(s(s(x0)), x1)) -> c23(RM(s(0), x1)) IF_RM(false, s(s(x0)), add(s(0), x1)) -> c23(RM(s(s(x0)), x1)) ---------------------------------------- (206) 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) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) if_min(true, z0, z1, z2) -> z2 if_min(false, z0, z1, z2) -> minIter(z0, z1, z2) minIter(add(z0, z1), z2, z3) -> if_min(le(z0, z3), z1, z2, z3) minIter(nil, add(z0, z1), z2) -> minIter(add(z0, z1), add(z0, z1), s(z2)) Tuples: LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) MIN(add(z0, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) RM(0, add(0, x2)) -> c21(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c21(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c21(IF_RM(false, s(z0), add(0, x2))) MINITER(add(s(s(z0)), x1), x2, s(0)) -> c12(IF_MIN(false, x1, x2, s(0)), LE(s(s(z0)), s(0))) MINITER(add(s(x0), x1), x2, s(x3)) -> c12(LE(s(x0), s(x3))) MINITER(add(s(0), x1), x2, s(z0)) -> c12(LE(s(0), s(z0))) RM(s(x0), add(s(x1), x2)) -> c21(EQ(s(x0), s(x1))) MINSORT(add(x0, x1), x2) -> c(MIN(add(x0, x1))) MINITER(add(s(s(s(z0))), x1), x2, s(s(0))) -> c12(IF_MIN(false, x1, x2, s(s(0))), LE(s(s(s(z0))), s(s(0)))) MINITER(add(s(s(x0)), x1), x2, s(s(x3))) -> c12(LE(s(s(x0)), s(s(x3)))) MINITER(add(s(s(0)), x1), x2, s(s(z0))) -> c12(LE(s(s(0)), s(s(z0)))) RM(s(s(0)), add(s(s(0)), x2)) -> c21(IF_RM(true, s(s(0)), add(s(s(0)), x2)), EQ(s(s(0)), s(s(0)))) RM(s(s(0)), add(s(s(s(z0))), x2)) -> c21(IF_RM(false, s(s(0)), add(s(s(s(z0))), x2)), EQ(s(s(0)), s(s(s(z0))))) RM(s(s(s(z0))), add(s(s(0)), x2)) -> c21(IF_RM(false, s(s(s(z0))), add(s(s(0)), x2)), EQ(s(s(s(z0))), s(s(0)))) RM(s(s(x0)), add(s(s(x1)), x2)) -> c21(EQ(s(s(x0)), s(s(x1)))) IF_MINSORT(true, add(0, add(0, x0)), x1) -> c26(RM(0, add(0, x0))) IF_MINSORT(true, add(0, add(s(x0), x1)), x2) -> c26(RM(0, add(s(x0), x1))) IF_MINSORT(true, add(s(x0), add(0, x1)), x2) -> c26(RM(s(x0), add(0, x1))) IF_MINSORT(true, add(s(x0), add(s(x1), x2)), x3) -> c1(RM(s(x0), add(s(x1), x2))) IF_MINSORT(true, add(z0, nil), x1) -> c26(MINSORT(x1, nil)) MINSORT(add(x0, x1), x2) -> c2(MIN(add(x0, x1))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(0)))) -> c12(IF_MIN(false, x1, x2, s(s(s(0)))), LE(s(s(s(s(z0)))), s(s(s(0))))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(s(z1))))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(s(s(z1))))), LE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MINITER(add(s(s(s(x0))), x1), x2, s(s(s(x3)))) -> c12(LE(s(s(s(x0))), s(s(s(x3))))) MINITER(add(s(s(s(0))), x1), x2, s(s(s(z0)))) -> c12(LE(s(s(s(0))), s(s(s(z0))))) RM(s(s(s(0))), add(s(s(s(0))), x2)) -> c21(IF_RM(true, s(s(s(0))), add(s(s(s(0))), x2)), EQ(s(s(s(0))), s(s(s(0))))) RM(s(s(s(0))), add(s(s(s(s(z0)))), x2)) -> c21(IF_RM(false, s(s(s(0))), add(s(s(s(s(z0)))), x2)), EQ(s(s(s(0))), s(s(s(s(z0)))))) RM(s(s(s(s(z0)))), add(s(s(s(0))), x2)) -> c21(IF_RM(false, s(s(s(s(z0)))), add(s(s(s(0))), x2)), EQ(s(s(s(s(z0)))), s(s(s(0))))) RM(s(s(s(s(z0)))), add(s(s(s(s(z1)))), x2)) -> c21(IF_RM(eq(z0, z1), s(s(s(s(z0)))), add(s(s(s(s(z1)))), x2)), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) RM(s(s(s(x0))), add(s(s(s(x1))), x2)) -> c21(EQ(s(s(s(x0))), s(s(s(x1))))) IF_MINSORT(true, add(0, add(0, add(z1, z2))), x1) -> c26(MINSORT(app(if_rm(eq(0, z1), 0, add(z1, z2)), x1), nil), APP(rm(0, add(0, add(z1, z2))), x1), RM(0, add(0, add(z1, z2)))) IF_MINSORT(true, add(0, add(0, x0)), x1) -> c4(APP(rm(0, add(0, x0)), x1)) IF_MINSORT(true, add(0, add(0, x0)), x1) -> c4(RM(0, add(0, x0))) IF_MINSORT(true, add(0, add(s(x0), x1)), z2) -> c26(MINSORT(add(s(x0), app(rm(0, x1), z2)), nil), APP(rm(0, add(s(x0), x1)), z2), RM(0, add(s(x0), x1))) IF_MINSORT(true, add(0, add(s(x0), nil)), x2) -> c26(MINSORT(app(add(s(x0), nil), x2), nil), APP(rm(0, add(s(x0), nil)), x2), RM(0, add(s(x0), nil))) IF_MINSORT(true, add(0, add(s(x0), add(z1, z2))), x2) -> c26(MINSORT(app(add(s(x0), if_rm(eq(0, z1), 0, add(z1, z2))), x2), nil), APP(rm(0, add(s(x0), add(z1, z2))), x2), RM(0, add(s(x0), add(z1, z2)))) IF_MINSORT(true, add(0, add(s(x0), x1)), x2) -> c5(APP(rm(0, add(s(x0), x1)), x2)) IF_MINSORT(true, add(0, add(s(x0), x1)), x2) -> c5(RM(0, add(s(x0), x1))) IF_MINSORT(true, add(s(x0), add(0, x1)), z2) -> c26(MINSORT(add(0, app(rm(s(x0), x1), z2)), nil), APP(rm(s(x0), add(0, x1)), z2), RM(s(x0), add(0, x1))) IF_MINSORT(true, add(s(x0), add(0, nil)), x2) -> c26(MINSORT(app(add(0, nil), x2), nil), APP(rm(s(x0), add(0, nil)), x2), RM(s(x0), add(0, nil))) IF_MINSORT(true, add(s(x0), add(0, add(z1, z2))), x2) -> c26(MINSORT(app(add(0, if_rm(eq(s(x0), z1), s(x0), add(z1, z2))), x2), nil), APP(rm(s(x0), add(0, add(z1, z2))), x2), RM(s(x0), add(0, add(z1, z2)))) IF_MINSORT(true, add(s(x0), add(0, x1)), x2) -> c7(APP(rm(s(x0), add(0, x1)), x2)) IF_MINSORT(true, add(s(x0), add(0, x1)), x2) -> c7(RM(s(x0), add(0, x1))) IF_MINSORT(true, add(s(0), add(s(0), z2)), x1) -> c26(MINSORT(app(rm(s(0), z2), x1), nil), APP(rm(s(0), add(s(0), z2)), x1), RM(s(0), add(s(0), z2))) IF_MINSORT(true, add(s(0), add(s(0), x0)), x1) -> c26(RM(s(0), add(s(0), x0))) IF_MINSORT(true, add(s(0), add(s(s(x0)), z2)), x2) -> c26(MINSORT(app(add(s(s(x0)), rm(s(0), z2)), x2), nil), APP(rm(s(0), add(s(s(x0)), z2)), x2), RM(s(0), add(s(s(x0)), z2))) IF_MINSORT(true, add(s(0), add(s(s(x0)), x1)), x2) -> c26(RM(s(0), add(s(s(x0)), x1))) IF_MINSORT(true, add(s(s(x0)), add(s(0), z2)), x2) -> c26(MINSORT(app(add(s(0), rm(s(s(x0)), z2)), x2), nil), APP(rm(s(s(x0)), add(s(0), z2)), x2), RM(s(s(x0)), add(s(0), z2))) IF_MINSORT(true, add(s(s(x0)), add(s(0), x1)), x2) -> c26(RM(s(s(x0)), add(s(0), x1))) IF_MINSORT(true, add(s(s(0)), add(s(s(0)), x2)), x3) -> c26(MINSORT(app(if_rm(true, s(s(0)), add(s(s(0)), x2)), x3), nil), APP(rm(s(s(0)), add(s(s(0)), x2)), x3), RM(s(s(0)), add(s(s(0)), x2))) IF_MINSORT(true, add(s(s(0)), add(s(s(s(z0))), x2)), x3) -> c26(MINSORT(app(if_rm(false, s(s(0)), add(s(s(s(z0))), x2)), x3), nil), APP(rm(s(s(0)), add(s(s(s(z0))), x2)), x3), RM(s(s(0)), add(s(s(s(z0))), x2))) IF_MINSORT(true, add(s(s(s(z0))), add(s(s(0)), x2)), x3) -> c26(MINSORT(app(if_rm(false, s(s(s(z0))), add(s(s(0)), x2)), x3), nil), APP(rm(s(s(s(z0))), add(s(s(0)), x2)), x3), RM(s(s(s(z0))), add(s(s(0)), x2))) IF_MINSORT(true, add(s(s(s(z0))), add(s(s(s(z1))), x2)), x3) -> c26(MINSORT(app(if_rm(eq(z0, z1), s(s(s(z0))), add(s(s(s(z1))), x2)), x3), nil), APP(rm(s(s(s(z0))), add(s(s(s(z1))), x2)), x3), RM(s(s(s(z0))), add(s(s(s(z1))), x2))) IF_MINSORT(true, add(s(s(x0)), add(s(s(x1)), x2)), x3) -> c9(APP(rm(s(s(x0)), add(s(s(x1)), x2)), x3)) IF_MINSORT(true, add(s(s(x0)), add(s(s(x1)), x2)), x3) -> c9(RM(s(s(x0)), add(s(s(x1)), x2))) IF_MINSORT(true, add(s(x0), add(s(x1), z2)), x3) -> c1(APP(if_rm(eq(s(x0), s(x1)), s(x0), add(s(x1), z2)), x3)) IF_MINSORT(true, add(0, add(0, x2)), x3) -> c26(APP(if_rm(true, 0, add(0, x2)), x3)) IF_MINSORT(true, add(0, add(s(z0), x2)), x3) -> c26(APP(if_rm(false, 0, add(s(z0), x2)), x3)) IF_MINSORT(true, add(s(z0), add(0, x2)), x3) -> c26(APP(if_rm(false, s(z0), add(0, x2)), x3)) IF_MINSORT(true, add(s(z0), add(s(z1), x2)), x3) -> c26(APP(if_rm(eq(z0, z1), s(z0), add(s(z1), x2)), x3)) MINSORT(add(s(x0), z0), x2) -> c25(IF_MINSORT(eq(s(x0), minIter(z0, add(s(x0), z0), 0)), add(s(x0), z0), x2), EQ(s(x0), minIter(add(s(x0), z0), add(s(x0), z0), 0)), MIN(add(s(x0), z0))) MINSORT(add(s(x0), x1), x2) -> c13(EQ(s(x0), minIter(add(s(x0), x1), add(s(x0), x1), 0))) MINSORT(add(s(x0), x1), x2) -> c13(MIN(add(s(x0), x1))) MINSORT(add(0, z0), x1) -> c25(IF_MINSORT(eq(0, 0), add(0, z0), x1), MIN(add(0, z0))) MINSORT(add(0, x0), x1) -> c25(MIN(add(0, x0))) MINSORT(add(z0, z1), x2) -> c2(EQ(z0, if_min(le(z0, 0), z1, add(z0, z1), 0))) MINSORT(add(s(z0), x1), x2) -> c(EQ(s(z0), if_min(false, x1, add(s(z0), x1), 0))) IF_MINSORT(true, add(0, add(0, nil)), z0) -> c26(MINSORT(app(nil, z0), nil), APP(if_rm(eq(0, 0), 0, add(0, nil)), z0), RM(0, add(0, nil))) IF_MIN(false, x1, x2, 0) -> c14(MINITER(x1, x2, 0)) IF_MIN(false, x1, x2, s(0)) -> c14(MINITER(x1, x2, s(0))) IF_MIN(false, x1, x2, s(s(0))) -> c14(MINITER(x1, x2, s(s(0)))) IF_MIN(false, x1, x2, s(s(s(0)))) -> c14(MINITER(x1, x2, s(s(s(0))))) IF_MIN(false, x1, x2, s(s(s(s(x3))))) -> c14(MINITER(x1, x2, s(s(s(s(x3)))))) MINITER(nil, add(z0, z1), 0) -> c11(MINITER(add(z0, z1), add(z0, z1), s(0))) MINITER(nil, add(z0, z1), s(0)) -> c11(MINITER(add(z0, z1), add(z0, z1), s(s(0)))) MINITER(nil, add(z0, z1), s(s(0))) -> c11(MINITER(add(z0, z1), add(z0, z1), s(s(s(0))))) MINITER(nil, add(z0, z1), s(s(s(0)))) -> c11(MINITER(add(z0, z1), add(z0, z1), s(s(s(s(0)))))) MINITER(nil, add(z0, z1), s(s(s(s(x2))))) -> c11(MINITER(add(z0, z1), add(z0, z1), s(s(s(s(s(x2))))))) EQ(s(s(y0)), s(s(y1))) -> c3(EQ(s(y0), s(y1))) RM(s(0), add(s(0), x2)) -> c21(IF_RM(true, s(0), add(s(0), x2))) RM(s(0), add(s(s(z0)), x2)) -> c21(IF_RM(false, s(0), add(s(s(z0)), x2))) RM(s(s(z0)), add(s(0), x2)) -> c21(IF_RM(false, s(s(z0)), add(s(0), x2))) IF_RM(true, 0, add(0, x0)) -> c22(RM(0, x0)) IF_RM(true, s(s(0)), add(s(s(0)), x0)) -> c22(RM(s(s(0)), x0)) IF_RM(true, s(s(s(0))), add(s(s(s(0))), x0)) -> c22(RM(s(s(s(0))), x0)) IF_RM(true, s(s(s(s(x0)))), add(s(s(s(s(x1)))), x2)) -> c22(RM(s(s(s(s(x0)))), x2)) IF_RM(true, s(0), add(s(0), x0)) -> c22(RM(s(0), x0)) IF_RM(false, 0, add(s(x0), x1)) -> c23(RM(0, x1)) IF_RM(false, s(x0), add(0, x1)) -> c23(RM(s(x0), x1)) IF_RM(false, s(s(0)), add(s(s(s(x0))), x1)) -> c23(RM(s(s(0)), x1)) IF_RM(false, s(s(s(x0))), add(s(s(0)), x1)) -> c23(RM(s(s(s(x0))), x1)) IF_RM(false, s(s(s(0))), add(s(s(s(s(x0)))), x1)) -> c23(RM(s(s(s(0))), x1)) IF_RM(false, s(s(s(s(x0)))), add(s(s(s(0))), x1)) -> c23(RM(s(s(s(s(x0)))), x1)) IF_RM(false, s(s(s(s(x0)))), add(s(s(s(s(x1)))), x2)) -> c23(RM(s(s(s(s(x0)))), x2)) IF_RM(false, s(0), add(s(s(x0)), x1)) -> c23(RM(s(0), x1)) IF_RM(false, s(s(x0)), add(s(0), x1)) -> c23(RM(s(s(x0)), x1)) S tuples: LE(s(z0), s(z1)) -> c6(LE(z0, z1)) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) MINITER(add(s(s(z0)), x1), x2, s(0)) -> c12(IF_MIN(false, x1, x2, s(0)), LE(s(s(z0)), s(0))) MINITER(add(s(x0), x1), x2, s(x3)) -> c12(LE(s(x0), s(x3))) MINITER(add(s(0), x1), x2, s(z0)) -> c12(LE(s(0), s(z0))) MINITER(add(s(s(s(z0))), x1), x2, s(s(0))) -> c12(IF_MIN(false, x1, x2, s(s(0))), LE(s(s(s(z0))), s(s(0)))) MINITER(add(s(s(x0)), x1), x2, s(s(x3))) -> c12(LE(s(s(x0)), s(s(x3)))) MINITER(add(s(s(0)), x1), x2, s(s(z0))) -> c12(LE(s(s(0)), s(s(z0)))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(0)))) -> c12(IF_MIN(false, x1, x2, s(s(s(0)))), LE(s(s(s(s(z0)))), s(s(s(0))))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(s(z1))))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(s(s(z1))))), LE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MINITER(add(s(s(s(x0))), x1), x2, s(s(s(x3)))) -> c12(LE(s(s(s(x0))), s(s(s(x3))))) MINITER(add(s(s(s(0))), x1), x2, s(s(s(z0)))) -> c12(LE(s(s(s(0))), s(s(s(z0))))) IF_MIN(false, x1, x2, 0) -> c14(MINITER(x1, x2, 0)) IF_MIN(false, x1, x2, s(0)) -> c14(MINITER(x1, x2, s(0))) IF_MIN(false, x1, x2, s(s(0))) -> c14(MINITER(x1, x2, s(s(0)))) IF_MIN(false, x1, x2, s(s(s(0)))) -> c14(MINITER(x1, x2, s(s(s(0))))) IF_MIN(false, x1, x2, s(s(s(s(x3))))) -> c14(MINITER(x1, x2, s(s(s(s(x3)))))) MINITER(nil, add(z0, z1), 0) -> c11(MINITER(add(z0, z1), add(z0, z1), s(0))) MINITER(nil, add(z0, z1), s(0)) -> c11(MINITER(add(z0, z1), add(z0, z1), s(s(0)))) MINITER(nil, add(z0, z1), s(s(0))) -> c11(MINITER(add(z0, z1), add(z0, z1), s(s(s(0))))) MINITER(nil, add(z0, z1), s(s(s(0)))) -> c11(MINITER(add(z0, z1), add(z0, z1), s(s(s(s(0)))))) MINITER(nil, add(z0, z1), s(s(s(s(x2))))) -> c11(MINITER(add(z0, z1), add(z0, z1), s(s(s(s(s(x2))))))) EQ(s(s(y0)), s(s(y1))) -> c3(EQ(s(y0), s(y1))) K tuples: IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) MIN(add(z0, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_RM(true, 0, add(0, x0)) -> c22(RM(0, x0)) IF_RM(true, s(s(0)), add(s(s(0)), x0)) -> c22(RM(s(s(0)), x0)) IF_RM(true, s(s(s(0))), add(s(s(s(0))), x0)) -> c22(RM(s(s(s(0))), x0)) IF_RM(true, s(s(s(s(x0)))), add(s(s(s(s(x1)))), x2)) -> c22(RM(s(s(s(s(x0)))), x2)) IF_RM(true, s(0), add(s(0), x0)) -> c22(RM(s(0), x0)) IF_RM(false, 0, add(s(x0), x1)) -> c23(RM(0, x1)) IF_RM(false, s(x0), add(0, x1)) -> c23(RM(s(x0), x1)) IF_RM(false, s(s(0)), add(s(s(s(x0))), x1)) -> c23(RM(s(s(0)), x1)) IF_RM(false, s(s(s(x0))), add(s(s(0)), x1)) -> c23(RM(s(s(s(x0))), x1)) IF_RM(false, s(s(s(0))), add(s(s(s(s(x0)))), x1)) -> c23(RM(s(s(s(0))), x1)) IF_RM(false, s(s(s(s(x0)))), add(s(s(s(0))), x1)) -> c23(RM(s(s(s(s(x0)))), x1)) IF_RM(false, s(s(s(s(x0)))), add(s(s(s(s(x1)))), x2)) -> c23(RM(s(s(s(s(x0)))), x2)) IF_RM(false, s(0), add(s(s(x0)), x1)) -> c23(RM(s(0), x1)) IF_RM(false, s(s(x0)), add(s(0), x1)) -> c23(RM(s(s(x0)), x1)) Defined Rule Symbols: le_2, eq_2, app_2, if_rm_3, rm_2, if_min_4, minIter_3 Defined Pair Symbols: LE_2, APP_2, MIN_1, IF_MINSORT_3, MINITER_3, RM_2, MINSORT_2, IF_MIN_4, EQ_2, IF_RM_3 Compound Symbols: c6_1, c8_1, c10_1, c27_1, c12_1, c21_1, c12_2, c_1, c21_2, c26_1, c1_1, c2_1, c26_3, c4_1, c5_1, c7_1, c9_1, c25_3, c13_1, c25_2, c25_1, c14_1, c11_1, c3_1, c22_1, c23_1 ---------------------------------------- (207) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IF_MINSORT(false, add(z0, z1), z2) -> c27(MINSORT(z1, add(z0, z2))) by IF_MINSORT(false, add(s(x0), x1), x2) -> c27(MINSORT(x1, add(s(x0), x2))) IF_MINSORT(false, add(0, x0), x1) -> c27(MINSORT(x0, add(0, x1))) ---------------------------------------- (208) 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) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) if_min(true, z0, z1, z2) -> z2 if_min(false, z0, z1, z2) -> minIter(z0, z1, z2) minIter(add(z0, z1), z2, z3) -> if_min(le(z0, z3), z1, z2, z3) minIter(nil, add(z0, z1), z2) -> minIter(add(z0, z1), add(z0, z1), s(z2)) Tuples: LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) MIN(add(z0, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) RM(0, add(0, x2)) -> c21(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c21(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c21(IF_RM(false, s(z0), add(0, x2))) MINITER(add(s(s(z0)), x1), x2, s(0)) -> c12(IF_MIN(false, x1, x2, s(0)), LE(s(s(z0)), s(0))) MINITER(add(s(x0), x1), x2, s(x3)) -> c12(LE(s(x0), s(x3))) MINITER(add(s(0), x1), x2, s(z0)) -> c12(LE(s(0), s(z0))) RM(s(x0), add(s(x1), x2)) -> c21(EQ(s(x0), s(x1))) MINSORT(add(x0, x1), x2) -> c(MIN(add(x0, x1))) MINITER(add(s(s(s(z0))), x1), x2, s(s(0))) -> c12(IF_MIN(false, x1, x2, s(s(0))), LE(s(s(s(z0))), s(s(0)))) MINITER(add(s(s(x0)), x1), x2, s(s(x3))) -> c12(LE(s(s(x0)), s(s(x3)))) MINITER(add(s(s(0)), x1), x2, s(s(z0))) -> c12(LE(s(s(0)), s(s(z0)))) RM(s(s(0)), add(s(s(0)), x2)) -> c21(IF_RM(true, s(s(0)), add(s(s(0)), x2)), EQ(s(s(0)), s(s(0)))) RM(s(s(0)), add(s(s(s(z0))), x2)) -> c21(IF_RM(false, s(s(0)), add(s(s(s(z0))), x2)), EQ(s(s(0)), s(s(s(z0))))) RM(s(s(s(z0))), add(s(s(0)), x2)) -> c21(IF_RM(false, s(s(s(z0))), add(s(s(0)), x2)), EQ(s(s(s(z0))), s(s(0)))) RM(s(s(x0)), add(s(s(x1)), x2)) -> c21(EQ(s(s(x0)), s(s(x1)))) IF_MINSORT(true, add(0, add(0, x0)), x1) -> c26(RM(0, add(0, x0))) IF_MINSORT(true, add(0, add(s(x0), x1)), x2) -> c26(RM(0, add(s(x0), x1))) IF_MINSORT(true, add(s(x0), add(0, x1)), x2) -> c26(RM(s(x0), add(0, x1))) IF_MINSORT(true, add(s(x0), add(s(x1), x2)), x3) -> c1(RM(s(x0), add(s(x1), x2))) IF_MINSORT(true, add(z0, nil), x1) -> c26(MINSORT(x1, nil)) MINSORT(add(x0, x1), x2) -> c2(MIN(add(x0, x1))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(0)))) -> c12(IF_MIN(false, x1, x2, s(s(s(0)))), LE(s(s(s(s(z0)))), s(s(s(0))))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(s(z1))))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(s(s(z1))))), LE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MINITER(add(s(s(s(x0))), x1), x2, s(s(s(x3)))) -> c12(LE(s(s(s(x0))), s(s(s(x3))))) MINITER(add(s(s(s(0))), x1), x2, s(s(s(z0)))) -> c12(LE(s(s(s(0))), s(s(s(z0))))) RM(s(s(s(0))), add(s(s(s(0))), x2)) -> c21(IF_RM(true, s(s(s(0))), add(s(s(s(0))), x2)), EQ(s(s(s(0))), s(s(s(0))))) RM(s(s(s(0))), add(s(s(s(s(z0)))), x2)) -> c21(IF_RM(false, s(s(s(0))), add(s(s(s(s(z0)))), x2)), EQ(s(s(s(0))), s(s(s(s(z0)))))) RM(s(s(s(s(z0)))), add(s(s(s(0))), x2)) -> c21(IF_RM(false, s(s(s(s(z0)))), add(s(s(s(0))), x2)), EQ(s(s(s(s(z0)))), s(s(s(0))))) RM(s(s(s(s(z0)))), add(s(s(s(s(z1)))), x2)) -> c21(IF_RM(eq(z0, z1), s(s(s(s(z0)))), add(s(s(s(s(z1)))), x2)), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) RM(s(s(s(x0))), add(s(s(s(x1))), x2)) -> c21(EQ(s(s(s(x0))), s(s(s(x1))))) IF_MINSORT(true, add(0, add(0, add(z1, z2))), x1) -> c26(MINSORT(app(if_rm(eq(0, z1), 0, add(z1, z2)), x1), nil), APP(rm(0, add(0, add(z1, z2))), x1), RM(0, add(0, add(z1, z2)))) IF_MINSORT(true, add(0, add(0, x0)), x1) -> c4(APP(rm(0, add(0, x0)), x1)) IF_MINSORT(true, add(0, add(0, x0)), x1) -> c4(RM(0, add(0, x0))) IF_MINSORT(true, add(0, add(s(x0), x1)), z2) -> c26(MINSORT(add(s(x0), app(rm(0, x1), z2)), nil), APP(rm(0, add(s(x0), x1)), z2), RM(0, add(s(x0), x1))) IF_MINSORT(true, add(0, add(s(x0), nil)), x2) -> c26(MINSORT(app(add(s(x0), nil), x2), nil), APP(rm(0, add(s(x0), nil)), x2), RM(0, add(s(x0), nil))) IF_MINSORT(true, add(0, add(s(x0), add(z1, z2))), x2) -> c26(MINSORT(app(add(s(x0), if_rm(eq(0, z1), 0, add(z1, z2))), x2), nil), APP(rm(0, add(s(x0), add(z1, z2))), x2), RM(0, add(s(x0), add(z1, z2)))) IF_MINSORT(true, add(0, add(s(x0), x1)), x2) -> c5(APP(rm(0, add(s(x0), x1)), x2)) IF_MINSORT(true, add(0, add(s(x0), x1)), x2) -> c5(RM(0, add(s(x0), x1))) IF_MINSORT(true, add(s(x0), add(0, x1)), z2) -> c26(MINSORT(add(0, app(rm(s(x0), x1), z2)), nil), APP(rm(s(x0), add(0, x1)), z2), RM(s(x0), add(0, x1))) IF_MINSORT(true, add(s(x0), add(0, nil)), x2) -> c26(MINSORT(app(add(0, nil), x2), nil), APP(rm(s(x0), add(0, nil)), x2), RM(s(x0), add(0, nil))) IF_MINSORT(true, add(s(x0), add(0, add(z1, z2))), x2) -> c26(MINSORT(app(add(0, if_rm(eq(s(x0), z1), s(x0), add(z1, z2))), x2), nil), APP(rm(s(x0), add(0, add(z1, z2))), x2), RM(s(x0), add(0, add(z1, z2)))) IF_MINSORT(true, add(s(x0), add(0, x1)), x2) -> c7(APP(rm(s(x0), add(0, x1)), x2)) IF_MINSORT(true, add(s(x0), add(0, x1)), x2) -> c7(RM(s(x0), add(0, x1))) IF_MINSORT(true, add(s(0), add(s(0), z2)), x1) -> c26(MINSORT(app(rm(s(0), z2), x1), nil), APP(rm(s(0), add(s(0), z2)), x1), RM(s(0), add(s(0), z2))) IF_MINSORT(true, add(s(0), add(s(0), x0)), x1) -> c26(RM(s(0), add(s(0), x0))) IF_MINSORT(true, add(s(0), add(s(s(x0)), z2)), x2) -> c26(MINSORT(app(add(s(s(x0)), rm(s(0), z2)), x2), nil), APP(rm(s(0), add(s(s(x0)), z2)), x2), RM(s(0), add(s(s(x0)), z2))) IF_MINSORT(true, add(s(0), add(s(s(x0)), x1)), x2) -> c26(RM(s(0), add(s(s(x0)), x1))) IF_MINSORT(true, add(s(s(x0)), add(s(0), z2)), x2) -> c26(MINSORT(app(add(s(0), rm(s(s(x0)), z2)), x2), nil), APP(rm(s(s(x0)), add(s(0), z2)), x2), RM(s(s(x0)), add(s(0), z2))) IF_MINSORT(true, add(s(s(x0)), add(s(0), x1)), x2) -> c26(RM(s(s(x0)), add(s(0), x1))) IF_MINSORT(true, add(s(s(0)), add(s(s(0)), x2)), x3) -> c26(MINSORT(app(if_rm(true, s(s(0)), add(s(s(0)), x2)), x3), nil), APP(rm(s(s(0)), add(s(s(0)), x2)), x3), RM(s(s(0)), add(s(s(0)), x2))) IF_MINSORT(true, add(s(s(0)), add(s(s(s(z0))), x2)), x3) -> c26(MINSORT(app(if_rm(false, s(s(0)), add(s(s(s(z0))), x2)), x3), nil), APP(rm(s(s(0)), add(s(s(s(z0))), x2)), x3), RM(s(s(0)), add(s(s(s(z0))), x2))) IF_MINSORT(true, add(s(s(s(z0))), add(s(s(0)), x2)), x3) -> c26(MINSORT(app(if_rm(false, s(s(s(z0))), add(s(s(0)), x2)), x3), nil), APP(rm(s(s(s(z0))), add(s(s(0)), x2)), x3), RM(s(s(s(z0))), add(s(s(0)), x2))) IF_MINSORT(true, add(s(s(s(z0))), add(s(s(s(z1))), x2)), x3) -> c26(MINSORT(app(if_rm(eq(z0, z1), s(s(s(z0))), add(s(s(s(z1))), x2)), x3), nil), APP(rm(s(s(s(z0))), add(s(s(s(z1))), x2)), x3), RM(s(s(s(z0))), add(s(s(s(z1))), x2))) IF_MINSORT(true, add(s(s(x0)), add(s(s(x1)), x2)), x3) -> c9(APP(rm(s(s(x0)), add(s(s(x1)), x2)), x3)) IF_MINSORT(true, add(s(s(x0)), add(s(s(x1)), x2)), x3) -> c9(RM(s(s(x0)), add(s(s(x1)), x2))) IF_MINSORT(true, add(s(x0), add(s(x1), z2)), x3) -> c1(APP(if_rm(eq(s(x0), s(x1)), s(x0), add(s(x1), z2)), x3)) IF_MINSORT(true, add(0, add(0, x2)), x3) -> c26(APP(if_rm(true, 0, add(0, x2)), x3)) IF_MINSORT(true, add(0, add(s(z0), x2)), x3) -> c26(APP(if_rm(false, 0, add(s(z0), x2)), x3)) IF_MINSORT(true, add(s(z0), add(0, x2)), x3) -> c26(APP(if_rm(false, s(z0), add(0, x2)), x3)) IF_MINSORT(true, add(s(z0), add(s(z1), x2)), x3) -> c26(APP(if_rm(eq(z0, z1), s(z0), add(s(z1), x2)), x3)) MINSORT(add(s(x0), z0), x2) -> c25(IF_MINSORT(eq(s(x0), minIter(z0, add(s(x0), z0), 0)), add(s(x0), z0), x2), EQ(s(x0), minIter(add(s(x0), z0), add(s(x0), z0), 0)), MIN(add(s(x0), z0))) MINSORT(add(s(x0), x1), x2) -> c13(EQ(s(x0), minIter(add(s(x0), x1), add(s(x0), x1), 0))) MINSORT(add(s(x0), x1), x2) -> c13(MIN(add(s(x0), x1))) MINSORT(add(0, z0), x1) -> c25(IF_MINSORT(eq(0, 0), add(0, z0), x1), MIN(add(0, z0))) MINSORT(add(0, x0), x1) -> c25(MIN(add(0, x0))) MINSORT(add(z0, z1), x2) -> c2(EQ(z0, if_min(le(z0, 0), z1, add(z0, z1), 0))) MINSORT(add(s(z0), x1), x2) -> c(EQ(s(z0), if_min(false, x1, add(s(z0), x1), 0))) IF_MINSORT(true, add(0, add(0, nil)), z0) -> c26(MINSORT(app(nil, z0), nil), APP(if_rm(eq(0, 0), 0, add(0, nil)), z0), RM(0, add(0, nil))) IF_MIN(false, x1, x2, 0) -> c14(MINITER(x1, x2, 0)) IF_MIN(false, x1, x2, s(0)) -> c14(MINITER(x1, x2, s(0))) IF_MIN(false, x1, x2, s(s(0))) -> c14(MINITER(x1, x2, s(s(0)))) IF_MIN(false, x1, x2, s(s(s(0)))) -> c14(MINITER(x1, x2, s(s(s(0))))) IF_MIN(false, x1, x2, s(s(s(s(x3))))) -> c14(MINITER(x1, x2, s(s(s(s(x3)))))) MINITER(nil, add(z0, z1), 0) -> c11(MINITER(add(z0, z1), add(z0, z1), s(0))) MINITER(nil, add(z0, z1), s(0)) -> c11(MINITER(add(z0, z1), add(z0, z1), s(s(0)))) MINITER(nil, add(z0, z1), s(s(0))) -> c11(MINITER(add(z0, z1), add(z0, z1), s(s(s(0))))) MINITER(nil, add(z0, z1), s(s(s(0)))) -> c11(MINITER(add(z0, z1), add(z0, z1), s(s(s(s(0)))))) MINITER(nil, add(z0, z1), s(s(s(s(x2))))) -> c11(MINITER(add(z0, z1), add(z0, z1), s(s(s(s(s(x2))))))) EQ(s(s(y0)), s(s(y1))) -> c3(EQ(s(y0), s(y1))) RM(s(0), add(s(0), x2)) -> c21(IF_RM(true, s(0), add(s(0), x2))) RM(s(0), add(s(s(z0)), x2)) -> c21(IF_RM(false, s(0), add(s(s(z0)), x2))) RM(s(s(z0)), add(s(0), x2)) -> c21(IF_RM(false, s(s(z0)), add(s(0), x2))) IF_RM(true, 0, add(0, x0)) -> c22(RM(0, x0)) IF_RM(true, s(s(0)), add(s(s(0)), x0)) -> c22(RM(s(s(0)), x0)) IF_RM(true, s(s(s(0))), add(s(s(s(0))), x0)) -> c22(RM(s(s(s(0))), x0)) IF_RM(true, s(s(s(s(x0)))), add(s(s(s(s(x1)))), x2)) -> c22(RM(s(s(s(s(x0)))), x2)) IF_RM(true, s(0), add(s(0), x0)) -> c22(RM(s(0), x0)) IF_RM(false, 0, add(s(x0), x1)) -> c23(RM(0, x1)) IF_RM(false, s(x0), add(0, x1)) -> c23(RM(s(x0), x1)) IF_RM(false, s(s(0)), add(s(s(s(x0))), x1)) -> c23(RM(s(s(0)), x1)) IF_RM(false, s(s(s(x0))), add(s(s(0)), x1)) -> c23(RM(s(s(s(x0))), x1)) IF_RM(false, s(s(s(0))), add(s(s(s(s(x0)))), x1)) -> c23(RM(s(s(s(0))), x1)) IF_RM(false, s(s(s(s(x0)))), add(s(s(s(0))), x1)) -> c23(RM(s(s(s(s(x0)))), x1)) IF_RM(false, s(s(s(s(x0)))), add(s(s(s(s(x1)))), x2)) -> c23(RM(s(s(s(s(x0)))), x2)) IF_RM(false, s(0), add(s(s(x0)), x1)) -> c23(RM(s(0), x1)) IF_RM(false, s(s(x0)), add(s(0), x1)) -> c23(RM(s(s(x0)), x1)) IF_MINSORT(false, add(s(x0), x1), x2) -> c27(MINSORT(x1, add(s(x0), x2))) IF_MINSORT(false, add(0, x0), x1) -> c27(MINSORT(x0, add(0, x1))) S tuples: LE(s(z0), s(z1)) -> c6(LE(z0, z1)) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) MINITER(add(s(s(z0)), x1), x2, s(0)) -> c12(IF_MIN(false, x1, x2, s(0)), LE(s(s(z0)), s(0))) MINITER(add(s(x0), x1), x2, s(x3)) -> c12(LE(s(x0), s(x3))) MINITER(add(s(0), x1), x2, s(z0)) -> c12(LE(s(0), s(z0))) MINITER(add(s(s(s(z0))), x1), x2, s(s(0))) -> c12(IF_MIN(false, x1, x2, s(s(0))), LE(s(s(s(z0))), s(s(0)))) MINITER(add(s(s(x0)), x1), x2, s(s(x3))) -> c12(LE(s(s(x0)), s(s(x3)))) MINITER(add(s(s(0)), x1), x2, s(s(z0))) -> c12(LE(s(s(0)), s(s(z0)))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(0)))) -> c12(IF_MIN(false, x1, x2, s(s(s(0)))), LE(s(s(s(s(z0)))), s(s(s(0))))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(s(z1))))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(s(s(z1))))), LE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MINITER(add(s(s(s(x0))), x1), x2, s(s(s(x3)))) -> c12(LE(s(s(s(x0))), s(s(s(x3))))) MINITER(add(s(s(s(0))), x1), x2, s(s(s(z0)))) -> c12(LE(s(s(s(0))), s(s(s(z0))))) IF_MIN(false, x1, x2, 0) -> c14(MINITER(x1, x2, 0)) IF_MIN(false, x1, x2, s(0)) -> c14(MINITER(x1, x2, s(0))) IF_MIN(false, x1, x2, s(s(0))) -> c14(MINITER(x1, x2, s(s(0)))) IF_MIN(false, x1, x2, s(s(s(0)))) -> c14(MINITER(x1, x2, s(s(s(0))))) IF_MIN(false, x1, x2, s(s(s(s(x3))))) -> c14(MINITER(x1, x2, s(s(s(s(x3)))))) MINITER(nil, add(z0, z1), 0) -> c11(MINITER(add(z0, z1), add(z0, z1), s(0))) MINITER(nil, add(z0, z1), s(0)) -> c11(MINITER(add(z0, z1), add(z0, z1), s(s(0)))) MINITER(nil, add(z0, z1), s(s(0))) -> c11(MINITER(add(z0, z1), add(z0, z1), s(s(s(0))))) MINITER(nil, add(z0, z1), s(s(s(0)))) -> c11(MINITER(add(z0, z1), add(z0, z1), s(s(s(s(0)))))) MINITER(nil, add(z0, z1), s(s(s(s(x2))))) -> c11(MINITER(add(z0, z1), add(z0, z1), s(s(s(s(s(x2))))))) EQ(s(s(y0)), s(s(y1))) -> c3(EQ(s(y0), s(y1))) K tuples: MIN(add(z0, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_RM(true, 0, add(0, x0)) -> c22(RM(0, x0)) IF_RM(true, s(s(0)), add(s(s(0)), x0)) -> c22(RM(s(s(0)), x0)) IF_RM(true, s(s(s(0))), add(s(s(s(0))), x0)) -> c22(RM(s(s(s(0))), x0)) IF_RM(true, s(s(s(s(x0)))), add(s(s(s(s(x1)))), x2)) -> c22(RM(s(s(s(s(x0)))), x2)) IF_RM(true, s(0), add(s(0), x0)) -> c22(RM(s(0), x0)) IF_RM(false, 0, add(s(x0), x1)) -> c23(RM(0, x1)) IF_RM(false, s(x0), add(0, x1)) -> c23(RM(s(x0), x1)) IF_RM(false, s(s(0)), add(s(s(s(x0))), x1)) -> c23(RM(s(s(0)), x1)) IF_RM(false, s(s(s(x0))), add(s(s(0)), x1)) -> c23(RM(s(s(s(x0))), x1)) IF_RM(false, s(s(s(0))), add(s(s(s(s(x0)))), x1)) -> c23(RM(s(s(s(0))), x1)) IF_RM(false, s(s(s(s(x0)))), add(s(s(s(0))), x1)) -> c23(RM(s(s(s(s(x0)))), x1)) IF_RM(false, s(s(s(s(x0)))), add(s(s(s(s(x1)))), x2)) -> c23(RM(s(s(s(s(x0)))), x2)) IF_RM(false, s(0), add(s(s(x0)), x1)) -> c23(RM(s(0), x1)) IF_RM(false, s(s(x0)), add(s(0), x1)) -> c23(RM(s(s(x0)), x1)) IF_MINSORT(false, add(s(x0), x1), x2) -> c27(MINSORT(x1, add(s(x0), x2))) IF_MINSORT(false, add(0, x0), x1) -> c27(MINSORT(x0, add(0, x1))) Defined Rule Symbols: le_2, eq_2, app_2, if_rm_3, rm_2, if_min_4, minIter_3 Defined Pair Symbols: LE_2, APP_2, MIN_1, MINITER_3, RM_2, MINSORT_2, IF_MINSORT_3, IF_MIN_4, EQ_2, IF_RM_3 Compound Symbols: c6_1, c8_1, c10_1, c12_1, c21_1, c12_2, c_1, c21_2, c26_1, c1_1, c2_1, c26_3, c4_1, c5_1, c7_1, c9_1, c25_3, c13_1, c25_2, c25_1, c14_1, c11_1, c3_1, c22_1, c23_1, c27_1 ---------------------------------------- (209) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MINSORT(add(0, z0), x1) -> c25(IF_MINSORT(eq(0, 0), add(0, z0), x1), MIN(add(0, z0))) by MINSORT(add(0, x0), x1) -> c25(IF_MINSORT(true, add(0, x0), x1), MIN(add(0, x0))) ---------------------------------------- (210) 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) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) if_min(true, z0, z1, z2) -> z2 if_min(false, z0, z1, z2) -> minIter(z0, z1, z2) minIter(add(z0, z1), z2, z3) -> if_min(le(z0, z3), z1, z2, z3) minIter(nil, add(z0, z1), z2) -> minIter(add(z0, z1), add(z0, z1), s(z2)) Tuples: LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) MIN(add(z0, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) RM(0, add(0, x2)) -> c21(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c21(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c21(IF_RM(false, s(z0), add(0, x2))) MINITER(add(s(s(z0)), x1), x2, s(0)) -> c12(IF_MIN(false, x1, x2, s(0)), LE(s(s(z0)), s(0))) MINITER(add(s(x0), x1), x2, s(x3)) -> c12(LE(s(x0), s(x3))) MINITER(add(s(0), x1), x2, s(z0)) -> c12(LE(s(0), s(z0))) RM(s(x0), add(s(x1), x2)) -> c21(EQ(s(x0), s(x1))) MINSORT(add(x0, x1), x2) -> c(MIN(add(x0, x1))) MINITER(add(s(s(s(z0))), x1), x2, s(s(0))) -> c12(IF_MIN(false, x1, x2, s(s(0))), LE(s(s(s(z0))), s(s(0)))) MINITER(add(s(s(x0)), x1), x2, s(s(x3))) -> c12(LE(s(s(x0)), s(s(x3)))) MINITER(add(s(s(0)), x1), x2, s(s(z0))) -> c12(LE(s(s(0)), s(s(z0)))) RM(s(s(0)), add(s(s(0)), x2)) -> c21(IF_RM(true, s(s(0)), add(s(s(0)), x2)), EQ(s(s(0)), s(s(0)))) RM(s(s(0)), add(s(s(s(z0))), x2)) -> c21(IF_RM(false, s(s(0)), add(s(s(s(z0))), x2)), EQ(s(s(0)), s(s(s(z0))))) RM(s(s(s(z0))), add(s(s(0)), x2)) -> c21(IF_RM(false, s(s(s(z0))), add(s(s(0)), x2)), EQ(s(s(s(z0))), s(s(0)))) RM(s(s(x0)), add(s(s(x1)), x2)) -> c21(EQ(s(s(x0)), s(s(x1)))) IF_MINSORT(true, add(0, add(0, x0)), x1) -> c26(RM(0, add(0, x0))) IF_MINSORT(true, add(0, add(s(x0), x1)), x2) -> c26(RM(0, add(s(x0), x1))) IF_MINSORT(true, add(s(x0), add(0, x1)), x2) -> c26(RM(s(x0), add(0, x1))) IF_MINSORT(true, add(s(x0), add(s(x1), x2)), x3) -> c1(RM(s(x0), add(s(x1), x2))) IF_MINSORT(true, add(z0, nil), x1) -> c26(MINSORT(x1, nil)) MINSORT(add(x0, x1), x2) -> c2(MIN(add(x0, x1))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(0)))) -> c12(IF_MIN(false, x1, x2, s(s(s(0)))), LE(s(s(s(s(z0)))), s(s(s(0))))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(s(z1))))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(s(s(z1))))), LE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MINITER(add(s(s(s(x0))), x1), x2, s(s(s(x3)))) -> c12(LE(s(s(s(x0))), s(s(s(x3))))) MINITER(add(s(s(s(0))), x1), x2, s(s(s(z0)))) -> c12(LE(s(s(s(0))), s(s(s(z0))))) RM(s(s(s(0))), add(s(s(s(0))), x2)) -> c21(IF_RM(true, s(s(s(0))), add(s(s(s(0))), x2)), EQ(s(s(s(0))), s(s(s(0))))) RM(s(s(s(0))), add(s(s(s(s(z0)))), x2)) -> c21(IF_RM(false, s(s(s(0))), add(s(s(s(s(z0)))), x2)), EQ(s(s(s(0))), s(s(s(s(z0)))))) RM(s(s(s(s(z0)))), add(s(s(s(0))), x2)) -> c21(IF_RM(false, s(s(s(s(z0)))), add(s(s(s(0))), x2)), EQ(s(s(s(s(z0)))), s(s(s(0))))) RM(s(s(s(s(z0)))), add(s(s(s(s(z1)))), x2)) -> c21(IF_RM(eq(z0, z1), s(s(s(s(z0)))), add(s(s(s(s(z1)))), x2)), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) RM(s(s(s(x0))), add(s(s(s(x1))), x2)) -> c21(EQ(s(s(s(x0))), s(s(s(x1))))) IF_MINSORT(true, add(0, add(0, add(z1, z2))), x1) -> c26(MINSORT(app(if_rm(eq(0, z1), 0, add(z1, z2)), x1), nil), APP(rm(0, add(0, add(z1, z2))), x1), RM(0, add(0, add(z1, z2)))) IF_MINSORT(true, add(0, add(0, x0)), x1) -> c4(APP(rm(0, add(0, x0)), x1)) IF_MINSORT(true, add(0, add(0, x0)), x1) -> c4(RM(0, add(0, x0))) IF_MINSORT(true, add(0, add(s(x0), x1)), z2) -> c26(MINSORT(add(s(x0), app(rm(0, x1), z2)), nil), APP(rm(0, add(s(x0), x1)), z2), RM(0, add(s(x0), x1))) IF_MINSORT(true, add(0, add(s(x0), nil)), x2) -> c26(MINSORT(app(add(s(x0), nil), x2), nil), APP(rm(0, add(s(x0), nil)), x2), RM(0, add(s(x0), nil))) IF_MINSORT(true, add(0, add(s(x0), add(z1, z2))), x2) -> c26(MINSORT(app(add(s(x0), if_rm(eq(0, z1), 0, add(z1, z2))), x2), nil), APP(rm(0, add(s(x0), add(z1, z2))), x2), RM(0, add(s(x0), add(z1, z2)))) IF_MINSORT(true, add(0, add(s(x0), x1)), x2) -> c5(APP(rm(0, add(s(x0), x1)), x2)) IF_MINSORT(true, add(0, add(s(x0), x1)), x2) -> c5(RM(0, add(s(x0), x1))) IF_MINSORT(true, add(s(x0), add(0, x1)), z2) -> c26(MINSORT(add(0, app(rm(s(x0), x1), z2)), nil), APP(rm(s(x0), add(0, x1)), z2), RM(s(x0), add(0, x1))) IF_MINSORT(true, add(s(x0), add(0, nil)), x2) -> c26(MINSORT(app(add(0, nil), x2), nil), APP(rm(s(x0), add(0, nil)), x2), RM(s(x0), add(0, nil))) IF_MINSORT(true, add(s(x0), add(0, add(z1, z2))), x2) -> c26(MINSORT(app(add(0, if_rm(eq(s(x0), z1), s(x0), add(z1, z2))), x2), nil), APP(rm(s(x0), add(0, add(z1, z2))), x2), RM(s(x0), add(0, add(z1, z2)))) IF_MINSORT(true, add(s(x0), add(0, x1)), x2) -> c7(APP(rm(s(x0), add(0, x1)), x2)) IF_MINSORT(true, add(s(x0), add(0, x1)), x2) -> c7(RM(s(x0), add(0, x1))) IF_MINSORT(true, add(s(0), add(s(0), z2)), x1) -> c26(MINSORT(app(rm(s(0), z2), x1), nil), APP(rm(s(0), add(s(0), z2)), x1), RM(s(0), add(s(0), z2))) IF_MINSORT(true, add(s(0), add(s(0), x0)), x1) -> c26(RM(s(0), add(s(0), x0))) IF_MINSORT(true, add(s(0), add(s(s(x0)), z2)), x2) -> c26(MINSORT(app(add(s(s(x0)), rm(s(0), z2)), x2), nil), APP(rm(s(0), add(s(s(x0)), z2)), x2), RM(s(0), add(s(s(x0)), z2))) IF_MINSORT(true, add(s(0), add(s(s(x0)), x1)), x2) -> c26(RM(s(0), add(s(s(x0)), x1))) IF_MINSORT(true, add(s(s(x0)), add(s(0), z2)), x2) -> c26(MINSORT(app(add(s(0), rm(s(s(x0)), z2)), x2), nil), APP(rm(s(s(x0)), add(s(0), z2)), x2), RM(s(s(x0)), add(s(0), z2))) IF_MINSORT(true, add(s(s(x0)), add(s(0), x1)), x2) -> c26(RM(s(s(x0)), add(s(0), x1))) IF_MINSORT(true, add(s(s(0)), add(s(s(0)), x2)), x3) -> c26(MINSORT(app(if_rm(true, s(s(0)), add(s(s(0)), x2)), x3), nil), APP(rm(s(s(0)), add(s(s(0)), x2)), x3), RM(s(s(0)), add(s(s(0)), x2))) IF_MINSORT(true, add(s(s(0)), add(s(s(s(z0))), x2)), x3) -> c26(MINSORT(app(if_rm(false, s(s(0)), add(s(s(s(z0))), x2)), x3), nil), APP(rm(s(s(0)), add(s(s(s(z0))), x2)), x3), RM(s(s(0)), add(s(s(s(z0))), x2))) IF_MINSORT(true, add(s(s(s(z0))), add(s(s(0)), x2)), x3) -> c26(MINSORT(app(if_rm(false, s(s(s(z0))), add(s(s(0)), x2)), x3), nil), APP(rm(s(s(s(z0))), add(s(s(0)), x2)), x3), RM(s(s(s(z0))), add(s(s(0)), x2))) IF_MINSORT(true, add(s(s(s(z0))), add(s(s(s(z1))), x2)), x3) -> c26(MINSORT(app(if_rm(eq(z0, z1), s(s(s(z0))), add(s(s(s(z1))), x2)), x3), nil), APP(rm(s(s(s(z0))), add(s(s(s(z1))), x2)), x3), RM(s(s(s(z0))), add(s(s(s(z1))), x2))) IF_MINSORT(true, add(s(s(x0)), add(s(s(x1)), x2)), x3) -> c9(APP(rm(s(s(x0)), add(s(s(x1)), x2)), x3)) IF_MINSORT(true, add(s(s(x0)), add(s(s(x1)), x2)), x3) -> c9(RM(s(s(x0)), add(s(s(x1)), x2))) IF_MINSORT(true, add(s(x0), add(s(x1), z2)), x3) -> c1(APP(if_rm(eq(s(x0), s(x1)), s(x0), add(s(x1), z2)), x3)) IF_MINSORT(true, add(0, add(0, x2)), x3) -> c26(APP(if_rm(true, 0, add(0, x2)), x3)) IF_MINSORT(true, add(0, add(s(z0), x2)), x3) -> c26(APP(if_rm(false, 0, add(s(z0), x2)), x3)) IF_MINSORT(true, add(s(z0), add(0, x2)), x3) -> c26(APP(if_rm(false, s(z0), add(0, x2)), x3)) IF_MINSORT(true, add(s(z0), add(s(z1), x2)), x3) -> c26(APP(if_rm(eq(z0, z1), s(z0), add(s(z1), x2)), x3)) MINSORT(add(s(x0), z0), x2) -> c25(IF_MINSORT(eq(s(x0), minIter(z0, add(s(x0), z0), 0)), add(s(x0), z0), x2), EQ(s(x0), minIter(add(s(x0), z0), add(s(x0), z0), 0)), MIN(add(s(x0), z0))) MINSORT(add(s(x0), x1), x2) -> c13(EQ(s(x0), minIter(add(s(x0), x1), add(s(x0), x1), 0))) MINSORT(add(s(x0), x1), x2) -> c13(MIN(add(s(x0), x1))) MINSORT(add(0, x0), x1) -> c25(MIN(add(0, x0))) MINSORT(add(z0, z1), x2) -> c2(EQ(z0, if_min(le(z0, 0), z1, add(z0, z1), 0))) MINSORT(add(s(z0), x1), x2) -> c(EQ(s(z0), if_min(false, x1, add(s(z0), x1), 0))) IF_MINSORT(true, add(0, add(0, nil)), z0) -> c26(MINSORT(app(nil, z0), nil), APP(if_rm(eq(0, 0), 0, add(0, nil)), z0), RM(0, add(0, nil))) IF_MIN(false, x1, x2, 0) -> c14(MINITER(x1, x2, 0)) IF_MIN(false, x1, x2, s(0)) -> c14(MINITER(x1, x2, s(0))) IF_MIN(false, x1, x2, s(s(0))) -> c14(MINITER(x1, x2, s(s(0)))) IF_MIN(false, x1, x2, s(s(s(0)))) -> c14(MINITER(x1, x2, s(s(s(0))))) IF_MIN(false, x1, x2, s(s(s(s(x3))))) -> c14(MINITER(x1, x2, s(s(s(s(x3)))))) MINITER(nil, add(z0, z1), 0) -> c11(MINITER(add(z0, z1), add(z0, z1), s(0))) MINITER(nil, add(z0, z1), s(0)) -> c11(MINITER(add(z0, z1), add(z0, z1), s(s(0)))) MINITER(nil, add(z0, z1), s(s(0))) -> c11(MINITER(add(z0, z1), add(z0, z1), s(s(s(0))))) MINITER(nil, add(z0, z1), s(s(s(0)))) -> c11(MINITER(add(z0, z1), add(z0, z1), s(s(s(s(0)))))) MINITER(nil, add(z0, z1), s(s(s(s(x2))))) -> c11(MINITER(add(z0, z1), add(z0, z1), s(s(s(s(s(x2))))))) EQ(s(s(y0)), s(s(y1))) -> c3(EQ(s(y0), s(y1))) RM(s(0), add(s(0), x2)) -> c21(IF_RM(true, s(0), add(s(0), x2))) RM(s(0), add(s(s(z0)), x2)) -> c21(IF_RM(false, s(0), add(s(s(z0)), x2))) RM(s(s(z0)), add(s(0), x2)) -> c21(IF_RM(false, s(s(z0)), add(s(0), x2))) IF_RM(true, 0, add(0, x0)) -> c22(RM(0, x0)) IF_RM(true, s(s(0)), add(s(s(0)), x0)) -> c22(RM(s(s(0)), x0)) IF_RM(true, s(s(s(0))), add(s(s(s(0))), x0)) -> c22(RM(s(s(s(0))), x0)) IF_RM(true, s(s(s(s(x0)))), add(s(s(s(s(x1)))), x2)) -> c22(RM(s(s(s(s(x0)))), x2)) IF_RM(true, s(0), add(s(0), x0)) -> c22(RM(s(0), x0)) IF_RM(false, 0, add(s(x0), x1)) -> c23(RM(0, x1)) IF_RM(false, s(x0), add(0, x1)) -> c23(RM(s(x0), x1)) IF_RM(false, s(s(0)), add(s(s(s(x0))), x1)) -> c23(RM(s(s(0)), x1)) IF_RM(false, s(s(s(x0))), add(s(s(0)), x1)) -> c23(RM(s(s(s(x0))), x1)) IF_RM(false, s(s(s(0))), add(s(s(s(s(x0)))), x1)) -> c23(RM(s(s(s(0))), x1)) IF_RM(false, s(s(s(s(x0)))), add(s(s(s(0))), x1)) -> c23(RM(s(s(s(s(x0)))), x1)) IF_RM(false, s(s(s(s(x0)))), add(s(s(s(s(x1)))), x2)) -> c23(RM(s(s(s(s(x0)))), x2)) IF_RM(false, s(0), add(s(s(x0)), x1)) -> c23(RM(s(0), x1)) IF_RM(false, s(s(x0)), add(s(0), x1)) -> c23(RM(s(s(x0)), x1)) IF_MINSORT(false, add(s(x0), x1), x2) -> c27(MINSORT(x1, add(s(x0), x2))) IF_MINSORT(false, add(0, x0), x1) -> c27(MINSORT(x0, add(0, x1))) MINSORT(add(0, x0), x1) -> c25(IF_MINSORT(true, add(0, x0), x1), MIN(add(0, x0))) S tuples: LE(s(z0), s(z1)) -> c6(LE(z0, z1)) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) MINITER(add(s(s(z0)), x1), x2, s(0)) -> c12(IF_MIN(false, x1, x2, s(0)), LE(s(s(z0)), s(0))) MINITER(add(s(x0), x1), x2, s(x3)) -> c12(LE(s(x0), s(x3))) MINITER(add(s(0), x1), x2, s(z0)) -> c12(LE(s(0), s(z0))) MINITER(add(s(s(s(z0))), x1), x2, s(s(0))) -> c12(IF_MIN(false, x1, x2, s(s(0))), LE(s(s(s(z0))), s(s(0)))) MINITER(add(s(s(x0)), x1), x2, s(s(x3))) -> c12(LE(s(s(x0)), s(s(x3)))) MINITER(add(s(s(0)), x1), x2, s(s(z0))) -> c12(LE(s(s(0)), s(s(z0)))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(0)))) -> c12(IF_MIN(false, x1, x2, s(s(s(0)))), LE(s(s(s(s(z0)))), s(s(s(0))))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(s(z1))))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(s(s(z1))))), LE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MINITER(add(s(s(s(x0))), x1), x2, s(s(s(x3)))) -> c12(LE(s(s(s(x0))), s(s(s(x3))))) MINITER(add(s(s(s(0))), x1), x2, s(s(s(z0)))) -> c12(LE(s(s(s(0))), s(s(s(z0))))) IF_MIN(false, x1, x2, 0) -> c14(MINITER(x1, x2, 0)) IF_MIN(false, x1, x2, s(0)) -> c14(MINITER(x1, x2, s(0))) IF_MIN(false, x1, x2, s(s(0))) -> c14(MINITER(x1, x2, s(s(0)))) IF_MIN(false, x1, x2, s(s(s(0)))) -> c14(MINITER(x1, x2, s(s(s(0))))) IF_MIN(false, x1, x2, s(s(s(s(x3))))) -> c14(MINITER(x1, x2, s(s(s(s(x3)))))) MINITER(nil, add(z0, z1), 0) -> c11(MINITER(add(z0, z1), add(z0, z1), s(0))) MINITER(nil, add(z0, z1), s(0)) -> c11(MINITER(add(z0, z1), add(z0, z1), s(s(0)))) MINITER(nil, add(z0, z1), s(s(0))) -> c11(MINITER(add(z0, z1), add(z0, z1), s(s(s(0))))) MINITER(nil, add(z0, z1), s(s(s(0)))) -> c11(MINITER(add(z0, z1), add(z0, z1), s(s(s(s(0)))))) MINITER(nil, add(z0, z1), s(s(s(s(x2))))) -> c11(MINITER(add(z0, z1), add(z0, z1), s(s(s(s(s(x2))))))) EQ(s(s(y0)), s(s(y1))) -> c3(EQ(s(y0), s(y1))) K tuples: MIN(add(z0, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_RM(true, 0, add(0, x0)) -> c22(RM(0, x0)) IF_RM(true, s(s(0)), add(s(s(0)), x0)) -> c22(RM(s(s(0)), x0)) IF_RM(true, s(s(s(0))), add(s(s(s(0))), x0)) -> c22(RM(s(s(s(0))), x0)) IF_RM(true, s(s(s(s(x0)))), add(s(s(s(s(x1)))), x2)) -> c22(RM(s(s(s(s(x0)))), x2)) IF_RM(true, s(0), add(s(0), x0)) -> c22(RM(s(0), x0)) IF_RM(false, 0, add(s(x0), x1)) -> c23(RM(0, x1)) IF_RM(false, s(x0), add(0, x1)) -> c23(RM(s(x0), x1)) IF_RM(false, s(s(0)), add(s(s(s(x0))), x1)) -> c23(RM(s(s(0)), x1)) IF_RM(false, s(s(s(x0))), add(s(s(0)), x1)) -> c23(RM(s(s(s(x0))), x1)) IF_RM(false, s(s(s(0))), add(s(s(s(s(x0)))), x1)) -> c23(RM(s(s(s(0))), x1)) IF_RM(false, s(s(s(s(x0)))), add(s(s(s(0))), x1)) -> c23(RM(s(s(s(s(x0)))), x1)) IF_RM(false, s(s(s(s(x0)))), add(s(s(s(s(x1)))), x2)) -> c23(RM(s(s(s(s(x0)))), x2)) IF_RM(false, s(0), add(s(s(x0)), x1)) -> c23(RM(s(0), x1)) IF_RM(false, s(s(x0)), add(s(0), x1)) -> c23(RM(s(s(x0)), x1)) IF_MINSORT(false, add(s(x0), x1), x2) -> c27(MINSORT(x1, add(s(x0), x2))) IF_MINSORT(false, add(0, x0), x1) -> c27(MINSORT(x0, add(0, x1))) Defined Rule Symbols: le_2, eq_2, app_2, if_rm_3, rm_2, if_min_4, minIter_3 Defined Pair Symbols: LE_2, APP_2, MIN_1, MINITER_3, RM_2, MINSORT_2, IF_MINSORT_3, IF_MIN_4, EQ_2, IF_RM_3 Compound Symbols: c6_1, c8_1, c10_1, c12_1, c21_1, c12_2, c_1, c21_2, c26_1, c1_1, c2_1, c26_3, c4_1, c5_1, c7_1, c9_1, c25_3, c13_1, c25_1, c14_1, c11_1, c3_1, c22_1, c23_1, c27_1, c25_2 ---------------------------------------- (211) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 1 leading nodes: IF_MINSORT(false, add(0, x0), x1) -> c27(MINSORT(x0, add(0, x1))) ---------------------------------------- (212) 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) app(nil, z0) -> z0 app(add(z0, z1), z2) -> add(z0, app(z1, z2)) if_rm(true, z0, add(z1, z2)) -> rm(z0, z2) if_rm(false, z0, add(z1, z2)) -> add(z1, rm(z0, z2)) rm(z0, nil) -> nil rm(z0, add(z1, z2)) -> if_rm(eq(z0, z1), z0, add(z1, z2)) if_min(true, z0, z1, z2) -> z2 if_min(false, z0, z1, z2) -> minIter(z0, z1, z2) minIter(add(z0, z1), z2, z3) -> if_min(le(z0, z3), z1, z2, z3) minIter(nil, add(z0, z1), z2) -> minIter(add(z0, z1), add(z0, z1), s(z2)) Tuples: LE(s(z0), s(z1)) -> c6(LE(z0, z1)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) MIN(add(z0, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) RM(0, add(0, x2)) -> c21(IF_RM(true, 0, add(0, x2))) RM(0, add(s(z0), x2)) -> c21(IF_RM(false, 0, add(s(z0), x2))) RM(s(z0), add(0, x2)) -> c21(IF_RM(false, s(z0), add(0, x2))) MINITER(add(s(s(z0)), x1), x2, s(0)) -> c12(IF_MIN(false, x1, x2, s(0)), LE(s(s(z0)), s(0))) MINITER(add(s(x0), x1), x2, s(x3)) -> c12(LE(s(x0), s(x3))) MINITER(add(s(0), x1), x2, s(z0)) -> c12(LE(s(0), s(z0))) RM(s(x0), add(s(x1), x2)) -> c21(EQ(s(x0), s(x1))) MINSORT(add(x0, x1), x2) -> c(MIN(add(x0, x1))) MINITER(add(s(s(s(z0))), x1), x2, s(s(0))) -> c12(IF_MIN(false, x1, x2, s(s(0))), LE(s(s(s(z0))), s(s(0)))) MINITER(add(s(s(x0)), x1), x2, s(s(x3))) -> c12(LE(s(s(x0)), s(s(x3)))) MINITER(add(s(s(0)), x1), x2, s(s(z0))) -> c12(LE(s(s(0)), s(s(z0)))) RM(s(s(0)), add(s(s(0)), x2)) -> c21(IF_RM(true, s(s(0)), add(s(s(0)), x2)), EQ(s(s(0)), s(s(0)))) RM(s(s(0)), add(s(s(s(z0))), x2)) -> c21(IF_RM(false, s(s(0)), add(s(s(s(z0))), x2)), EQ(s(s(0)), s(s(s(z0))))) RM(s(s(s(z0))), add(s(s(0)), x2)) -> c21(IF_RM(false, s(s(s(z0))), add(s(s(0)), x2)), EQ(s(s(s(z0))), s(s(0)))) RM(s(s(x0)), add(s(s(x1)), x2)) -> c21(EQ(s(s(x0)), s(s(x1)))) IF_MINSORT(true, add(0, add(0, x0)), x1) -> c26(RM(0, add(0, x0))) IF_MINSORT(true, add(0, add(s(x0), x1)), x2) -> c26(RM(0, add(s(x0), x1))) IF_MINSORT(true, add(s(x0), add(0, x1)), x2) -> c26(RM(s(x0), add(0, x1))) IF_MINSORT(true, add(s(x0), add(s(x1), x2)), x3) -> c1(RM(s(x0), add(s(x1), x2))) IF_MINSORT(true, add(z0, nil), x1) -> c26(MINSORT(x1, nil)) MINSORT(add(x0, x1), x2) -> c2(MIN(add(x0, x1))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(0)))) -> c12(IF_MIN(false, x1, x2, s(s(s(0)))), LE(s(s(s(s(z0)))), s(s(s(0))))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(s(z1))))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(s(s(z1))))), LE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MINITER(add(s(s(s(x0))), x1), x2, s(s(s(x3)))) -> c12(LE(s(s(s(x0))), s(s(s(x3))))) MINITER(add(s(s(s(0))), x1), x2, s(s(s(z0)))) -> c12(LE(s(s(s(0))), s(s(s(z0))))) RM(s(s(s(0))), add(s(s(s(0))), x2)) -> c21(IF_RM(true, s(s(s(0))), add(s(s(s(0))), x2)), EQ(s(s(s(0))), s(s(s(0))))) RM(s(s(s(0))), add(s(s(s(s(z0)))), x2)) -> c21(IF_RM(false, s(s(s(0))), add(s(s(s(s(z0)))), x2)), EQ(s(s(s(0))), s(s(s(s(z0)))))) RM(s(s(s(s(z0)))), add(s(s(s(0))), x2)) -> c21(IF_RM(false, s(s(s(s(z0)))), add(s(s(s(0))), x2)), EQ(s(s(s(s(z0)))), s(s(s(0))))) RM(s(s(s(s(z0)))), add(s(s(s(s(z1)))), x2)) -> c21(IF_RM(eq(z0, z1), s(s(s(s(z0)))), add(s(s(s(s(z1)))), x2)), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) RM(s(s(s(x0))), add(s(s(s(x1))), x2)) -> c21(EQ(s(s(s(x0))), s(s(s(x1))))) IF_MINSORT(true, add(0, add(0, add(z1, z2))), x1) -> c26(MINSORT(app(if_rm(eq(0, z1), 0, add(z1, z2)), x1), nil), APP(rm(0, add(0, add(z1, z2))), x1), RM(0, add(0, add(z1, z2)))) IF_MINSORT(true, add(0, add(0, x0)), x1) -> c4(APP(rm(0, add(0, x0)), x1)) IF_MINSORT(true, add(0, add(0, x0)), x1) -> c4(RM(0, add(0, x0))) IF_MINSORT(true, add(0, add(s(x0), x1)), z2) -> c26(MINSORT(add(s(x0), app(rm(0, x1), z2)), nil), APP(rm(0, add(s(x0), x1)), z2), RM(0, add(s(x0), x1))) IF_MINSORT(true, add(0, add(s(x0), nil)), x2) -> c26(MINSORT(app(add(s(x0), nil), x2), nil), APP(rm(0, add(s(x0), nil)), x2), RM(0, add(s(x0), nil))) IF_MINSORT(true, add(0, add(s(x0), add(z1, z2))), x2) -> c26(MINSORT(app(add(s(x0), if_rm(eq(0, z1), 0, add(z1, z2))), x2), nil), APP(rm(0, add(s(x0), add(z1, z2))), x2), RM(0, add(s(x0), add(z1, z2)))) IF_MINSORT(true, add(0, add(s(x0), x1)), x2) -> c5(APP(rm(0, add(s(x0), x1)), x2)) IF_MINSORT(true, add(0, add(s(x0), x1)), x2) -> c5(RM(0, add(s(x0), x1))) IF_MINSORT(true, add(s(x0), add(0, x1)), z2) -> c26(MINSORT(add(0, app(rm(s(x0), x1), z2)), nil), APP(rm(s(x0), add(0, x1)), z2), RM(s(x0), add(0, x1))) IF_MINSORT(true, add(s(x0), add(0, nil)), x2) -> c26(MINSORT(app(add(0, nil), x2), nil), APP(rm(s(x0), add(0, nil)), x2), RM(s(x0), add(0, nil))) IF_MINSORT(true, add(s(x0), add(0, add(z1, z2))), x2) -> c26(MINSORT(app(add(0, if_rm(eq(s(x0), z1), s(x0), add(z1, z2))), x2), nil), APP(rm(s(x0), add(0, add(z1, z2))), x2), RM(s(x0), add(0, add(z1, z2)))) IF_MINSORT(true, add(s(x0), add(0, x1)), x2) -> c7(APP(rm(s(x0), add(0, x1)), x2)) IF_MINSORT(true, add(s(x0), add(0, x1)), x2) -> c7(RM(s(x0), add(0, x1))) IF_MINSORT(true, add(s(0), add(s(0), z2)), x1) -> c26(MINSORT(app(rm(s(0), z2), x1), nil), APP(rm(s(0), add(s(0), z2)), x1), RM(s(0), add(s(0), z2))) IF_MINSORT(true, add(s(0), add(s(0), x0)), x1) -> c26(RM(s(0), add(s(0), x0))) IF_MINSORT(true, add(s(0), add(s(s(x0)), z2)), x2) -> c26(MINSORT(app(add(s(s(x0)), rm(s(0), z2)), x2), nil), APP(rm(s(0), add(s(s(x0)), z2)), x2), RM(s(0), add(s(s(x0)), z2))) IF_MINSORT(true, add(s(0), add(s(s(x0)), x1)), x2) -> c26(RM(s(0), add(s(s(x0)), x1))) IF_MINSORT(true, add(s(s(x0)), add(s(0), z2)), x2) -> c26(MINSORT(app(add(s(0), rm(s(s(x0)), z2)), x2), nil), APP(rm(s(s(x0)), add(s(0), z2)), x2), RM(s(s(x0)), add(s(0), z2))) IF_MINSORT(true, add(s(s(x0)), add(s(0), x1)), x2) -> c26(RM(s(s(x0)), add(s(0), x1))) IF_MINSORT(true, add(s(s(0)), add(s(s(0)), x2)), x3) -> c26(MINSORT(app(if_rm(true, s(s(0)), add(s(s(0)), x2)), x3), nil), APP(rm(s(s(0)), add(s(s(0)), x2)), x3), RM(s(s(0)), add(s(s(0)), x2))) IF_MINSORT(true, add(s(s(0)), add(s(s(s(z0))), x2)), x3) -> c26(MINSORT(app(if_rm(false, s(s(0)), add(s(s(s(z0))), x2)), x3), nil), APP(rm(s(s(0)), add(s(s(s(z0))), x2)), x3), RM(s(s(0)), add(s(s(s(z0))), x2))) IF_MINSORT(true, add(s(s(s(z0))), add(s(s(0)), x2)), x3) -> c26(MINSORT(app(if_rm(false, s(s(s(z0))), add(s(s(0)), x2)), x3), nil), APP(rm(s(s(s(z0))), add(s(s(0)), x2)), x3), RM(s(s(s(z0))), add(s(s(0)), x2))) IF_MINSORT(true, add(s(s(s(z0))), add(s(s(s(z1))), x2)), x3) -> c26(MINSORT(app(if_rm(eq(z0, z1), s(s(s(z0))), add(s(s(s(z1))), x2)), x3), nil), APP(rm(s(s(s(z0))), add(s(s(s(z1))), x2)), x3), RM(s(s(s(z0))), add(s(s(s(z1))), x2))) IF_MINSORT(true, add(s(s(x0)), add(s(s(x1)), x2)), x3) -> c9(APP(rm(s(s(x0)), add(s(s(x1)), x2)), x3)) IF_MINSORT(true, add(s(s(x0)), add(s(s(x1)), x2)), x3) -> c9(RM(s(s(x0)), add(s(s(x1)), x2))) IF_MINSORT(true, add(s(x0), add(s(x1), z2)), x3) -> c1(APP(if_rm(eq(s(x0), s(x1)), s(x0), add(s(x1), z2)), x3)) IF_MINSORT(true, add(0, add(0, x2)), x3) -> c26(APP(if_rm(true, 0, add(0, x2)), x3)) IF_MINSORT(true, add(0, add(s(z0), x2)), x3) -> c26(APP(if_rm(false, 0, add(s(z0), x2)), x3)) IF_MINSORT(true, add(s(z0), add(0, x2)), x3) -> c26(APP(if_rm(false, s(z0), add(0, x2)), x3)) IF_MINSORT(true, add(s(z0), add(s(z1), x2)), x3) -> c26(APP(if_rm(eq(z0, z1), s(z0), add(s(z1), x2)), x3)) MINSORT(add(s(x0), z0), x2) -> c25(IF_MINSORT(eq(s(x0), minIter(z0, add(s(x0), z0), 0)), add(s(x0), z0), x2), EQ(s(x0), minIter(add(s(x0), z0), add(s(x0), z0), 0)), MIN(add(s(x0), z0))) MINSORT(add(s(x0), x1), x2) -> c13(EQ(s(x0), minIter(add(s(x0), x1), add(s(x0), x1), 0))) MINSORT(add(s(x0), x1), x2) -> c13(MIN(add(s(x0), x1))) MINSORT(add(0, x0), x1) -> c25(MIN(add(0, x0))) MINSORT(add(z0, z1), x2) -> c2(EQ(z0, if_min(le(z0, 0), z1, add(z0, z1), 0))) MINSORT(add(s(z0), x1), x2) -> c(EQ(s(z0), if_min(false, x1, add(s(z0), x1), 0))) IF_MINSORT(true, add(0, add(0, nil)), z0) -> c26(MINSORT(app(nil, z0), nil), APP(if_rm(eq(0, 0), 0, add(0, nil)), z0), RM(0, add(0, nil))) IF_MIN(false, x1, x2, 0) -> c14(MINITER(x1, x2, 0)) IF_MIN(false, x1, x2, s(0)) -> c14(MINITER(x1, x2, s(0))) IF_MIN(false, x1, x2, s(s(0))) -> c14(MINITER(x1, x2, s(s(0)))) IF_MIN(false, x1, x2, s(s(s(0)))) -> c14(MINITER(x1, x2, s(s(s(0))))) IF_MIN(false, x1, x2, s(s(s(s(x3))))) -> c14(MINITER(x1, x2, s(s(s(s(x3)))))) MINITER(nil, add(z0, z1), 0) -> c11(MINITER(add(z0, z1), add(z0, z1), s(0))) MINITER(nil, add(z0, z1), s(0)) -> c11(MINITER(add(z0, z1), add(z0, z1), s(s(0)))) MINITER(nil, add(z0, z1), s(s(0))) -> c11(MINITER(add(z0, z1), add(z0, z1), s(s(s(0))))) MINITER(nil, add(z0, z1), s(s(s(0)))) -> c11(MINITER(add(z0, z1), add(z0, z1), s(s(s(s(0)))))) MINITER(nil, add(z0, z1), s(s(s(s(x2))))) -> c11(MINITER(add(z0, z1), add(z0, z1), s(s(s(s(s(x2))))))) EQ(s(s(y0)), s(s(y1))) -> c3(EQ(s(y0), s(y1))) RM(s(0), add(s(0), x2)) -> c21(IF_RM(true, s(0), add(s(0), x2))) RM(s(0), add(s(s(z0)), x2)) -> c21(IF_RM(false, s(0), add(s(s(z0)), x2))) RM(s(s(z0)), add(s(0), x2)) -> c21(IF_RM(false, s(s(z0)), add(s(0), x2))) IF_RM(true, 0, add(0, x0)) -> c22(RM(0, x0)) IF_RM(true, s(s(0)), add(s(s(0)), x0)) -> c22(RM(s(s(0)), x0)) IF_RM(true, s(s(s(0))), add(s(s(s(0))), x0)) -> c22(RM(s(s(s(0))), x0)) IF_RM(true, s(s(s(s(x0)))), add(s(s(s(s(x1)))), x2)) -> c22(RM(s(s(s(s(x0)))), x2)) IF_RM(true, s(0), add(s(0), x0)) -> c22(RM(s(0), x0)) IF_RM(false, 0, add(s(x0), x1)) -> c23(RM(0, x1)) IF_RM(false, s(x0), add(0, x1)) -> c23(RM(s(x0), x1)) IF_RM(false, s(s(0)), add(s(s(s(x0))), x1)) -> c23(RM(s(s(0)), x1)) IF_RM(false, s(s(s(x0))), add(s(s(0)), x1)) -> c23(RM(s(s(s(x0))), x1)) IF_RM(false, s(s(s(0))), add(s(s(s(s(x0)))), x1)) -> c23(RM(s(s(s(0))), x1)) IF_RM(false, s(s(s(s(x0)))), add(s(s(s(0))), x1)) -> c23(RM(s(s(s(s(x0)))), x1)) IF_RM(false, s(s(s(s(x0)))), add(s(s(s(s(x1)))), x2)) -> c23(RM(s(s(s(s(x0)))), x2)) IF_RM(false, s(0), add(s(s(x0)), x1)) -> c23(RM(s(0), x1)) IF_RM(false, s(s(x0)), add(s(0), x1)) -> c23(RM(s(s(x0)), x1)) IF_MINSORT(false, add(s(x0), x1), x2) -> c27(MINSORT(x1, add(s(x0), x2))) MINSORT(add(0, x0), x1) -> c25(IF_MINSORT(true, add(0, x0), x1), MIN(add(0, x0))) S tuples: LE(s(z0), s(z1)) -> c6(LE(z0, z1)) MINITER(add(s(z0), x1), x2, 0) -> c12(IF_MIN(false, x1, x2, 0)) MINITER(add(s(s(z0)), x1), x2, s(0)) -> c12(IF_MIN(false, x1, x2, s(0)), LE(s(s(z0)), s(0))) MINITER(add(s(x0), x1), x2, s(x3)) -> c12(LE(s(x0), s(x3))) MINITER(add(s(0), x1), x2, s(z0)) -> c12(LE(s(0), s(z0))) MINITER(add(s(s(s(z0))), x1), x2, s(s(0))) -> c12(IF_MIN(false, x1, x2, s(s(0))), LE(s(s(s(z0))), s(s(0)))) MINITER(add(s(s(x0)), x1), x2, s(s(x3))) -> c12(LE(s(s(x0)), s(s(x3)))) MINITER(add(s(s(0)), x1), x2, s(s(z0))) -> c12(LE(s(s(0)), s(s(z0)))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(0)))) -> c12(IF_MIN(false, x1, x2, s(s(s(0)))), LE(s(s(s(s(z0)))), s(s(s(0))))) MINITER(add(s(s(s(s(z0)))), x1), x2, s(s(s(s(z1))))) -> c12(IF_MIN(le(z0, z1), x1, x2, s(s(s(s(z1))))), LE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MINITER(add(s(s(s(x0))), x1), x2, s(s(s(x3)))) -> c12(LE(s(s(s(x0))), s(s(s(x3))))) MINITER(add(s(s(s(0))), x1), x2, s(s(s(z0)))) -> c12(LE(s(s(s(0))), s(s(s(z0))))) IF_MIN(false, x1, x2, 0) -> c14(MINITER(x1, x2, 0)) IF_MIN(false, x1, x2, s(0)) -> c14(MINITER(x1, x2, s(0))) IF_MIN(false, x1, x2, s(s(0))) -> c14(MINITER(x1, x2, s(s(0)))) IF_MIN(false, x1, x2, s(s(s(0)))) -> c14(MINITER(x1, x2, s(s(s(0))))) IF_MIN(false, x1, x2, s(s(s(s(x3))))) -> c14(MINITER(x1, x2, s(s(s(s(x3)))))) MINITER(nil, add(z0, z1), 0) -> c11(MINITER(add(z0, z1), add(z0, z1), s(0))) MINITER(nil, add(z0, z1), s(0)) -> c11(MINITER(add(z0, z1), add(z0, z1), s(s(0)))) MINITER(nil, add(z0, z1), s(s(0))) -> c11(MINITER(add(z0, z1), add(z0, z1), s(s(s(0))))) MINITER(nil, add(z0, z1), s(s(s(0)))) -> c11(MINITER(add(z0, z1), add(z0, z1), s(s(s(s(0)))))) MINITER(nil, add(z0, z1), s(s(s(s(x2))))) -> c11(MINITER(add(z0, z1), add(z0, z1), s(s(s(s(s(x2))))))) EQ(s(s(y0)), s(s(y1))) -> c3(EQ(s(y0), s(y1))) K tuples: MIN(add(z0, z1)) -> c10(MINITER(add(z0, z1), add(z0, z1), 0)) APP(add(z0, z1), z2) -> c8(APP(z1, z2)) IF_RM(true, 0, add(0, x0)) -> c22(RM(0, x0)) IF_RM(true, s(s(0)), add(s(s(0)), x0)) -> c22(RM(s(s(0)), x0)) IF_RM(true, s(s(s(0))), add(s(s(s(0))), x0)) -> c22(RM(s(s(s(0))), x0)) IF_RM(true, s(s(s(s(x0)))), add(s(s(s(s(x1)))), x2)) -> c22(RM(s(s(s(s(x0)))), x2)) IF_RM(true, s(0), add(s(0), x0)) -> c22(RM(s(0), x0)) IF_RM(false, 0, add(s(x0), x1)) -> c23(RM(0, x1)) IF_RM(false, s(x0), add(0, x1)) -> c23(RM(s(x0), x1)) IF_RM(false, s(s(0)), add(s(s(s(x0))), x1)) -> c23(RM(s(s(0)), x1)) IF_RM(false, s(s(s(x0))), add(s(s(0)), x1)) -> c23(RM(s(s(s(x0))), x1)) IF_RM(false, s(s(s(0))), add(s(s(s(s(x0)))), x1)) -> c23(RM(s(s(s(0))), x1)) IF_RM(false, s(s(s(s(x0)))), add(s(s(s(0))), x1)) -> c23(RM(s(s(s(s(x0)))), x1)) IF_RM(false, s(s(s(s(x0)))), add(s(s(s(s(x1)))), x2)) -> c23(RM(s(s(s(s(x0)))), x2)) IF_RM(false, s(0), add(s(s(x0)), x1)) -> c23(RM(s(0), x1)) IF_RM(false, s(s(x0)), add(s(0), x1)) -> c23(RM(s(s(x0)), x1)) IF_MINSORT(false, add(s(x0), x1), x2) -> c27(MINSORT(x1, add(s(x0), x2))) Defined Rule Symbols: le_2, eq_2, app_2, if_rm_3, rm_2, if_min_4, minIter_3 Defined Pair Symbols: LE_2, APP_2, MIN_1, MINITER_3, RM_2, MINSORT_2, IF_MINSORT_3, IF_MIN_4, EQ_2, IF_RM_3 Compound Symbols: c6_1, c8_1, c10_1, c12_1, c21_1, c12_2, c_1, c21_2, c26_1, c1_1, c2_1, c26_3, c4_1, c5_1, c7_1, c9_1, c25_3, c13_1, c25_1, c14_1, c11_1, c3_1, c22_1, c23_1, c27_1, c25_2