KILLED proof of input_ry1emHW4V0.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).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) CpxTrsToCdtProof [BOTH BOUNDS(ID, ID), 0 ms] (6) CdtProblem (7) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms] (8) CpxRelTRS (9) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (10) CpxRelTRS (11) TypeInferenceProof [BOTH BOUNDS(ID, ID), 9 ms] (12) typed CpxTrs (13) OrderProof [LOWER BOUND(ID), 0 ms] (14) typed CpxTrs (15) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] (16) TRS for Loop Detection (17) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (18) CdtProblem (19) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (20) CdtProblem (21) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (22) CdtProblem (23) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (24) CdtProblem (25) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms] (26) CpxRelTRS (27) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (28) CpxTRS (29) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms] (30) CpxWeightedTrs (31) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (32) CpxTypedWeightedTrs (33) CompletionProof [UPPER BOUND(ID), 0 ms] (34) CpxTypedWeightedCompleteTrs (35) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (36) CpxTypedWeightedCompleteTrs (37) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (38) CpxRNTS (39) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (40) CpxRNTS (41) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms] (42) CpxRNTS (43) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (44) CpxRNTS (45) IntTrsBoundProof [UPPER BOUND(ID), 724 ms] (46) CpxRNTS (47) IntTrsBoundProof [UPPER BOUND(ID), 244 ms] (48) CpxRNTS (49) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (50) CpxRNTS (51) IntTrsBoundProof [UPPER BOUND(ID), 18.5 s] (52) CpxRNTS (53) IntTrsBoundProof [UPPER BOUND(ID), 9841 ms] (54) CpxRNTS (55) CompletionProof [UPPER BOUND(ID), 0 ms] (56) CpxTypedWeightedCompleteTrs (57) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (58) CpxRNTS (59) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (60) CdtProblem (61) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (62) CdtProblem (63) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 534 ms] (64) CdtProblem (65) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 1 ms] (66) CdtProblem (67) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (68) CdtProblem (69) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (70) CdtProblem (71) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (72) CdtProblem (73) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (74) CdtProblem (75) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (76) CdtProblem (77) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 25 ms] (78) CdtProblem (79) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 454 ms] (80) CdtProblem (81) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (82) CdtProblem (83) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (84) CdtProblem (85) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (86) CdtProblem (87) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 747 ms] (88) CdtProblem (89) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (90) CdtProblem (91) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (92) CdtProblem (93) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (94) CdtProblem (95) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (96) CdtProblem (97) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (98) CdtProblem (99) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 15 ms] (100) CdtProblem (101) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 836 ms] (102) CdtProblem (103) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (104) CdtProblem (105) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (106) CdtProblem (107) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 971 ms] (108) CdtProblem (109) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (110) CdtProblem (111) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (112) CdtProblem (113) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 0 ms] (114) CdtProblem (115) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 929 ms] (116) CdtProblem (117) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (118) CdtProblem (119) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (120) CdtProblem (121) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 1019 ms] (122) CdtProblem (123) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (124) CdtProblem (125) CdtRewritingProof [BOTH BOUNDS(ID, ID), 1 ms] (126) CdtProblem (127) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (128) CdtProblem (129) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (130) CdtProblem (131) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (132) CdtProblem (133) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (134) CdtProblem (135) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (136) CdtProblem (137) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (138) CdtProblem (139) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (140) CdtProblem (141) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (142) CdtProblem (143) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 15 ms] (144) CdtProblem (145) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (146) CdtProblem (147) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (148) CdtProblem (149) CdtInstantiationProof [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) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 3 ms] (156) CdtProblem (157) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (158) CdtProblem (159) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 25 ms] (160) CdtProblem (161) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (162) CdtProblem (163) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 7 ms] (164) CdtProblem (165) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (166) CdtProblem (167) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 700 ms] (168) CdtProblem (169) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (170) CdtProblem (171) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms] (172) CdtProblem (173) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 627 ms] (174) CdtProblem (175) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (176) CdtProblem (177) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms] (178) CdtProblem (179) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (180) CdtProblem (181) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (182) CdtProblem (183) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 1029 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) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (192) CdtProblem (193) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 845 ms] (194) CdtProblem (195) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (196) CdtProblem (197) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (198) CdtProblem (199) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (200) CdtProblem (201) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms] (202) CdtProblem (203) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (204) CdtProblem (205) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms] (206) CdtProblem (207) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (208) CdtProblem (209) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (210) CdtProblem (211) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (212) CdtProblem (213) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (214) CdtProblem (215) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms] (216) CdtProblem (217) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (218) CdtProblem (219) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms] (220) CpxWeightedTrs (221) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (222) CpxTypedWeightedTrs (223) CompletionProof [UPPER BOUND(ID), 0 ms] (224) CpxTypedWeightedCompleteTrs (225) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (226) CpxTypedWeightedCompleteTrs (227) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (228) CpxRNTS (229) SimplificationProof [BOTH BOUNDS(ID, ID), 19 ms] (230) CpxRNTS (231) CompletionProof [UPPER BOUND(ID), 0 ms] (232) CpxTypedWeightedCompleteTrs (233) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (234) CpxRNTS ---------------------------------------- (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: function(iszero, 0, dummy, dummy2) -> true function(iszero, s(x), dummy, dummy2) -> false function(p, 0, dummy, dummy2) -> 0 function(p, s(0), dummy, dummy2) -> 0 function(p, s(s(x)), dummy, dummy2) -> s(function(p, s(x), x, x)) function(plus, dummy, x, y) -> function(if, function(iszero, x, x, x), x, y) function(if, true, x, y) -> y function(if, false, x, y) -> function(plus, function(third, x, y, y), function(p, x, x, y), s(y)) function(third, x, y, z) -> z S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (1) RenamingProof (BOTH BOUNDS(ID, ID)) Renamed function symbols to avoid clashes with predefined symbol. ---------------------------------------- (2) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: function(iszero, 0', dummy, dummy2) -> true function(iszero, s(x), dummy, dummy2) -> false function(p, 0', dummy, dummy2) -> 0' function(p, s(0'), dummy, dummy2) -> 0' function(p, s(s(x)), dummy, dummy2) -> s(function(p, s(x), x, x)) function(plus, dummy, x, y) -> function(if, function(iszero, x, x, x), x, y) function(if, true, x, y) -> y function(if, false, x, y) -> function(plus, function(third, x, y, y), function(p, x, x, y), s(y)) function(third, x, y, z) -> z S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (3) RelTrsToTrsProof (UPPER BOUND(ID)) transformed relative TRS to TRS ---------------------------------------- (4) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: function(iszero, 0, dummy, dummy2) -> true function(iszero, s(x), dummy, dummy2) -> false function(p, 0, dummy, dummy2) -> 0 function(p, s(0), dummy, dummy2) -> 0 function(p, s(s(x)), dummy, dummy2) -> s(function(p, s(x), x, x)) function(plus, dummy, x, y) -> function(if, function(iszero, x, x, x), x, y) function(if, true, x, y) -> y function(if, false, x, y) -> function(plus, function(third, x, y, y), function(p, x, x, y), s(y)) function(third, x, y, z) -> z S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (5) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (6) Obligation: Complexity Dependency Tuples Problem Rules: function(iszero, 0, z0, z1) -> true function(iszero, s(z0), z1, z2) -> false function(p, 0, z0, z1) -> 0 function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) function(plus, z0, z1, z2) -> function(if, function(iszero, z1, z1, z1), z1, z2) function(if, true, z0, z1) -> z1 function(if, false, z0, z1) -> function(plus, function(third, z0, z1, z1), function(p, z0, z0, z1), s(z1)) function(third, z0, z1, z2) -> z2 Tuples: FUNCTION(iszero, 0, z0, z1) -> c FUNCTION(iszero, s(z0), z1, z2) -> c1 FUNCTION(p, 0, z0, z1) -> c2 FUNCTION(p, s(0), z0, z1) -> c3 FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, z0, z1, z2) -> c5(FUNCTION(if, function(iszero, z1, z1, z1), z1, z2), FUNCTION(iszero, z1, z1, z1)) FUNCTION(if, true, z0, z1) -> c6 FUNCTION(if, false, z0, z1) -> c7(FUNCTION(plus, function(third, z0, z1, z1), function(p, z0, z0, z1), s(z1)), FUNCTION(third, z0, z1, z1)) FUNCTION(if, false, z0, z1) -> c8(FUNCTION(plus, function(third, z0, z1, z1), function(p, z0, z0, z1), s(z1)), FUNCTION(p, z0, z0, z1)) FUNCTION(third, z0, z1, z2) -> c9 S tuples: FUNCTION(iszero, 0, z0, z1) -> c FUNCTION(iszero, s(z0), z1, z2) -> c1 FUNCTION(p, 0, z0, z1) -> c2 FUNCTION(p, s(0), z0, z1) -> c3 FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, z0, z1, z2) -> c5(FUNCTION(if, function(iszero, z1, z1, z1), z1, z2), FUNCTION(iszero, z1, z1, z1)) FUNCTION(if, true, z0, z1) -> c6 FUNCTION(if, false, z0, z1) -> c7(FUNCTION(plus, function(third, z0, z1, z1), function(p, z0, z0, z1), s(z1)), FUNCTION(third, z0, z1, z1)) FUNCTION(if, false, z0, z1) -> c8(FUNCTION(plus, function(third, z0, z1, z1), function(p, z0, z0, z1), s(z1)), FUNCTION(p, z0, z0, z1)) FUNCTION(third, z0, z1, z2) -> c9 K tuples:none Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c, c1, c2, c3, c4_1, c5_2, c6, c7_2, c8_2, c9 ---------------------------------------- (7) CdtToCpxRelTrsProof (BOTH BOUNDS(ID, ID)) Converted S to standard rules, and D \ S as well as R to relative rules. ---------------------------------------- (8) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: FUNCTION(iszero, 0, z0, z1) -> c FUNCTION(iszero, s(z0), z1, z2) -> c1 FUNCTION(p, 0, z0, z1) -> c2 FUNCTION(p, s(0), z0, z1) -> c3 FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, z0, z1, z2) -> c5(FUNCTION(if, function(iszero, z1, z1, z1), z1, z2), FUNCTION(iszero, z1, z1, z1)) FUNCTION(if, true, z0, z1) -> c6 FUNCTION(if, false, z0, z1) -> c7(FUNCTION(plus, function(third, z0, z1, z1), function(p, z0, z0, z1), s(z1)), FUNCTION(third, z0, z1, z1)) FUNCTION(if, false, z0, z1) -> c8(FUNCTION(plus, function(third, z0, z1, z1), function(p, z0, z0, z1), s(z1)), FUNCTION(p, z0, z0, z1)) FUNCTION(third, z0, z1, z2) -> c9 The (relative) TRS S consists of the following rules: function(iszero, 0, z0, z1) -> true function(iszero, s(z0), z1, z2) -> false function(p, 0, z0, z1) -> 0 function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) function(plus, z0, z1, z2) -> function(if, function(iszero, z1, z1, z1), z1, z2) function(if, true, z0, z1) -> z1 function(if, false, z0, z1) -> function(plus, function(third, z0, z1, z1), function(p, z0, z0, z1), s(z1)) function(third, z0, z1, z2) -> z2 Rewrite Strategy: INNERMOST ---------------------------------------- (9) RenamingProof (BOTH BOUNDS(ID, ID)) Renamed function symbols to avoid clashes with predefined symbol. ---------------------------------------- (10) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: FUNCTION(iszero, 0', z0, z1) -> c FUNCTION(iszero, s(z0), z1, z2) -> c1 FUNCTION(p, 0', z0, z1) -> c2 FUNCTION(p, s(0'), z0, z1) -> c3 FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, z0, z1, z2) -> c5(FUNCTION(if, function(iszero, z1, z1, z1), z1, z2), FUNCTION(iszero, z1, z1, z1)) FUNCTION(if, true, z0, z1) -> c6 FUNCTION(if, false, z0, z1) -> c7(FUNCTION(plus, function(third, z0, z1, z1), function(p, z0, z0, z1), s(z1)), FUNCTION(third, z0, z1, z1)) FUNCTION(if, false, z0, z1) -> c8(FUNCTION(plus, function(third, z0, z1, z1), function(p, z0, z0, z1), s(z1)), FUNCTION(p, z0, z0, z1)) FUNCTION(third, z0, z1, z2) -> c9 The (relative) TRS S consists of the following rules: function(iszero, 0', z0, z1) -> true function(iszero, s(z0), z1, z2) -> false function(p, 0', z0, z1) -> 0' function(p, s(0'), z0, z1) -> 0' function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) function(plus, z0, z1, z2) -> function(if, function(iszero, z1, z1, z1), z1, z2) function(if, true, z0, z1) -> z1 function(if, false, z0, z1) -> function(plus, function(third, z0, z1, z1), function(p, z0, z0, z1), s(z1)) function(third, z0, z1, z2) -> z2 Rewrite Strategy: INNERMOST ---------------------------------------- (11) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Inferred types. ---------------------------------------- (12) Obligation: Innermost TRS: Rules: FUNCTION(iszero, 0', z0, z1) -> c FUNCTION(iszero, s(z0), z1, z2) -> c1 FUNCTION(p, 0', z0, z1) -> c2 FUNCTION(p, s(0'), z0, z1) -> c3 FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, z0, z1, z2) -> c5(FUNCTION(if, function(iszero, z1, z1, z1), z1, z2), FUNCTION(iszero, z1, z1, z1)) FUNCTION(if, true, z0, z1) -> c6 FUNCTION(if, false, z0, z1) -> c7(FUNCTION(plus, function(third, z0, z1, z1), function(p, z0, z0, z1), s(z1)), FUNCTION(third, z0, z1, z1)) FUNCTION(if, false, z0, z1) -> c8(FUNCTION(plus, function(third, z0, z1, z1), function(p, z0, z0, z1), s(z1)), FUNCTION(p, z0, z0, z1)) FUNCTION(third, z0, z1, z2) -> c9 function(iszero, 0', z0, z1) -> true function(iszero, s(z0), z1, z2) -> false function(p, 0', z0, z1) -> 0' function(p, s(0'), z0, z1) -> 0' function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) function(plus, z0, z1, z2) -> function(if, function(iszero, z1, z1, z1), z1, z2) function(if, true, z0, z1) -> z1 function(if, false, z0, z1) -> function(plus, function(third, z0, z1, z1), function(p, z0, z0, z1), s(z1)) function(third, z0, z1, z2) -> z2 Types: FUNCTION :: iszero:p:plus:if:third -> 0':s:true:false -> 0':s:true:false -> 0':s:true:false -> c:c1:c2:c3:c4:c5:c6:c7:c8:c9 iszero :: iszero:p:plus:if:third 0' :: 0':s:true:false c :: c:c1:c2:c3:c4:c5:c6:c7:c8:c9 s :: 0':s:true:false -> 0':s:true:false c1 :: c:c1:c2:c3:c4:c5:c6:c7:c8:c9 p :: iszero:p:plus:if:third c2 :: c:c1:c2:c3:c4:c5:c6:c7:c8:c9 c3 :: c:c1:c2:c3:c4:c5:c6:c7:c8:c9 c4 :: c:c1:c2:c3:c4:c5:c6:c7:c8:c9 -> c:c1:c2:c3:c4:c5:c6:c7:c8:c9 plus :: iszero:p:plus:if:third c5 :: c:c1:c2:c3:c4:c5:c6:c7:c8:c9 -> c:c1:c2:c3:c4:c5:c6:c7:c8:c9 -> c:c1:c2:c3:c4:c5:c6:c7:c8:c9 if :: iszero:p:plus:if:third function :: iszero:p:plus:if:third -> 0':s:true:false -> 0':s:true:false -> 0':s:true:false -> 0':s:true:false true :: 0':s:true:false c6 :: c:c1:c2:c3:c4:c5:c6:c7:c8:c9 false :: 0':s:true:false c7 :: c:c1:c2:c3:c4:c5:c6:c7:c8:c9 -> c:c1:c2:c3:c4:c5:c6:c7:c8:c9 -> c:c1:c2:c3:c4:c5:c6:c7:c8:c9 third :: iszero:p:plus:if:third c8 :: c:c1:c2:c3:c4:c5:c6:c7:c8:c9 -> c:c1:c2:c3:c4:c5:c6:c7:c8:c9 -> c:c1:c2:c3:c4:c5:c6:c7:c8:c9 c9 :: c:c1:c2:c3:c4:c5:c6:c7:c8:c9 hole_c:c1:c2:c3:c4:c5:c6:c7:c8:c91_10 :: c:c1:c2:c3:c4:c5:c6:c7:c8:c9 hole_iszero:p:plus:if:third2_10 :: iszero:p:plus:if:third hole_0':s:true:false3_10 :: 0':s:true:false gen_c:c1:c2:c3:c4:c5:c6:c7:c8:c94_10 :: Nat -> c:c1:c2:c3:c4:c5:c6:c7:c8:c9 gen_0':s:true:false5_10 :: Nat -> 0':s:true:false ---------------------------------------- (13) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: FUNCTION, function They will be analysed ascendingly in the following order: function < FUNCTION ---------------------------------------- (14) Obligation: Innermost TRS: Rules: FUNCTION(iszero, 0', z0, z1) -> c FUNCTION(iszero, s(z0), z1, z2) -> c1 FUNCTION(p, 0', z0, z1) -> c2 FUNCTION(p, s(0'), z0, z1) -> c3 FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, z0, z1, z2) -> c5(FUNCTION(if, function(iszero, z1, z1, z1), z1, z2), FUNCTION(iszero, z1, z1, z1)) FUNCTION(if, true, z0, z1) -> c6 FUNCTION(if, false, z0, z1) -> c7(FUNCTION(plus, function(third, z0, z1, z1), function(p, z0, z0, z1), s(z1)), FUNCTION(third, z0, z1, z1)) FUNCTION(if, false, z0, z1) -> c8(FUNCTION(plus, function(third, z0, z1, z1), function(p, z0, z0, z1), s(z1)), FUNCTION(p, z0, z0, z1)) FUNCTION(third, z0, z1, z2) -> c9 function(iszero, 0', z0, z1) -> true function(iszero, s(z0), z1, z2) -> false function(p, 0', z0, z1) -> 0' function(p, s(0'), z0, z1) -> 0' function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) function(plus, z0, z1, z2) -> function(if, function(iszero, z1, z1, z1), z1, z2) function(if, true, z0, z1) -> z1 function(if, false, z0, z1) -> function(plus, function(third, z0, z1, z1), function(p, z0, z0, z1), s(z1)) function(third, z0, z1, z2) -> z2 Types: FUNCTION :: iszero:p:plus:if:third -> 0':s:true:false -> 0':s:true:false -> 0':s:true:false -> c:c1:c2:c3:c4:c5:c6:c7:c8:c9 iszero :: iszero:p:plus:if:third 0' :: 0':s:true:false c :: c:c1:c2:c3:c4:c5:c6:c7:c8:c9 s :: 0':s:true:false -> 0':s:true:false c1 :: c:c1:c2:c3:c4:c5:c6:c7:c8:c9 p :: iszero:p:plus:if:third c2 :: c:c1:c2:c3:c4:c5:c6:c7:c8:c9 c3 :: c:c1:c2:c3:c4:c5:c6:c7:c8:c9 c4 :: c:c1:c2:c3:c4:c5:c6:c7:c8:c9 -> c:c1:c2:c3:c4:c5:c6:c7:c8:c9 plus :: iszero:p:plus:if:third c5 :: c:c1:c2:c3:c4:c5:c6:c7:c8:c9 -> c:c1:c2:c3:c4:c5:c6:c7:c8:c9 -> c:c1:c2:c3:c4:c5:c6:c7:c8:c9 if :: iszero:p:plus:if:third function :: iszero:p:plus:if:third -> 0':s:true:false -> 0':s:true:false -> 0':s:true:false -> 0':s:true:false true :: 0':s:true:false c6 :: c:c1:c2:c3:c4:c5:c6:c7:c8:c9 false :: 0':s:true:false c7 :: c:c1:c2:c3:c4:c5:c6:c7:c8:c9 -> c:c1:c2:c3:c4:c5:c6:c7:c8:c9 -> c:c1:c2:c3:c4:c5:c6:c7:c8:c9 third :: iszero:p:plus:if:third c8 :: c:c1:c2:c3:c4:c5:c6:c7:c8:c9 -> c:c1:c2:c3:c4:c5:c6:c7:c8:c9 -> c:c1:c2:c3:c4:c5:c6:c7:c8:c9 c9 :: c:c1:c2:c3:c4:c5:c6:c7:c8:c9 hole_c:c1:c2:c3:c4:c5:c6:c7:c8:c91_10 :: c:c1:c2:c3:c4:c5:c6:c7:c8:c9 hole_iszero:p:plus:if:third2_10 :: iszero:p:plus:if:third hole_0':s:true:false3_10 :: 0':s:true:false gen_c:c1:c2:c3:c4:c5:c6:c7:c8:c94_10 :: Nat -> c:c1:c2:c3:c4:c5:c6:c7:c8:c9 gen_0':s:true:false5_10 :: Nat -> 0':s:true:false Generator Equations: gen_c:c1:c2:c3:c4:c5:c6:c7:c8:c94_10(0) <=> c gen_c:c1:c2:c3:c4:c5:c6:c7:c8:c94_10(+(x, 1)) <=> c4(gen_c:c1:c2:c3:c4:c5:c6:c7:c8:c94_10(x)) gen_0':s:true:false5_10(0) <=> 0' gen_0':s:true:false5_10(+(x, 1)) <=> s(gen_0':s:true:false5_10(x)) The following defined symbols remain to be analysed: function, FUNCTION They will be analysed ascendingly in the following order: function < FUNCTION ---------------------------------------- (15) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) Transformed a relative TRS into a decreasing-loop problem. ---------------------------------------- (16) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: FUNCTION(iszero, 0, z0, z1) -> c FUNCTION(iszero, s(z0), z1, z2) -> c1 FUNCTION(p, 0, z0, z1) -> c2 FUNCTION(p, s(0), z0, z1) -> c3 FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, z0, z1, z2) -> c5(FUNCTION(if, function(iszero, z1, z1, z1), z1, z2), FUNCTION(iszero, z1, z1, z1)) FUNCTION(if, true, z0, z1) -> c6 FUNCTION(if, false, z0, z1) -> c7(FUNCTION(plus, function(third, z0, z1, z1), function(p, z0, z0, z1), s(z1)), FUNCTION(third, z0, z1, z1)) FUNCTION(if, false, z0, z1) -> c8(FUNCTION(plus, function(third, z0, z1, z1), function(p, z0, z0, z1), s(z1)), FUNCTION(p, z0, z0, z1)) FUNCTION(third, z0, z1, z2) -> c9 The (relative) TRS S consists of the following rules: function(iszero, 0, z0, z1) -> true function(iszero, s(z0), z1, z2) -> false function(p, 0, z0, z1) -> 0 function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) function(plus, z0, z1, z2) -> function(if, function(iszero, z1, z1, z1), z1, z2) function(if, true, z0, z1) -> z1 function(if, false, z0, z1) -> function(plus, function(third, z0, z1, z1), function(p, z0, z0, z1), s(z1)) function(third, z0, z1, z2) -> z2 Rewrite Strategy: INNERMOST ---------------------------------------- (17) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (18) Obligation: Complexity Dependency Tuples Problem Rules: function(iszero, 0, z0, z1) -> true function(iszero, s(z0), z1, z2) -> false function(p, 0, z0, z1) -> 0 function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) function(plus, z0, z1, z2) -> function(if, function(iszero, z1, z1, z1), z1, z2) function(if, true, z0, z1) -> z1 function(if, false, z0, z1) -> function(plus, function(third, z0, z1, z1), function(p, z0, z0, z1), s(z1)) function(third, z0, z1, z2) -> z2 Tuples: FUNCTION(iszero, 0, z0, z1) -> c FUNCTION(iszero, s(z0), z1, z2) -> c1 FUNCTION(p, 0, z0, z1) -> c2 FUNCTION(p, s(0), z0, z1) -> c3 FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, z0, z1, z2) -> c5(FUNCTION(if, function(iszero, z1, z1, z1), z1, z2), FUNCTION(iszero, z1, z1, z1)) FUNCTION(if, true, z0, z1) -> c6 FUNCTION(if, false, z0, z1) -> c7(FUNCTION(plus, function(third, z0, z1, z1), function(p, z0, z0, z1), s(z1)), FUNCTION(third, z0, z1, z1)) FUNCTION(if, false, z0, z1) -> c8(FUNCTION(plus, function(third, z0, z1, z1), function(p, z0, z0, z1), s(z1)), FUNCTION(p, z0, z0, z1)) FUNCTION(third, z0, z1, z2) -> c9 S tuples: FUNCTION(iszero, 0, z0, z1) -> c FUNCTION(iszero, s(z0), z1, z2) -> c1 FUNCTION(p, 0, z0, z1) -> c2 FUNCTION(p, s(0), z0, z1) -> c3 FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, z0, z1, z2) -> c5(FUNCTION(if, function(iszero, z1, z1, z1), z1, z2), FUNCTION(iszero, z1, z1, z1)) FUNCTION(if, true, z0, z1) -> c6 FUNCTION(if, false, z0, z1) -> c7(FUNCTION(plus, function(third, z0, z1, z1), function(p, z0, z0, z1), s(z1)), FUNCTION(third, z0, z1, z1)) FUNCTION(if, false, z0, z1) -> c8(FUNCTION(plus, function(third, z0, z1, z1), function(p, z0, z0, z1), s(z1)), FUNCTION(p, z0, z0, z1)) FUNCTION(third, z0, z1, z2) -> c9 K tuples:none Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c, c1, c2, c3, c4_1, c5_2, c6, c7_2, c8_2, c9 ---------------------------------------- (19) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 6 trailing nodes: FUNCTION(iszero, 0, z0, z1) -> c FUNCTION(p, 0, z0, z1) -> c2 FUNCTION(p, s(0), z0, z1) -> c3 FUNCTION(if, true, z0, z1) -> c6 FUNCTION(iszero, s(z0), z1, z2) -> c1 FUNCTION(third, z0, z1, z2) -> c9 ---------------------------------------- (20) Obligation: Complexity Dependency Tuples Problem Rules: function(iszero, 0, z0, z1) -> true function(iszero, s(z0), z1, z2) -> false function(p, 0, z0, z1) -> 0 function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) function(plus, z0, z1, z2) -> function(if, function(iszero, z1, z1, z1), z1, z2) function(if, true, z0, z1) -> z1 function(if, false, z0, z1) -> function(plus, function(third, z0, z1, z1), function(p, z0, z0, z1), s(z1)) function(third, z0, z1, z2) -> z2 Tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, z0, z1, z2) -> c5(FUNCTION(if, function(iszero, z1, z1, z1), z1, z2), FUNCTION(iszero, z1, z1, z1)) FUNCTION(if, false, z0, z1) -> c7(FUNCTION(plus, function(third, z0, z1, z1), function(p, z0, z0, z1), s(z1)), FUNCTION(third, z0, z1, z1)) FUNCTION(if, false, z0, z1) -> c8(FUNCTION(plus, function(third, z0, z1, z1), function(p, z0, z0, z1), s(z1)), FUNCTION(p, z0, z0, z1)) S tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, z0, z1, z2) -> c5(FUNCTION(if, function(iszero, z1, z1, z1), z1, z2), FUNCTION(iszero, z1, z1, z1)) FUNCTION(if, false, z0, z1) -> c7(FUNCTION(plus, function(third, z0, z1, z1), function(p, z0, z0, z1), s(z1)), FUNCTION(third, z0, z1, z1)) FUNCTION(if, false, z0, z1) -> c8(FUNCTION(plus, function(third, z0, z1, z1), function(p, z0, z0, z1), s(z1)), FUNCTION(p, z0, z0, z1)) K tuples:none Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c4_1, c5_2, c7_2, c8_2 ---------------------------------------- (21) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (22) Obligation: Complexity Dependency Tuples Problem Rules: function(iszero, 0, z0, z1) -> true function(iszero, s(z0), z1, z2) -> false function(p, 0, z0, z1) -> 0 function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) function(plus, z0, z1, z2) -> function(if, function(iszero, z1, z1, z1), z1, z2) function(if, true, z0, z1) -> z1 function(if, false, z0, z1) -> function(plus, function(third, z0, z1, z1), function(p, z0, z0, z1), s(z1)) function(third, z0, z1, z2) -> z2 Tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(if, false, z0, z1) -> c8(FUNCTION(plus, function(third, z0, z1, z1), function(p, z0, z0, z1), s(z1)), FUNCTION(p, z0, z0, z1)) FUNCTION(plus, z0, z1, z2) -> c5(FUNCTION(if, function(iszero, z1, z1, z1), z1, z2)) FUNCTION(if, false, z0, z1) -> c7(FUNCTION(plus, function(third, z0, z1, z1), function(p, z0, z0, z1), s(z1))) S tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(if, false, z0, z1) -> c8(FUNCTION(plus, function(third, z0, z1, z1), function(p, z0, z0, z1), s(z1)), FUNCTION(p, z0, z0, z1)) FUNCTION(plus, z0, z1, z2) -> c5(FUNCTION(if, function(iszero, z1, z1, z1), z1, z2)) FUNCTION(if, false, z0, z1) -> c7(FUNCTION(plus, function(third, z0, z1, z1), function(p, z0, z0, z1), s(z1))) K tuples:none Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c4_1, c8_2, c5_1, c7_1 ---------------------------------------- (23) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: function(plus, z0, z1, z2) -> function(if, function(iszero, z1, z1, z1), z1, z2) function(if, true, z0, z1) -> z1 function(if, false, z0, z1) -> function(plus, function(third, z0, z1, z1), function(p, z0, z0, z1), s(z1)) ---------------------------------------- (24) Obligation: Complexity Dependency Tuples Problem Rules: function(third, z0, z1, z2) -> z2 function(p, 0, z0, z1) -> 0 function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) function(iszero, 0, z0, z1) -> true function(iszero, s(z0), z1, z2) -> false Tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(if, false, z0, z1) -> c8(FUNCTION(plus, function(third, z0, z1, z1), function(p, z0, z0, z1), s(z1)), FUNCTION(p, z0, z0, z1)) FUNCTION(plus, z0, z1, z2) -> c5(FUNCTION(if, function(iszero, z1, z1, z1), z1, z2)) FUNCTION(if, false, z0, z1) -> c7(FUNCTION(plus, function(third, z0, z1, z1), function(p, z0, z0, z1), s(z1))) S tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(if, false, z0, z1) -> c8(FUNCTION(plus, function(third, z0, z1, z1), function(p, z0, z0, z1), s(z1)), FUNCTION(p, z0, z0, z1)) FUNCTION(plus, z0, z1, z2) -> c5(FUNCTION(if, function(iszero, z1, z1, z1), z1, z2)) FUNCTION(if, false, z0, z1) -> c7(FUNCTION(plus, function(third, z0, z1, z1), function(p, z0, z0, z1), s(z1))) K tuples:none Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c4_1, c8_2, c5_1, c7_1 ---------------------------------------- (25) CdtToCpxRelTrsProof (BOTH BOUNDS(ID, ID)) Converted S to standard rules, and D \ S as well as R to relative rules. ---------------------------------------- (26) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(if, false, z0, z1) -> c8(FUNCTION(plus, function(third, z0, z1, z1), function(p, z0, z0, z1), s(z1)), FUNCTION(p, z0, z0, z1)) FUNCTION(plus, z0, z1, z2) -> c5(FUNCTION(if, function(iszero, z1, z1, z1), z1, z2)) FUNCTION(if, false, z0, z1) -> c7(FUNCTION(plus, function(third, z0, z1, z1), function(p, z0, z0, z1), s(z1))) The (relative) TRS S consists of the following rules: function(third, z0, z1, z2) -> z2 function(p, 0, z0, z1) -> 0 function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) function(iszero, 0, z0, z1) -> true function(iszero, s(z0), z1, z2) -> false Rewrite Strategy: INNERMOST ---------------------------------------- (27) RelTrsToTrsProof (UPPER BOUND(ID)) transformed relative TRS to TRS ---------------------------------------- (28) Obligation: The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(if, false, z0, z1) -> c8(FUNCTION(plus, function(third, z0, z1, z1), function(p, z0, z0, z1), s(z1)), FUNCTION(p, z0, z0, z1)) FUNCTION(plus, z0, z1, z2) -> c5(FUNCTION(if, function(iszero, z1, z1, z1), z1, z2)) FUNCTION(if, false, z0, z1) -> c7(FUNCTION(plus, function(third, z0, z1, z1), function(p, z0, z0, z1), s(z1))) function(third, z0, z1, z2) -> z2 function(p, 0, z0, z1) -> 0 function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) function(iszero, 0, z0, z1) -> true function(iszero, s(z0), z1, z2) -> false S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (29) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID)) Transformed relative TRS to weighted TRS ---------------------------------------- (30) 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: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) [1] FUNCTION(if, false, z0, z1) -> c8(FUNCTION(plus, function(third, z0, z1, z1), function(p, z0, z0, z1), s(z1)), FUNCTION(p, z0, z0, z1)) [1] FUNCTION(plus, z0, z1, z2) -> c5(FUNCTION(if, function(iszero, z1, z1, z1), z1, z2)) [1] FUNCTION(if, false, z0, z1) -> c7(FUNCTION(plus, function(third, z0, z1, z1), function(p, z0, z0, z1), s(z1))) [1] function(third, z0, z1, z2) -> z2 [0] function(p, 0, z0, z1) -> 0 [0] function(p, s(0), z0, z1) -> 0 [0] function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) [0] function(iszero, 0, z0, z1) -> true [0] function(iszero, s(z0), z1, z2) -> false [0] Rewrite Strategy: INNERMOST ---------------------------------------- (31) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (32) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) [1] FUNCTION(if, false, z0, z1) -> c8(FUNCTION(plus, function(third, z0, z1, z1), function(p, z0, z0, z1), s(z1)), FUNCTION(p, z0, z0, z1)) [1] FUNCTION(plus, z0, z1, z2) -> c5(FUNCTION(if, function(iszero, z1, z1, z1), z1, z2)) [1] FUNCTION(if, false, z0, z1) -> c7(FUNCTION(plus, function(third, z0, z1, z1), function(p, z0, z0, z1), s(z1))) [1] function(third, z0, z1, z2) -> z2 [0] function(p, 0, z0, z1) -> 0 [0] function(p, s(0), z0, z1) -> 0 [0] function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) [0] function(iszero, 0, z0, z1) -> true [0] function(iszero, s(z0), z1, z2) -> false [0] The TRS has the following type information: FUNCTION :: p:if:plus:third:iszero -> s:false:0:true -> s:false:0:true -> s:false:0:true -> c4:c8:c5:c7 p :: p:if:plus:third:iszero s :: s:false:0:true -> s:false:0:true c4 :: c4:c8:c5:c7 -> c4:c8:c5:c7 if :: p:if:plus:third:iszero false :: s:false:0:true c8 :: c4:c8:c5:c7 -> c4:c8:c5:c7 -> c4:c8:c5:c7 plus :: p:if:plus:third:iszero function :: p:if:plus:third:iszero -> s:false:0:true -> s:false:0:true -> s:false:0:true -> s:false:0:true third :: p:if:plus:third:iszero c5 :: c4:c8:c5:c7 -> c4:c8:c5:c7 iszero :: p:if:plus:third:iszero c7 :: c4:c8:c5:c7 -> c4:c8:c5:c7 0 :: s:false:0:true true :: s:false:0:true Rewrite Strategy: INNERMOST ---------------------------------------- (33) 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: FUNCTION_4 (c) The following functions are completely defined: function_4 Due to the following rules being added: function(v0, v1, v2, v3) -> null_function [0] And the following fresh constants: null_function, const ---------------------------------------- (34) 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: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) [1] FUNCTION(if, false, z0, z1) -> c8(FUNCTION(plus, function(third, z0, z1, z1), function(p, z0, z0, z1), s(z1)), FUNCTION(p, z0, z0, z1)) [1] FUNCTION(plus, z0, z1, z2) -> c5(FUNCTION(if, function(iszero, z1, z1, z1), z1, z2)) [1] FUNCTION(if, false, z0, z1) -> c7(FUNCTION(plus, function(third, z0, z1, z1), function(p, z0, z0, z1), s(z1))) [1] function(third, z0, z1, z2) -> z2 [0] function(p, 0, z0, z1) -> 0 [0] function(p, s(0), z0, z1) -> 0 [0] function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) [0] function(iszero, 0, z0, z1) -> true [0] function(iszero, s(z0), z1, z2) -> false [0] function(v0, v1, v2, v3) -> null_function [0] The TRS has the following type information: FUNCTION :: p:if:plus:third:iszero -> s:false:0:true:null_function -> s:false:0:true:null_function -> s:false:0:true:null_function -> c4:c8:c5:c7 p :: p:if:plus:third:iszero s :: s:false:0:true:null_function -> s:false:0:true:null_function c4 :: c4:c8:c5:c7 -> c4:c8:c5:c7 if :: p:if:plus:third:iszero false :: s:false:0:true:null_function c8 :: c4:c8:c5:c7 -> c4:c8:c5:c7 -> c4:c8:c5:c7 plus :: p:if:plus:third:iszero function :: p:if:plus:third:iszero -> s:false:0:true:null_function -> s:false:0:true:null_function -> s:false:0:true:null_function -> s:false:0:true:null_function third :: p:if:plus:third:iszero c5 :: c4:c8:c5:c7 -> c4:c8:c5:c7 iszero :: p:if:plus:third:iszero c7 :: c4:c8:c5:c7 -> c4:c8:c5:c7 0 :: s:false:0:true:null_function true :: s:false:0:true:null_function null_function :: s:false:0:true:null_function const :: c4:c8:c5:c7 Rewrite Strategy: INNERMOST ---------------------------------------- (35) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (36) 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: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) [1] FUNCTION(if, false, 0, z1) -> c8(FUNCTION(plus, z1, 0, s(z1)), FUNCTION(p, 0, 0, z1)) [1] FUNCTION(if, false, s(0), z1) -> c8(FUNCTION(plus, z1, 0, s(z1)), FUNCTION(p, s(0), s(0), z1)) [1] FUNCTION(if, false, s(s(z0')), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(z0'), z0', z0')), s(z1)), FUNCTION(p, s(s(z0')), s(s(z0')), z1)) [1] FUNCTION(if, false, z0, z1) -> c8(FUNCTION(plus, z1, null_function, s(z1)), FUNCTION(p, z0, z0, z1)) [1] FUNCTION(if, false, 0, z1) -> c8(FUNCTION(plus, null_function, 0, s(z1)), FUNCTION(p, 0, 0, z1)) [1] FUNCTION(if, false, s(0), z1) -> c8(FUNCTION(plus, null_function, 0, s(z1)), FUNCTION(p, s(0), s(0), z1)) [1] FUNCTION(if, false, s(s(z0'')), z1) -> c8(FUNCTION(plus, null_function, s(function(p, s(z0''), z0'', z0'')), s(z1)), FUNCTION(p, s(s(z0'')), s(s(z0'')), z1)) [1] FUNCTION(if, false, z0, z1) -> c8(FUNCTION(plus, null_function, null_function, s(z1)), FUNCTION(p, z0, z0, z1)) [1] FUNCTION(plus, z0, 0, z2) -> c5(FUNCTION(if, true, 0, z2)) [1] FUNCTION(plus, z0, s(z01), z2) -> c5(FUNCTION(if, false, s(z01), z2)) [1] FUNCTION(plus, z0, z1, z2) -> c5(FUNCTION(if, null_function, z1, z2)) [1] FUNCTION(if, false, 0, z1) -> c7(FUNCTION(plus, z1, 0, s(z1))) [1] FUNCTION(if, false, s(0), z1) -> c7(FUNCTION(plus, z1, 0, s(z1))) [1] FUNCTION(if, false, s(s(z02)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(z02), z02, z02)), s(z1))) [1] FUNCTION(if, false, z0, z1) -> c7(FUNCTION(plus, z1, null_function, s(z1))) [1] FUNCTION(if, false, 0, z1) -> c7(FUNCTION(plus, null_function, 0, s(z1))) [1] FUNCTION(if, false, s(0), z1) -> c7(FUNCTION(plus, null_function, 0, s(z1))) [1] FUNCTION(if, false, s(s(z03)), z1) -> c7(FUNCTION(plus, null_function, s(function(p, s(z03), z03, z03)), s(z1))) [1] FUNCTION(if, false, z0, z1) -> c7(FUNCTION(plus, null_function, null_function, s(z1))) [1] function(third, z0, z1, z2) -> z2 [0] function(p, 0, z0, z1) -> 0 [0] function(p, s(0), z0, z1) -> 0 [0] function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) [0] function(iszero, 0, z0, z1) -> true [0] function(iszero, s(z0), z1, z2) -> false [0] function(v0, v1, v2, v3) -> null_function [0] The TRS has the following type information: FUNCTION :: p:if:plus:third:iszero -> s:false:0:true:null_function -> s:false:0:true:null_function -> s:false:0:true:null_function -> c4:c8:c5:c7 p :: p:if:plus:third:iszero s :: s:false:0:true:null_function -> s:false:0:true:null_function c4 :: c4:c8:c5:c7 -> c4:c8:c5:c7 if :: p:if:plus:third:iszero false :: s:false:0:true:null_function c8 :: c4:c8:c5:c7 -> c4:c8:c5:c7 -> c4:c8:c5:c7 plus :: p:if:plus:third:iszero function :: p:if:plus:third:iszero -> s:false:0:true:null_function -> s:false:0:true:null_function -> s:false:0:true:null_function -> s:false:0:true:null_function third :: p:if:plus:third:iszero c5 :: c4:c8:c5:c7 -> c4:c8:c5:c7 iszero :: p:if:plus:third:iszero c7 :: c4:c8:c5:c7 -> c4:c8:c5:c7 0 :: s:false:0:true:null_function true :: s:false:0:true:null_function null_function :: s:false:0:true:null_function const :: c4:c8:c5:c7 Rewrite Strategy: INNERMOST ---------------------------------------- (37) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: p => 2 if => 0 false => 1 plus => 3 third => 4 iszero => 1 0 => 0 true => 2 null_function => 0 const => 0 ---------------------------------------- (38) Obligation: Complexity RNTS consisting of the following rules: FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, z1, 0, 1 + z1) :|: z'' = 0, z1 >= 0, z3 = z1, z' = 1, z = 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, z1, 0, 1 + z1) :|: z1 >= 0, z3 = z1, z' = 1, z = 0, z'' = 1 + 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, z1, 0, 1 + z1) :|: z1 >= 0, z'' = z0, z3 = z1, z0 >= 0, z' = 1, z = 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, z1, 1 + function(2, 1 + z02, z02, z02), 1 + z1) :|: z1 >= 0, z'' = 1 + (1 + z02), z02 >= 0, z3 = z1, z' = 1, z = 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, 0, 0, 1 + z1) :|: z'' = 0, z1 >= 0, z3 = z1, z' = 1, z = 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, 0, 0, 1 + z1) :|: z1 >= 0, z3 = z1, z' = 1, z = 0, z'' = 1 + 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, 0, 0, 1 + z1) :|: z1 >= 0, z'' = z0, z3 = z1, z0 >= 0, z' = 1, z = 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, 0, 1 + function(2, 1 + z03, z03, z03), 1 + z1) :|: z1 >= 0, z3 = z1, z' = 1, z03 >= 0, z = 0, z'' = 1 + (1 + z03) FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(2, 1 + z0, z0, z0) :|: z = 2, z1 >= 0, z0 >= 0, z3 = z2, z2 >= 0, z' = 1 + (1 + z0), z'' = z1 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(0, 2, 0, z2) :|: z'' = 0, z = 3, z0 >= 0, z3 = z2, z' = z0, z2 >= 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(0, 1, 1 + z01, z2) :|: z = 3, z01 >= 0, z0 >= 0, z3 = z2, z' = z0, z2 >= 0, z'' = 1 + z01 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(0, 0, z1, z2) :|: z = 3, z1 >= 0, z0 >= 0, z3 = z2, z' = z0, z2 >= 0, z'' = z1 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, z1, 0, 1 + z1) + FUNCTION(2, z0, z0, z1) :|: z1 >= 0, z'' = z0, z3 = z1, z0 >= 0, z' = 1, z = 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, z1, 0, 1 + z1) + FUNCTION(2, 0, 0, z1) :|: z'' = 0, z1 >= 0, z3 = z1, z' = 1, z = 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, z1, 0, 1 + z1) + FUNCTION(2, 1 + 0, 1 + 0, z1) :|: z1 >= 0, z3 = z1, z' = 1, z = 0, z'' = 1 + 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, z1, 1 + function(2, 1 + z0', z0', z0'), 1 + z1) + FUNCTION(2, 1 + (1 + z0'), 1 + (1 + z0'), z1) :|: z1 >= 0, z0' >= 0, z'' = 1 + (1 + z0'), z3 = z1, z' = 1, z = 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, 0, 0, 1 + z1) + FUNCTION(2, z0, z0, z1) :|: z1 >= 0, z'' = z0, z3 = z1, z0 >= 0, z' = 1, z = 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, 0, 0, 1 + z1) + FUNCTION(2, 0, 0, z1) :|: z'' = 0, z1 >= 0, z3 = z1, z' = 1, z = 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, 0, 0, 1 + z1) + FUNCTION(2, 1 + 0, 1 + 0, z1) :|: z1 >= 0, z3 = z1, z' = 1, z = 0, z'' = 1 + 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, 0, 1 + function(2, 1 + z0'', z0'', z0''), 1 + z1) + FUNCTION(2, 1 + (1 + z0''), 1 + (1 + z0''), z1) :|: z1 >= 0, z'' = 1 + (1 + z0''), z3 = z1, z' = 1, z0'' >= 0, z = 0 function(z, z', z'', z3) -{ 0 }-> z2 :|: z1 >= 0, z0 >= 0, z = 4, z3 = z2, z' = z0, z2 >= 0, z'' = z1 function(z, z', z'', z3) -{ 0 }-> 2 :|: z1 >= 0, z = 1, z'' = z0, z3 = z1, z0 >= 0, z' = 0 function(z, z', z'', z3) -{ 0 }-> 1 :|: z1 >= 0, z = 1, z0 >= 0, z' = 1 + z0, z3 = z2, z2 >= 0, z'' = z1 function(z, z', z'', z3) -{ 0 }-> 0 :|: z = 2, z1 >= 0, z'' = z0, z3 = z1, z0 >= 0, z' = 0 function(z, z', z'', z3) -{ 0 }-> 0 :|: z = 2, z1 >= 0, z'' = z0, z' = 1 + 0, z3 = z1, z0 >= 0 function(z, z', z'', z3) -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, z3 = v3, v2 >= 0, v3 >= 0 function(z, z', z'', z3) -{ 0 }-> 1 + function(2, 1 + z0, z0, z0) :|: z = 2, z1 >= 0, z0 >= 0, z3 = z2, z2 >= 0, z' = 1 + (1 + z0), z'' = z1 ---------------------------------------- (39) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (40) Obligation: Complexity RNTS consisting of the following rules: FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, z3, 0, 1 + z3) :|: z'' = 0, z3 >= 0, z' = 1, z = 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, z3, 0, 1 + z3) :|: z3 >= 0, z' = 1, z = 0, z'' = 1 + 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, z3, 0, 1 + z3) :|: z3 >= 0, z'' >= 0, z' = 1, z = 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, z3, 1 + function(2, 1 + (z'' - 2), z'' - 2, z'' - 2), 1 + z3) :|: z3 >= 0, z'' - 2 >= 0, z' = 1, z = 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, 0, 0, 1 + z3) :|: z'' = 0, z3 >= 0, z' = 1, z = 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, 0, 0, 1 + z3) :|: z3 >= 0, z' = 1, z = 0, z'' = 1 + 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, 0, 0, 1 + z3) :|: z3 >= 0, z'' >= 0, z' = 1, z = 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, 0, 1 + function(2, 1 + (z'' - 2), z'' - 2, z'' - 2), 1 + z3) :|: z3 >= 0, z' = 1, z'' - 2 >= 0, z = 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(2, 1 + (z' - 2), z' - 2, z' - 2) :|: z = 2, z'' >= 0, z' - 2 >= 0, z3 >= 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(0, 2, 0, z3) :|: z'' = 0, z = 3, z' >= 0, z3 >= 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(0, 1, 1 + (z'' - 1), z3) :|: z = 3, z'' - 1 >= 0, z' >= 0, z3 >= 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(0, 0, z'', z3) :|: z = 3, z'' >= 0, z' >= 0, z3 >= 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, z3, 0, 1 + z3) + FUNCTION(2, z'', z'', z3) :|: z3 >= 0, z'' >= 0, z' = 1, z = 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, z3, 0, 1 + z3) + FUNCTION(2, 0, 0, z3) :|: z'' = 0, z3 >= 0, z' = 1, z = 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, z3, 0, 1 + z3) + FUNCTION(2, 1 + 0, 1 + 0, z3) :|: z3 >= 0, z' = 1, z = 0, z'' = 1 + 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, z3, 1 + function(2, 1 + (z'' - 2), z'' - 2, z'' - 2), 1 + z3) + FUNCTION(2, 1 + (1 + (z'' - 2)), 1 + (1 + (z'' - 2)), z3) :|: z3 >= 0, z'' - 2 >= 0, z' = 1, z = 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, 0, 0, 1 + z3) + FUNCTION(2, z'', z'', z3) :|: z3 >= 0, z'' >= 0, z' = 1, z = 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, 0, 0, 1 + z3) + FUNCTION(2, 0, 0, z3) :|: z'' = 0, z3 >= 0, z' = 1, z = 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, 0, 0, 1 + z3) + FUNCTION(2, 1 + 0, 1 + 0, z3) :|: z3 >= 0, z' = 1, z = 0, z'' = 1 + 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, 0, 1 + function(2, 1 + (z'' - 2), z'' - 2, z'' - 2), 1 + z3) + FUNCTION(2, 1 + (1 + (z'' - 2)), 1 + (1 + (z'' - 2)), z3) :|: z3 >= 0, z' = 1, z'' - 2 >= 0, z = 0 function(z, z', z'', z3) -{ 0 }-> z3 :|: z'' >= 0, z' >= 0, z = 4, z3 >= 0 function(z, z', z'', z3) -{ 0 }-> 2 :|: z3 >= 0, z = 1, z'' >= 0, z' = 0 function(z, z', z'', z3) -{ 0 }-> 1 :|: z'' >= 0, z = 1, z' - 1 >= 0, z3 >= 0 function(z, z', z'', z3) -{ 0 }-> 0 :|: z = 2, z3 >= 0, z'' >= 0, z' = 0 function(z, z', z'', z3) -{ 0 }-> 0 :|: z = 2, z3 >= 0, z' = 1 + 0, z'' >= 0 function(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0 function(z, z', z'', z3) -{ 0 }-> 1 + function(2, 1 + (z' - 2), z' - 2, z' - 2) :|: z = 2, z'' >= 0, z' - 2 >= 0, z3 >= 0 ---------------------------------------- (41) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { function } { FUNCTION } ---------------------------------------- (42) Obligation: Complexity RNTS consisting of the following rules: FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, z3, 0, 1 + z3) :|: z'' = 0, z3 >= 0, z' = 1, z = 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, z3, 0, 1 + z3) :|: z3 >= 0, z' = 1, z = 0, z'' = 1 + 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, z3, 0, 1 + z3) :|: z3 >= 0, z'' >= 0, z' = 1, z = 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, z3, 1 + function(2, 1 + (z'' - 2), z'' - 2, z'' - 2), 1 + z3) :|: z3 >= 0, z'' - 2 >= 0, z' = 1, z = 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, 0, 0, 1 + z3) :|: z'' = 0, z3 >= 0, z' = 1, z = 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, 0, 0, 1 + z3) :|: z3 >= 0, z' = 1, z = 0, z'' = 1 + 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, 0, 0, 1 + z3) :|: z3 >= 0, z'' >= 0, z' = 1, z = 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, 0, 1 + function(2, 1 + (z'' - 2), z'' - 2, z'' - 2), 1 + z3) :|: z3 >= 0, z' = 1, z'' - 2 >= 0, z = 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(2, 1 + (z' - 2), z' - 2, z' - 2) :|: z = 2, z'' >= 0, z' - 2 >= 0, z3 >= 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(0, 2, 0, z3) :|: z'' = 0, z = 3, z' >= 0, z3 >= 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(0, 1, 1 + (z'' - 1), z3) :|: z = 3, z'' - 1 >= 0, z' >= 0, z3 >= 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(0, 0, z'', z3) :|: z = 3, z'' >= 0, z' >= 0, z3 >= 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, z3, 0, 1 + z3) + FUNCTION(2, z'', z'', z3) :|: z3 >= 0, z'' >= 0, z' = 1, z = 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, z3, 0, 1 + z3) + FUNCTION(2, 0, 0, z3) :|: z'' = 0, z3 >= 0, z' = 1, z = 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, z3, 0, 1 + z3) + FUNCTION(2, 1 + 0, 1 + 0, z3) :|: z3 >= 0, z' = 1, z = 0, z'' = 1 + 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, z3, 1 + function(2, 1 + (z'' - 2), z'' - 2, z'' - 2), 1 + z3) + FUNCTION(2, 1 + (1 + (z'' - 2)), 1 + (1 + (z'' - 2)), z3) :|: z3 >= 0, z'' - 2 >= 0, z' = 1, z = 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, 0, 0, 1 + z3) + FUNCTION(2, z'', z'', z3) :|: z3 >= 0, z'' >= 0, z' = 1, z = 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, 0, 0, 1 + z3) + FUNCTION(2, 0, 0, z3) :|: z'' = 0, z3 >= 0, z' = 1, z = 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, 0, 0, 1 + z3) + FUNCTION(2, 1 + 0, 1 + 0, z3) :|: z3 >= 0, z' = 1, z = 0, z'' = 1 + 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, 0, 1 + function(2, 1 + (z'' - 2), z'' - 2, z'' - 2), 1 + z3) + FUNCTION(2, 1 + (1 + (z'' - 2)), 1 + (1 + (z'' - 2)), z3) :|: z3 >= 0, z' = 1, z'' - 2 >= 0, z = 0 function(z, z', z'', z3) -{ 0 }-> z3 :|: z'' >= 0, z' >= 0, z = 4, z3 >= 0 function(z, z', z'', z3) -{ 0 }-> 2 :|: z3 >= 0, z = 1, z'' >= 0, z' = 0 function(z, z', z'', z3) -{ 0 }-> 1 :|: z'' >= 0, z = 1, z' - 1 >= 0, z3 >= 0 function(z, z', z'', z3) -{ 0 }-> 0 :|: z = 2, z3 >= 0, z'' >= 0, z' = 0 function(z, z', z'', z3) -{ 0 }-> 0 :|: z = 2, z3 >= 0, z' = 1 + 0, z'' >= 0 function(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0 function(z, z', z'', z3) -{ 0 }-> 1 + function(2, 1 + (z' - 2), z' - 2, z' - 2) :|: z = 2, z'' >= 0, z' - 2 >= 0, z3 >= 0 Function symbols to be analyzed: {function}, {FUNCTION} ---------------------------------------- (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: FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, z3, 0, 1 + z3) :|: z'' = 0, z3 >= 0, z' = 1, z = 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, z3, 0, 1 + z3) :|: z3 >= 0, z' = 1, z = 0, z'' = 1 + 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, z3, 0, 1 + z3) :|: z3 >= 0, z'' >= 0, z' = 1, z = 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, z3, 1 + function(2, 1 + (z'' - 2), z'' - 2, z'' - 2), 1 + z3) :|: z3 >= 0, z'' - 2 >= 0, z' = 1, z = 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, 0, 0, 1 + z3) :|: z'' = 0, z3 >= 0, z' = 1, z = 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, 0, 0, 1 + z3) :|: z3 >= 0, z' = 1, z = 0, z'' = 1 + 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, 0, 0, 1 + z3) :|: z3 >= 0, z'' >= 0, z' = 1, z = 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, 0, 1 + function(2, 1 + (z'' - 2), z'' - 2, z'' - 2), 1 + z3) :|: z3 >= 0, z' = 1, z'' - 2 >= 0, z = 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(2, 1 + (z' - 2), z' - 2, z' - 2) :|: z = 2, z'' >= 0, z' - 2 >= 0, z3 >= 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(0, 2, 0, z3) :|: z'' = 0, z = 3, z' >= 0, z3 >= 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(0, 1, 1 + (z'' - 1), z3) :|: z = 3, z'' - 1 >= 0, z' >= 0, z3 >= 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(0, 0, z'', z3) :|: z = 3, z'' >= 0, z' >= 0, z3 >= 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, z3, 0, 1 + z3) + FUNCTION(2, z'', z'', z3) :|: z3 >= 0, z'' >= 0, z' = 1, z = 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, z3, 0, 1 + z3) + FUNCTION(2, 0, 0, z3) :|: z'' = 0, z3 >= 0, z' = 1, z = 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, z3, 0, 1 + z3) + FUNCTION(2, 1 + 0, 1 + 0, z3) :|: z3 >= 0, z' = 1, z = 0, z'' = 1 + 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, z3, 1 + function(2, 1 + (z'' - 2), z'' - 2, z'' - 2), 1 + z3) + FUNCTION(2, 1 + (1 + (z'' - 2)), 1 + (1 + (z'' - 2)), z3) :|: z3 >= 0, z'' - 2 >= 0, z' = 1, z = 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, 0, 0, 1 + z3) + FUNCTION(2, z'', z'', z3) :|: z3 >= 0, z'' >= 0, z' = 1, z = 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, 0, 0, 1 + z3) + FUNCTION(2, 0, 0, z3) :|: z'' = 0, z3 >= 0, z' = 1, z = 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, 0, 0, 1 + z3) + FUNCTION(2, 1 + 0, 1 + 0, z3) :|: z3 >= 0, z' = 1, z = 0, z'' = 1 + 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, 0, 1 + function(2, 1 + (z'' - 2), z'' - 2, z'' - 2), 1 + z3) + FUNCTION(2, 1 + (1 + (z'' - 2)), 1 + (1 + (z'' - 2)), z3) :|: z3 >= 0, z' = 1, z'' - 2 >= 0, z = 0 function(z, z', z'', z3) -{ 0 }-> z3 :|: z'' >= 0, z' >= 0, z = 4, z3 >= 0 function(z, z', z'', z3) -{ 0 }-> 2 :|: z3 >= 0, z = 1, z'' >= 0, z' = 0 function(z, z', z'', z3) -{ 0 }-> 1 :|: z'' >= 0, z = 1, z' - 1 >= 0, z3 >= 0 function(z, z', z'', z3) -{ 0 }-> 0 :|: z = 2, z3 >= 0, z'' >= 0, z' = 0 function(z, z', z'', z3) -{ 0 }-> 0 :|: z = 2, z3 >= 0, z' = 1 + 0, z'' >= 0 function(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0 function(z, z', z'', z3) -{ 0 }-> 1 + function(2, 1 + (z' - 2), z' - 2, z' - 2) :|: z = 2, z'' >= 0, z' - 2 >= 0, z3 >= 0 Function symbols to be analyzed: {function}, {FUNCTION} ---------------------------------------- (45) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: function after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2 + z' + z3 ---------------------------------------- (46) Obligation: Complexity RNTS consisting of the following rules: FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, z3, 0, 1 + z3) :|: z'' = 0, z3 >= 0, z' = 1, z = 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, z3, 0, 1 + z3) :|: z3 >= 0, z' = 1, z = 0, z'' = 1 + 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, z3, 0, 1 + z3) :|: z3 >= 0, z'' >= 0, z' = 1, z = 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, z3, 1 + function(2, 1 + (z'' - 2), z'' - 2, z'' - 2), 1 + z3) :|: z3 >= 0, z'' - 2 >= 0, z' = 1, z = 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, 0, 0, 1 + z3) :|: z'' = 0, z3 >= 0, z' = 1, z = 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, 0, 0, 1 + z3) :|: z3 >= 0, z' = 1, z = 0, z'' = 1 + 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, 0, 0, 1 + z3) :|: z3 >= 0, z'' >= 0, z' = 1, z = 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, 0, 1 + function(2, 1 + (z'' - 2), z'' - 2, z'' - 2), 1 + z3) :|: z3 >= 0, z' = 1, z'' - 2 >= 0, z = 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(2, 1 + (z' - 2), z' - 2, z' - 2) :|: z = 2, z'' >= 0, z' - 2 >= 0, z3 >= 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(0, 2, 0, z3) :|: z'' = 0, z = 3, z' >= 0, z3 >= 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(0, 1, 1 + (z'' - 1), z3) :|: z = 3, z'' - 1 >= 0, z' >= 0, z3 >= 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(0, 0, z'', z3) :|: z = 3, z'' >= 0, z' >= 0, z3 >= 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, z3, 0, 1 + z3) + FUNCTION(2, z'', z'', z3) :|: z3 >= 0, z'' >= 0, z' = 1, z = 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, z3, 0, 1 + z3) + FUNCTION(2, 0, 0, z3) :|: z'' = 0, z3 >= 0, z' = 1, z = 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, z3, 0, 1 + z3) + FUNCTION(2, 1 + 0, 1 + 0, z3) :|: z3 >= 0, z' = 1, z = 0, z'' = 1 + 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, z3, 1 + function(2, 1 + (z'' - 2), z'' - 2, z'' - 2), 1 + z3) + FUNCTION(2, 1 + (1 + (z'' - 2)), 1 + (1 + (z'' - 2)), z3) :|: z3 >= 0, z'' - 2 >= 0, z' = 1, z = 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, 0, 0, 1 + z3) + FUNCTION(2, z'', z'', z3) :|: z3 >= 0, z'' >= 0, z' = 1, z = 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, 0, 0, 1 + z3) + FUNCTION(2, 0, 0, z3) :|: z'' = 0, z3 >= 0, z' = 1, z = 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, 0, 0, 1 + z3) + FUNCTION(2, 1 + 0, 1 + 0, z3) :|: z3 >= 0, z' = 1, z = 0, z'' = 1 + 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, 0, 1 + function(2, 1 + (z'' - 2), z'' - 2, z'' - 2), 1 + z3) + FUNCTION(2, 1 + (1 + (z'' - 2)), 1 + (1 + (z'' - 2)), z3) :|: z3 >= 0, z' = 1, z'' - 2 >= 0, z = 0 function(z, z', z'', z3) -{ 0 }-> z3 :|: z'' >= 0, z' >= 0, z = 4, z3 >= 0 function(z, z', z'', z3) -{ 0 }-> 2 :|: z3 >= 0, z = 1, z'' >= 0, z' = 0 function(z, z', z'', z3) -{ 0 }-> 1 :|: z'' >= 0, z = 1, z' - 1 >= 0, z3 >= 0 function(z, z', z'', z3) -{ 0 }-> 0 :|: z = 2, z3 >= 0, z'' >= 0, z' = 0 function(z, z', z'', z3) -{ 0 }-> 0 :|: z = 2, z3 >= 0, z' = 1 + 0, z'' >= 0 function(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0 function(z, z', z'', z3) -{ 0 }-> 1 + function(2, 1 + (z' - 2), z' - 2, z' - 2) :|: z = 2, z'' >= 0, z' - 2 >= 0, z3 >= 0 Function symbols to be analyzed: {function}, {FUNCTION} Previous analysis results are: function: runtime: ?, size: O(n^1) [2 + z' + z3] ---------------------------------------- (47) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: function after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (48) Obligation: Complexity RNTS consisting of the following rules: FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, z3, 0, 1 + z3) :|: z'' = 0, z3 >= 0, z' = 1, z = 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, z3, 0, 1 + z3) :|: z3 >= 0, z' = 1, z = 0, z'' = 1 + 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, z3, 0, 1 + z3) :|: z3 >= 0, z'' >= 0, z' = 1, z = 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, z3, 1 + function(2, 1 + (z'' - 2), z'' - 2, z'' - 2), 1 + z3) :|: z3 >= 0, z'' - 2 >= 0, z' = 1, z = 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, 0, 0, 1 + z3) :|: z'' = 0, z3 >= 0, z' = 1, z = 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, 0, 0, 1 + z3) :|: z3 >= 0, z' = 1, z = 0, z'' = 1 + 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, 0, 0, 1 + z3) :|: z3 >= 0, z'' >= 0, z' = 1, z = 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, 0, 1 + function(2, 1 + (z'' - 2), z'' - 2, z'' - 2), 1 + z3) :|: z3 >= 0, z' = 1, z'' - 2 >= 0, z = 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(2, 1 + (z' - 2), z' - 2, z' - 2) :|: z = 2, z'' >= 0, z' - 2 >= 0, z3 >= 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(0, 2, 0, z3) :|: z'' = 0, z = 3, z' >= 0, z3 >= 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(0, 1, 1 + (z'' - 1), z3) :|: z = 3, z'' - 1 >= 0, z' >= 0, z3 >= 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(0, 0, z'', z3) :|: z = 3, z'' >= 0, z' >= 0, z3 >= 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, z3, 0, 1 + z3) + FUNCTION(2, z'', z'', z3) :|: z3 >= 0, z'' >= 0, z' = 1, z = 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, z3, 0, 1 + z3) + FUNCTION(2, 0, 0, z3) :|: z'' = 0, z3 >= 0, z' = 1, z = 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, z3, 0, 1 + z3) + FUNCTION(2, 1 + 0, 1 + 0, z3) :|: z3 >= 0, z' = 1, z = 0, z'' = 1 + 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, z3, 1 + function(2, 1 + (z'' - 2), z'' - 2, z'' - 2), 1 + z3) + FUNCTION(2, 1 + (1 + (z'' - 2)), 1 + (1 + (z'' - 2)), z3) :|: z3 >= 0, z'' - 2 >= 0, z' = 1, z = 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, 0, 0, 1 + z3) + FUNCTION(2, z'', z'', z3) :|: z3 >= 0, z'' >= 0, z' = 1, z = 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, 0, 0, 1 + z3) + FUNCTION(2, 0, 0, z3) :|: z'' = 0, z3 >= 0, z' = 1, z = 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, 0, 0, 1 + z3) + FUNCTION(2, 1 + 0, 1 + 0, z3) :|: z3 >= 0, z' = 1, z = 0, z'' = 1 + 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, 0, 1 + function(2, 1 + (z'' - 2), z'' - 2, z'' - 2), 1 + z3) + FUNCTION(2, 1 + (1 + (z'' - 2)), 1 + (1 + (z'' - 2)), z3) :|: z3 >= 0, z' = 1, z'' - 2 >= 0, z = 0 function(z, z', z'', z3) -{ 0 }-> z3 :|: z'' >= 0, z' >= 0, z = 4, z3 >= 0 function(z, z', z'', z3) -{ 0 }-> 2 :|: z3 >= 0, z = 1, z'' >= 0, z' = 0 function(z, z', z'', z3) -{ 0 }-> 1 :|: z'' >= 0, z = 1, z' - 1 >= 0, z3 >= 0 function(z, z', z'', z3) -{ 0 }-> 0 :|: z = 2, z3 >= 0, z'' >= 0, z' = 0 function(z, z', z'', z3) -{ 0 }-> 0 :|: z = 2, z3 >= 0, z' = 1 + 0, z'' >= 0 function(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0 function(z, z', z'', z3) -{ 0 }-> 1 + function(2, 1 + (z' - 2), z' - 2, z' - 2) :|: z = 2, z'' >= 0, z' - 2 >= 0, z3 >= 0 Function symbols to be analyzed: {FUNCTION} Previous analysis results are: function: runtime: O(1) [0], size: O(n^1) [2 + z' + z3] ---------------------------------------- (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: FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, z3, 0, 1 + z3) :|: z'' = 0, z3 >= 0, z' = 1, z = 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, z3, 0, 1 + z3) :|: z3 >= 0, z' = 1, z = 0, z'' = 1 + 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, z3, 0, 1 + z3) :|: z3 >= 0, z'' >= 0, z' = 1, z = 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, z3, 1 + s'', 1 + z3) :|: s'' >= 0, s'' <= z'' - 2 + 2 + (1 + (z'' - 2)), z3 >= 0, z'' - 2 >= 0, z' = 1, z = 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, 0, 0, 1 + z3) :|: z'' = 0, z3 >= 0, z' = 1, z = 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, 0, 0, 1 + z3) :|: z3 >= 0, z' = 1, z = 0, z'' = 1 + 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, 0, 0, 1 + z3) :|: z3 >= 0, z'' >= 0, z' = 1, z = 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, 0, 1 + s1, 1 + z3) :|: s1 >= 0, s1 <= z'' - 2 + 2 + (1 + (z'' - 2)), z3 >= 0, z' = 1, z'' - 2 >= 0, z = 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(2, 1 + (z' - 2), z' - 2, z' - 2) :|: z = 2, z'' >= 0, z' - 2 >= 0, z3 >= 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(0, 2, 0, z3) :|: z'' = 0, z = 3, z' >= 0, z3 >= 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(0, 1, 1 + (z'' - 1), z3) :|: z = 3, z'' - 1 >= 0, z' >= 0, z3 >= 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(0, 0, z'', z3) :|: z = 3, z'' >= 0, z' >= 0, z3 >= 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, z3, 0, 1 + z3) + FUNCTION(2, z'', z'', z3) :|: z3 >= 0, z'' >= 0, z' = 1, z = 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, z3, 0, 1 + z3) + FUNCTION(2, 0, 0, z3) :|: z'' = 0, z3 >= 0, z' = 1, z = 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, z3, 0, 1 + z3) + FUNCTION(2, 1 + 0, 1 + 0, z3) :|: z3 >= 0, z' = 1, z = 0, z'' = 1 + 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, z3, 1 + s, 1 + z3) + FUNCTION(2, 1 + (1 + (z'' - 2)), 1 + (1 + (z'' - 2)), z3) :|: s >= 0, s <= z'' - 2 + 2 + (1 + (z'' - 2)), z3 >= 0, z'' - 2 >= 0, z' = 1, z = 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, 0, 0, 1 + z3) + FUNCTION(2, z'', z'', z3) :|: z3 >= 0, z'' >= 0, z' = 1, z = 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, 0, 0, 1 + z3) + FUNCTION(2, 0, 0, z3) :|: z'' = 0, z3 >= 0, z' = 1, z = 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, 0, 0, 1 + z3) + FUNCTION(2, 1 + 0, 1 + 0, z3) :|: z3 >= 0, z' = 1, z = 0, z'' = 1 + 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, 0, 1 + s', 1 + z3) + FUNCTION(2, 1 + (1 + (z'' - 2)), 1 + (1 + (z'' - 2)), z3) :|: s' >= 0, s' <= z'' - 2 + 2 + (1 + (z'' - 2)), z3 >= 0, z' = 1, z'' - 2 >= 0, z = 0 function(z, z', z'', z3) -{ 0 }-> z3 :|: z'' >= 0, z' >= 0, z = 4, z3 >= 0 function(z, z', z'', z3) -{ 0 }-> 2 :|: z3 >= 0, z = 1, z'' >= 0, z' = 0 function(z, z', z'', z3) -{ 0 }-> 1 :|: z'' >= 0, z = 1, z' - 1 >= 0, z3 >= 0 function(z, z', z'', z3) -{ 0 }-> 0 :|: z = 2, z3 >= 0, z'' >= 0, z' = 0 function(z, z', z'', z3) -{ 0 }-> 0 :|: z = 2, z3 >= 0, z' = 1 + 0, z'' >= 0 function(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0 function(z, z', z'', z3) -{ 0 }-> 1 + s2 :|: s2 >= 0, s2 <= z' - 2 + 2 + (1 + (z' - 2)), z = 2, z'' >= 0, z' - 2 >= 0, z3 >= 0 Function symbols to be analyzed: {FUNCTION} Previous analysis results are: function: runtime: O(1) [0], size: O(n^1) [2 + z' + z3] ---------------------------------------- (51) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: FUNCTION after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (52) Obligation: Complexity RNTS consisting of the following rules: FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, z3, 0, 1 + z3) :|: z'' = 0, z3 >= 0, z' = 1, z = 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, z3, 0, 1 + z3) :|: z3 >= 0, z' = 1, z = 0, z'' = 1 + 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, z3, 0, 1 + z3) :|: z3 >= 0, z'' >= 0, z' = 1, z = 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, z3, 1 + s'', 1 + z3) :|: s'' >= 0, s'' <= z'' - 2 + 2 + (1 + (z'' - 2)), z3 >= 0, z'' - 2 >= 0, z' = 1, z = 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, 0, 0, 1 + z3) :|: z'' = 0, z3 >= 0, z' = 1, z = 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, 0, 0, 1 + z3) :|: z3 >= 0, z' = 1, z = 0, z'' = 1 + 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, 0, 0, 1 + z3) :|: z3 >= 0, z'' >= 0, z' = 1, z = 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, 0, 1 + s1, 1 + z3) :|: s1 >= 0, s1 <= z'' - 2 + 2 + (1 + (z'' - 2)), z3 >= 0, z' = 1, z'' - 2 >= 0, z = 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(2, 1 + (z' - 2), z' - 2, z' - 2) :|: z = 2, z'' >= 0, z' - 2 >= 0, z3 >= 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(0, 2, 0, z3) :|: z'' = 0, z = 3, z' >= 0, z3 >= 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(0, 1, 1 + (z'' - 1), z3) :|: z = 3, z'' - 1 >= 0, z' >= 0, z3 >= 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(0, 0, z'', z3) :|: z = 3, z'' >= 0, z' >= 0, z3 >= 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, z3, 0, 1 + z3) + FUNCTION(2, z'', z'', z3) :|: z3 >= 0, z'' >= 0, z' = 1, z = 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, z3, 0, 1 + z3) + FUNCTION(2, 0, 0, z3) :|: z'' = 0, z3 >= 0, z' = 1, z = 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, z3, 0, 1 + z3) + FUNCTION(2, 1 + 0, 1 + 0, z3) :|: z3 >= 0, z' = 1, z = 0, z'' = 1 + 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, z3, 1 + s, 1 + z3) + FUNCTION(2, 1 + (1 + (z'' - 2)), 1 + (1 + (z'' - 2)), z3) :|: s >= 0, s <= z'' - 2 + 2 + (1 + (z'' - 2)), z3 >= 0, z'' - 2 >= 0, z' = 1, z = 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, 0, 0, 1 + z3) + FUNCTION(2, z'', z'', z3) :|: z3 >= 0, z'' >= 0, z' = 1, z = 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, 0, 0, 1 + z3) + FUNCTION(2, 0, 0, z3) :|: z'' = 0, z3 >= 0, z' = 1, z = 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, 0, 0, 1 + z3) + FUNCTION(2, 1 + 0, 1 + 0, z3) :|: z3 >= 0, z' = 1, z = 0, z'' = 1 + 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, 0, 1 + s', 1 + z3) + FUNCTION(2, 1 + (1 + (z'' - 2)), 1 + (1 + (z'' - 2)), z3) :|: s' >= 0, s' <= z'' - 2 + 2 + (1 + (z'' - 2)), z3 >= 0, z' = 1, z'' - 2 >= 0, z = 0 function(z, z', z'', z3) -{ 0 }-> z3 :|: z'' >= 0, z' >= 0, z = 4, z3 >= 0 function(z, z', z'', z3) -{ 0 }-> 2 :|: z3 >= 0, z = 1, z'' >= 0, z' = 0 function(z, z', z'', z3) -{ 0 }-> 1 :|: z'' >= 0, z = 1, z' - 1 >= 0, z3 >= 0 function(z, z', z'', z3) -{ 0 }-> 0 :|: z = 2, z3 >= 0, z'' >= 0, z' = 0 function(z, z', z'', z3) -{ 0 }-> 0 :|: z = 2, z3 >= 0, z' = 1 + 0, z'' >= 0 function(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0 function(z, z', z'', z3) -{ 0 }-> 1 + s2 :|: s2 >= 0, s2 <= z' - 2 + 2 + (1 + (z' - 2)), z = 2, z'' >= 0, z' - 2 >= 0, z3 >= 0 Function symbols to be analyzed: {FUNCTION} Previous analysis results are: function: runtime: O(1) [0], size: O(n^1) [2 + z' + z3] FUNCTION: runtime: ?, size: O(1) [0] ---------------------------------------- (53) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: FUNCTION after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (54) Obligation: Complexity RNTS consisting of the following rules: FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, z3, 0, 1 + z3) :|: z'' = 0, z3 >= 0, z' = 1, z = 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, z3, 0, 1 + z3) :|: z3 >= 0, z' = 1, z = 0, z'' = 1 + 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, z3, 0, 1 + z3) :|: z3 >= 0, z'' >= 0, z' = 1, z = 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, z3, 1 + s'', 1 + z3) :|: s'' >= 0, s'' <= z'' - 2 + 2 + (1 + (z'' - 2)), z3 >= 0, z'' - 2 >= 0, z' = 1, z = 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, 0, 0, 1 + z3) :|: z'' = 0, z3 >= 0, z' = 1, z = 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, 0, 0, 1 + z3) :|: z3 >= 0, z' = 1, z = 0, z'' = 1 + 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, 0, 0, 1 + z3) :|: z3 >= 0, z'' >= 0, z' = 1, z = 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, 0, 1 + s1, 1 + z3) :|: s1 >= 0, s1 <= z'' - 2 + 2 + (1 + (z'' - 2)), z3 >= 0, z' = 1, z'' - 2 >= 0, z = 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(2, 1 + (z' - 2), z' - 2, z' - 2) :|: z = 2, z'' >= 0, z' - 2 >= 0, z3 >= 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(0, 2, 0, z3) :|: z'' = 0, z = 3, z' >= 0, z3 >= 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(0, 1, 1 + (z'' - 1), z3) :|: z = 3, z'' - 1 >= 0, z' >= 0, z3 >= 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(0, 0, z'', z3) :|: z = 3, z'' >= 0, z' >= 0, z3 >= 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, z3, 0, 1 + z3) + FUNCTION(2, z'', z'', z3) :|: z3 >= 0, z'' >= 0, z' = 1, z = 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, z3, 0, 1 + z3) + FUNCTION(2, 0, 0, z3) :|: z'' = 0, z3 >= 0, z' = 1, z = 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, z3, 0, 1 + z3) + FUNCTION(2, 1 + 0, 1 + 0, z3) :|: z3 >= 0, z' = 1, z = 0, z'' = 1 + 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, z3, 1 + s, 1 + z3) + FUNCTION(2, 1 + (1 + (z'' - 2)), 1 + (1 + (z'' - 2)), z3) :|: s >= 0, s <= z'' - 2 + 2 + (1 + (z'' - 2)), z3 >= 0, z'' - 2 >= 0, z' = 1, z = 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, 0, 0, 1 + z3) + FUNCTION(2, z'', z'', z3) :|: z3 >= 0, z'' >= 0, z' = 1, z = 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, 0, 0, 1 + z3) + FUNCTION(2, 0, 0, z3) :|: z'' = 0, z3 >= 0, z' = 1, z = 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, 0, 0, 1 + z3) + FUNCTION(2, 1 + 0, 1 + 0, z3) :|: z3 >= 0, z' = 1, z = 0, z'' = 1 + 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, 0, 1 + s', 1 + z3) + FUNCTION(2, 1 + (1 + (z'' - 2)), 1 + (1 + (z'' - 2)), z3) :|: s' >= 0, s' <= z'' - 2 + 2 + (1 + (z'' - 2)), z3 >= 0, z' = 1, z'' - 2 >= 0, z = 0 function(z, z', z'', z3) -{ 0 }-> z3 :|: z'' >= 0, z' >= 0, z = 4, z3 >= 0 function(z, z', z'', z3) -{ 0 }-> 2 :|: z3 >= 0, z = 1, z'' >= 0, z' = 0 function(z, z', z'', z3) -{ 0 }-> 1 :|: z'' >= 0, z = 1, z' - 1 >= 0, z3 >= 0 function(z, z', z'', z3) -{ 0 }-> 0 :|: z = 2, z3 >= 0, z'' >= 0, z' = 0 function(z, z', z'', z3) -{ 0 }-> 0 :|: z = 2, z3 >= 0, z' = 1 + 0, z'' >= 0 function(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0 function(z, z', z'', z3) -{ 0 }-> 1 + s2 :|: s2 >= 0, s2 <= z' - 2 + 2 + (1 + (z' - 2)), z = 2, z'' >= 0, z' - 2 >= 0, z3 >= 0 Function symbols to be analyzed: {FUNCTION} Previous analysis results are: function: runtime: O(1) [0], size: O(n^1) [2 + z' + z3] FUNCTION: runtime: INF, size: O(1) [0] ---------------------------------------- (55) 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: function(v0, v1, v2, v3) -> null_function [0] FUNCTION(v0, v1, v2, v3) -> null_FUNCTION [0] And the following fresh constants: null_function, null_FUNCTION ---------------------------------------- (56) 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: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) [1] FUNCTION(if, false, z0, z1) -> c8(FUNCTION(plus, function(third, z0, z1, z1), function(p, z0, z0, z1), s(z1)), FUNCTION(p, z0, z0, z1)) [1] FUNCTION(plus, z0, z1, z2) -> c5(FUNCTION(if, function(iszero, z1, z1, z1), z1, z2)) [1] FUNCTION(if, false, z0, z1) -> c7(FUNCTION(plus, function(third, z0, z1, z1), function(p, z0, z0, z1), s(z1))) [1] function(third, z0, z1, z2) -> z2 [0] function(p, 0, z0, z1) -> 0 [0] function(p, s(0), z0, z1) -> 0 [0] function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) [0] function(iszero, 0, z0, z1) -> true [0] function(iszero, s(z0), z1, z2) -> false [0] function(v0, v1, v2, v3) -> null_function [0] FUNCTION(v0, v1, v2, v3) -> null_FUNCTION [0] The TRS has the following type information: FUNCTION :: p:if:plus:third:iszero -> s:false:0:true:null_function -> s:false:0:true:null_function -> s:false:0:true:null_function -> c4:c8:c5:c7:null_FUNCTION p :: p:if:plus:third:iszero s :: s:false:0:true:null_function -> s:false:0:true:null_function c4 :: c4:c8:c5:c7:null_FUNCTION -> c4:c8:c5:c7:null_FUNCTION if :: p:if:plus:third:iszero false :: s:false:0:true:null_function c8 :: c4:c8:c5:c7:null_FUNCTION -> c4:c8:c5:c7:null_FUNCTION -> c4:c8:c5:c7:null_FUNCTION plus :: p:if:plus:third:iszero function :: p:if:plus:third:iszero -> s:false:0:true:null_function -> s:false:0:true:null_function -> s:false:0:true:null_function -> s:false:0:true:null_function third :: p:if:plus:third:iszero c5 :: c4:c8:c5:c7:null_FUNCTION -> c4:c8:c5:c7:null_FUNCTION iszero :: p:if:plus:third:iszero c7 :: c4:c8:c5:c7:null_FUNCTION -> c4:c8:c5:c7:null_FUNCTION 0 :: s:false:0:true:null_function true :: s:false:0:true:null_function null_function :: s:false:0:true:null_function null_FUNCTION :: c4:c8:c5:c7:null_FUNCTION Rewrite Strategy: INNERMOST ---------------------------------------- (57) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: p => 2 if => 0 false => 1 plus => 3 third => 4 iszero => 1 0 => 0 true => 2 null_function => 0 null_FUNCTION => 0 ---------------------------------------- (58) Obligation: Complexity RNTS consisting of the following rules: FUNCTION(z, z', z'', z3) -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, z3 = v3, v2 >= 0, v3 >= 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, function(4, z0, z1, z1), function(2, z0, z0, z1), 1 + z1) :|: z1 >= 0, z'' = z0, z3 = z1, z0 >= 0, z' = 1, z = 0 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(2, 1 + z0, z0, z0) :|: z = 2, z1 >= 0, z0 >= 0, z3 = z2, z2 >= 0, z' = 1 + (1 + z0), z'' = z1 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(0, function(1, z1, z1, z1), z1, z2) :|: z = 3, z1 >= 0, z0 >= 0, z3 = z2, z' = z0, z2 >= 0, z'' = z1 FUNCTION(z, z', z'', z3) -{ 1 }-> 1 + FUNCTION(3, function(4, z0, z1, z1), function(2, z0, z0, z1), 1 + z1) + FUNCTION(2, z0, z0, z1) :|: z1 >= 0, z'' = z0, z3 = z1, z0 >= 0, z' = 1, z = 0 function(z, z', z'', z3) -{ 0 }-> z2 :|: z1 >= 0, z0 >= 0, z = 4, z3 = z2, z' = z0, z2 >= 0, z'' = z1 function(z, z', z'', z3) -{ 0 }-> 2 :|: z1 >= 0, z = 1, z'' = z0, z3 = z1, z0 >= 0, z' = 0 function(z, z', z'', z3) -{ 0 }-> 1 :|: z1 >= 0, z = 1, z0 >= 0, z' = 1 + z0, z3 = z2, z2 >= 0, z'' = z1 function(z, z', z'', z3) -{ 0 }-> 0 :|: z = 2, z1 >= 0, z'' = z0, z3 = z1, z0 >= 0, z' = 0 function(z, z', z'', z3) -{ 0 }-> 0 :|: z = 2, z1 >= 0, z'' = z0, z' = 1 + 0, z3 = z1, z0 >= 0 function(z, z', z'', z3) -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, z3 = v3, v2 >= 0, v3 >= 0 function(z, z', z'', z3) -{ 0 }-> 1 + function(2, 1 + z0, z0, z0) :|: z = 2, z1 >= 0, z0 >= 0, z3 = z2, z2 >= 0, z' = 1 + (1 + z0), z'' = z1 Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (59) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FUNCTION(if, false, z0, z1) -> c8(FUNCTION(plus, function(third, z0, z1, z1), function(p, z0, z0, z1), s(z1)), FUNCTION(p, z0, z0, z1)) by FUNCTION(if, false, 0, z1) -> c8(FUNCTION(plus, function(third, 0, z1, z1), 0, s(z1)), FUNCTION(p, 0, 0, z1)) FUNCTION(if, false, s(0), z1) -> c8(FUNCTION(plus, function(third, s(0), z1, z1), 0, s(z1)), FUNCTION(p, s(0), s(0), z1)) FUNCTION(if, false, s(s(z0)), z2) -> c8(FUNCTION(plus, function(third, s(s(z0)), z2, z2), s(function(p, s(z0), z0, z0)), s(z2)), FUNCTION(p, s(s(z0)), s(s(z0)), z2)) FUNCTION(if, false, z0, z1) -> c8(FUNCTION(plus, z1, function(p, z0, z0, z1), s(z1)), FUNCTION(p, z0, z0, z1)) ---------------------------------------- (60) Obligation: Complexity Dependency Tuples Problem Rules: function(third, z0, z1, z2) -> z2 function(p, 0, z0, z1) -> 0 function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) function(iszero, 0, z0, z1) -> true function(iszero, s(z0), z1, z2) -> false Tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, z0, z1, z2) -> c5(FUNCTION(if, function(iszero, z1, z1, z1), z1, z2)) FUNCTION(if, false, z0, z1) -> c7(FUNCTION(plus, function(third, z0, z1, z1), function(p, z0, z0, z1), s(z1))) FUNCTION(if, false, 0, z1) -> c8(FUNCTION(plus, function(third, 0, z1, z1), 0, s(z1)), FUNCTION(p, 0, 0, z1)) FUNCTION(if, false, s(0), z1) -> c8(FUNCTION(plus, function(third, s(0), z1, z1), 0, s(z1)), FUNCTION(p, s(0), s(0), z1)) FUNCTION(if, false, s(s(z0)), z2) -> c8(FUNCTION(plus, function(third, s(s(z0)), z2, z2), s(function(p, s(z0), z0, z0)), s(z2)), FUNCTION(p, s(s(z0)), s(s(z0)), z2)) FUNCTION(if, false, z0, z1) -> c8(FUNCTION(plus, z1, function(p, z0, z0, z1), s(z1)), FUNCTION(p, z0, z0, z1)) S tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, z0, z1, z2) -> c5(FUNCTION(if, function(iszero, z1, z1, z1), z1, z2)) FUNCTION(if, false, z0, z1) -> c7(FUNCTION(plus, function(third, z0, z1, z1), function(p, z0, z0, z1), s(z1))) FUNCTION(if, false, 0, z1) -> c8(FUNCTION(plus, function(third, 0, z1, z1), 0, s(z1)), FUNCTION(p, 0, 0, z1)) FUNCTION(if, false, s(0), z1) -> c8(FUNCTION(plus, function(third, s(0), z1, z1), 0, s(z1)), FUNCTION(p, s(0), s(0), z1)) FUNCTION(if, false, s(s(z0)), z2) -> c8(FUNCTION(plus, function(third, s(s(z0)), z2, z2), s(function(p, s(z0), z0, z0)), s(z2)), FUNCTION(p, s(s(z0)), s(s(z0)), z2)) FUNCTION(if, false, z0, z1) -> c8(FUNCTION(plus, z1, function(p, z0, z0, z1), s(z1)), FUNCTION(p, z0, z0, z1)) K tuples:none Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c4_1, c5_1, c7_1, c8_2 ---------------------------------------- (61) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (62) Obligation: Complexity Dependency Tuples Problem Rules: function(third, z0, z1, z2) -> z2 function(p, 0, z0, z1) -> 0 function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) function(iszero, 0, z0, z1) -> true function(iszero, s(z0), z1, z2) -> false Tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, z0, z1, z2) -> c5(FUNCTION(if, function(iszero, z1, z1, z1), z1, z2)) FUNCTION(if, false, z0, z1) -> c7(FUNCTION(plus, function(third, z0, z1, z1), function(p, z0, z0, z1), s(z1))) FUNCTION(if, false, s(s(z0)), z2) -> c8(FUNCTION(plus, function(third, s(s(z0)), z2, z2), s(function(p, s(z0), z0, z0)), s(z2)), FUNCTION(p, s(s(z0)), s(s(z0)), z2)) FUNCTION(if, false, z0, z1) -> c8(FUNCTION(plus, z1, function(p, z0, z0, z1), s(z1)), FUNCTION(p, z0, z0, z1)) FUNCTION(if, false, 0, z1) -> c8(FUNCTION(plus, function(third, 0, z1, z1), 0, s(z1))) FUNCTION(if, false, s(0), z1) -> c8(FUNCTION(plus, function(third, s(0), z1, z1), 0, s(z1))) S tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, z0, z1, z2) -> c5(FUNCTION(if, function(iszero, z1, z1, z1), z1, z2)) FUNCTION(if, false, z0, z1) -> c7(FUNCTION(plus, function(third, z0, z1, z1), function(p, z0, z0, z1), s(z1))) FUNCTION(if, false, s(s(z0)), z2) -> c8(FUNCTION(plus, function(third, s(s(z0)), z2, z2), s(function(p, s(z0), z0, z0)), s(z2)), FUNCTION(p, s(s(z0)), s(s(z0)), z2)) FUNCTION(if, false, z0, z1) -> c8(FUNCTION(plus, z1, function(p, z0, z0, z1), s(z1)), FUNCTION(p, z0, z0, z1)) FUNCTION(if, false, 0, z1) -> c8(FUNCTION(plus, function(third, 0, z1, z1), 0, s(z1))) FUNCTION(if, false, s(0), z1) -> c8(FUNCTION(plus, function(third, s(0), z1, z1), 0, s(z1))) K tuples:none Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c4_1, c5_1, c7_1, c8_2, c8_1 ---------------------------------------- (63) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. FUNCTION(if, false, s(0), z1) -> c8(FUNCTION(plus, function(third, s(0), z1, z1), 0, s(z1))) We considered the (Usable) Rules: function(p, 0, z0, z1) -> 0 function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) And the Tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, z0, z1, z2) -> c5(FUNCTION(if, function(iszero, z1, z1, z1), z1, z2)) FUNCTION(if, false, z0, z1) -> c7(FUNCTION(plus, function(third, z0, z1, z1), function(p, z0, z0, z1), s(z1))) FUNCTION(if, false, s(s(z0)), z2) -> c8(FUNCTION(plus, function(third, s(s(z0)), z2, z2), s(function(p, s(z0), z0, z0)), s(z2)), FUNCTION(p, s(s(z0)), s(s(z0)), z2)) FUNCTION(if, false, z0, z1) -> c8(FUNCTION(plus, z1, function(p, z0, z0, z1), s(z1)), FUNCTION(p, z0, z0, z1)) FUNCTION(if, false, 0, z1) -> c8(FUNCTION(plus, function(third, 0, z1, z1), 0, s(z1))) FUNCTION(if, false, s(0), z1) -> c8(FUNCTION(plus, function(third, s(0), z1, z1), 0, s(z1))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(FUNCTION(x_1, x_2, x_3, x_4)) = [2]x_1*x_3 POL(c4(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c8(x_1, x_2)) = x_1 + x_2 POL(false) = 0 POL(function(x_1, x_2, x_3, x_4)) = x_2 POL(if) = [2] POL(iszero) = 0 POL(p) = 0 POL(plus) = [2] POL(s(x_1)) = [1] POL(third) = 0 POL(true) = [1] ---------------------------------------- (64) Obligation: Complexity Dependency Tuples Problem Rules: function(third, z0, z1, z2) -> z2 function(p, 0, z0, z1) -> 0 function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) function(iszero, 0, z0, z1) -> true function(iszero, s(z0), z1, z2) -> false Tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, z0, z1, z2) -> c5(FUNCTION(if, function(iszero, z1, z1, z1), z1, z2)) FUNCTION(if, false, z0, z1) -> c7(FUNCTION(plus, function(third, z0, z1, z1), function(p, z0, z0, z1), s(z1))) FUNCTION(if, false, s(s(z0)), z2) -> c8(FUNCTION(plus, function(third, s(s(z0)), z2, z2), s(function(p, s(z0), z0, z0)), s(z2)), FUNCTION(p, s(s(z0)), s(s(z0)), z2)) FUNCTION(if, false, z0, z1) -> c8(FUNCTION(plus, z1, function(p, z0, z0, z1), s(z1)), FUNCTION(p, z0, z0, z1)) FUNCTION(if, false, 0, z1) -> c8(FUNCTION(plus, function(third, 0, z1, z1), 0, s(z1))) FUNCTION(if, false, s(0), z1) -> c8(FUNCTION(plus, function(third, s(0), z1, z1), 0, s(z1))) S tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, z0, z1, z2) -> c5(FUNCTION(if, function(iszero, z1, z1, z1), z1, z2)) FUNCTION(if, false, z0, z1) -> c7(FUNCTION(plus, function(third, z0, z1, z1), function(p, z0, z0, z1), s(z1))) FUNCTION(if, false, s(s(z0)), z2) -> c8(FUNCTION(plus, function(third, s(s(z0)), z2, z2), s(function(p, s(z0), z0, z0)), s(z2)), FUNCTION(p, s(s(z0)), s(s(z0)), z2)) FUNCTION(if, false, z0, z1) -> c8(FUNCTION(plus, z1, function(p, z0, z0, z1), s(z1)), FUNCTION(p, z0, z0, z1)) FUNCTION(if, false, 0, z1) -> c8(FUNCTION(plus, function(third, 0, z1, z1), 0, s(z1))) K tuples: FUNCTION(if, false, s(0), z1) -> c8(FUNCTION(plus, function(third, s(0), z1, z1), 0, s(z1))) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c4_1, c5_1, c7_1, c8_2, c8_1 ---------------------------------------- (65) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FUNCTION(plus, z0, z1, z2) -> c5(FUNCTION(if, function(iszero, z1, z1, z1), z1, z2)) by FUNCTION(plus, x0, 0, x2) -> c5(FUNCTION(if, true, 0, x2)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) ---------------------------------------- (66) Obligation: Complexity Dependency Tuples Problem Rules: function(third, z0, z1, z2) -> z2 function(p, 0, z0, z1) -> 0 function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) function(iszero, 0, z0, z1) -> true function(iszero, s(z0), z1, z2) -> false Tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(if, false, z0, z1) -> c7(FUNCTION(plus, function(third, z0, z1, z1), function(p, z0, z0, z1), s(z1))) FUNCTION(if, false, s(s(z0)), z2) -> c8(FUNCTION(plus, function(third, s(s(z0)), z2, z2), s(function(p, s(z0), z0, z0)), s(z2)), FUNCTION(p, s(s(z0)), s(s(z0)), z2)) FUNCTION(if, false, z0, z1) -> c8(FUNCTION(plus, z1, function(p, z0, z0, z1), s(z1)), FUNCTION(p, z0, z0, z1)) FUNCTION(if, false, 0, z1) -> c8(FUNCTION(plus, function(third, 0, z1, z1), 0, s(z1))) FUNCTION(if, false, s(0), z1) -> c8(FUNCTION(plus, function(third, s(0), z1, z1), 0, s(z1))) FUNCTION(plus, x0, 0, x2) -> c5(FUNCTION(if, true, 0, x2)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) S tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(if, false, z0, z1) -> c7(FUNCTION(plus, function(third, z0, z1, z1), function(p, z0, z0, z1), s(z1))) FUNCTION(if, false, s(s(z0)), z2) -> c8(FUNCTION(plus, function(third, s(s(z0)), z2, z2), s(function(p, s(z0), z0, z0)), s(z2)), FUNCTION(p, s(s(z0)), s(s(z0)), z2)) FUNCTION(if, false, z0, z1) -> c8(FUNCTION(plus, z1, function(p, z0, z0, z1), s(z1)), FUNCTION(p, z0, z0, z1)) FUNCTION(if, false, 0, z1) -> c8(FUNCTION(plus, function(third, 0, z1, z1), 0, s(z1))) FUNCTION(plus, x0, 0, x2) -> c5(FUNCTION(if, true, 0, x2)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) K tuples: FUNCTION(if, false, s(0), z1) -> c8(FUNCTION(plus, function(third, s(0), z1, z1), 0, s(z1))) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c4_1, c7_1, c8_2, c8_1, c5_1 ---------------------------------------- (67) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing nodes: FUNCTION(if, false, 0, z1) -> c8(FUNCTION(plus, function(third, 0, z1, z1), 0, s(z1))) FUNCTION(if, false, s(0), z1) -> c8(FUNCTION(plus, function(third, s(0), z1, z1), 0, s(z1))) FUNCTION(plus, x0, 0, x2) -> c5(FUNCTION(if, true, 0, x2)) ---------------------------------------- (68) Obligation: Complexity Dependency Tuples Problem Rules: function(third, z0, z1, z2) -> z2 function(p, 0, z0, z1) -> 0 function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) function(iszero, 0, z0, z1) -> true function(iszero, s(z0), z1, z2) -> false Tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(if, false, z0, z1) -> c7(FUNCTION(plus, function(third, z0, z1, z1), function(p, z0, z0, z1), s(z1))) FUNCTION(if, false, s(s(z0)), z2) -> c8(FUNCTION(plus, function(third, s(s(z0)), z2, z2), s(function(p, s(z0), z0, z0)), s(z2)), FUNCTION(p, s(s(z0)), s(s(z0)), z2)) FUNCTION(if, false, z0, z1) -> c8(FUNCTION(plus, z1, function(p, z0, z0, z1), s(z1)), FUNCTION(p, z0, z0, z1)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) S tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(if, false, z0, z1) -> c7(FUNCTION(plus, function(third, z0, z1, z1), function(p, z0, z0, z1), s(z1))) FUNCTION(if, false, s(s(z0)), z2) -> c8(FUNCTION(plus, function(third, s(s(z0)), z2, z2), s(function(p, s(z0), z0, z0)), s(z2)), FUNCTION(p, s(s(z0)), s(s(z0)), z2)) FUNCTION(if, false, z0, z1) -> c8(FUNCTION(plus, z1, function(p, z0, z0, z1), s(z1)), FUNCTION(p, z0, z0, z1)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) K tuples:none Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c4_1, c7_1, c8_2, c5_1 ---------------------------------------- (69) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: function(iszero, 0, z0, z1) -> true function(iszero, s(z0), z1, z2) -> false ---------------------------------------- (70) Obligation: Complexity Dependency Tuples Problem Rules: function(third, z0, z1, z2) -> z2 function(p, 0, z0, z1) -> 0 function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(if, false, z0, z1) -> c7(FUNCTION(plus, function(third, z0, z1, z1), function(p, z0, z0, z1), s(z1))) FUNCTION(if, false, s(s(z0)), z2) -> c8(FUNCTION(plus, function(third, s(s(z0)), z2, z2), s(function(p, s(z0), z0, z0)), s(z2)), FUNCTION(p, s(s(z0)), s(s(z0)), z2)) FUNCTION(if, false, z0, z1) -> c8(FUNCTION(plus, z1, function(p, z0, z0, z1), s(z1)), FUNCTION(p, z0, z0, z1)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) S tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(if, false, z0, z1) -> c7(FUNCTION(plus, function(third, z0, z1, z1), function(p, z0, z0, z1), s(z1))) FUNCTION(if, false, s(s(z0)), z2) -> c8(FUNCTION(plus, function(third, s(s(z0)), z2, z2), s(function(p, s(z0), z0, z0)), s(z2)), FUNCTION(p, s(s(z0)), s(s(z0)), z2)) FUNCTION(if, false, z0, z1) -> c8(FUNCTION(plus, z1, function(p, z0, z0, z1), s(z1)), FUNCTION(p, z0, z0, z1)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) K tuples:none Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c4_1, c7_1, c8_2, c5_1 ---------------------------------------- (71) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FUNCTION(if, false, z0, z1) -> c7(FUNCTION(plus, function(third, z0, z1, z1), function(p, z0, z0, z1), s(z1))) by FUNCTION(if, false, 0, z1) -> c7(FUNCTION(plus, function(third, 0, z1, z1), 0, s(z1))) FUNCTION(if, false, s(0), z1) -> c7(FUNCTION(plus, function(third, s(0), z1, z1), 0, s(z1))) FUNCTION(if, false, s(s(z0)), z2) -> c7(FUNCTION(plus, function(third, s(s(z0)), z2, z2), s(function(p, s(z0), z0, z0)), s(z2))) FUNCTION(if, false, z0, z1) -> c7(FUNCTION(plus, z1, function(p, z0, z0, z1), s(z1))) ---------------------------------------- (72) Obligation: Complexity Dependency Tuples Problem Rules: function(third, z0, z1, z2) -> z2 function(p, 0, z0, z1) -> 0 function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(if, false, s(s(z0)), z2) -> c8(FUNCTION(plus, function(third, s(s(z0)), z2, z2), s(function(p, s(z0), z0, z0)), s(z2)), FUNCTION(p, s(s(z0)), s(s(z0)), z2)) FUNCTION(if, false, z0, z1) -> c8(FUNCTION(plus, z1, function(p, z0, z0, z1), s(z1)), FUNCTION(p, z0, z0, z1)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, 0, z1) -> c7(FUNCTION(plus, function(third, 0, z1, z1), 0, s(z1))) FUNCTION(if, false, s(0), z1) -> c7(FUNCTION(plus, function(third, s(0), z1, z1), 0, s(z1))) FUNCTION(if, false, s(s(z0)), z2) -> c7(FUNCTION(plus, function(third, s(s(z0)), z2, z2), s(function(p, s(z0), z0, z0)), s(z2))) FUNCTION(if, false, z0, z1) -> c7(FUNCTION(plus, z1, function(p, z0, z0, z1), s(z1))) S tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(if, false, s(s(z0)), z2) -> c8(FUNCTION(plus, function(third, s(s(z0)), z2, z2), s(function(p, s(z0), z0, z0)), s(z2)), FUNCTION(p, s(s(z0)), s(s(z0)), z2)) FUNCTION(if, false, z0, z1) -> c8(FUNCTION(plus, z1, function(p, z0, z0, z1), s(z1)), FUNCTION(p, z0, z0, z1)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, 0, z1) -> c7(FUNCTION(plus, function(third, 0, z1, z1), 0, s(z1))) FUNCTION(if, false, s(0), z1) -> c7(FUNCTION(plus, function(third, s(0), z1, z1), 0, s(z1))) FUNCTION(if, false, s(s(z0)), z2) -> c7(FUNCTION(plus, function(third, s(s(z0)), z2, z2), s(function(p, s(z0), z0, z0)), s(z2))) FUNCTION(if, false, z0, z1) -> c7(FUNCTION(plus, z1, function(p, z0, z0, z1), s(z1))) K tuples:none Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c4_1, c8_2, c5_1, c7_1 ---------------------------------------- (73) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing nodes: FUNCTION(if, false, 0, z1) -> c7(FUNCTION(plus, function(third, 0, z1, z1), 0, s(z1))) FUNCTION(if, false, s(0), z1) -> c7(FUNCTION(plus, function(third, s(0), z1, z1), 0, s(z1))) ---------------------------------------- (74) Obligation: Complexity Dependency Tuples Problem Rules: function(third, z0, z1, z2) -> z2 function(p, 0, z0, z1) -> 0 function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(if, false, s(s(z0)), z2) -> c8(FUNCTION(plus, function(third, s(s(z0)), z2, z2), s(function(p, s(z0), z0, z0)), s(z2)), FUNCTION(p, s(s(z0)), s(s(z0)), z2)) FUNCTION(if, false, z0, z1) -> c8(FUNCTION(plus, z1, function(p, z0, z0, z1), s(z1)), FUNCTION(p, z0, z0, z1)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(z0)), z2) -> c7(FUNCTION(plus, function(third, s(s(z0)), z2, z2), s(function(p, s(z0), z0, z0)), s(z2))) FUNCTION(if, false, z0, z1) -> c7(FUNCTION(plus, z1, function(p, z0, z0, z1), s(z1))) S tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(if, false, s(s(z0)), z2) -> c8(FUNCTION(plus, function(third, s(s(z0)), z2, z2), s(function(p, s(z0), z0, z0)), s(z2)), FUNCTION(p, s(s(z0)), s(s(z0)), z2)) FUNCTION(if, false, z0, z1) -> c8(FUNCTION(plus, z1, function(p, z0, z0, z1), s(z1)), FUNCTION(p, z0, z0, z1)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(z0)), z2) -> c7(FUNCTION(plus, function(third, s(s(z0)), z2, z2), s(function(p, s(z0), z0, z0)), s(z2))) FUNCTION(if, false, z0, z1) -> c7(FUNCTION(plus, z1, function(p, z0, z0, z1), s(z1))) K tuples:none Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c4_1, c8_2, c5_1, c7_1 ---------------------------------------- (75) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FUNCTION(if, false, s(s(z0)), z2) -> c8(FUNCTION(plus, function(third, s(s(z0)), z2, z2), s(function(p, s(z0), z0, z0)), s(z2)), FUNCTION(p, s(s(z0)), s(s(z0)), z2)) by FUNCTION(if, false, s(s(0)), x1) -> c8(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1)), FUNCTION(p, s(s(0)), s(s(0)), x1)) FUNCTION(if, false, s(s(s(z0))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(z0))), x1, x1), s(s(function(p, s(z0), z0, z0))), s(x1)), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), x1)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) ---------------------------------------- (76) Obligation: Complexity Dependency Tuples Problem Rules: function(third, z0, z1, z2) -> z2 function(p, 0, z0, z1) -> 0 function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(if, false, z0, z1) -> c8(FUNCTION(plus, z1, function(p, z0, z0, z1), s(z1)), FUNCTION(p, z0, z0, z1)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(z0)), z2) -> c7(FUNCTION(plus, function(third, s(s(z0)), z2, z2), s(function(p, s(z0), z0, z0)), s(z2))) FUNCTION(if, false, z0, z1) -> c7(FUNCTION(plus, z1, function(p, z0, z0, z1), s(z1))) FUNCTION(if, false, s(s(0)), x1) -> c8(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1)), FUNCTION(p, s(s(0)), s(s(0)), x1)) FUNCTION(if, false, s(s(s(z0))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(z0))), x1, x1), s(s(function(p, s(z0), z0, z0))), s(x1)), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), x1)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) S tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(if, false, z0, z1) -> c8(FUNCTION(plus, z1, function(p, z0, z0, z1), s(z1)), FUNCTION(p, z0, z0, z1)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(z0)), z2) -> c7(FUNCTION(plus, function(third, s(s(z0)), z2, z2), s(function(p, s(z0), z0, z0)), s(z2))) FUNCTION(if, false, z0, z1) -> c7(FUNCTION(plus, z1, function(p, z0, z0, z1), s(z1))) FUNCTION(if, false, s(s(0)), x1) -> c8(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1)), FUNCTION(p, s(s(0)), s(s(0)), x1)) FUNCTION(if, false, s(s(s(z0))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(z0))), x1, x1), s(s(function(p, s(z0), z0, z0))), s(x1)), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), x1)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) K tuples:none Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c4_1, c8_2, c5_1, c7_1, c8_1 ---------------------------------------- (77) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) We considered the (Usable) Rules:none And the Tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(if, false, z0, z1) -> c8(FUNCTION(plus, z1, function(p, z0, z0, z1), s(z1)), FUNCTION(p, z0, z0, z1)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(z0)), z2) -> c7(FUNCTION(plus, function(third, s(s(z0)), z2, z2), s(function(p, s(z0), z0, z0)), s(z2))) FUNCTION(if, false, z0, z1) -> c7(FUNCTION(plus, z1, function(p, z0, z0, z1), s(z1))) FUNCTION(if, false, s(s(0)), x1) -> c8(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1)), FUNCTION(p, s(s(0)), s(s(0)), x1)) FUNCTION(if, false, s(s(s(z0))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(z0))), x1, x1), s(s(function(p, s(z0), z0, z0))), s(x1)), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), x1)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = [1] POL(FUNCTION(x_1, x_2, x_3, x_4)) = x_1 POL(c4(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c8(x_1, x_2)) = x_1 + x_2 POL(false) = 0 POL(function(x_1, x_2, x_3, x_4)) = [1] + x_1 + x_2 POL(if) = [1] POL(p) = 0 POL(plus) = [1] POL(s(x_1)) = [1] + x_1 POL(third) = 0 ---------------------------------------- (78) Obligation: Complexity Dependency Tuples Problem Rules: function(third, z0, z1, z2) -> z2 function(p, 0, z0, z1) -> 0 function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(if, false, z0, z1) -> c8(FUNCTION(plus, z1, function(p, z0, z0, z1), s(z1)), FUNCTION(p, z0, z0, z1)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(z0)), z2) -> c7(FUNCTION(plus, function(third, s(s(z0)), z2, z2), s(function(p, s(z0), z0, z0)), s(z2))) FUNCTION(if, false, z0, z1) -> c7(FUNCTION(plus, z1, function(p, z0, z0, z1), s(z1))) FUNCTION(if, false, s(s(0)), x1) -> c8(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1)), FUNCTION(p, s(s(0)), s(s(0)), x1)) FUNCTION(if, false, s(s(s(z0))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(z0))), x1, x1), s(s(function(p, s(z0), z0, z0))), s(x1)), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), x1)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) S tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(if, false, z0, z1) -> c8(FUNCTION(plus, z1, function(p, z0, z0, z1), s(z1)), FUNCTION(p, z0, z0, z1)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(z0)), z2) -> c7(FUNCTION(plus, function(third, s(s(z0)), z2, z2), s(function(p, s(z0), z0, z0)), s(z2))) FUNCTION(if, false, z0, z1) -> c7(FUNCTION(plus, z1, function(p, z0, z0, z1), s(z1))) FUNCTION(if, false, s(s(0)), x1) -> c8(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1)), FUNCTION(p, s(s(0)), s(s(0)), x1)) FUNCTION(if, false, s(s(s(z0))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(z0))), x1, x1), s(s(function(p, s(z0), z0, z0))), s(x1)), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), x1)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) K tuples: FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c4_1, c8_2, c5_1, c7_1, c8_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. FUNCTION(if, false, s(s(0)), x1) -> c8(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1)), FUNCTION(p, s(s(0)), s(s(0)), x1)) We considered the (Usable) Rules: function(p, 0, z0, z1) -> 0 function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) And the Tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(if, false, z0, z1) -> c8(FUNCTION(plus, z1, function(p, z0, z0, z1), s(z1)), FUNCTION(p, z0, z0, z1)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(z0)), z2) -> c7(FUNCTION(plus, function(third, s(s(z0)), z2, z2), s(function(p, s(z0), z0, z0)), s(z2))) FUNCTION(if, false, z0, z1) -> c7(FUNCTION(plus, z1, function(p, z0, z0, z1), s(z1))) FUNCTION(if, false, s(s(0)), x1) -> c8(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1)), FUNCTION(p, s(s(0)), s(s(0)), x1)) FUNCTION(if, false, s(s(s(z0))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(z0))), x1, x1), s(s(function(p, s(z0), z0, z0))), s(x1)), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), x1)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(FUNCTION(x_1, x_2, x_3, x_4)) = [2]x_1*x_3 POL(c4(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c8(x_1, x_2)) = x_1 + x_2 POL(false) = 0 POL(function(x_1, x_2, x_3, x_4)) = x_2 POL(if) = [1] POL(p) = 0 POL(plus) = [1] POL(s(x_1)) = [1] + x_1 POL(third) = 0 ---------------------------------------- (80) Obligation: Complexity Dependency Tuples Problem Rules: function(third, z0, z1, z2) -> z2 function(p, 0, z0, z1) -> 0 function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(if, false, z0, z1) -> c8(FUNCTION(plus, z1, function(p, z0, z0, z1), s(z1)), FUNCTION(p, z0, z0, z1)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(z0)), z2) -> c7(FUNCTION(plus, function(third, s(s(z0)), z2, z2), s(function(p, s(z0), z0, z0)), s(z2))) FUNCTION(if, false, z0, z1) -> c7(FUNCTION(plus, z1, function(p, z0, z0, z1), s(z1))) FUNCTION(if, false, s(s(0)), x1) -> c8(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1)), FUNCTION(p, s(s(0)), s(s(0)), x1)) FUNCTION(if, false, s(s(s(z0))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(z0))), x1, x1), s(s(function(p, s(z0), z0, z0))), s(x1)), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), x1)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) S tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(if, false, z0, z1) -> c8(FUNCTION(plus, z1, function(p, z0, z0, z1), s(z1)), FUNCTION(p, z0, z0, z1)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(z0)), z2) -> c7(FUNCTION(plus, function(third, s(s(z0)), z2, z2), s(function(p, s(z0), z0, z0)), s(z2))) FUNCTION(if, false, z0, z1) -> c7(FUNCTION(plus, z1, function(p, z0, z0, z1), s(z1))) FUNCTION(if, false, s(s(s(z0))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(z0))), x1, x1), s(s(function(p, s(z0), z0, z0))), s(x1)), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), x1)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) K tuples: FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(0)), x1) -> c8(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1)), FUNCTION(p, s(s(0)), s(s(0)), x1)) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c4_1, c8_2, c5_1, c7_1, c8_1 ---------------------------------------- (81) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FUNCTION(if, false, z0, z1) -> c8(FUNCTION(plus, z1, function(p, z0, z0, z1), s(z1)), FUNCTION(p, z0, z0, z1)) by FUNCTION(if, false, 0, z1) -> c8(FUNCTION(plus, z1, 0, s(z1)), FUNCTION(p, 0, 0, z1)) FUNCTION(if, false, s(0), z1) -> c8(FUNCTION(plus, z1, 0, s(z1)), FUNCTION(p, s(0), s(0), z1)) FUNCTION(if, false, s(s(z0)), z2) -> c8(FUNCTION(plus, z2, s(function(p, s(z0), z0, z0)), s(z2)), FUNCTION(p, s(s(z0)), s(s(z0)), z2)) ---------------------------------------- (82) Obligation: Complexity Dependency Tuples Problem Rules: function(third, z0, z1, z2) -> z2 function(p, 0, z0, z1) -> 0 function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(z0)), z2) -> c7(FUNCTION(plus, function(third, s(s(z0)), z2, z2), s(function(p, s(z0), z0, z0)), s(z2))) FUNCTION(if, false, z0, z1) -> c7(FUNCTION(plus, z1, function(p, z0, z0, z1), s(z1))) FUNCTION(if, false, s(s(0)), x1) -> c8(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1)), FUNCTION(p, s(s(0)), s(s(0)), x1)) FUNCTION(if, false, s(s(s(z0))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(z0))), x1, x1), s(s(function(p, s(z0), z0, z0))), s(x1)), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), x1)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, 0, z1) -> c8(FUNCTION(plus, z1, 0, s(z1)), FUNCTION(p, 0, 0, z1)) FUNCTION(if, false, s(0), z1) -> c8(FUNCTION(plus, z1, 0, s(z1)), FUNCTION(p, s(0), s(0), z1)) S tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(z0)), z2) -> c7(FUNCTION(plus, function(third, s(s(z0)), z2, z2), s(function(p, s(z0), z0, z0)), s(z2))) FUNCTION(if, false, z0, z1) -> c7(FUNCTION(plus, z1, function(p, z0, z0, z1), s(z1))) FUNCTION(if, false, s(s(s(z0))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(z0))), x1, x1), s(s(function(p, s(z0), z0, z0))), s(x1)), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), x1)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, 0, z1) -> c8(FUNCTION(plus, z1, 0, s(z1)), FUNCTION(p, 0, 0, z1)) FUNCTION(if, false, s(0), z1) -> c8(FUNCTION(plus, z1, 0, s(z1)), FUNCTION(p, s(0), s(0), z1)) K tuples: FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(0)), x1) -> c8(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1)), FUNCTION(p, s(s(0)), s(s(0)), x1)) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c4_1, c5_1, c7_1, c8_2, c8_1 ---------------------------------------- (83) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing nodes: FUNCTION(if, false, s(0), z1) -> c8(FUNCTION(plus, z1, 0, s(z1)), FUNCTION(p, s(0), s(0), z1)) FUNCTION(if, false, 0, z1) -> c8(FUNCTION(plus, z1, 0, s(z1)), FUNCTION(p, 0, 0, z1)) ---------------------------------------- (84) Obligation: Complexity Dependency Tuples Problem Rules: function(third, z0, z1, z2) -> z2 function(p, 0, z0, z1) -> 0 function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(z0)), z2) -> c7(FUNCTION(plus, function(third, s(s(z0)), z2, z2), s(function(p, s(z0), z0, z0)), s(z2))) FUNCTION(if, false, z0, z1) -> c7(FUNCTION(plus, z1, function(p, z0, z0, z1), s(z1))) FUNCTION(if, false, s(s(0)), x1) -> c8(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1)), FUNCTION(p, s(s(0)), s(s(0)), x1)) FUNCTION(if, false, s(s(s(z0))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(z0))), x1, x1), s(s(function(p, s(z0), z0, z0))), s(x1)), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), x1)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) S tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(z0)), z2) -> c7(FUNCTION(plus, function(third, s(s(z0)), z2, z2), s(function(p, s(z0), z0, z0)), s(z2))) FUNCTION(if, false, z0, z1) -> c7(FUNCTION(plus, z1, function(p, z0, z0, z1), s(z1))) FUNCTION(if, false, s(s(s(z0))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(z0))), x1, x1), s(s(function(p, s(z0), z0, z0))), s(x1)), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), x1)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) K tuples: FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(0)), x1) -> c8(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1)), FUNCTION(p, s(s(0)), s(s(0)), x1)) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c4_1, c5_1, c7_1, c8_2, c8_1 ---------------------------------------- (85) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FUNCTION(if, false, s(s(z0)), z2) -> c7(FUNCTION(plus, function(third, s(s(z0)), z2, z2), s(function(p, s(z0), z0, z0)), s(z2))) by FUNCTION(if, false, s(s(0)), x1) -> c7(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1))) FUNCTION(if, false, s(s(s(z0))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(z0))), x1, x1), s(s(function(p, s(z0), z0, z0))), s(x1))) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) ---------------------------------------- (86) Obligation: Complexity Dependency Tuples Problem Rules: function(third, z0, z1, z2) -> z2 function(p, 0, z0, z1) -> 0 function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, z0, z1) -> c7(FUNCTION(plus, z1, function(p, z0, z0, z1), s(z1))) FUNCTION(if, false, s(s(0)), x1) -> c8(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1)), FUNCTION(p, s(s(0)), s(s(0)), x1)) FUNCTION(if, false, s(s(s(z0))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(z0))), x1, x1), s(s(function(p, s(z0), z0, z0))), s(x1)), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), x1)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(0)), x1) -> c7(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1))) FUNCTION(if, false, s(s(s(z0))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(z0))), x1, x1), s(s(function(p, s(z0), z0, z0))), s(x1))) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) S tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, z0, z1) -> c7(FUNCTION(plus, z1, function(p, z0, z0, z1), s(z1))) FUNCTION(if, false, s(s(s(z0))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(z0))), x1, x1), s(s(function(p, s(z0), z0, z0))), s(x1)), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), x1)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(0)), x1) -> c7(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1))) FUNCTION(if, false, s(s(s(z0))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(z0))), x1, x1), s(s(function(p, s(z0), z0, z0))), s(x1))) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) K tuples: FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(0)), x1) -> c8(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1)), FUNCTION(p, s(s(0)), s(s(0)), x1)) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c4_1, c5_1, c7_1, c8_2, c8_1 ---------------------------------------- (87) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. FUNCTION(if, false, s(s(0)), x1) -> c7(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1))) We considered the (Usable) Rules: function(p, 0, z0, z1) -> 0 function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) And the Tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, z0, z1) -> c7(FUNCTION(plus, z1, function(p, z0, z0, z1), s(z1))) FUNCTION(if, false, s(s(0)), x1) -> c8(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1)), FUNCTION(p, s(s(0)), s(s(0)), x1)) FUNCTION(if, false, s(s(s(z0))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(z0))), x1, x1), s(s(function(p, s(z0), z0, z0))), s(x1)), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), x1)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(0)), x1) -> c7(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1))) FUNCTION(if, false, s(s(s(z0))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(z0))), x1, x1), s(s(function(p, s(z0), z0, z0))), s(x1))) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = [2] POL(FUNCTION(x_1, x_2, x_3, x_4)) = [2]x_1*x_3 POL(c4(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c8(x_1, x_2)) = x_1 + x_2 POL(false) = 0 POL(function(x_1, x_2, x_3, x_4)) = x_2 POL(if) = [2] POL(p) = 0 POL(plus) = [2] POL(s(x_1)) = [2] + x_1 POL(third) = 0 ---------------------------------------- (88) Obligation: Complexity Dependency Tuples Problem Rules: function(third, z0, z1, z2) -> z2 function(p, 0, z0, z1) -> 0 function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, z0, z1) -> c7(FUNCTION(plus, z1, function(p, z0, z0, z1), s(z1))) FUNCTION(if, false, s(s(0)), x1) -> c8(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1)), FUNCTION(p, s(s(0)), s(s(0)), x1)) FUNCTION(if, false, s(s(s(z0))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(z0))), x1, x1), s(s(function(p, s(z0), z0, z0))), s(x1)), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), x1)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(0)), x1) -> c7(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1))) FUNCTION(if, false, s(s(s(z0))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(z0))), x1, x1), s(s(function(p, s(z0), z0, z0))), s(x1))) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) S tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, z0, z1) -> c7(FUNCTION(plus, z1, function(p, z0, z0, z1), s(z1))) FUNCTION(if, false, s(s(s(z0))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(z0))), x1, x1), s(s(function(p, s(z0), z0, z0))), s(x1)), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), x1)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(s(z0))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(z0))), x1, x1), s(s(function(p, s(z0), z0, z0))), s(x1))) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) K tuples: FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(0)), x1) -> c8(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1)), FUNCTION(p, s(s(0)), s(s(0)), x1)) FUNCTION(if, false, s(s(0)), x1) -> c7(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1))) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c4_1, c5_1, c7_1, c8_2, c8_1 ---------------------------------------- (89) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FUNCTION(if, false, z0, z1) -> c7(FUNCTION(plus, z1, function(p, z0, z0, z1), s(z1))) by FUNCTION(if, false, 0, z1) -> c7(FUNCTION(plus, z1, 0, s(z1))) FUNCTION(if, false, s(0), z1) -> c7(FUNCTION(plus, z1, 0, s(z1))) FUNCTION(if, false, s(s(z0)), z2) -> c7(FUNCTION(plus, z2, s(function(p, s(z0), z0, z0)), s(z2))) ---------------------------------------- (90) Obligation: Complexity Dependency Tuples Problem Rules: function(third, z0, z1, z2) -> z2 function(p, 0, z0, z1) -> 0 function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(0)), x1) -> c8(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1)), FUNCTION(p, s(s(0)), s(s(0)), x1)) FUNCTION(if, false, s(s(s(z0))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(z0))), x1, x1), s(s(function(p, s(z0), z0, z0))), s(x1)), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), x1)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(0)), x1) -> c7(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1))) FUNCTION(if, false, s(s(s(z0))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(z0))), x1, x1), s(s(function(p, s(z0), z0, z0))), s(x1))) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, 0, z1) -> c7(FUNCTION(plus, z1, 0, s(z1))) FUNCTION(if, false, s(0), z1) -> c7(FUNCTION(plus, z1, 0, s(z1))) S tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(s(z0))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(z0))), x1, x1), s(s(function(p, s(z0), z0, z0))), s(x1)), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), x1)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(s(z0))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(z0))), x1, x1), s(s(function(p, s(z0), z0, z0))), s(x1))) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, 0, z1) -> c7(FUNCTION(plus, z1, 0, s(z1))) FUNCTION(if, false, s(0), z1) -> c7(FUNCTION(plus, z1, 0, s(z1))) K tuples: FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(0)), x1) -> c8(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1)), FUNCTION(p, s(s(0)), s(s(0)), x1)) FUNCTION(if, false, s(s(0)), x1) -> c7(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1))) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c4_1, c5_1, c8_2, c8_1, c7_1 ---------------------------------------- (91) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing nodes: FUNCTION(if, false, s(0), z1) -> c7(FUNCTION(plus, z1, 0, s(z1))) FUNCTION(if, false, 0, z1) -> c7(FUNCTION(plus, z1, 0, s(z1))) ---------------------------------------- (92) Obligation: Complexity Dependency Tuples Problem Rules: function(third, z0, z1, z2) -> z2 function(p, 0, z0, z1) -> 0 function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(0)), x1) -> c8(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1)), FUNCTION(p, s(s(0)), s(s(0)), x1)) FUNCTION(if, false, s(s(s(z0))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(z0))), x1, x1), s(s(function(p, s(z0), z0, z0))), s(x1)), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), x1)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(0)), x1) -> c7(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1))) FUNCTION(if, false, s(s(s(z0))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(z0))), x1, x1), s(s(function(p, s(z0), z0, z0))), s(x1))) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) S tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(s(z0))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(z0))), x1, x1), s(s(function(p, s(z0), z0, z0))), s(x1)), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), x1)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(s(z0))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(z0))), x1, x1), s(s(function(p, s(z0), z0, z0))), s(x1))) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) K tuples: FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(0)), x1) -> c8(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1)), FUNCTION(p, s(s(0)), s(s(0)), x1)) FUNCTION(if, false, s(s(0)), x1) -> c7(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1))) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c4_1, c5_1, c8_2, c8_1, c7_1 ---------------------------------------- (93) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: function(p, 0, z0, z1) -> 0 ---------------------------------------- (94) Obligation: Complexity Dependency Tuples Problem Rules: function(third, z0, z1, z2) -> z2 function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(0)), x1) -> c8(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1)), FUNCTION(p, s(s(0)), s(s(0)), x1)) FUNCTION(if, false, s(s(s(z0))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(z0))), x1, x1), s(s(function(p, s(z0), z0, z0))), s(x1)), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), x1)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(0)), x1) -> c7(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1))) FUNCTION(if, false, s(s(s(z0))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(z0))), x1, x1), s(s(function(p, s(z0), z0, z0))), s(x1))) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) S tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(s(z0))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(z0))), x1, x1), s(s(function(p, s(z0), z0, z0))), s(x1)), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), x1)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(s(z0))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(z0))), x1, x1), s(s(function(p, s(z0), z0, z0))), s(x1))) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) K tuples: FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(0)), x1) -> c8(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1)), FUNCTION(p, s(s(0)), s(s(0)), x1)) FUNCTION(if, false, s(s(0)), x1) -> c7(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1))) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c4_1, c5_1, c8_2, c8_1, c7_1 ---------------------------------------- (95) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FUNCTION(if, false, s(s(0)), x1) -> c8(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1)), FUNCTION(p, s(s(0)), s(s(0)), x1)) by FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1)), FUNCTION(p, s(s(0)), s(s(0)), z1)) FUNCTION(if, false, s(s(0)), x0) -> c8(FUNCTION(p, s(s(0)), s(s(0)), x0)) ---------------------------------------- (96) Obligation: Complexity Dependency Tuples Problem Rules: function(third, z0, z1, z2) -> z2 function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(s(z0))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(z0))), x1, x1), s(s(function(p, s(z0), z0, z0))), s(x1)), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), x1)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(0)), x1) -> c7(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1))) FUNCTION(if, false, s(s(s(z0))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(z0))), x1, x1), s(s(function(p, s(z0), z0, z0))), s(x1))) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1)), FUNCTION(p, s(s(0)), s(s(0)), z1)) FUNCTION(if, false, s(s(0)), x0) -> c8(FUNCTION(p, s(s(0)), s(s(0)), x0)) S tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(s(z0))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(z0))), x1, x1), s(s(function(p, s(z0), z0, z0))), s(x1)), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), x1)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(s(z0))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(z0))), x1, x1), s(s(function(p, s(z0), z0, z0))), s(x1))) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) K tuples: FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(0)), x1) -> c8(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1)), FUNCTION(p, s(s(0)), s(s(0)), x1)) FUNCTION(if, false, s(s(0)), x1) -> c7(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1))) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c4_1, c5_1, c8_2, c8_1, c7_1 ---------------------------------------- (97) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FUNCTION(if, false, s(s(s(z0))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(z0))), x1, x1), s(s(function(p, s(z0), z0, z0))), s(x1)), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), x1)) by FUNCTION(if, false, s(s(s(0))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(0))), x1, x1), s(s(0)), s(x1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), x1)) FUNCTION(if, false, s(s(s(s(z0)))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(z0)))), x1, x1), s(s(s(function(p, s(z0), z0, z0)))), s(x1)), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), x1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) ---------------------------------------- (98) Obligation: Complexity Dependency Tuples Problem Rules: function(third, z0, z1, z2) -> z2 function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(0)), x1) -> c7(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1))) FUNCTION(if, false, s(s(s(z0))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(z0))), x1, x1), s(s(function(p, s(z0), z0, z0))), s(x1))) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1)), FUNCTION(p, s(s(0)), s(s(0)), z1)) FUNCTION(if, false, s(s(0)), x0) -> c8(FUNCTION(p, s(s(0)), s(s(0)), x0)) FUNCTION(if, false, s(s(s(0))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(0))), x1, x1), s(s(0)), s(x1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), x1)) FUNCTION(if, false, s(s(s(s(z0)))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(z0)))), x1, x1), s(s(s(function(p, s(z0), z0, z0)))), s(x1)), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), x1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) S tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(s(z0))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(z0))), x1, x1), s(s(function(p, s(z0), z0, z0))), s(x1))) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(0))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(0))), x1, x1), s(s(0)), s(x1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), x1)) FUNCTION(if, false, s(s(s(s(z0)))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(z0)))), x1, x1), s(s(s(function(p, s(z0), z0, z0)))), s(x1)), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), x1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) K tuples: FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(0)), x1) -> c8(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1)), FUNCTION(p, s(s(0)), s(s(0)), x1)) FUNCTION(if, false, s(s(0)), x1) -> c7(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1))) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c4_1, c5_1, c8_2, c8_1, c7_1 ---------------------------------------- (99) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) We considered the (Usable) Rules:none And the Tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(0)), x1) -> c7(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1))) FUNCTION(if, false, s(s(s(z0))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(z0))), x1, x1), s(s(function(p, s(z0), z0, z0))), s(x1))) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1)), FUNCTION(p, s(s(0)), s(s(0)), z1)) FUNCTION(if, false, s(s(0)), x0) -> c8(FUNCTION(p, s(s(0)), s(s(0)), x0)) FUNCTION(if, false, s(s(s(0))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(0))), x1, x1), s(s(0)), s(x1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), x1)) FUNCTION(if, false, s(s(s(s(z0)))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(z0)))), x1, x1), s(s(s(function(p, s(z0), z0, z0)))), s(x1)), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), x1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(FUNCTION(x_1, x_2, x_3, x_4)) = x_1 POL(c4(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c8(x_1, x_2)) = x_1 + x_2 POL(false) = 0 POL(function(x_1, x_2, x_3, x_4)) = [1] + [3]x_2 POL(if) = [1] POL(p) = 0 POL(plus) = [1] POL(s(x_1)) = [3] + x_1 POL(third) = 0 ---------------------------------------- (100) Obligation: Complexity Dependency Tuples Problem Rules: function(third, z0, z1, z2) -> z2 function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(0)), x1) -> c7(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1))) FUNCTION(if, false, s(s(s(z0))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(z0))), x1, x1), s(s(function(p, s(z0), z0, z0))), s(x1))) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1)), FUNCTION(p, s(s(0)), s(s(0)), z1)) FUNCTION(if, false, s(s(0)), x0) -> c8(FUNCTION(p, s(s(0)), s(s(0)), x0)) FUNCTION(if, false, s(s(s(0))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(0))), x1, x1), s(s(0)), s(x1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), x1)) FUNCTION(if, false, s(s(s(s(z0)))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(z0)))), x1, x1), s(s(s(function(p, s(z0), z0, z0)))), s(x1)), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), x1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) S tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(s(z0))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(z0))), x1, x1), s(s(function(p, s(z0), z0, z0))), s(x1))) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(0))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(0))), x1, x1), s(s(0)), s(x1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), x1)) FUNCTION(if, false, s(s(s(s(z0)))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(z0)))), x1, x1), s(s(s(function(p, s(z0), z0, z0)))), s(x1)), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), x1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) K tuples: FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(0)), x1) -> c8(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1)), FUNCTION(p, s(s(0)), s(s(0)), x1)) FUNCTION(if, false, s(s(0)), x1) -> c7(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1))) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c4_1, c5_1, c8_2, c8_1, c7_1 ---------------------------------------- (101) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. FUNCTION(if, false, s(s(s(0))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(0))), x1, x1), s(s(0)), s(x1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), x1)) We considered the (Usable) Rules: function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) And the Tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(0)), x1) -> c7(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1))) FUNCTION(if, false, s(s(s(z0))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(z0))), x1, x1), s(s(function(p, s(z0), z0, z0))), s(x1))) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1)), FUNCTION(p, s(s(0)), s(s(0)), z1)) FUNCTION(if, false, s(s(0)), x0) -> c8(FUNCTION(p, s(s(0)), s(s(0)), x0)) FUNCTION(if, false, s(s(s(0))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(0))), x1, x1), s(s(0)), s(x1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), x1)) FUNCTION(if, false, s(s(s(s(z0)))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(z0)))), x1, x1), s(s(s(function(p, s(z0), z0, z0)))), s(x1)), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), x1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(FUNCTION(x_1, x_2, x_3, x_4)) = [2]x_1*x_3 POL(c4(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c8(x_1, x_2)) = x_1 + x_2 POL(false) = 0 POL(function(x_1, x_2, x_3, x_4)) = x_2 POL(if) = [1] POL(p) = 0 POL(plus) = [1] POL(s(x_1)) = [2] + x_1 POL(third) = 0 ---------------------------------------- (102) Obligation: Complexity Dependency Tuples Problem Rules: function(third, z0, z1, z2) -> z2 function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(0)), x1) -> c7(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1))) FUNCTION(if, false, s(s(s(z0))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(z0))), x1, x1), s(s(function(p, s(z0), z0, z0))), s(x1))) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1)), FUNCTION(p, s(s(0)), s(s(0)), z1)) FUNCTION(if, false, s(s(0)), x0) -> c8(FUNCTION(p, s(s(0)), s(s(0)), x0)) FUNCTION(if, false, s(s(s(0))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(0))), x1, x1), s(s(0)), s(x1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), x1)) FUNCTION(if, false, s(s(s(s(z0)))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(z0)))), x1, x1), s(s(s(function(p, s(z0), z0, z0)))), s(x1)), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), x1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) S tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(s(z0))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(z0))), x1, x1), s(s(function(p, s(z0), z0, z0))), s(x1))) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(s(z0)))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(z0)))), x1, x1), s(s(s(function(p, s(z0), z0, z0)))), s(x1)), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), x1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) K tuples: FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(0)), x1) -> c8(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1)), FUNCTION(p, s(s(0)), s(s(0)), x1)) FUNCTION(if, false, s(s(0)), x1) -> c7(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1))) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(s(0))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(0))), x1, x1), s(s(0)), s(x1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), x1)) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c4_1, c5_1, c8_2, c8_1, c7_1 ---------------------------------------- (103) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FUNCTION(if, false, s(s(0)), x1) -> c7(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1))) by FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) ---------------------------------------- (104) Obligation: Complexity Dependency Tuples Problem Rules: function(third, z0, z1, z2) -> z2 function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(s(z0))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(z0))), x1, x1), s(s(function(p, s(z0), z0, z0))), s(x1))) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1)), FUNCTION(p, s(s(0)), s(s(0)), z1)) FUNCTION(if, false, s(s(0)), x0) -> c8(FUNCTION(p, s(s(0)), s(s(0)), x0)) FUNCTION(if, false, s(s(s(0))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(0))), x1, x1), s(s(0)), s(x1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), x1)) FUNCTION(if, false, s(s(s(s(z0)))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(z0)))), x1, x1), s(s(s(function(p, s(z0), z0, z0)))), s(x1)), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), x1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) S tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(s(z0))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(z0))), x1, x1), s(s(function(p, s(z0), z0, z0))), s(x1))) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(s(z0)))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(z0)))), x1, x1), s(s(s(function(p, s(z0), z0, z0)))), s(x1)), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), x1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) K tuples: FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(0)), x1) -> c8(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1)), FUNCTION(p, s(s(0)), s(s(0)), x1)) FUNCTION(if, false, s(s(0)), x1) -> c7(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1))) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(s(0))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(0))), x1, x1), s(s(0)), s(x1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), x1)) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c4_1, c5_1, c8_2, c8_1, c7_1 ---------------------------------------- (105) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FUNCTION(if, false, s(s(s(z0))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(z0))), x1, x1), s(s(function(p, s(z0), z0, z0))), s(x1))) by FUNCTION(if, false, s(s(s(0))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(0))), x1, x1), s(s(0)), s(x1))) FUNCTION(if, false, s(s(s(s(z0)))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(z0)))), x1, x1), s(s(s(function(p, s(z0), z0, z0)))), s(x1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) ---------------------------------------- (106) Obligation: Complexity Dependency Tuples Problem Rules: function(third, z0, z1, z2) -> z2 function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1)), FUNCTION(p, s(s(0)), s(s(0)), z1)) FUNCTION(if, false, s(s(0)), x0) -> c8(FUNCTION(p, s(s(0)), s(s(0)), x0)) FUNCTION(if, false, s(s(s(0))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(0))), x1, x1), s(s(0)), s(x1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), x1)) FUNCTION(if, false, s(s(s(s(z0)))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(z0)))), x1, x1), s(s(s(function(p, s(z0), z0, z0)))), s(x1)), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), x1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(0))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(0))), x1, x1), s(s(0)), s(x1))) FUNCTION(if, false, s(s(s(s(z0)))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(z0)))), x1, x1), s(s(s(function(p, s(z0), z0, z0)))), s(x1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) S tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(s(z0)))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(z0)))), x1, x1), s(s(s(function(p, s(z0), z0, z0)))), s(x1)), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), x1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(0))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(0))), x1, x1), s(s(0)), s(x1))) FUNCTION(if, false, s(s(s(s(z0)))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(z0)))), x1, x1), s(s(s(function(p, s(z0), z0, z0)))), s(x1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) K tuples: FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(0)), x1) -> c8(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1)), FUNCTION(p, s(s(0)), s(s(0)), x1)) FUNCTION(if, false, s(s(0)), x1) -> c7(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1))) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(s(0))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(0))), x1, x1), s(s(0)), s(x1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), x1)) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c4_1, c5_1, c8_2, c8_1, c7_1 ---------------------------------------- (107) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. FUNCTION(if, false, s(s(s(0))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(0))), x1, x1), s(s(0)), s(x1))) We considered the (Usable) Rules: function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) And the Tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1)), FUNCTION(p, s(s(0)), s(s(0)), z1)) FUNCTION(if, false, s(s(0)), x0) -> c8(FUNCTION(p, s(s(0)), s(s(0)), x0)) FUNCTION(if, false, s(s(s(0))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(0))), x1, x1), s(s(0)), s(x1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), x1)) FUNCTION(if, false, s(s(s(s(z0)))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(z0)))), x1, x1), s(s(s(function(p, s(z0), z0, z0)))), s(x1)), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), x1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(0))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(0))), x1, x1), s(s(0)), s(x1))) FUNCTION(if, false, s(s(s(s(z0)))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(z0)))), x1, x1), s(s(s(function(p, s(z0), z0, z0)))), s(x1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = [1] POL(FUNCTION(x_1, x_2, x_3, x_4)) = x_1*x_3 POL(c4(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c8(x_1, x_2)) = x_1 + x_2 POL(false) = 0 POL(function(x_1, x_2, x_3, x_4)) = x_2 POL(if) = [2] POL(p) = 0 POL(plus) = [2] POL(s(x_1)) = [1] + x_1 POL(third) = 0 ---------------------------------------- (108) Obligation: Complexity Dependency Tuples Problem Rules: function(third, z0, z1, z2) -> z2 function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1)), FUNCTION(p, s(s(0)), s(s(0)), z1)) FUNCTION(if, false, s(s(0)), x0) -> c8(FUNCTION(p, s(s(0)), s(s(0)), x0)) FUNCTION(if, false, s(s(s(0))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(0))), x1, x1), s(s(0)), s(x1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), x1)) FUNCTION(if, false, s(s(s(s(z0)))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(z0)))), x1, x1), s(s(s(function(p, s(z0), z0, z0)))), s(x1)), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), x1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(0))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(0))), x1, x1), s(s(0)), s(x1))) FUNCTION(if, false, s(s(s(s(z0)))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(z0)))), x1, x1), s(s(s(function(p, s(z0), z0, z0)))), s(x1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) S tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(s(z0)))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(z0)))), x1, x1), s(s(s(function(p, s(z0), z0, z0)))), s(x1)), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), x1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(s(z0)))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(z0)))), x1, x1), s(s(s(function(p, s(z0), z0, z0)))), s(x1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) K tuples: FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(0)), x1) -> c8(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1)), FUNCTION(p, s(s(0)), s(s(0)), x1)) FUNCTION(if, false, s(s(0)), x1) -> c7(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1))) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(s(0))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(0))), x1, x1), s(s(0)), s(x1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), x1)) FUNCTION(if, false, s(s(s(0))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(0))), x1, x1), s(s(0)), s(x1))) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c4_1, c5_1, c8_2, c8_1, c7_1 ---------------------------------------- (109) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FUNCTION(if, false, s(s(s(0))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(0))), x1, x1), s(s(0)), s(x1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), x1)) by FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), z1)) FUNCTION(if, false, s(s(s(0))), x0) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), x0)) ---------------------------------------- (110) Obligation: Complexity Dependency Tuples Problem Rules: function(third, z0, z1, z2) -> z2 function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1)), FUNCTION(p, s(s(0)), s(s(0)), z1)) FUNCTION(if, false, s(s(0)), x0) -> c8(FUNCTION(p, s(s(0)), s(s(0)), x0)) FUNCTION(if, false, s(s(s(s(z0)))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(z0)))), x1, x1), s(s(s(function(p, s(z0), z0, z0)))), s(x1)), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), x1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(0))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(0))), x1, x1), s(s(0)), s(x1))) FUNCTION(if, false, s(s(s(s(z0)))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(z0)))), x1, x1), s(s(s(function(p, s(z0), z0, z0)))), s(x1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), z1)) FUNCTION(if, false, s(s(s(0))), x0) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), x0)) S tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(s(z0)))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(z0)))), x1, x1), s(s(s(function(p, s(z0), z0, z0)))), s(x1)), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), x1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(s(z0)))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(z0)))), x1, x1), s(s(s(function(p, s(z0), z0, z0)))), s(x1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) K tuples: FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(0)), x1) -> c8(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1)), FUNCTION(p, s(s(0)), s(s(0)), x1)) FUNCTION(if, false, s(s(0)), x1) -> c7(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1))) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(s(0))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(0))), x1, x1), s(s(0)), s(x1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), x1)) FUNCTION(if, false, s(s(s(0))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(0))), x1, x1), s(s(0)), s(x1))) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c4_1, c5_1, c8_2, c8_1, c7_1 ---------------------------------------- (111) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FUNCTION(if, false, s(s(s(s(z0)))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(z0)))), x1, x1), s(s(s(function(p, s(z0), z0, z0)))), s(x1)), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), x1)) by FUNCTION(if, false, s(s(s(s(0)))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(0)))), x1, x1), s(s(s(0))), s(x1)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), x1)) FUNCTION(if, false, s(s(s(s(s(z0))))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(s(z0))))), x1, x1), s(s(s(s(function(p, s(z0), z0, z0))))), s(x1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), x1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) ---------------------------------------- (112) Obligation: Complexity Dependency Tuples Problem Rules: function(third, z0, z1, z2) -> z2 function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1)), FUNCTION(p, s(s(0)), s(s(0)), z1)) FUNCTION(if, false, s(s(0)), x0) -> c8(FUNCTION(p, s(s(0)), s(s(0)), x0)) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(0))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(0))), x1, x1), s(s(0)), s(x1))) FUNCTION(if, false, s(s(s(s(z0)))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(z0)))), x1, x1), s(s(s(function(p, s(z0), z0, z0)))), s(x1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), z1)) FUNCTION(if, false, s(s(s(0))), x0) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), x0)) FUNCTION(if, false, s(s(s(s(0)))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(0)))), x1, x1), s(s(s(0))), s(x1)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), x1)) FUNCTION(if, false, s(s(s(s(s(z0))))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(s(z0))))), x1, x1), s(s(s(s(function(p, s(z0), z0, z0))))), s(x1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), x1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) S tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(s(z0)))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(z0)))), x1, x1), s(s(s(function(p, s(z0), z0, z0)))), s(x1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(0)))), x1, x1), s(s(s(0))), s(x1)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), x1)) FUNCTION(if, false, s(s(s(s(s(z0))))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(s(z0))))), x1, x1), s(s(s(s(function(p, s(z0), z0, z0))))), s(x1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), x1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) K tuples: FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(0)), x1) -> c8(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1)), FUNCTION(p, s(s(0)), s(s(0)), x1)) FUNCTION(if, false, s(s(0)), x1) -> c7(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1))) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(s(0))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(0))), x1, x1), s(s(0)), s(x1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), x1)) FUNCTION(if, false, s(s(s(0))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(0))), x1, x1), s(s(0)), s(x1))) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c4_1, c5_1, c8_2, c8_1, c7_1 ---------------------------------------- (113) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) We considered the (Usable) Rules:none And the Tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1)), FUNCTION(p, s(s(0)), s(s(0)), z1)) FUNCTION(if, false, s(s(0)), x0) -> c8(FUNCTION(p, s(s(0)), s(s(0)), x0)) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(0))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(0))), x1, x1), s(s(0)), s(x1))) FUNCTION(if, false, s(s(s(s(z0)))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(z0)))), x1, x1), s(s(s(function(p, s(z0), z0, z0)))), s(x1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), z1)) FUNCTION(if, false, s(s(s(0))), x0) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), x0)) FUNCTION(if, false, s(s(s(s(0)))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(0)))), x1, x1), s(s(s(0))), s(x1)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), x1)) FUNCTION(if, false, s(s(s(s(s(z0))))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(s(z0))))), x1, x1), s(s(s(s(function(p, s(z0), z0, z0))))), s(x1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), x1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = [1] POL(FUNCTION(x_1, x_2, x_3, x_4)) = x_1 POL(c4(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c8(x_1, x_2)) = x_1 + x_2 POL(false) = 0 POL(function(x_1, x_2, x_3, x_4)) = [1] + x_1 + x_2 POL(if) = [1] POL(p) = 0 POL(plus) = [1] POL(s(x_1)) = [1] + x_1 POL(third) = 0 ---------------------------------------- (114) Obligation: Complexity Dependency Tuples Problem Rules: function(third, z0, z1, z2) -> z2 function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1)), FUNCTION(p, s(s(0)), s(s(0)), z1)) FUNCTION(if, false, s(s(0)), x0) -> c8(FUNCTION(p, s(s(0)), s(s(0)), x0)) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(0))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(0))), x1, x1), s(s(0)), s(x1))) FUNCTION(if, false, s(s(s(s(z0)))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(z0)))), x1, x1), s(s(s(function(p, s(z0), z0, z0)))), s(x1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), z1)) FUNCTION(if, false, s(s(s(0))), x0) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), x0)) FUNCTION(if, false, s(s(s(s(0)))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(0)))), x1, x1), s(s(s(0))), s(x1)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), x1)) FUNCTION(if, false, s(s(s(s(s(z0))))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(s(z0))))), x1, x1), s(s(s(s(function(p, s(z0), z0, z0))))), s(x1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), x1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) S tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(s(z0)))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(z0)))), x1, x1), s(s(s(function(p, s(z0), z0, z0)))), s(x1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(0)))), x1, x1), s(s(s(0))), s(x1)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), x1)) FUNCTION(if, false, s(s(s(s(s(z0))))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(s(z0))))), x1, x1), s(s(s(s(function(p, s(z0), z0, z0))))), s(x1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), x1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) K tuples: FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(0)), x1) -> c8(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1)), FUNCTION(p, s(s(0)), s(s(0)), x1)) FUNCTION(if, false, s(s(0)), x1) -> c7(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1))) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(s(0))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(0))), x1, x1), s(s(0)), s(x1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), x1)) FUNCTION(if, false, s(s(s(0))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(0))), x1, x1), s(s(0)), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c4_1, c5_1, c8_2, c8_1, c7_1 ---------------------------------------- (115) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. FUNCTION(if, false, s(s(s(s(0)))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(0)))), x1, x1), s(s(s(0))), s(x1)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), x1)) We considered the (Usable) Rules: function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) And the Tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1)), FUNCTION(p, s(s(0)), s(s(0)), z1)) FUNCTION(if, false, s(s(0)), x0) -> c8(FUNCTION(p, s(s(0)), s(s(0)), x0)) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(0))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(0))), x1, x1), s(s(0)), s(x1))) FUNCTION(if, false, s(s(s(s(z0)))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(z0)))), x1, x1), s(s(s(function(p, s(z0), z0, z0)))), s(x1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), z1)) FUNCTION(if, false, s(s(s(0))), x0) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), x0)) FUNCTION(if, false, s(s(s(s(0)))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(0)))), x1, x1), s(s(s(0))), s(x1)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), x1)) FUNCTION(if, false, s(s(s(s(s(z0))))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(s(z0))))), x1, x1), s(s(s(s(function(p, s(z0), z0, z0))))), s(x1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), x1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = [2] POL(FUNCTION(x_1, x_2, x_3, x_4)) = [2]x_1*x_3 POL(c4(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c8(x_1, x_2)) = x_1 + x_2 POL(false) = 0 POL(function(x_1, x_2, x_3, x_4)) = x_2 POL(if) = [1] POL(p) = 0 POL(plus) = [1] POL(s(x_1)) = [2] + x_1 POL(third) = 0 ---------------------------------------- (116) Obligation: Complexity Dependency Tuples Problem Rules: function(third, z0, z1, z2) -> z2 function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1)), FUNCTION(p, s(s(0)), s(s(0)), z1)) FUNCTION(if, false, s(s(0)), x0) -> c8(FUNCTION(p, s(s(0)), s(s(0)), x0)) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(0))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(0))), x1, x1), s(s(0)), s(x1))) FUNCTION(if, false, s(s(s(s(z0)))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(z0)))), x1, x1), s(s(s(function(p, s(z0), z0, z0)))), s(x1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), z1)) FUNCTION(if, false, s(s(s(0))), x0) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), x0)) FUNCTION(if, false, s(s(s(s(0)))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(0)))), x1, x1), s(s(s(0))), s(x1)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), x1)) FUNCTION(if, false, s(s(s(s(s(z0))))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(s(z0))))), x1, x1), s(s(s(s(function(p, s(z0), z0, z0))))), s(x1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), x1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) S tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(s(z0)))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(z0)))), x1, x1), s(s(s(function(p, s(z0), z0, z0)))), s(x1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(s(z0))))), x1, x1), s(s(s(s(function(p, s(z0), z0, z0))))), s(x1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), x1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) K tuples: FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(0)), x1) -> c8(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1)), FUNCTION(p, s(s(0)), s(s(0)), x1)) FUNCTION(if, false, s(s(0)), x1) -> c7(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1))) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(s(0))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(0))), x1, x1), s(s(0)), s(x1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), x1)) FUNCTION(if, false, s(s(s(0))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(0))), x1, x1), s(s(0)), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(s(0)))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(0)))), x1, x1), s(s(s(0))), s(x1)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), x1)) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c4_1, c5_1, c8_2, c8_1, c7_1 ---------------------------------------- (117) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FUNCTION(if, false, s(s(s(0))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(0))), x1, x1), s(s(0)), s(x1))) by FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) ---------------------------------------- (118) Obligation: Complexity Dependency Tuples Problem Rules: function(third, z0, z1, z2) -> z2 function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1)), FUNCTION(p, s(s(0)), s(s(0)), z1)) FUNCTION(if, false, s(s(0)), x0) -> c8(FUNCTION(p, s(s(0)), s(s(0)), x0)) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(s(z0)))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(z0)))), x1, x1), s(s(s(function(p, s(z0), z0, z0)))), s(x1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), z1)) FUNCTION(if, false, s(s(s(0))), x0) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), x0)) FUNCTION(if, false, s(s(s(s(0)))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(0)))), x1, x1), s(s(s(0))), s(x1)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), x1)) FUNCTION(if, false, s(s(s(s(s(z0))))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(s(z0))))), x1, x1), s(s(s(s(function(p, s(z0), z0, z0))))), s(x1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), x1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) S tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(s(z0)))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(z0)))), x1, x1), s(s(s(function(p, s(z0), z0, z0)))), s(x1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(s(z0))))), x1, x1), s(s(s(s(function(p, s(z0), z0, z0))))), s(x1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), x1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) K tuples: FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(0)), x1) -> c8(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1)), FUNCTION(p, s(s(0)), s(s(0)), x1)) FUNCTION(if, false, s(s(0)), x1) -> c7(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1))) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(s(0))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(0))), x1, x1), s(s(0)), s(x1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), x1)) FUNCTION(if, false, s(s(s(0))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(0))), x1, x1), s(s(0)), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(s(0)))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(0)))), x1, x1), s(s(s(0))), s(x1)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), x1)) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c4_1, c5_1, c8_2, c8_1, c7_1 ---------------------------------------- (119) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FUNCTION(if, false, s(s(s(s(z0)))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(z0)))), x1, x1), s(s(s(function(p, s(z0), z0, z0)))), s(x1))) by FUNCTION(if, false, s(s(s(s(0)))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(0)))), x1, x1), s(s(s(0))), s(x1))) FUNCTION(if, false, s(s(s(s(s(z0))))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(s(z0))))), x1, x1), s(s(s(s(function(p, s(z0), z0, z0))))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) ---------------------------------------- (120) Obligation: Complexity Dependency Tuples Problem Rules: function(third, z0, z1, z2) -> z2 function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1)), FUNCTION(p, s(s(0)), s(s(0)), z1)) FUNCTION(if, false, s(s(0)), x0) -> c8(FUNCTION(p, s(s(0)), s(s(0)), x0)) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), z1)) FUNCTION(if, false, s(s(s(0))), x0) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), x0)) FUNCTION(if, false, s(s(s(s(0)))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(0)))), x1, x1), s(s(s(0))), s(x1)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), x1)) FUNCTION(if, false, s(s(s(s(s(z0))))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(s(z0))))), x1, x1), s(s(s(s(function(p, s(z0), z0, z0))))), s(x1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), x1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(0)))), x1, x1), s(s(s(0))), s(x1))) FUNCTION(if, false, s(s(s(s(s(z0))))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(s(z0))))), x1, x1), s(s(s(s(function(p, s(z0), z0, z0))))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) S tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(s(z0))))), x1, x1), s(s(s(s(function(p, s(z0), z0, z0))))), s(x1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), x1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(0)))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(0)))), x1, x1), s(s(s(0))), s(x1))) FUNCTION(if, false, s(s(s(s(s(z0))))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(s(z0))))), x1, x1), s(s(s(s(function(p, s(z0), z0, z0))))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) K tuples: FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(0)), x1) -> c8(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1)), FUNCTION(p, s(s(0)), s(s(0)), x1)) FUNCTION(if, false, s(s(0)), x1) -> c7(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1))) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(s(0))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(0))), x1, x1), s(s(0)), s(x1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), x1)) FUNCTION(if, false, s(s(s(0))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(0))), x1, x1), s(s(0)), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(s(0)))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(0)))), x1, x1), s(s(s(0))), s(x1)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), x1)) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c4_1, c5_1, c8_2, c8_1, c7_1 ---------------------------------------- (121) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. FUNCTION(if, false, s(s(s(s(0)))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(0)))), x1, x1), s(s(s(0))), s(x1))) We considered the (Usable) Rules: function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) And the Tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1)), FUNCTION(p, s(s(0)), s(s(0)), z1)) FUNCTION(if, false, s(s(0)), x0) -> c8(FUNCTION(p, s(s(0)), s(s(0)), x0)) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), z1)) FUNCTION(if, false, s(s(s(0))), x0) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), x0)) FUNCTION(if, false, s(s(s(s(0)))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(0)))), x1, x1), s(s(s(0))), s(x1)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), x1)) FUNCTION(if, false, s(s(s(s(s(z0))))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(s(z0))))), x1, x1), s(s(s(s(function(p, s(z0), z0, z0))))), s(x1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), x1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(0)))), x1, x1), s(s(s(0))), s(x1))) FUNCTION(if, false, s(s(s(s(s(z0))))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(s(z0))))), x1, x1), s(s(s(s(function(p, s(z0), z0, z0))))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(FUNCTION(x_1, x_2, x_3, x_4)) = x_1*x_3 POL(c4(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c8(x_1, x_2)) = x_1 + x_2 POL(false) = 0 POL(function(x_1, x_2, x_3, x_4)) = x_2 POL(if) = [1] POL(p) = 0 POL(plus) = [1] POL(s(x_1)) = [1] + x_1 POL(third) = 0 ---------------------------------------- (122) Obligation: Complexity Dependency Tuples Problem Rules: function(third, z0, z1, z2) -> z2 function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1)), FUNCTION(p, s(s(0)), s(s(0)), z1)) FUNCTION(if, false, s(s(0)), x0) -> c8(FUNCTION(p, s(s(0)), s(s(0)), x0)) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), z1)) FUNCTION(if, false, s(s(s(0))), x0) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), x0)) FUNCTION(if, false, s(s(s(s(0)))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(0)))), x1, x1), s(s(s(0))), s(x1)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), x1)) FUNCTION(if, false, s(s(s(s(s(z0))))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(s(z0))))), x1, x1), s(s(s(s(function(p, s(z0), z0, z0))))), s(x1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), x1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(0)))), x1, x1), s(s(s(0))), s(x1))) FUNCTION(if, false, s(s(s(s(s(z0))))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(s(z0))))), x1, x1), s(s(s(s(function(p, s(z0), z0, z0))))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) S tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(s(z0))))), x1, x1), s(s(s(s(function(p, s(z0), z0, z0))))), s(x1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), x1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(s(z0))))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(s(z0))))), x1, x1), s(s(s(s(function(p, s(z0), z0, z0))))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) K tuples: FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(0)), x1) -> c8(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1)), FUNCTION(p, s(s(0)), s(s(0)), x1)) FUNCTION(if, false, s(s(0)), x1) -> c7(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1))) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(s(0))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(0))), x1, x1), s(s(0)), s(x1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), x1)) FUNCTION(if, false, s(s(s(0))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(0))), x1, x1), s(s(0)), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(s(0)))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(0)))), x1, x1), s(s(s(0))), s(x1)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), x1)) FUNCTION(if, false, s(s(s(s(0)))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(0)))), x1, x1), s(s(s(0))), s(x1))) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c4_1, c5_1, c8_2, c8_1, c7_1 ---------------------------------------- (123) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FUNCTION(if, false, s(s(s(s(0)))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(0)))), x1, x1), s(s(s(0))), s(x1)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), x1)) by FUNCTION(if, false, s(s(s(s(0)))), z0) -> c8(FUNCTION(plus, z0, s(s(s(0))), s(z0)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), z0)) ---------------------------------------- (124) Obligation: Complexity Dependency Tuples Problem Rules: function(third, z0, z1, z2) -> z2 function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1)), FUNCTION(p, s(s(0)), s(s(0)), z1)) FUNCTION(if, false, s(s(0)), x0) -> c8(FUNCTION(p, s(s(0)), s(s(0)), x0)) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), z1)) FUNCTION(if, false, s(s(s(0))), x0) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), x0)) FUNCTION(if, false, s(s(s(s(s(z0))))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(s(z0))))), x1, x1), s(s(s(s(function(p, s(z0), z0, z0))))), s(x1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), x1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(0)))), x1, x1), s(s(s(0))), s(x1))) FUNCTION(if, false, s(s(s(s(s(z0))))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(s(z0))))), x1, x1), s(s(s(s(function(p, s(z0), z0, z0))))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c8(FUNCTION(plus, z0, s(s(s(0))), s(z0)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), z0)) S tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(s(z0))))), x1, x1), s(s(s(s(function(p, s(z0), z0, z0))))), s(x1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), x1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(s(z0))))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(s(z0))))), x1, x1), s(s(s(s(function(p, s(z0), z0, z0))))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) K tuples: FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(0)), x1) -> c8(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1)), FUNCTION(p, s(s(0)), s(s(0)), x1)) FUNCTION(if, false, s(s(0)), x1) -> c7(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1))) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(s(0))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(0))), x1, x1), s(s(0)), s(x1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), x1)) FUNCTION(if, false, s(s(s(0))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(0))), x1, x1), s(s(0)), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(s(0)))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(0)))), x1, x1), s(s(s(0))), s(x1)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), x1)) FUNCTION(if, false, s(s(s(s(0)))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(0)))), x1, x1), s(s(s(0))), s(x1))) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c4_1, c5_1, c8_2, c8_1, c7_1 ---------------------------------------- (125) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FUNCTION(if, false, s(s(s(s(s(z0))))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(s(z0))))), x1, x1), s(s(s(s(function(p, s(z0), z0, z0))))), s(x1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), x1)) by FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) ---------------------------------------- (126) Obligation: Complexity Dependency Tuples Problem Rules: function(third, z0, z1, z2) -> z2 function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1)), FUNCTION(p, s(s(0)), s(s(0)), z1)) FUNCTION(if, false, s(s(0)), x0) -> c8(FUNCTION(p, s(s(0)), s(s(0)), x0)) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), z1)) FUNCTION(if, false, s(s(s(0))), x0) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), x0)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(0)))), x1, x1), s(s(s(0))), s(x1))) FUNCTION(if, false, s(s(s(s(s(z0))))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(s(z0))))), x1, x1), s(s(s(s(function(p, s(z0), z0, z0))))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c8(FUNCTION(plus, z0, s(s(s(0))), s(z0)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), z0)) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) S tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(s(z0))))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(s(z0))))), x1, x1), s(s(s(s(function(p, s(z0), z0, z0))))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) K tuples: FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(0)), x1) -> c8(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1)), FUNCTION(p, s(s(0)), s(s(0)), x1)) FUNCTION(if, false, s(s(0)), x1) -> c7(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1))) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(s(0))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(0))), x1, x1), s(s(0)), s(x1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), x1)) FUNCTION(if, false, s(s(s(0))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(0))), x1, x1), s(s(0)), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(s(0)))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(0)))), x1, x1), s(s(s(0))), s(x1)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), x1)) FUNCTION(if, false, s(s(s(s(0)))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(0)))), x1, x1), s(s(s(0))), s(x1))) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c4_1, c5_1, c8_2, c8_1, c7_1 ---------------------------------------- (127) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) by FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) ---------------------------------------- (128) Obligation: Complexity Dependency Tuples Problem Rules: function(third, z0, z1, z2) -> z2 function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1)), FUNCTION(p, s(s(0)), s(s(0)), z1)) FUNCTION(if, false, s(s(0)), x0) -> c8(FUNCTION(p, s(s(0)), s(s(0)), x0)) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), z1)) FUNCTION(if, false, s(s(s(0))), x0) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), x0)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(0)))), x1, x1), s(s(s(0))), s(x1))) FUNCTION(if, false, s(s(s(s(s(z0))))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(s(z0))))), x1, x1), s(s(s(s(function(p, s(z0), z0, z0))))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c8(FUNCTION(plus, z0, s(s(s(0))), s(z0)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), z0)) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) S tuples: FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(s(z0))))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(s(z0))))), x1, x1), s(s(s(s(function(p, s(z0), z0, z0))))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) K tuples: FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(0)), x1) -> c8(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1)), FUNCTION(p, s(s(0)), s(s(0)), x1)) FUNCTION(if, false, s(s(0)), x1) -> c7(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1))) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(s(0))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(0))), x1, x1), s(s(0)), s(x1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), x1)) FUNCTION(if, false, s(s(s(0))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(0))), x1, x1), s(s(0)), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(s(0)))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(0)))), x1, x1), s(s(s(0))), s(x1)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), x1)) FUNCTION(if, false, s(s(s(s(0)))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(0)))), x1, x1), s(s(s(0))), s(x1))) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c5_1, c8_2, c8_1, c7_1, c4_1 ---------------------------------------- (129) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: FUNCTION(if, false, s(s(0)), x0) -> c8(FUNCTION(p, s(s(0)), s(s(0)), x0)) ---------------------------------------- (130) Obligation: Complexity Dependency Tuples Problem Rules: function(third, z0, z1, z2) -> z2 function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1)), FUNCTION(p, s(s(0)), s(s(0)), z1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), z1)) FUNCTION(if, false, s(s(s(0))), x0) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), x0)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(0)))), x1, x1), s(s(s(0))), s(x1))) FUNCTION(if, false, s(s(s(s(s(z0))))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(s(z0))))), x1, x1), s(s(s(s(function(p, s(z0), z0, z0))))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c8(FUNCTION(plus, z0, s(s(s(0))), s(z0)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), z0)) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) S tuples: FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(s(z0))))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(s(z0))))), x1, x1), s(s(s(s(function(p, s(z0), z0, z0))))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) K tuples: FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(s(0)))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(0)))), x1, x1), s(s(s(0))), s(x1))) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c5_1, c8_2, c8_1, c7_1, c4_1 ---------------------------------------- (131) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (132) Obligation: Complexity Dependency Tuples Problem Rules: function(third, z0, z1, z2) -> z2 function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), z1)) FUNCTION(if, false, s(s(s(0))), x0) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), x0)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(0)))), x1, x1), s(s(s(0))), s(x1))) FUNCTION(if, false, s(s(s(s(s(z0))))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(s(z0))))), x1, x1), s(s(s(s(function(p, s(z0), z0, z0))))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c8(FUNCTION(plus, z0, s(s(s(0))), s(z0)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), z0)) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1))) S tuples: FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(s(z0))))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(s(z0))))), x1, x1), s(s(s(s(function(p, s(z0), z0, z0))))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) K tuples: FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(s(0)))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(0)))), x1, x1), s(s(s(0))), s(x1))) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c5_1, c8_2, c8_1, c7_1, c4_1 ---------------------------------------- (133) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FUNCTION(if, false, s(s(s(s(0)))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(0)))), x1, x1), s(s(s(0))), s(x1))) by FUNCTION(if, false, s(s(s(s(0)))), z0) -> c7(FUNCTION(plus, z0, s(s(s(0))), s(z0))) ---------------------------------------- (134) Obligation: Complexity Dependency Tuples Problem Rules: function(third, z0, z1, z2) -> z2 function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), z1)) FUNCTION(if, false, s(s(s(0))), x0) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), x0)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(s(z0))))), x1, x1), s(s(s(s(function(p, s(z0), z0, z0))))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c8(FUNCTION(plus, z0, s(s(s(0))), s(z0)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), z0)) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c7(FUNCTION(plus, z0, s(s(s(0))), s(z0))) S tuples: FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(s(z0))))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(s(z0))))), x1, x1), s(s(s(s(function(p, s(z0), z0, z0))))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) K tuples: FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(s(0)))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(0)))), x1, x1), s(s(s(0))), s(x1))) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c5_1, c8_2, c8_1, c7_1, c4_1 ---------------------------------------- (135) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FUNCTION(if, false, s(s(s(s(s(z0))))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(s(z0))))), x1, x1), s(s(s(s(function(p, s(z0), z0, z0))))), s(x1))) by FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) ---------------------------------------- (136) Obligation: Complexity Dependency Tuples Problem Rules: function(third, z0, z1, z2) -> z2 function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), z1)) FUNCTION(if, false, s(s(s(0))), x0) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), x0)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c8(FUNCTION(plus, z0, s(s(s(0))), s(z0)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), z0)) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c7(FUNCTION(plus, z0, s(s(s(0))), s(z0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) S tuples: FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) K tuples: FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(s(0)))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(0)))), x1, x1), s(s(s(0))), s(x1))) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c5_1, c8_2, c8_1, c7_1, c4_1 ---------------------------------------- (137) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: function(third, z0, z1, z2) -> z2 ---------------------------------------- (138) Obligation: Complexity Dependency Tuples Problem Rules: function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), z1)) FUNCTION(if, false, s(s(s(0))), x0) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), x0)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c8(FUNCTION(plus, z0, s(s(s(0))), s(z0)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), z0)) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c7(FUNCTION(plus, z0, s(s(s(0))), s(z0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) S tuples: FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) K tuples: FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(s(0)))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(0)))), x1, x1), s(s(s(0))), s(x1))) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c5_1, c8_2, c8_1, c7_1, c4_1 ---------------------------------------- (139) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) by FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(0), s(x0)) -> c5(FUNCTION(if, false, s(0), s(x0))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) ---------------------------------------- (140) Obligation: Complexity Dependency Tuples Problem Rules: function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), z1)) FUNCTION(if, false, s(s(s(0))), x0) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), x0)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c8(FUNCTION(plus, z0, s(s(s(0))), s(z0)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), z0)) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c7(FUNCTION(plus, z0, s(s(s(0))), s(z0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(0), s(x0)) -> c5(FUNCTION(if, false, s(0), s(x0))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) S tuples: FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(0), s(x0)) -> c5(FUNCTION(if, false, s(0), s(x0))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) K tuples: FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(s(0)))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(0)))), x1, x1), s(s(s(0))), s(x1))) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c8_2, c8_1, c7_1, c4_1, c5_1 ---------------------------------------- (141) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (142) Obligation: Complexity Dependency Tuples Problem Rules: function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), z1)) FUNCTION(if, false, s(s(s(0))), x0) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), x0)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c8(FUNCTION(plus, z0, s(s(s(0))), s(z0)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), z0)) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c7(FUNCTION(plus, z0, s(s(s(0))), s(z0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(plus, x0, s(0), s(x0)) -> c5 S tuples: FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(plus, x0, s(0), s(x0)) -> c5 K tuples: FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(s(0)))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(0)))), x1, x1), s(s(s(0))), s(x1))) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c8_2, c8_1, c7_1, c4_1, c5_1, c5 ---------------------------------------- (143) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. FUNCTION(plus, x0, s(0), s(x0)) -> c5 We considered the (Usable) Rules:none And the Tuples: FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), z1)) FUNCTION(if, false, s(s(s(0))), x0) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), x0)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c8(FUNCTION(plus, z0, s(s(s(0))), s(z0)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), z0)) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c7(FUNCTION(plus, z0, s(s(s(0))), s(z0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(plus, x0, s(0), s(x0)) -> c5 The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = [1] POL(FUNCTION(x_1, x_2, x_3, x_4)) = x_1 POL(c4(x_1)) = x_1 POL(c5) = 0 POL(c5(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c8(x_1, x_2)) = x_1 + x_2 POL(false) = 0 POL(function(x_1, x_2, x_3, x_4)) = [1] + x_1 + x_2 POL(if) = [1] POL(p) = 0 POL(plus) = [1] POL(s(x_1)) = [1] + x_1 ---------------------------------------- (144) Obligation: Complexity Dependency Tuples Problem Rules: function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), z1)) FUNCTION(if, false, s(s(s(0))), x0) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), x0)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c8(FUNCTION(plus, z0, s(s(s(0))), s(z0)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), z0)) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c7(FUNCTION(plus, z0, s(s(s(0))), s(z0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(plus, x0, s(0), s(x0)) -> c5 S tuples: FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) K tuples: FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(s(0)))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(0)))), x1, x1), s(s(s(0))), s(x1))) FUNCTION(plus, x0, s(0), s(x0)) -> c5 Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c8_2, c8_1, c7_1, c4_1, c5_1, c5 ---------------------------------------- (145) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) by FUNCTION(if, false, s(s(s(y0))), z1) -> c8(FUNCTION(p, s(s(s(y0))), s(s(s(y0))), z1)) ---------------------------------------- (146) Obligation: Complexity Dependency Tuples Problem Rules: function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), z1)) FUNCTION(if, false, s(s(s(0))), x0) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), x0)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c8(FUNCTION(plus, z0, s(s(s(0))), s(z0)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), z0)) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c7(FUNCTION(plus, z0, s(s(s(0))), s(z0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(plus, x0, s(0), s(x0)) -> c5 S tuples: FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) K tuples: FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(s(0)))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(0)))), x1, x1), s(s(s(0))), s(x1))) FUNCTION(plus, x0, s(0), s(x0)) -> c5 Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c8_2, c7_1, c8_1, c4_1, c5_1, c5 ---------------------------------------- (147) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: FUNCTION(plus, x0, s(0), s(x0)) -> c5 ---------------------------------------- (148) Obligation: Complexity Dependency Tuples Problem Rules: function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), z1)) FUNCTION(if, false, s(s(s(0))), x0) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), x0)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c8(FUNCTION(plus, z0, s(s(s(0))), s(z0)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), z0)) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c7(FUNCTION(plus, z0, s(s(s(0))), s(z0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) S tuples: FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) K tuples: FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c8_2, c7_1, c8_1, c4_1, c5_1 ---------------------------------------- (149) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) by FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(0), 0, 0)), s(s(x0))), FUNCTION(p, s(s(0)), s(s(0)), s(x0))) FUNCTION(if, false, s(s(s(x1))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(s(x1)), s(x1), s(x1))), s(s(x0))), FUNCTION(p, s(s(s(x1))), s(s(s(x1))), s(x0))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(s(0)), s(0), s(0))), s(s(x0))), FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(s(x1)))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(s(s(x1))), s(s(x1)), s(s(x1)))), s(s(x0))), FUNCTION(p, s(s(s(s(x1)))), s(s(s(s(x1)))), s(x0))) ---------------------------------------- (150) Obligation: Complexity Dependency Tuples Problem Rules: function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), z1)) FUNCTION(if, false, s(s(s(0))), x0) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), x0)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c8(FUNCTION(plus, z0, s(s(s(0))), s(z0)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), z0)) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c7(FUNCTION(plus, z0, s(s(s(0))), s(z0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(0), 0, 0)), s(s(x0))), FUNCTION(p, s(s(0)), s(s(0)), s(x0))) FUNCTION(if, false, s(s(s(x1))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(s(x1)), s(x1), s(x1))), s(s(x0))), FUNCTION(p, s(s(s(x1))), s(s(s(x1))), s(x0))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(s(0)), s(0), s(0))), s(s(x0))), FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(s(x1)))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(s(s(x1))), s(s(x1)), s(s(x1)))), s(s(x0))), FUNCTION(p, s(s(s(s(x1)))), s(s(s(s(x1)))), s(x0))) S tuples: FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(0), 0, 0)), s(s(x0))), FUNCTION(p, s(s(0)), s(s(0)), s(x0))) FUNCTION(if, false, s(s(s(x1))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(s(x1)), s(x1), s(x1))), s(s(x0))), FUNCTION(p, s(s(s(x1))), s(s(s(x1))), s(x0))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(s(0)), s(0), s(0))), s(s(x0))), FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(s(x1)))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(s(s(x1))), s(s(x1)), s(s(x1)))), s(s(x0))), FUNCTION(p, s(s(s(s(x1)))), s(s(s(s(x1)))), s(x0))) K tuples: FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c7_1, c8_2, c8_1, c4_1, c5_1 ---------------------------------------- (151) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (152) Obligation: Complexity Dependency Tuples Problem Rules: function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), z1)) FUNCTION(if, false, s(s(s(0))), x0) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), x0)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c8(FUNCTION(plus, z0, s(s(s(0))), s(z0)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), z0)) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c7(FUNCTION(plus, z0, s(s(s(0))), s(z0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(x1))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(s(x1)), s(x1), s(x1))), s(s(x0))), FUNCTION(p, s(s(s(x1))), s(s(s(x1))), s(x0))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(s(0)), s(0), s(0))), s(s(x0))), FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(s(x1)))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(s(s(x1))), s(s(x1)), s(s(x1)))), s(s(x0))), FUNCTION(p, s(s(s(s(x1)))), s(s(s(s(x1)))), s(x0))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(0), 0, 0)), s(s(x0)))) S tuples: FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(x1))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(s(x1)), s(x1), s(x1))), s(s(x0))), FUNCTION(p, s(s(s(x1))), s(s(s(x1))), s(x0))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(s(0)), s(0), s(0))), s(s(x0))), FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(s(x1)))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(s(s(x1))), s(s(x1)), s(s(x1)))), s(s(x0))), FUNCTION(p, s(s(s(s(x1)))), s(s(s(s(x1)))), s(x0))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(0), 0, 0)), s(s(x0)))) K tuples: FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c7_1, c8_2, c8_1, c4_1, c5_1 ---------------------------------------- (153) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FUNCTION(if, false, s(s(s(x1))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(s(x1)), s(x1), s(x1))), s(s(x0))), FUNCTION(p, s(s(s(x1))), s(s(s(x1))), s(x0))) by FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) ---------------------------------------- (154) Obligation: Complexity Dependency Tuples Problem Rules: function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), z1)) FUNCTION(if, false, s(s(s(0))), x0) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), x0)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c8(FUNCTION(plus, z0, s(s(s(0))), s(z0)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), z0)) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c7(FUNCTION(plus, z0, s(s(s(0))), s(z0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(s(0)), s(0), s(0))), s(s(x0))), FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(s(x1)))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(s(s(x1))), s(s(x1)), s(s(x1)))), s(s(x0))), FUNCTION(p, s(s(s(s(x1)))), s(s(s(s(x1)))), s(x0))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(0), 0, 0)), s(s(x0)))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) S tuples: FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(s(0)), s(0), s(0))), s(s(x0))), FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(s(x1)))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(s(s(x1))), s(s(x1)), s(s(x1)))), s(s(x0))), FUNCTION(p, s(s(s(s(x1)))), s(s(s(s(x1)))), s(x0))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(0), 0, 0)), s(s(x0)))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) K tuples: FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c7_1, c8_2, c8_1, c4_1, c5_1 ---------------------------------------- (155) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) We considered the (Usable) Rules:none And the Tuples: FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), z1)) FUNCTION(if, false, s(s(s(0))), x0) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), x0)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c8(FUNCTION(plus, z0, s(s(s(0))), s(z0)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), z0)) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c7(FUNCTION(plus, z0, s(s(s(0))), s(z0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(s(0)), s(0), s(0))), s(s(x0))), FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(s(x1)))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(s(s(x1))), s(s(x1)), s(s(x1)))), s(s(x0))), FUNCTION(p, s(s(s(s(x1)))), s(s(s(s(x1)))), s(x0))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(0), 0, 0)), s(s(x0)))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = [3] POL(FUNCTION(x_1, x_2, x_3, x_4)) = x_1 POL(c4(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c8(x_1, x_2)) = x_1 + x_2 POL(false) = 0 POL(function(x_1, x_2, x_3, x_4)) = [3] + [3]x_3 + [3]x_4 POL(if) = [2] POL(p) = 0 POL(plus) = [2] POL(s(x_1)) = [3] ---------------------------------------- (156) Obligation: Complexity Dependency Tuples Problem Rules: function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), z1)) FUNCTION(if, false, s(s(s(0))), x0) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), x0)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c8(FUNCTION(plus, z0, s(s(s(0))), s(z0)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), z0)) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c7(FUNCTION(plus, z0, s(s(s(0))), s(z0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(s(0)), s(0), s(0))), s(s(x0))), FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(s(x1)))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(s(s(x1))), s(s(x1)), s(s(x1)))), s(s(x0))), FUNCTION(p, s(s(s(s(x1)))), s(s(s(s(x1)))), s(x0))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(0), 0, 0)), s(s(x0)))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) S tuples: FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(s(0)), s(0), s(0))), s(s(x0))), FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(s(x1)))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(s(s(x1))), s(s(x1)), s(s(x1)))), s(s(x0))), FUNCTION(p, s(s(s(s(x1)))), s(s(s(s(x1)))), s(x0))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(0), 0, 0)), s(s(x0)))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) K tuples: FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c7_1, c8_2, c8_1, c4_1, c5_1 ---------------------------------------- (157) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(s(0)), s(0), s(0))), s(s(x0))), FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) by FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(function(p, s(0), 0, 0))), s(s(x0))), FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) ---------------------------------------- (158) Obligation: Complexity Dependency Tuples Problem Rules: function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), z1)) FUNCTION(if, false, s(s(s(0))), x0) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), x0)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c8(FUNCTION(plus, z0, s(s(s(0))), s(z0)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), z0)) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c7(FUNCTION(plus, z0, s(s(s(0))), s(z0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(s(x1)))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(s(s(x1))), s(s(x1)), s(s(x1)))), s(s(x0))), FUNCTION(p, s(s(s(s(x1)))), s(s(s(s(x1)))), s(x0))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(0), 0, 0)), s(s(x0)))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(function(p, s(0), 0, 0))), s(s(x0))), FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) S tuples: FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(s(x1)))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(s(s(x1))), s(s(x1)), s(s(x1)))), s(s(x0))), FUNCTION(p, s(s(s(s(x1)))), s(s(s(s(x1)))), s(x0))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(0), 0, 0)), s(s(x0)))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(function(p, s(0), 0, 0))), s(s(x0))), FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) K tuples: FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c7_1, c8_2, c8_1, c4_1, c5_1 ---------------------------------------- (159) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) We considered the (Usable) Rules:none And the Tuples: FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), z1)) FUNCTION(if, false, s(s(s(0))), x0) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), x0)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c8(FUNCTION(plus, z0, s(s(s(0))), s(z0)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), z0)) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c7(FUNCTION(plus, z0, s(s(s(0))), s(z0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(s(x1)))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(s(s(x1))), s(s(x1)), s(s(x1)))), s(s(x0))), FUNCTION(p, s(s(s(s(x1)))), s(s(s(s(x1)))), s(x0))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(0), 0, 0)), s(s(x0)))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(function(p, s(0), 0, 0))), s(s(x0))), FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = [3] POL(FUNCTION(x_1, x_2, x_3, x_4)) = x_1 POL(c4(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c8(x_1, x_2)) = x_1 + x_2 POL(false) = 0 POL(function(x_1, x_2, x_3, x_4)) = [3] + [3]x_3 + [3]x_4 POL(if) = [2] POL(p) = 0 POL(plus) = [2] POL(s(x_1)) = [3] ---------------------------------------- (160) Obligation: Complexity Dependency Tuples Problem Rules: function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), z1)) FUNCTION(if, false, s(s(s(0))), x0) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), x0)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c8(FUNCTION(plus, z0, s(s(s(0))), s(z0)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), z0)) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c7(FUNCTION(plus, z0, s(s(s(0))), s(z0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(s(x1)))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(s(s(x1))), s(s(x1)), s(s(x1)))), s(s(x0))), FUNCTION(p, s(s(s(s(x1)))), s(s(s(s(x1)))), s(x0))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(0), 0, 0)), s(s(x0)))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(function(p, s(0), 0, 0))), s(s(x0))), FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) S tuples: FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(s(x1)))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(s(s(x1))), s(s(x1)), s(s(x1)))), s(s(x0))), FUNCTION(p, s(s(s(s(x1)))), s(s(s(s(x1)))), s(x0))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(0), 0, 0)), s(s(x0)))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(function(p, s(0), 0, 0))), s(s(x0))), FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) K tuples: FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c7_1, c8_2, c8_1, c4_1, c5_1 ---------------------------------------- (161) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FUNCTION(if, false, s(s(s(s(x1)))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(s(s(x1))), s(s(x1)), s(s(x1)))), s(s(x0))), FUNCTION(p, s(s(s(s(x1)))), s(s(s(s(x1)))), s(x0))) by FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(s(x0)), s(x0), s(x0)))), s(s(x1))), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) ---------------------------------------- (162) Obligation: Complexity Dependency Tuples Problem Rules: function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), z1)) FUNCTION(if, false, s(s(s(0))), x0) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), x0)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c8(FUNCTION(plus, z0, s(s(s(0))), s(z0)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), z0)) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c7(FUNCTION(plus, z0, s(s(s(0))), s(z0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(0), 0, 0)), s(s(x0)))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(function(p, s(0), 0, 0))), s(s(x0))), FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(s(x0)), s(x0), s(x0)))), s(s(x1))), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) S tuples: FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(0), 0, 0)), s(s(x0)))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(function(p, s(0), 0, 0))), s(s(x0))), FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(s(x0)), s(x0), s(x0)))), s(s(x1))), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) K tuples: FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c7_1, c8_2, c8_1, c4_1, c5_1 ---------------------------------------- (163) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) We considered the (Usable) Rules:none And the Tuples: FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), z1)) FUNCTION(if, false, s(s(s(0))), x0) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), x0)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c8(FUNCTION(plus, z0, s(s(s(0))), s(z0)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), z0)) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c7(FUNCTION(plus, z0, s(s(s(0))), s(z0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(0), 0, 0)), s(s(x0)))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(function(p, s(0), 0, 0))), s(s(x0))), FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(s(x0)), s(x0), s(x0)))), s(s(x1))), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = [1] POL(FUNCTION(x_1, x_2, x_3, x_4)) = x_1 POL(c4(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c8(x_1, x_2)) = x_1 + x_2 POL(false) = 0 POL(function(x_1, x_2, x_3, x_4)) = [1] + x_1 + x_2 POL(if) = [1] POL(p) = 0 POL(plus) = [1] POL(s(x_1)) = [1] + x_1 ---------------------------------------- (164) Obligation: Complexity Dependency Tuples Problem Rules: function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), z1)) FUNCTION(if, false, s(s(s(0))), x0) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), x0)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c8(FUNCTION(plus, z0, s(s(s(0))), s(z0)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), z0)) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c7(FUNCTION(plus, z0, s(s(s(0))), s(z0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(0), 0, 0)), s(s(x0)))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(function(p, s(0), 0, 0))), s(s(x0))), FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(s(x0)), s(x0), s(x0)))), s(s(x1))), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) S tuples: FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(0), 0, 0)), s(s(x0)))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(function(p, s(0), 0, 0))), s(s(x0))), FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(s(x0)), s(x0), s(x0)))), s(s(x1))), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) K tuples: FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c7_1, c8_2, c8_1, c4_1, c5_1 ---------------------------------------- (165) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(0), 0, 0)), s(s(x0)))) by FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) ---------------------------------------- (166) Obligation: Complexity Dependency Tuples Problem Rules: function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), z1)) FUNCTION(if, false, s(s(s(0))), x0) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), x0)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c8(FUNCTION(plus, z0, s(s(s(0))), s(z0)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), z0)) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c7(FUNCTION(plus, z0, s(s(s(0))), s(z0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(function(p, s(0), 0, 0))), s(s(x0))), FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(s(x0)), s(x0), s(x0)))), s(s(x1))), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) S tuples: FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(function(p, s(0), 0, 0))), s(s(x0))), FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(s(x0)), s(x0), s(x0)))), s(s(x1))), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) K tuples: FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c7_1, c8_2, c8_1, c4_1, c5_1 ---------------------------------------- (167) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) We considered the (Usable) Rules: function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) And the Tuples: FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), z1)) FUNCTION(if, false, s(s(s(0))), x0) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), x0)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c8(FUNCTION(plus, z0, s(s(s(0))), s(z0)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), z0)) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c7(FUNCTION(plus, z0, s(s(s(0))), s(z0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(function(p, s(0), 0, 0))), s(s(x0))), FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(s(x0)), s(x0), s(x0)))), s(s(x1))), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = [2] POL(FUNCTION(x_1, x_2, x_3, x_4)) = [2]x_1^2 + [2]x_1*x_3 POL(c4(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c8(x_1, x_2)) = x_1 + x_2 POL(false) = 0 POL(function(x_1, x_2, x_3, x_4)) = x_2 POL(if) = [2] POL(p) = 0 POL(plus) = [2] POL(s(x_1)) = [2] + x_1 ---------------------------------------- (168) Obligation: Complexity Dependency Tuples Problem Rules: function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), z1)) FUNCTION(if, false, s(s(s(0))), x0) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), x0)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c8(FUNCTION(plus, z0, s(s(s(0))), s(z0)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), z0)) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c7(FUNCTION(plus, z0, s(s(s(0))), s(z0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(function(p, s(0), 0, 0))), s(s(x0))), FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(s(x0)), s(x0), s(x0)))), s(s(x1))), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) S tuples: FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(function(p, s(0), 0, 0))), s(s(x0))), FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(s(x0)), s(x0), s(x0)))), s(s(x1))), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) K tuples: FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c7_1, c8_2, c8_1, c4_1, c5_1 ---------------------------------------- (169) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(function(p, s(0), 0, 0))), s(s(x0))), FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) by FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(0)), s(s(x0))), FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) ---------------------------------------- (170) Obligation: Complexity Dependency Tuples Problem Rules: function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), z1)) FUNCTION(if, false, s(s(s(0))), x0) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), x0)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c8(FUNCTION(plus, z0, s(s(s(0))), s(z0)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), z0)) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c7(FUNCTION(plus, z0, s(s(s(0))), s(z0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(s(x0)), s(x0), s(x0)))), s(s(x1))), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(0)), s(s(x0))), FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) S tuples: FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(s(x0)), s(x0), s(x0)))), s(s(x1))), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(0)), s(s(x0))), FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) K tuples: FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c7_1, c8_2, c8_1, c4_1, c5_1 ---------------------------------------- (171) CdtKnowledgeProof (BOTH BOUNDS(ID, ID)) The following tuples could be moved from S to K by knowledge propagation: FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) ---------------------------------------- (172) Obligation: Complexity Dependency Tuples Problem Rules: function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), z1)) FUNCTION(if, false, s(s(s(0))), x0) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), x0)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c8(FUNCTION(plus, z0, s(s(s(0))), s(z0)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), z0)) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c7(FUNCTION(plus, z0, s(s(s(0))), s(z0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(s(x0)), s(x0), s(x0)))), s(s(x1))), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(0)), s(s(x0))), FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) S tuples: FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(s(x0)), s(x0), s(x0)))), s(s(x1))), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(0)), s(s(x0))), FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) K tuples: FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c7_1, c8_2, c8_1, c4_1, c5_1 ---------------------------------------- (173) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(0)), s(s(x0))), FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) We considered the (Usable) Rules: function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) And the Tuples: FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), z1)) FUNCTION(if, false, s(s(s(0))), x0) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), x0)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c8(FUNCTION(plus, z0, s(s(s(0))), s(z0)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), z0)) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c7(FUNCTION(plus, z0, s(s(s(0))), s(z0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(s(x0)), s(x0), s(x0)))), s(s(x1))), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(0)), s(s(x0))), FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(FUNCTION(x_1, x_2, x_3, x_4)) = [2]x_1*x_3 POL(c4(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c8(x_1, x_2)) = x_1 + x_2 POL(false) = 0 POL(function(x_1, x_2, x_3, x_4)) = x_2 POL(if) = [1] POL(p) = 0 POL(plus) = [1] POL(s(x_1)) = [2] + x_1 ---------------------------------------- (174) Obligation: Complexity Dependency Tuples Problem Rules: function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), z1)) FUNCTION(if, false, s(s(s(0))), x0) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), x0)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c8(FUNCTION(plus, z0, s(s(s(0))), s(z0)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), z0)) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c7(FUNCTION(plus, z0, s(s(s(0))), s(z0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(s(x0)), s(x0), s(x0)))), s(s(x1))), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(0)), s(s(x0))), FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) S tuples: FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(s(x0)), s(x0), s(x0)))), s(s(x1))), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) K tuples: FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(0)), s(s(x0))), FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c7_1, c8_2, c8_1, c4_1, c5_1 ---------------------------------------- (175) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(s(x0)), s(x0), s(x0)))), s(s(x1))), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) by FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1))), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) ---------------------------------------- (176) Obligation: Complexity Dependency Tuples Problem Rules: function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), z1)) FUNCTION(if, false, s(s(s(0))), x0) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), x0)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c8(FUNCTION(plus, z0, s(s(s(0))), s(z0)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), z0)) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c7(FUNCTION(plus, z0, s(s(s(0))), s(z0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(0)), s(s(x0))), FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1))), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), s(x1))) S tuples: FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1))), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) K tuples: FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(0)), s(s(x0))), FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c7_1, c8_2, c8_1, c4_1, c5_1 ---------------------------------------- (177) CdtKnowledgeProof (BOTH BOUNDS(ID, ID)) The following tuples could be moved from S to K by knowledge propagation: FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) ---------------------------------------- (178) Obligation: Complexity Dependency Tuples Problem Rules: function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), z1)) FUNCTION(if, false, s(s(s(0))), x0) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), x0)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c8(FUNCTION(plus, z0, s(s(s(0))), s(z0)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), z0)) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c7(FUNCTION(plus, z0, s(s(s(0))), s(z0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(0)), s(s(x0))), FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1))), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), s(x1))) S tuples: FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1))), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), s(x1))) K tuples: FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(0)), s(s(x0))), FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c7_1, c8_2, c8_1, c4_1, c5_1 ---------------------------------------- (179) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) by FUNCTION(if, false, s(s(z0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0)))) FUNCTION(if, false, s(s(0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(0), 0, 0)), s(s(x0)))) FUNCTION(if, false, s(s(s(x1))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(s(x1)), s(x1), s(x1))), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(s(0)), s(0), s(0))), s(s(x0)))) FUNCTION(if, false, s(s(s(s(x1)))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(s(s(x1))), s(s(x1)), s(s(x1)))), s(s(x0)))) ---------------------------------------- (180) Obligation: Complexity Dependency Tuples Problem Rules: function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), z1)) FUNCTION(if, false, s(s(s(0))), x0) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), x0)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c8(FUNCTION(plus, z0, s(s(s(0))), s(z0)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), z0)) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c7(FUNCTION(plus, z0, s(s(s(0))), s(z0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(0)), s(s(x0))), FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1))), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0)))) FUNCTION(if, false, s(s(0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(0), 0, 0)), s(s(x0)))) FUNCTION(if, false, s(s(s(x1))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(s(x1)), s(x1), s(x1))), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(s(0)), s(0), s(0))), s(s(x0)))) FUNCTION(if, false, s(s(s(s(x1)))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(s(s(x1))), s(s(x1)), s(s(x1)))), s(s(x0)))) S tuples: FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1))), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0)))) FUNCTION(if, false, s(s(0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(0), 0, 0)), s(s(x0)))) FUNCTION(if, false, s(s(s(x1))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(s(x1)), s(x1), s(x1))), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(s(0)), s(0), s(0))), s(s(x0)))) FUNCTION(if, false, s(s(s(s(x1)))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(s(s(x1))), s(s(x1)), s(s(x1)))), s(s(x0)))) K tuples: FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(0)), s(s(x0))), FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c8_2, c8_1, c7_1, c4_1, c5_1 ---------------------------------------- (181) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FUNCTION(if, false, s(s(0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(0), 0, 0)), s(s(x0)))) by FUNCTION(if, false, s(s(0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(0), s(s(x0)))) ---------------------------------------- (182) Obligation: Complexity Dependency Tuples Problem Rules: function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), z1)) FUNCTION(if, false, s(s(s(0))), x0) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), x0)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c8(FUNCTION(plus, z0, s(s(s(0))), s(z0)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), z0)) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c7(FUNCTION(plus, z0, s(s(s(0))), s(z0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(0)), s(s(x0))), FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1))), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0)))) FUNCTION(if, false, s(s(s(x1))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(s(x1)), s(x1), s(x1))), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(s(0)), s(0), s(0))), s(s(x0)))) FUNCTION(if, false, s(s(s(s(x1)))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(s(s(x1))), s(s(x1)), s(s(x1)))), s(s(x0)))) FUNCTION(if, false, s(s(0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(0), s(s(x0)))) S tuples: FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1))), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0)))) FUNCTION(if, false, s(s(s(x1))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(s(x1)), s(x1), s(x1))), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(s(0)), s(0), s(0))), s(s(x0)))) FUNCTION(if, false, s(s(s(s(x1)))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(s(s(x1))), s(s(x1)), s(s(x1)))), s(s(x0)))) FUNCTION(if, false, s(s(0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(0), s(s(x0)))) K tuples: FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(0)), s(s(x0))), FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c8_2, c8_1, c7_1, c4_1, c5_1 ---------------------------------------- (183) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. FUNCTION(if, false, s(s(0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(0), s(s(x0)))) We considered the (Usable) Rules: function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) And the Tuples: FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), z1)) FUNCTION(if, false, s(s(s(0))), x0) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), x0)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c8(FUNCTION(plus, z0, s(s(s(0))), s(z0)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), z0)) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c7(FUNCTION(plus, z0, s(s(s(0))), s(z0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(0)), s(s(x0))), FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1))), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0)))) FUNCTION(if, false, s(s(s(x1))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(s(x1)), s(x1), s(x1))), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(s(0)), s(0), s(0))), s(s(x0)))) FUNCTION(if, false, s(s(s(s(x1)))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(s(s(x1))), s(s(x1)), s(s(x1)))), s(s(x0)))) FUNCTION(if, false, s(s(0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(0), s(s(x0)))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(FUNCTION(x_1, x_2, x_3, x_4)) = x_1*x_3 POL(c4(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c8(x_1, x_2)) = x_1 + x_2 POL(false) = 0 POL(function(x_1, x_2, x_3, x_4)) = x_2 POL(if) = [1] POL(p) = 0 POL(plus) = [1] POL(s(x_1)) = [1] + x_1 ---------------------------------------- (184) Obligation: Complexity Dependency Tuples Problem Rules: function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), z1)) FUNCTION(if, false, s(s(s(0))), x0) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), x0)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c8(FUNCTION(plus, z0, s(s(s(0))), s(z0)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), z0)) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c7(FUNCTION(plus, z0, s(s(s(0))), s(z0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(0)), s(s(x0))), FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1))), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0)))) FUNCTION(if, false, s(s(s(x1))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(s(x1)), s(x1), s(x1))), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(s(0)), s(0), s(0))), s(s(x0)))) FUNCTION(if, false, s(s(s(s(x1)))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(s(s(x1))), s(s(x1)), s(s(x1)))), s(s(x0)))) FUNCTION(if, false, s(s(0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(0), s(s(x0)))) S tuples: FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1))), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0)))) FUNCTION(if, false, s(s(s(x1))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(s(x1)), s(x1), s(x1))), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(s(0)), s(0), s(0))), s(s(x0)))) FUNCTION(if, false, s(s(s(s(x1)))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(s(s(x1))), s(s(x1)), s(s(x1)))), s(s(x0)))) K tuples: FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(0)), s(s(x0))), FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(0), s(s(x0)))) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c8_2, c8_1, c7_1, c4_1, c5_1 ---------------------------------------- (185) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FUNCTION(if, false, s(s(s(x1))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(s(x1)), s(x1), s(x1))), s(s(x0)))) by FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1)))) ---------------------------------------- (186) Obligation: Complexity Dependency Tuples Problem Rules: function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), z1)) FUNCTION(if, false, s(s(s(0))), x0) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), x0)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c8(FUNCTION(plus, z0, s(s(s(0))), s(z0)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), z0)) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c7(FUNCTION(plus, z0, s(s(s(0))), s(z0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(0)), s(s(x0))), FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1))), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(s(0)), s(0), s(0))), s(s(x0)))) FUNCTION(if, false, s(s(s(s(x1)))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(s(s(x1))), s(s(x1)), s(s(x1)))), s(s(x0)))) FUNCTION(if, false, s(s(0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1)))) S tuples: FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1))), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(s(0)), s(0), s(0))), s(s(x0)))) FUNCTION(if, false, s(s(s(s(x1)))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(s(s(x1))), s(s(x1)), s(s(x1)))), s(s(x0)))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1)))) K tuples: FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(0)), s(s(x0))), FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(0), s(s(x0)))) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c8_2, c8_1, c7_1, c4_1, c5_1 ---------------------------------------- (187) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FUNCTION(if, false, s(s(s(0))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(s(0)), s(0), s(0))), s(s(x0)))) by FUNCTION(if, false, s(s(s(0))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(s(function(p, s(0), 0, 0))), s(s(x0)))) ---------------------------------------- (188) Obligation: Complexity Dependency Tuples Problem Rules: function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), z1)) FUNCTION(if, false, s(s(s(0))), x0) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), x0)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c8(FUNCTION(plus, z0, s(s(s(0))), s(z0)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), z0)) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c7(FUNCTION(plus, z0, s(s(s(0))), s(z0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(0)), s(s(x0))), FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1))), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0)))) FUNCTION(if, false, s(s(s(s(x1)))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(s(s(x1))), s(s(x1)), s(s(x1)))), s(s(x0)))) FUNCTION(if, false, s(s(0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(s(function(p, s(0), 0, 0))), s(s(x0)))) S tuples: FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1))), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0)))) FUNCTION(if, false, s(s(s(s(x1)))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(s(s(x1))), s(s(x1)), s(s(x1)))), s(s(x0)))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(s(function(p, s(0), 0, 0))), s(s(x0)))) K tuples: FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(0)), s(s(x0))), FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(0), s(s(x0)))) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c8_2, c8_1, c7_1, c4_1, c5_1 ---------------------------------------- (189) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FUNCTION(if, false, s(s(s(s(x1)))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(s(s(x1))), s(s(x1)), s(s(x1)))), s(s(x0)))) by FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(function(p, s(s(x0)), s(x0), s(x0)))), s(s(x1)))) ---------------------------------------- (190) Obligation: Complexity Dependency Tuples Problem Rules: function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), z1)) FUNCTION(if, false, s(s(s(0))), x0) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), x0)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c8(FUNCTION(plus, z0, s(s(s(0))), s(z0)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), z0)) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c7(FUNCTION(plus, z0, s(s(s(0))), s(z0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(0)), s(s(x0))), FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1))), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0)))) FUNCTION(if, false, s(s(0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(s(function(p, s(0), 0, 0))), s(s(x0)))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(function(p, s(s(x0)), s(x0), s(x0)))), s(s(x1)))) S tuples: FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1))), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0)))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(s(function(p, s(0), 0, 0))), s(s(x0)))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(function(p, s(s(x0)), s(x0), s(x0)))), s(s(x1)))) K tuples: FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(0)), s(s(x0))), FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(0), s(s(x0)))) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c8_2, c8_1, c7_1, c4_1, c5_1 ---------------------------------------- (191) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FUNCTION(if, false, s(s(s(0))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(s(function(p, s(0), 0, 0))), s(s(x0)))) by FUNCTION(if, false, s(s(s(0))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) ---------------------------------------- (192) Obligation: Complexity Dependency Tuples Problem Rules: function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), z1)) FUNCTION(if, false, s(s(s(0))), x0) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), x0)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c8(FUNCTION(plus, z0, s(s(s(0))), s(z0)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), z0)) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c7(FUNCTION(plus, z0, s(s(s(0))), s(z0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(0)), s(s(x0))), FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1))), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0)))) FUNCTION(if, false, s(s(0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1)))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(function(p, s(s(x0)), s(x0), s(x0)))), s(s(x1)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) S tuples: FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1))), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0)))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1)))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(function(p, s(s(x0)), s(x0), s(x0)))), s(s(x1)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) K tuples: FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(0)), s(s(x0))), FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(0), s(s(x0)))) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c8_2, c8_1, c7_1, c4_1, c5_1 ---------------------------------------- (193) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. FUNCTION(if, false, s(s(s(0))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) We considered the (Usable) Rules: function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) And the Tuples: FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), z1)) FUNCTION(if, false, s(s(s(0))), x0) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), x0)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c8(FUNCTION(plus, z0, s(s(s(0))), s(z0)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), z0)) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c7(FUNCTION(plus, z0, s(s(s(0))), s(z0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(0)), s(s(x0))), FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1))), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0)))) FUNCTION(if, false, s(s(0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1)))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(function(p, s(s(x0)), s(x0), s(x0)))), s(s(x1)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(FUNCTION(x_1, x_2, x_3, x_4)) = x_1*x_3 POL(c4(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c8(x_1, x_2)) = x_1 + x_2 POL(false) = 0 POL(function(x_1, x_2, x_3, x_4)) = x_2 POL(if) = [1] POL(p) = 0 POL(plus) = [1] POL(s(x_1)) = [1] + x_1 ---------------------------------------- (194) Obligation: Complexity Dependency Tuples Problem Rules: function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), z1)) FUNCTION(if, false, s(s(s(0))), x0) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), x0)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c8(FUNCTION(plus, z0, s(s(s(0))), s(z0)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), z0)) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c7(FUNCTION(plus, z0, s(s(s(0))), s(z0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(0)), s(s(x0))), FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1))), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0)))) FUNCTION(if, false, s(s(0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1)))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(function(p, s(s(x0)), s(x0), s(x0)))), s(s(x1)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) S tuples: FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1))), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0)))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1)))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(function(p, s(s(x0)), s(x0), s(x0)))), s(s(x1)))) K tuples: FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(0)), s(s(x0))), FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c8_2, c8_1, c7_1, c4_1, c5_1 ---------------------------------------- (195) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(function(p, s(s(x0)), s(x0), s(x0)))), s(s(x1)))) by FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1)))) ---------------------------------------- (196) Obligation: Complexity Dependency Tuples Problem Rules: function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), z1)) FUNCTION(if, false, s(s(s(0))), x0) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), x0)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c8(FUNCTION(plus, z0, s(s(s(0))), s(z0)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), z0)) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c7(FUNCTION(plus, z0, s(s(s(0))), s(z0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(0)), s(s(x0))), FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1))), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0)))) FUNCTION(if, false, s(s(0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1)))) S tuples: FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1))), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0)))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1)))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1)))) K tuples: FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(0)), s(s(x0))), FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c8_2, c8_1, c7_1, c4_1, c5_1 ---------------------------------------- (197) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) by FUNCTION(if, false, s(s(s(z0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(function(p, s(z0), z0, z0))), s(s(x0))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x0))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(function(p, s(0), 0, 0))), s(s(x0))), FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(s(x1)))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(function(p, s(s(x1)), s(x1), s(x1)))), s(s(x0))), FUNCTION(p, s(s(s(s(x1)))), s(s(s(s(x1)))), s(x0))) ---------------------------------------- (198) Obligation: Complexity Dependency Tuples Problem Rules: function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), z1)) FUNCTION(if, false, s(s(s(0))), x0) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), x0)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c8(FUNCTION(plus, z0, s(s(s(0))), s(z0)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), z0)) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c7(FUNCTION(plus, z0, s(s(s(0))), s(z0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(0)), s(s(x0))), FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1))), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0)))) FUNCTION(if, false, s(s(0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(function(p, s(0), 0, 0))), s(s(x0))), FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(s(x1)))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(function(p, s(s(x1)), s(x1), s(x1)))), s(s(x0))), FUNCTION(p, s(s(s(s(x1)))), s(s(s(s(x1)))), s(x0))) S tuples: FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1))), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0)))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1)))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(function(p, s(0), 0, 0))), s(s(x0))), FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(s(x1)))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(function(p, s(s(x1)), s(x1), s(x1)))), s(s(x0))), FUNCTION(p, s(s(s(s(x1)))), s(s(s(s(x1)))), s(x0))) K tuples: FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(0)), s(s(x0))), FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c8_1, c7_1, c8_2, c4_1, c5_1 ---------------------------------------- (199) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(function(p, s(0), 0, 0))), s(s(x0))), FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) by FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(0)), s(s(x0))), FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) ---------------------------------------- (200) Obligation: Complexity Dependency Tuples Problem Rules: function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), z1)) FUNCTION(if, false, s(s(s(0))), x0) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), x0)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c8(FUNCTION(plus, z0, s(s(s(0))), s(z0)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), z0)) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c7(FUNCTION(plus, z0, s(s(s(0))), s(z0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(0)), s(s(x0))), FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1))), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0)))) FUNCTION(if, false, s(s(0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1)))) FUNCTION(if, false, s(s(s(s(x1)))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(function(p, s(s(x1)), s(x1), s(x1)))), s(s(x0))), FUNCTION(p, s(s(s(s(x1)))), s(s(s(s(x1)))), s(x0))) S tuples: FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1))), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0)))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1)))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1)))) FUNCTION(if, false, s(s(s(s(x1)))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(function(p, s(s(x1)), s(x1), s(x1)))), s(s(x0))), FUNCTION(p, s(s(s(s(x1)))), s(s(s(s(x1)))), s(x0))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(0)), s(s(x0))), FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) K tuples: FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(0)), s(s(x0))), FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c8_1, c7_1, c8_2, c4_1, c5_1 ---------------------------------------- (201) CdtKnowledgeProof (BOTH BOUNDS(ID, ID)) The following tuples could be moved from S to K by knowledge propagation: FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(0)), s(s(x0))), FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(0)), s(s(x0))), FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) ---------------------------------------- (202) Obligation: Complexity Dependency Tuples Problem Rules: function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), z1)) FUNCTION(if, false, s(s(s(0))), x0) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), x0)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c8(FUNCTION(plus, z0, s(s(s(0))), s(z0)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), z0)) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c7(FUNCTION(plus, z0, s(s(s(0))), s(z0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(0)), s(s(x0))), FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1))), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0)))) FUNCTION(if, false, s(s(0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1)))) FUNCTION(if, false, s(s(s(s(x1)))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(function(p, s(s(x1)), s(x1), s(x1)))), s(s(x0))), FUNCTION(p, s(s(s(s(x1)))), s(s(s(s(x1)))), s(x0))) S tuples: FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1))), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0)))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1)))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1)))) FUNCTION(if, false, s(s(s(s(x1)))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(function(p, s(s(x1)), s(x1), s(x1)))), s(s(x0))), FUNCTION(p, s(s(s(s(x1)))), s(s(s(s(x1)))), s(x0))) K tuples: FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(0)), s(s(x0))), FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c8_1, c7_1, c8_2, c4_1, c5_1 ---------------------------------------- (203) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FUNCTION(if, false, s(s(s(s(x1)))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(function(p, s(s(x1)), s(x1), s(x1)))), s(s(x0))), FUNCTION(p, s(s(s(s(x1)))), s(s(s(s(x1)))), s(x0))) by FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1))), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) ---------------------------------------- (204) Obligation: Complexity Dependency Tuples Problem Rules: function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), z1)) FUNCTION(if, false, s(s(s(0))), x0) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), x0)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c8(FUNCTION(plus, z0, s(s(s(0))), s(z0)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), z0)) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c7(FUNCTION(plus, z0, s(s(s(0))), s(z0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(0)), s(s(x0))), FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1))), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0)))) FUNCTION(if, false, s(s(0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1)))) S tuples: FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1))), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0)))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1)))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1)))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) K tuples: FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(0)), s(s(x0))), FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c8_1, c7_1, c8_2, c4_1, c5_1 ---------------------------------------- (205) CdtKnowledgeProof (BOTH BOUNDS(ID, ID)) The following tuples could be moved from S to K by knowledge propagation: FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) ---------------------------------------- (206) Obligation: Complexity Dependency Tuples Problem Rules: function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), z1)) FUNCTION(if, false, s(s(s(0))), x0) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), x0)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c8(FUNCTION(plus, z0, s(s(s(0))), s(z0)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), z0)) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c7(FUNCTION(plus, z0, s(s(s(0))), s(z0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(0)), s(s(x0))), FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1))), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0)))) FUNCTION(if, false, s(s(0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1)))) S tuples: FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1))), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0)))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1)))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1)))) K tuples: FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(0)), s(s(x0))), FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c8_1, c7_1, c8_2, c4_1, c5_1 ---------------------------------------- (207) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) by FUNCTION(if, false, s(s(s(z0))), s(x0)) -> c8(FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x0))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(s(x1)))), s(x0)) -> c8(FUNCTION(p, s(s(s(s(x1)))), s(s(s(s(x1)))), s(x0))) ---------------------------------------- (208) Obligation: Complexity Dependency Tuples Problem Rules: function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), z1)) FUNCTION(if, false, s(s(s(0))), x0) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), x0)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c8(FUNCTION(plus, z0, s(s(s(0))), s(z0)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), z0)) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c7(FUNCTION(plus, z0, s(s(s(0))), s(z0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(0)), s(s(x0))), FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1))), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0)))) FUNCTION(if, false, s(s(0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1)))) S tuples: FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1))), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0)))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1)))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1)))) K tuples: FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(0)), s(s(x0))), FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c8_1, c7_1, c8_2, c4_1, c5_1 ---------------------------------------- (209) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) by FUNCTION(if, false, s(s(0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(0), s(s(x0)))) ---------------------------------------- (210) Obligation: Complexity Dependency Tuples Problem Rules: function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), z1)) FUNCTION(if, false, s(s(s(0))), x0) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), x0)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c8(FUNCTION(plus, z0, s(s(s(0))), s(z0)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), z0)) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c7(FUNCTION(plus, z0, s(s(s(0))), s(z0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(0)), s(s(x0))), FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1))), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0)))) FUNCTION(if, false, s(s(0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1)))) S tuples: FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1))), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0)))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1)))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1)))) K tuples: FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(0)), s(s(x0))), FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c8_1, c7_1, c8_2, c4_1, c5_1 ---------------------------------------- (211) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) by FUNCTION(if, false, s(s(s(z0))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(s(function(p, s(z0), z0, z0))), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(s(function(p, s(0), 0, 0))), s(s(x0)))) FUNCTION(if, false, s(s(s(s(x1)))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(s(function(p, s(s(x1)), s(x1), s(x1)))), s(s(x0)))) ---------------------------------------- (212) Obligation: Complexity Dependency Tuples Problem Rules: function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), z1)) FUNCTION(if, false, s(s(s(0))), x0) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), x0)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c8(FUNCTION(plus, z0, s(s(s(0))), s(z0)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), z0)) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c7(FUNCTION(plus, z0, s(s(s(0))), s(z0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(0)), s(s(x0))), FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1))), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0)))) FUNCTION(if, false, s(s(0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(s(function(p, s(0), 0, 0))), s(s(x0)))) FUNCTION(if, false, s(s(s(s(x1)))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(s(function(p, s(s(x1)), s(x1), s(x1)))), s(s(x0)))) S tuples: FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1))), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0)))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1)))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(s(function(p, s(0), 0, 0))), s(s(x0)))) FUNCTION(if, false, s(s(s(s(x1)))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(s(function(p, s(s(x1)), s(x1), s(x1)))), s(s(x0)))) K tuples: FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(0)), s(s(x0))), FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c8_1, c8_2, c7_1, c4_1, c5_1 ---------------------------------------- (213) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FUNCTION(if, false, s(s(s(0))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(s(function(p, s(0), 0, 0))), s(s(x0)))) by FUNCTION(if, false, s(s(s(0))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) ---------------------------------------- (214) Obligation: Complexity Dependency Tuples Problem Rules: function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), z1)) FUNCTION(if, false, s(s(s(0))), x0) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), x0)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c8(FUNCTION(plus, z0, s(s(s(0))), s(z0)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), z0)) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c7(FUNCTION(plus, z0, s(s(s(0))), s(z0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(0)), s(s(x0))), FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1))), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0)))) FUNCTION(if, false, s(s(0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1)))) FUNCTION(if, false, s(s(s(s(x1)))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(s(function(p, s(s(x1)), s(x1), s(x1)))), s(s(x0)))) S tuples: FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1))), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0)))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1)))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1)))) FUNCTION(if, false, s(s(s(s(x1)))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(s(function(p, s(s(x1)), s(x1), s(x1)))), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) K tuples: FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(0)), s(s(x0))), FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c8_1, c8_2, c7_1, c4_1, c5_1 ---------------------------------------- (215) CdtKnowledgeProof (BOTH BOUNDS(ID, ID)) The following tuples could be moved from S to K by knowledge propagation: FUNCTION(if, false, s(s(s(0))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) ---------------------------------------- (216) Obligation: Complexity Dependency Tuples Problem Rules: function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), z1)) FUNCTION(if, false, s(s(s(0))), x0) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), x0)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c8(FUNCTION(plus, z0, s(s(s(0))), s(z0)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), z0)) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c7(FUNCTION(plus, z0, s(s(s(0))), s(z0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(0)), s(s(x0))), FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1))), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0)))) FUNCTION(if, false, s(s(0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1)))) FUNCTION(if, false, s(s(s(s(x1)))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(s(function(p, s(s(x1)), s(x1), s(x1)))), s(s(x0)))) S tuples: FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1))), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0)))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1)))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1)))) FUNCTION(if, false, s(s(s(s(x1)))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(s(function(p, s(s(x1)), s(x1), s(x1)))), s(s(x0)))) K tuples: FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(0)), s(s(x0))), FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c8_1, c8_2, c7_1, c4_1, c5_1 ---------------------------------------- (217) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FUNCTION(if, false, s(s(s(s(x1)))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(s(function(p, s(s(x1)), s(x1), s(x1)))), s(s(x0)))) by FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1)))) ---------------------------------------- (218) Obligation: Complexity Dependency Tuples Problem Rules: function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), z1)) FUNCTION(if, false, s(s(s(0))), x0) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), x0)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c8(FUNCTION(plus, z0, s(s(s(0))), s(z0)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), z0)) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c7(FUNCTION(plus, z0, s(s(s(0))), s(z0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(0)), s(s(x0))), FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1))), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0)))) FUNCTION(if, false, s(s(0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1)))) S tuples: FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1))), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0)))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1)))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1)))) K tuples: FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(0)), s(s(x0))), FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c8_1, c8_2, c7_1, c4_1, c5_1 ---------------------------------------- (219) RelTrsToWeightedTrsProof (UPPER BOUND(ID)) Transformed relative TRS to weighted TRS ---------------------------------------- (220) 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: function(iszero, 0, dummy, dummy2) -> true [1] function(iszero, s(x), dummy, dummy2) -> false [1] function(p, 0, dummy, dummy2) -> 0 [1] function(p, s(0), dummy, dummy2) -> 0 [1] function(p, s(s(x)), dummy, dummy2) -> s(function(p, s(x), x, x)) [1] function(plus, dummy, x, y) -> function(if, function(iszero, x, x, x), x, y) [1] function(if, true, x, y) -> y [1] function(if, false, x, y) -> function(plus, function(third, x, y, y), function(p, x, x, y), s(y)) [1] function(third, x, y, z) -> z [1] Rewrite Strategy: INNERMOST ---------------------------------------- (221) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (222) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: function(iszero, 0, dummy, dummy2) -> true [1] function(iszero, s(x), dummy, dummy2) -> false [1] function(p, 0, dummy, dummy2) -> 0 [1] function(p, s(0), dummy, dummy2) -> 0 [1] function(p, s(s(x)), dummy, dummy2) -> s(function(p, s(x), x, x)) [1] function(plus, dummy, x, y) -> function(if, function(iszero, x, x, x), x, y) [1] function(if, true, x, y) -> y [1] function(if, false, x, y) -> function(plus, function(third, x, y, y), function(p, x, x, y), s(y)) [1] function(third, x, y, z) -> z [1] The TRS has the following type information: function :: iszero:p:plus:if:third -> 0:true:s:false -> 0:true:s:false -> 0:true:s:false -> 0:true:s:false iszero :: iszero:p:plus:if:third 0 :: 0:true:s:false true :: 0:true:s:false s :: 0:true:s:false -> 0:true:s:false false :: 0:true:s:false p :: iszero:p:plus:if:third plus :: iszero:p:plus:if:third if :: iszero:p:plus:if:third third :: iszero:p:plus:if:third Rewrite Strategy: INNERMOST ---------------------------------------- (223) 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: none (c) The following functions are completely defined: function_4 Due to the following rules being added: function(v0, v1, v2, v3) -> null_function [0] And the following fresh constants: null_function ---------------------------------------- (224) 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: function(iszero, 0, dummy, dummy2) -> true [1] function(iszero, s(x), dummy, dummy2) -> false [1] function(p, 0, dummy, dummy2) -> 0 [1] function(p, s(0), dummy, dummy2) -> 0 [1] function(p, s(s(x)), dummy, dummy2) -> s(function(p, s(x), x, x)) [1] function(plus, dummy, x, y) -> function(if, function(iszero, x, x, x), x, y) [1] function(if, true, x, y) -> y [1] function(if, false, x, y) -> function(plus, function(third, x, y, y), function(p, x, x, y), s(y)) [1] function(third, x, y, z) -> z [1] function(v0, v1, v2, v3) -> null_function [0] The TRS has the following type information: function :: iszero:p:plus:if:third -> 0:true:s:false:null_function -> 0:true:s:false:null_function -> 0:true:s:false:null_function -> 0:true:s:false:null_function iszero :: iszero:p:plus:if:third 0 :: 0:true:s:false:null_function true :: 0:true:s:false:null_function s :: 0:true:s:false:null_function -> 0:true:s:false:null_function false :: 0:true:s:false:null_function p :: iszero:p:plus:if:third plus :: iszero:p:plus:if:third if :: iszero:p:plus:if:third third :: iszero:p:plus:if:third null_function :: 0:true:s:false:null_function Rewrite Strategy: INNERMOST ---------------------------------------- (225) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (226) 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: function(iszero, 0, dummy, dummy2) -> true [1] function(iszero, s(x), dummy, dummy2) -> false [1] function(p, 0, dummy, dummy2) -> 0 [1] function(p, s(0), dummy, dummy2) -> 0 [1] function(p, s(s(x)), dummy, dummy2) -> s(function(p, s(x), x, x)) [1] function(plus, dummy, 0, y) -> function(if, true, 0, y) [2] function(plus, dummy, s(x'), y) -> function(if, false, s(x'), y) [2] function(plus, dummy, x, y) -> function(if, null_function, x, y) [1] function(if, true, x, y) -> y [1] function(if, false, 0, y) -> function(plus, y, 0, s(y)) [3] function(if, false, s(0), y) -> function(plus, y, 0, s(y)) [3] function(if, false, s(s(x'')), y) -> function(plus, y, s(function(p, s(x''), x'', x'')), s(y)) [3] function(if, false, x, y) -> function(plus, y, null_function, s(y)) [2] function(if, false, 0, y) -> function(plus, null_function, 0, s(y)) [2] function(if, false, s(0), y) -> function(plus, null_function, 0, s(y)) [2] function(if, false, s(s(x1)), y) -> function(plus, null_function, s(function(p, s(x1), x1, x1)), s(y)) [2] function(if, false, x, y) -> function(plus, null_function, null_function, s(y)) [1] function(third, x, y, z) -> z [1] function(v0, v1, v2, v3) -> null_function [0] The TRS has the following type information: function :: iszero:p:plus:if:third -> 0:true:s:false:null_function -> 0:true:s:false:null_function -> 0:true:s:false:null_function -> 0:true:s:false:null_function iszero :: iszero:p:plus:if:third 0 :: 0:true:s:false:null_function true :: 0:true:s:false:null_function s :: 0:true:s:false:null_function -> 0:true:s:false:null_function false :: 0:true:s:false:null_function p :: iszero:p:plus:if:third plus :: iszero:p:plus:if:third if :: iszero:p:plus:if:third third :: iszero:p:plus:if:third null_function :: 0:true:s:false:null_function Rewrite Strategy: INNERMOST ---------------------------------------- (227) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: iszero => 1 0 => 0 true => 2 false => 1 p => 2 plus => 3 if => 0 third => 4 null_function => 0 ---------------------------------------- (228) Obligation: Complexity RNTS consisting of the following rules: function(z', z'', z1, z2) -{ 1 }-> y :|: z2 = y, x >= 0, y >= 0, z'' = 2, z1 = x, z' = 0 function(z', z'', z1, z2) -{ 1 }-> z :|: z' = 4, z1 = y, z >= 0, z2 = z, x >= 0, y >= 0, z'' = x function(z', z'', z1, z2) -{ 3 }-> function(3, y, 0, 1 + y) :|: z1 = 0, z2 = y, y >= 0, z'' = 1, z' = 0 function(z', z'', z1, z2) -{ 3 }-> function(3, y, 0, 1 + y) :|: z2 = y, z1 = 1 + 0, y >= 0, z'' = 1, z' = 0 function(z', z'', z1, z2) -{ 2 }-> function(3, y, 0, 1 + y) :|: z2 = y, x >= 0, y >= 0, z'' = 1, z1 = x, z' = 0 function(z', z'', z1, z2) -{ 3 }-> function(3, y, 1 + function(2, 1 + x'', x'', x''), 1 + y) :|: z2 = y, y >= 0, x'' >= 0, z'' = 1, z1 = 1 + (1 + x''), z' = 0 function(z', z'', z1, z2) -{ 2 }-> function(3, 0, 0, 1 + y) :|: z1 = 0, z2 = y, y >= 0, z'' = 1, z' = 0 function(z', z'', z1, z2) -{ 2 }-> function(3, 0, 0, 1 + y) :|: z2 = y, z1 = 1 + 0, y >= 0, z'' = 1, z' = 0 function(z', z'', z1, z2) -{ 1 }-> function(3, 0, 0, 1 + y) :|: z2 = y, x >= 0, y >= 0, z'' = 1, z1 = x, z' = 0 function(z', z'', z1, z2) -{ 2 }-> function(3, 0, 1 + function(2, 1 + x1, x1, x1), 1 + y) :|: z2 = y, x1 >= 0, y >= 0, z'' = 1, z' = 0, z1 = 1 + (1 + x1) function(z', z'', z1, z2) -{ 2 }-> function(0, 2, 0, y) :|: z' = 3, z1 = 0, z2 = y, dummy >= 0, y >= 0, z'' = dummy function(z', z'', z1, z2) -{ 2 }-> function(0, 1, 1 + x', y) :|: z' = 3, z2 = y, dummy >= 0, x' >= 0, y >= 0, z'' = dummy, z1 = 1 + x' function(z', z'', z1, z2) -{ 1 }-> function(0, 0, x, y) :|: z' = 3, z2 = y, dummy >= 0, x >= 0, y >= 0, z'' = dummy, z1 = x function(z', z'', z1, z2) -{ 1 }-> 2 :|: z'' = 0, dummy2 >= 0, z2 = dummy2, dummy >= 0, z' = 1, z1 = dummy function(z', z'', z1, z2) -{ 1 }-> 1 :|: dummy2 >= 0, z2 = dummy2, x >= 0, dummy >= 0, z' = 1, z'' = 1 + x, z1 = dummy function(z', z'', z1, z2) -{ 1 }-> 0 :|: z'' = 0, dummy2 >= 0, z' = 2, z2 = dummy2, dummy >= 0, z1 = dummy function(z', z'', z1, z2) -{ 1 }-> 0 :|: dummy2 >= 0, z' = 2, z2 = dummy2, dummy >= 0, z'' = 1 + 0, z1 = dummy function(z', z'', z1, z2) -{ 0 }-> 0 :|: z2 = v3, v0 >= 0, z1 = v2, v1 >= 0, z'' = v1, v2 >= 0, v3 >= 0, z' = v0 function(z', z'', z1, z2) -{ 1 }-> 1 + function(2, 1 + x, x, x) :|: dummy2 >= 0, z' = 2, z2 = dummy2, x >= 0, dummy >= 0, z'' = 1 + (1 + x), z1 = dummy ---------------------------------------- (229) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (230) Obligation: Complexity RNTS consisting of the following rules: function(z', z'', z1, z2) -{ 1 }-> z2 :|: z1 >= 0, z2 >= 0, z'' = 2, z' = 0 function(z', z'', z1, z2) -{ 1 }-> z2 :|: z' = 4, z2 >= 0, z'' >= 0, z1 >= 0 function(z', z'', z1, z2) -{ 3 }-> function(3, z2, 0, 1 + z2) :|: z1 = 0, z2 >= 0, z'' = 1, z' = 0 function(z', z'', z1, z2) -{ 3 }-> function(3, z2, 0, 1 + z2) :|: z1 = 1 + 0, z2 >= 0, z'' = 1, z' = 0 function(z', z'', z1, z2) -{ 2 }-> function(3, z2, 0, 1 + z2) :|: z1 >= 0, z2 >= 0, z'' = 1, z' = 0 function(z', z'', z1, z2) -{ 3 }-> function(3, z2, 1 + function(2, 1 + (z1 - 2), z1 - 2, z1 - 2), 1 + z2) :|: z2 >= 0, z1 - 2 >= 0, z'' = 1, z' = 0 function(z', z'', z1, z2) -{ 2 }-> function(3, 0, 0, 1 + z2) :|: z1 = 0, z2 >= 0, z'' = 1, z' = 0 function(z', z'', z1, z2) -{ 2 }-> function(3, 0, 0, 1 + z2) :|: z1 = 1 + 0, z2 >= 0, z'' = 1, z' = 0 function(z', z'', z1, z2) -{ 1 }-> function(3, 0, 0, 1 + z2) :|: z1 >= 0, z2 >= 0, z'' = 1, z' = 0 function(z', z'', z1, z2) -{ 2 }-> function(3, 0, 1 + function(2, 1 + (z1 - 2), z1 - 2, z1 - 2), 1 + z2) :|: z1 - 2 >= 0, z2 >= 0, z'' = 1, z' = 0 function(z', z'', z1, z2) -{ 2 }-> function(0, 2, 0, z2) :|: z' = 3, z1 = 0, z'' >= 0, z2 >= 0 function(z', z'', z1, z2) -{ 2 }-> function(0, 1, 1 + (z1 - 1), z2) :|: z' = 3, z'' >= 0, z1 - 1 >= 0, z2 >= 0 function(z', z'', z1, z2) -{ 1 }-> function(0, 0, z1, z2) :|: z' = 3, z'' >= 0, z1 >= 0, z2 >= 0 function(z', z'', z1, z2) -{ 1 }-> 2 :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1 function(z', z'', z1, z2) -{ 1 }-> 1 :|: z2 >= 0, z'' - 1 >= 0, z1 >= 0, z' = 1 function(z', z'', z1, z2) -{ 1 }-> 0 :|: z'' = 0, z2 >= 0, z' = 2, z1 >= 0 function(z', z'', z1, z2) -{ 1 }-> 0 :|: z2 >= 0, z' = 2, z1 >= 0, z'' = 1 + 0 function(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0 function(z', z'', z1, z2) -{ 1 }-> 1 + function(2, 1 + (z'' - 2), z'' - 2, z'' - 2) :|: z2 >= 0, z' = 2, z'' - 2 >= 0, z1 >= 0 ---------------------------------------- (231) 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: function(v0, v1, v2, v3) -> null_function [0] And the following fresh constants: null_function ---------------------------------------- (232) 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: function(iszero, 0, dummy, dummy2) -> true [1] function(iszero, s(x), dummy, dummy2) -> false [1] function(p, 0, dummy, dummy2) -> 0 [1] function(p, s(0), dummy, dummy2) -> 0 [1] function(p, s(s(x)), dummy, dummy2) -> s(function(p, s(x), x, x)) [1] function(plus, dummy, x, y) -> function(if, function(iszero, x, x, x), x, y) [1] function(if, true, x, y) -> y [1] function(if, false, x, y) -> function(plus, function(third, x, y, y), function(p, x, x, y), s(y)) [1] function(third, x, y, z) -> z [1] function(v0, v1, v2, v3) -> null_function [0] The TRS has the following type information: function :: iszero:p:plus:if:third -> 0:true:s:false:null_function -> 0:true:s:false:null_function -> 0:true:s:false:null_function -> 0:true:s:false:null_function iszero :: iszero:p:plus:if:third 0 :: 0:true:s:false:null_function true :: 0:true:s:false:null_function s :: 0:true:s:false:null_function -> 0:true:s:false:null_function false :: 0:true:s:false:null_function p :: iszero:p:plus:if:third plus :: iszero:p:plus:if:third if :: iszero:p:plus:if:third third :: iszero:p:plus:if:third null_function :: 0:true:s:false:null_function Rewrite Strategy: INNERMOST ---------------------------------------- (233) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: iszero => 1 0 => 0 true => 2 false => 1 p => 2 plus => 3 if => 0 third => 4 null_function => 0 ---------------------------------------- (234) Obligation: Complexity RNTS consisting of the following rules: function(z', z'', z1, z2) -{ 1 }-> y :|: z2 = y, x >= 0, y >= 0, z'' = 2, z1 = x, z' = 0 function(z', z'', z1, z2) -{ 1 }-> z :|: z' = 4, z1 = y, z >= 0, z2 = z, x >= 0, y >= 0, z'' = x function(z', z'', z1, z2) -{ 1 }-> function(3, function(4, x, y, y), function(2, x, x, y), 1 + y) :|: z2 = y, x >= 0, y >= 0, z'' = 1, z1 = x, z' = 0 function(z', z'', z1, z2) -{ 1 }-> function(0, function(1, x, x, x), x, y) :|: z' = 3, z2 = y, dummy >= 0, x >= 0, y >= 0, z'' = dummy, z1 = x function(z', z'', z1, z2) -{ 1 }-> 2 :|: z'' = 0, dummy2 >= 0, z2 = dummy2, dummy >= 0, z' = 1, z1 = dummy function(z', z'', z1, z2) -{ 1 }-> 1 :|: dummy2 >= 0, z2 = dummy2, x >= 0, dummy >= 0, z' = 1, z'' = 1 + x, z1 = dummy function(z', z'', z1, z2) -{ 1 }-> 0 :|: z'' = 0, dummy2 >= 0, z' = 2, z2 = dummy2, dummy >= 0, z1 = dummy function(z', z'', z1, z2) -{ 1 }-> 0 :|: dummy2 >= 0, z' = 2, z2 = dummy2, dummy >= 0, z'' = 1 + 0, z1 = dummy function(z', z'', z1, z2) -{ 0 }-> 0 :|: z2 = v3, v0 >= 0, z1 = v2, v1 >= 0, z'' = v1, v2 >= 0, v3 >= 0, z' = v0 function(z', z'', z1, z2) -{ 1 }-> 1 + function(2, 1 + x, x, x) :|: dummy2 >= 0, z' = 2, z2 = dummy2, x >= 0, dummy >= 0, z'' = 1 + (1 + x), z1 = dummy Only complete derivations are relevant for the runtime complexity.