KILLED proof of input_R9D8M5jaCS.trs # AProVE Commit ID: 5b976082cb74a395683ed8cc7acf94bd611ab29f fuhs 20230524 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).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 [UPPER BOUND(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), 0 ms] (12) typed CpxTrs (13) OrderProof [LOWER BOUND(ID), 3 ms] (14) typed CpxTrs (15) RewriteLemmaProof [LOWER BOUND(ID), 289 ms] (16) typed CpxTrs (17) RewriteLemmaProof [LOWER BOUND(ID), 101 ms] (18) BEST (19) proven lower bound (20) LowerBoundPropagationProof [FINISHED, 0 ms] (21) BOUNDS(n^1, INF) (22) typed CpxTrs (23) RewriteLemmaProof [LOWER BOUND(ID), 12.0 s] (24) typed CpxTrs (25) RewriteLemmaProof [LOWER BOUND(ID), 650 ms] (26) BOUNDS(1, INF) (27) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (28) CdtProblem (29) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (30) CdtProblem (31) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (32) CdtProblem (33) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (34) CdtProblem (35) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms] (36) CpxRelTRS (37) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (38) CpxTRS (39) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms] (40) CpxWeightedTrs (41) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (42) CpxTypedWeightedTrs (43) CompletionProof [UPPER BOUND(ID), 0 ms] (44) CpxTypedWeightedCompleteTrs (45) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (46) CpxTypedWeightedCompleteTrs (47) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (48) CpxRNTS (49) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (50) CpxRNTS (51) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms] (52) CpxRNTS (53) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (54) CpxRNTS (55) IntTrsBoundProof [UPPER BOUND(ID), 271 ms] (56) CpxRNTS (57) IntTrsBoundProof [UPPER BOUND(ID), 115 ms] (58) CpxRNTS (59) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (60) CpxRNTS (61) IntTrsBoundProof [UPPER BOUND(ID), 209 ms] (62) CpxRNTS (63) IntTrsBoundProof [UPPER BOUND(ID), 84 ms] (64) CpxRNTS (65) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (66) CpxRNTS (67) IntTrsBoundProof [UPPER BOUND(ID), 130 ms] (68) CpxRNTS (69) IntTrsBoundProof [UPPER BOUND(ID), 4 ms] (70) CpxRNTS (71) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (72) CpxRNTS (73) IntTrsBoundProof [UPPER BOUND(ID), 178 ms] (74) CpxRNTS (75) IntTrsBoundProof [UPPER BOUND(ID), 2 ms] (76) CpxRNTS (77) ResultPropagationProof [UPPER BOUND(ID), 1 ms] (78) CpxRNTS (79) IntTrsBoundProof [UPPER BOUND(ID), 168 ms] (80) CpxRNTS (81) IntTrsBoundProof [UPPER BOUND(ID), 52 ms] (82) CpxRNTS (83) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (84) CpxRNTS (85) IntTrsBoundProof [UPPER BOUND(ID), 4329 ms] (86) CpxRNTS (87) IntTrsBoundProof [UPPER BOUND(ID), 550 ms] (88) CpxRNTS (89) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (90) CpxRNTS (91) IntTrsBoundProof [UPPER BOUND(ID), 1638 ms] (92) CpxRNTS (93) IntTrsBoundProof [UPPER BOUND(ID), 656 ms] (94) CpxRNTS (95) CompletionProof [UPPER BOUND(ID), 0 ms] (96) CpxTypedWeightedCompleteTrs (97) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (98) CpxRNTS (99) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (100) CdtProblem (101) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (102) CdtProblem (103) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (104) CdtProblem (105) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (106) CdtProblem (107) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (108) CdtProblem (109) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (110) CdtProblem (111) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (112) CdtProblem (113) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 63 ms] (114) CdtProblem (115) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (116) CdtProblem (117) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (118) CdtProblem (119) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (120) CdtProblem (121) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (122) CdtProblem (123) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (124) CdtProblem (125) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (126) CdtProblem (127) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (128) CdtProblem (129) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (130) CdtProblem (131) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (132) CdtProblem (133) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (134) CdtProblem (135) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms] (136) CdtProblem (137) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 51 ms] (138) CdtProblem (139) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (140) CdtProblem (141) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (142) CdtProblem (143) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms] (144) CdtProblem (145) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (146) CdtProblem (147) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 37 ms] (148) CdtProblem (149) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (150) CdtProblem (151) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (152) CdtProblem (153) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (154) CdtProblem (155) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (156) CdtProblem (157) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 2 ms] (158) CdtProblem (159) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (160) CdtProblem (161) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (162) CdtProblem (163) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (164) CdtProblem (165) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (166) CdtProblem (167) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (168) CdtProblem (169) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (170) CdtProblem (171) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (172) CdtProblem (173) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (174) CdtProblem (175) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (176) CdtProblem (177) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (178) CdtProblem (179) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (180) CdtProblem (181) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (182) CdtProblem (183) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (184) CdtProblem (185) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (186) CdtProblem (187) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (188) CdtProblem (189) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (190) CdtProblem (191) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (192) CdtProblem (193) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (194) CdtProblem (195) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (196) CdtProblem (197) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (198) CdtProblem (199) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (200) CdtProblem (201) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (202) CdtProblem (203) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (204) CdtProblem (205) CdtGraphSplitRhsProof [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) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (214) CdtProblem (215) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (216) CdtProblem (217) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (218) CdtProblem (219) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (220) CdtProblem (221) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (222) CdtProblem (223) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (224) CdtProblem (225) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (226) CdtProblem (227) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (228) CdtProblem (229) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms] (230) CpxWeightedTrs (231) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (232) CpxTypedWeightedTrs (233) CompletionProof [UPPER BOUND(ID), 0 ms] (234) CpxTypedWeightedCompleteTrs (235) NarrowingProof [BOTH BOUNDS(ID, ID), 3 ms] (236) CpxTypedWeightedCompleteTrs (237) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (238) CpxRNTS (239) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (240) CpxRNTS (241) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms] (242) CpxRNTS (243) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (244) CpxRNTS (245) IntTrsBoundProof [UPPER BOUND(ID), 360 ms] (246) CpxRNTS (247) IntTrsBoundProof [UPPER BOUND(ID), 120 ms] (248) CpxRNTS (249) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (250) CpxRNTS (251) IntTrsBoundProof [UPPER BOUND(ID), 90 ms] (252) CpxRNTS (253) IntTrsBoundProof [UPPER BOUND(ID), 63 ms] (254) CpxRNTS (255) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (256) CpxRNTS (257) IntTrsBoundProof [UPPER BOUND(ID), 201 ms] (258) CpxRNTS (259) IntTrsBoundProof [UPPER BOUND(ID), 65 ms] (260) CpxRNTS (261) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (262) CpxRNTS (263) IntTrsBoundProof [UPPER BOUND(ID), 199 ms] (264) CpxRNTS (265) IntTrsBoundProof [UPPER BOUND(ID), 2 ms] (266) CpxRNTS (267) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (268) CpxRNTS (269) IntTrsBoundProof [UPPER BOUND(ID), 97 ms] (270) CpxRNTS (271) IntTrsBoundProof [UPPER BOUND(ID), 1 ms] (272) CpxRNTS (273) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (274) CpxRNTS (275) IntTrsBoundProof [UPPER BOUND(ID), 4518 ms] (276) CpxRNTS (277) IntTrsBoundProof [UPPER BOUND(ID), 788 ms] (278) CpxRNTS (279) CompletionProof [UPPER BOUND(ID), 0 ms] (280) CpxTypedWeightedCompleteTrs (281) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (282) 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: plus(x, y) -> plusIter(x, y, 0) plusIter(x, y, z) -> ifPlus(le(x, z), x, y, z) ifPlus(true, x, y, z) -> y ifPlus(false, x, y, z) -> plusIter(x, s(y), s(z)) le(s(x), 0) -> false le(0, y) -> true le(s(x), s(y)) -> le(x, y) sum(xs) -> sumIter(xs, 0) sumIter(xs, x) -> ifSum(isempty(xs), xs, x, plus(x, head(xs))) ifSum(true, xs, x, y) -> x ifSum(false, xs, x, y) -> sumIter(tail(xs), y) isempty(nil) -> true isempty(cons(x, xs)) -> false head(nil) -> error head(cons(x, xs)) -> x tail(nil) -> nil tail(cons(x, xs)) -> xs a -> b a -> c 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: plus(x, y) -> plusIter(x, y, 0') plusIter(x, y, z) -> ifPlus(le(x, z), x, y, z) ifPlus(true, x, y, z) -> y ifPlus(false, x, y, z) -> plusIter(x, s(y), s(z)) le(s(x), 0') -> false le(0', y) -> true le(s(x), s(y)) -> le(x, y) sum(xs) -> sumIter(xs, 0') sumIter(xs, x) -> ifSum(isempty(xs), xs, x, plus(x, head(xs))) ifSum(true, xs, x, y) -> x ifSum(false, xs, x, y) -> sumIter(tail(xs), y) isempty(nil) -> true isempty(cons(x, xs)) -> false head(nil) -> error head(cons(x, xs)) -> x tail(nil) -> nil tail(cons(x, xs)) -> xs a -> b a -> c 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: plus(x, y) -> plusIter(x, y, 0) plusIter(x, y, z) -> ifPlus(le(x, z), x, y, z) ifPlus(true, x, y, z) -> y ifPlus(false, x, y, z) -> plusIter(x, s(y), s(z)) le(s(x), 0) -> false le(0, y) -> true le(s(x), s(y)) -> le(x, y) sum(xs) -> sumIter(xs, 0) sumIter(xs, x) -> ifSum(isempty(xs), xs, x, plus(x, head(xs))) ifSum(true, xs, x, y) -> x ifSum(false, xs, x, y) -> sumIter(tail(xs), y) isempty(nil) -> true isempty(cons(x, xs)) -> false head(nil) -> error head(cons(x, xs)) -> x tail(nil) -> nil tail(cons(x, xs)) -> xs a -> b a -> c S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (5) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (6) Obligation: Complexity Dependency Tuples Problem Rules: plus(z0, z1) -> plusIter(z0, z1, 0) plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2) ifPlus(true, z0, z1, z2) -> z1 ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2)) le(s(z0), 0) -> false le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) sum(z0) -> sumIter(z0, 0) sumIter(z0, z1) -> ifSum(isempty(z0), z0, z1, plus(z1, head(z0))) ifSum(true, z0, z1, z2) -> z1 ifSum(false, z0, z1, z2) -> sumIter(tail(z0), z2) isempty(nil) -> true isempty(cons(z0, z1)) -> false head(nil) -> error head(cons(z0, z1)) -> z0 tail(nil) -> nil tail(cons(z0, z1)) -> z1 a -> b a -> c Tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) PLUSITER(z0, z1, z2) -> c2(IFPLUS(le(z0, z2), z0, z1, z2), LE(z0, z2)) IFPLUS(true, z0, z1, z2) -> c3 IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2))) LE(s(z0), 0) -> c5 LE(0, z0) -> c6 LE(s(z0), s(z1)) -> c7(LE(z0, z1)) SUM(z0) -> c8(SUMITER(z0, 0)) SUMITER(z0, z1) -> c9(IFSUM(isempty(z0), z0, z1, plus(z1, head(z0))), ISEMPTY(z0)) SUMITER(z0, z1) -> c10(IFSUM(isempty(z0), z0, z1, plus(z1, head(z0))), PLUS(z1, head(z0)), HEAD(z0)) IFSUM(true, z0, z1, z2) -> c11 IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2), TAIL(z0)) ISEMPTY(nil) -> c13 ISEMPTY(cons(z0, z1)) -> c14 HEAD(nil) -> c15 HEAD(cons(z0, z1)) -> c16 TAIL(nil) -> c17 TAIL(cons(z0, z1)) -> c18 A -> c19 A -> c20 S tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) PLUSITER(z0, z1, z2) -> c2(IFPLUS(le(z0, z2), z0, z1, z2), LE(z0, z2)) IFPLUS(true, z0, z1, z2) -> c3 IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2))) LE(s(z0), 0) -> c5 LE(0, z0) -> c6 LE(s(z0), s(z1)) -> c7(LE(z0, z1)) SUM(z0) -> c8(SUMITER(z0, 0)) SUMITER(z0, z1) -> c9(IFSUM(isempty(z0), z0, z1, plus(z1, head(z0))), ISEMPTY(z0)) SUMITER(z0, z1) -> c10(IFSUM(isempty(z0), z0, z1, plus(z1, head(z0))), PLUS(z1, head(z0)), HEAD(z0)) IFSUM(true, z0, z1, z2) -> c11 IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2), TAIL(z0)) ISEMPTY(nil) -> c13 ISEMPTY(cons(z0, z1)) -> c14 HEAD(nil) -> c15 HEAD(cons(z0, z1)) -> c16 TAIL(nil) -> c17 TAIL(cons(z0, z1)) -> c18 A -> c19 A -> c20 K tuples:none Defined Rule Symbols: plus_2, plusIter_3, ifPlus_4, le_2, sum_1, sumIter_2, ifSum_4, isempty_1, head_1, tail_1, a Defined Pair Symbols: PLUS_2, PLUSITER_3, IFPLUS_4, LE_2, SUM_1, SUMITER_2, IFSUM_4, ISEMPTY_1, HEAD_1, TAIL_1, A Compound Symbols: c1_1, c2_2, c3, c4_1, c5, c6, c7_1, c8_1, c9_2, c10_3, c11, c12_2, c13, c14, c15, c16, c17, c18, c19, c20 ---------------------------------------- (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(n^1, INF). The TRS R consists of the following rules: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) PLUSITER(z0, z1, z2) -> c2(IFPLUS(le(z0, z2), z0, z1, z2), LE(z0, z2)) IFPLUS(true, z0, z1, z2) -> c3 IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2))) LE(s(z0), 0) -> c5 LE(0, z0) -> c6 LE(s(z0), s(z1)) -> c7(LE(z0, z1)) SUM(z0) -> c8(SUMITER(z0, 0)) SUMITER(z0, z1) -> c9(IFSUM(isempty(z0), z0, z1, plus(z1, head(z0))), ISEMPTY(z0)) SUMITER(z0, z1) -> c10(IFSUM(isempty(z0), z0, z1, plus(z1, head(z0))), PLUS(z1, head(z0)), HEAD(z0)) IFSUM(true, z0, z1, z2) -> c11 IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2), TAIL(z0)) ISEMPTY(nil) -> c13 ISEMPTY(cons(z0, z1)) -> c14 HEAD(nil) -> c15 HEAD(cons(z0, z1)) -> c16 TAIL(nil) -> c17 TAIL(cons(z0, z1)) -> c18 A -> c19 A -> c20 The (relative) TRS S consists of the following rules: plus(z0, z1) -> plusIter(z0, z1, 0) plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2) ifPlus(true, z0, z1, z2) -> z1 ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2)) le(s(z0), 0) -> false le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) sum(z0) -> sumIter(z0, 0) sumIter(z0, z1) -> ifSum(isempty(z0), z0, z1, plus(z1, head(z0))) ifSum(true, z0, z1, z2) -> z1 ifSum(false, z0, z1, z2) -> sumIter(tail(z0), z2) isempty(nil) -> true isempty(cons(z0, z1)) -> false head(nil) -> error head(cons(z0, z1)) -> z0 tail(nil) -> nil tail(cons(z0, z1)) -> z1 a -> b a -> c 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(n^1, INF). The TRS R consists of the following rules: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0')) PLUSITER(z0, z1, z2) -> c2(IFPLUS(le(z0, z2), z0, z1, z2), LE(z0, z2)) IFPLUS(true, z0, z1, z2) -> c3 IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2))) LE(s(z0), 0') -> c5 LE(0', z0) -> c6 LE(s(z0), s(z1)) -> c7(LE(z0, z1)) SUM(z0) -> c8(SUMITER(z0, 0')) SUMITER(z0, z1) -> c9(IFSUM(isempty(z0), z0, z1, plus(z1, head(z0))), ISEMPTY(z0)) SUMITER(z0, z1) -> c10(IFSUM(isempty(z0), z0, z1, plus(z1, head(z0))), PLUS(z1, head(z0)), HEAD(z0)) IFSUM(true, z0, z1, z2) -> c11 IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2), TAIL(z0)) ISEMPTY(nil) -> c13 ISEMPTY(cons(z0, z1)) -> c14 HEAD(nil) -> c15 HEAD(cons(z0, z1)) -> c16 TAIL(nil) -> c17 TAIL(cons(z0, z1)) -> c18 A -> c19 A -> c20 The (relative) TRS S consists of the following rules: plus(z0, z1) -> plusIter(z0, z1, 0') plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2) ifPlus(true, z0, z1, z2) -> z1 ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2)) le(s(z0), 0') -> false le(0', z0) -> true le(s(z0), s(z1)) -> le(z0, z1) sum(z0) -> sumIter(z0, 0') sumIter(z0, z1) -> ifSum(isempty(z0), z0, z1, plus(z1, head(z0))) ifSum(true, z0, z1, z2) -> z1 ifSum(false, z0, z1, z2) -> sumIter(tail(z0), z2) isempty(nil) -> true isempty(cons(z0, z1)) -> false head(nil) -> error head(cons(z0, z1)) -> z0 tail(nil) -> nil tail(cons(z0, z1)) -> z1 a -> b a -> c Rewrite Strategy: INNERMOST ---------------------------------------- (11) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Inferred types. ---------------------------------------- (12) Obligation: Innermost TRS: Rules: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0')) PLUSITER(z0, z1, z2) -> c2(IFPLUS(le(z0, z2), z0, z1, z2), LE(z0, z2)) IFPLUS(true, z0, z1, z2) -> c3 IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2))) LE(s(z0), 0') -> c5 LE(0', z0) -> c6 LE(s(z0), s(z1)) -> c7(LE(z0, z1)) SUM(z0) -> c8(SUMITER(z0, 0')) SUMITER(z0, z1) -> c9(IFSUM(isempty(z0), z0, z1, plus(z1, head(z0))), ISEMPTY(z0)) SUMITER(z0, z1) -> c10(IFSUM(isempty(z0), z0, z1, plus(z1, head(z0))), PLUS(z1, head(z0)), HEAD(z0)) IFSUM(true, z0, z1, z2) -> c11 IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2), TAIL(z0)) ISEMPTY(nil) -> c13 ISEMPTY(cons(z0, z1)) -> c14 HEAD(nil) -> c15 HEAD(cons(z0, z1)) -> c16 TAIL(nil) -> c17 TAIL(cons(z0, z1)) -> c18 A -> c19 A -> c20 plus(z0, z1) -> plusIter(z0, z1, 0') plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2) ifPlus(true, z0, z1, z2) -> z1 ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2)) le(s(z0), 0') -> false le(0', z0) -> true le(s(z0), s(z1)) -> le(z0, z1) sum(z0) -> sumIter(z0, 0') sumIter(z0, z1) -> ifSum(isempty(z0), z0, z1, plus(z1, head(z0))) ifSum(true, z0, z1, z2) -> z1 ifSum(false, z0, z1, z2) -> sumIter(tail(z0), z2) isempty(nil) -> true isempty(cons(z0, z1)) -> false head(nil) -> error head(cons(z0, z1)) -> z0 tail(nil) -> nil tail(cons(z0, z1)) -> z1 a -> b a -> c Types: PLUS :: 0':s:error -> 0':s:error -> c1 c1 :: c2 -> c1 PLUSITER :: 0':s:error -> 0':s:error -> 0':s:error -> c2 0' :: 0':s:error c2 :: c3:c4 -> c5:c6:c7 -> c2 IFPLUS :: true:false -> 0':s:error -> 0':s:error -> 0':s:error -> c3:c4 le :: 0':s:error -> 0':s:error -> true:false LE :: 0':s:error -> 0':s:error -> c5:c6:c7 true :: true:false c3 :: c3:c4 false :: true:false c4 :: c2 -> c3:c4 s :: 0':s:error -> 0':s:error c5 :: c5:c6:c7 c6 :: c5:c6:c7 c7 :: c5:c6:c7 -> c5:c6:c7 SUM :: nil:cons -> c8 c8 :: c9:c10 -> c8 SUMITER :: nil:cons -> 0':s:error -> c9:c10 c9 :: c11:c12 -> c13:c14 -> c9:c10 IFSUM :: true:false -> nil:cons -> 0':s:error -> 0':s:error -> c11:c12 isempty :: nil:cons -> true:false plus :: 0':s:error -> 0':s:error -> 0':s:error head :: nil:cons -> 0':s:error ISEMPTY :: nil:cons -> c13:c14 c10 :: c11:c12 -> c1 -> c15:c16 -> c9:c10 HEAD :: nil:cons -> c15:c16 c11 :: c11:c12 c12 :: c9:c10 -> c17:c18 -> c11:c12 tail :: nil:cons -> nil:cons TAIL :: nil:cons -> c17:c18 nil :: nil:cons c13 :: c13:c14 cons :: 0':s:error -> nil:cons -> nil:cons c14 :: c13:c14 c15 :: c15:c16 c16 :: c15:c16 c17 :: c17:c18 c18 :: c17:c18 A :: c19:c20 c19 :: c19:c20 c20 :: c19:c20 plusIter :: 0':s:error -> 0':s:error -> 0':s:error -> 0':s:error ifPlus :: true:false -> 0':s:error -> 0':s:error -> 0':s:error -> 0':s:error sum :: nil:cons -> 0':s:error sumIter :: nil:cons -> 0':s:error -> 0':s:error ifSum :: true:false -> nil:cons -> 0':s:error -> 0':s:error -> 0':s:error error :: 0':s:error a :: b:c b :: b:c c :: b:c hole_c11_21 :: c1 hole_0':s:error2_21 :: 0':s:error hole_c23_21 :: c2 hole_c3:c44_21 :: c3:c4 hole_c5:c6:c75_21 :: c5:c6:c7 hole_true:false6_21 :: true:false hole_c87_21 :: c8 hole_nil:cons8_21 :: nil:cons hole_c9:c109_21 :: c9:c10 hole_c11:c1210_21 :: c11:c12 hole_c13:c1411_21 :: c13:c14 hole_c15:c1612_21 :: c15:c16 hole_c17:c1813_21 :: c17:c18 hole_c19:c2014_21 :: c19:c20 hole_b:c15_21 :: b:c gen_0':s:error16_21 :: Nat -> 0':s:error gen_c5:c6:c717_21 :: Nat -> c5:c6:c7 gen_nil:cons18_21 :: Nat -> nil:cons ---------------------------------------- (13) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: PLUSITER, le, LE, SUMITER, plusIter, sumIter They will be analysed ascendingly in the following order: le < PLUSITER LE < PLUSITER le < plusIter ---------------------------------------- (14) Obligation: Innermost TRS: Rules: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0')) PLUSITER(z0, z1, z2) -> c2(IFPLUS(le(z0, z2), z0, z1, z2), LE(z0, z2)) IFPLUS(true, z0, z1, z2) -> c3 IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2))) LE(s(z0), 0') -> c5 LE(0', z0) -> c6 LE(s(z0), s(z1)) -> c7(LE(z0, z1)) SUM(z0) -> c8(SUMITER(z0, 0')) SUMITER(z0, z1) -> c9(IFSUM(isempty(z0), z0, z1, plus(z1, head(z0))), ISEMPTY(z0)) SUMITER(z0, z1) -> c10(IFSUM(isempty(z0), z0, z1, plus(z1, head(z0))), PLUS(z1, head(z0)), HEAD(z0)) IFSUM(true, z0, z1, z2) -> c11 IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2), TAIL(z0)) ISEMPTY(nil) -> c13 ISEMPTY(cons(z0, z1)) -> c14 HEAD(nil) -> c15 HEAD(cons(z0, z1)) -> c16 TAIL(nil) -> c17 TAIL(cons(z0, z1)) -> c18 A -> c19 A -> c20 plus(z0, z1) -> plusIter(z0, z1, 0') plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2) ifPlus(true, z0, z1, z2) -> z1 ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2)) le(s(z0), 0') -> false le(0', z0) -> true le(s(z0), s(z1)) -> le(z0, z1) sum(z0) -> sumIter(z0, 0') sumIter(z0, z1) -> ifSum(isempty(z0), z0, z1, plus(z1, head(z0))) ifSum(true, z0, z1, z2) -> z1 ifSum(false, z0, z1, z2) -> sumIter(tail(z0), z2) isempty(nil) -> true isempty(cons(z0, z1)) -> false head(nil) -> error head(cons(z0, z1)) -> z0 tail(nil) -> nil tail(cons(z0, z1)) -> z1 a -> b a -> c Types: PLUS :: 0':s:error -> 0':s:error -> c1 c1 :: c2 -> c1 PLUSITER :: 0':s:error -> 0':s:error -> 0':s:error -> c2 0' :: 0':s:error c2 :: c3:c4 -> c5:c6:c7 -> c2 IFPLUS :: true:false -> 0':s:error -> 0':s:error -> 0':s:error -> c3:c4 le :: 0':s:error -> 0':s:error -> true:false LE :: 0':s:error -> 0':s:error -> c5:c6:c7 true :: true:false c3 :: c3:c4 false :: true:false c4 :: c2 -> c3:c4 s :: 0':s:error -> 0':s:error c5 :: c5:c6:c7 c6 :: c5:c6:c7 c7 :: c5:c6:c7 -> c5:c6:c7 SUM :: nil:cons -> c8 c8 :: c9:c10 -> c8 SUMITER :: nil:cons -> 0':s:error -> c9:c10 c9 :: c11:c12 -> c13:c14 -> c9:c10 IFSUM :: true:false -> nil:cons -> 0':s:error -> 0':s:error -> c11:c12 isempty :: nil:cons -> true:false plus :: 0':s:error -> 0':s:error -> 0':s:error head :: nil:cons -> 0':s:error ISEMPTY :: nil:cons -> c13:c14 c10 :: c11:c12 -> c1 -> c15:c16 -> c9:c10 HEAD :: nil:cons -> c15:c16 c11 :: c11:c12 c12 :: c9:c10 -> c17:c18 -> c11:c12 tail :: nil:cons -> nil:cons TAIL :: nil:cons -> c17:c18 nil :: nil:cons c13 :: c13:c14 cons :: 0':s:error -> nil:cons -> nil:cons c14 :: c13:c14 c15 :: c15:c16 c16 :: c15:c16 c17 :: c17:c18 c18 :: c17:c18 A :: c19:c20 c19 :: c19:c20 c20 :: c19:c20 plusIter :: 0':s:error -> 0':s:error -> 0':s:error -> 0':s:error ifPlus :: true:false -> 0':s:error -> 0':s:error -> 0':s:error -> 0':s:error sum :: nil:cons -> 0':s:error sumIter :: nil:cons -> 0':s:error -> 0':s:error ifSum :: true:false -> nil:cons -> 0':s:error -> 0':s:error -> 0':s:error error :: 0':s:error a :: b:c b :: b:c c :: b:c hole_c11_21 :: c1 hole_0':s:error2_21 :: 0':s:error hole_c23_21 :: c2 hole_c3:c44_21 :: c3:c4 hole_c5:c6:c75_21 :: c5:c6:c7 hole_true:false6_21 :: true:false hole_c87_21 :: c8 hole_nil:cons8_21 :: nil:cons hole_c9:c109_21 :: c9:c10 hole_c11:c1210_21 :: c11:c12 hole_c13:c1411_21 :: c13:c14 hole_c15:c1612_21 :: c15:c16 hole_c17:c1813_21 :: c17:c18 hole_c19:c2014_21 :: c19:c20 hole_b:c15_21 :: b:c gen_0':s:error16_21 :: Nat -> 0':s:error gen_c5:c6:c717_21 :: Nat -> c5:c6:c7 gen_nil:cons18_21 :: Nat -> nil:cons Generator Equations: gen_0':s:error16_21(0) <=> 0' gen_0':s:error16_21(+(x, 1)) <=> s(gen_0':s:error16_21(x)) gen_c5:c6:c717_21(0) <=> c5 gen_c5:c6:c717_21(+(x, 1)) <=> c7(gen_c5:c6:c717_21(x)) gen_nil:cons18_21(0) <=> nil gen_nil:cons18_21(+(x, 1)) <=> cons(0', gen_nil:cons18_21(x)) The following defined symbols remain to be analysed: le, PLUSITER, LE, SUMITER, plusIter, sumIter They will be analysed ascendingly in the following order: le < PLUSITER LE < PLUSITER le < plusIter ---------------------------------------- (15) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: le(gen_0':s:error16_21(+(1, n20_21)), gen_0':s:error16_21(n20_21)) -> false, rt in Omega(0) Induction Base: le(gen_0':s:error16_21(+(1, 0)), gen_0':s:error16_21(0)) ->_R^Omega(0) false Induction Step: le(gen_0':s:error16_21(+(1, +(n20_21, 1))), gen_0':s:error16_21(+(n20_21, 1))) ->_R^Omega(0) le(gen_0':s:error16_21(+(1, n20_21)), gen_0':s:error16_21(n20_21)) ->_IH false We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0). ---------------------------------------- (16) Obligation: Innermost TRS: Rules: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0')) PLUSITER(z0, z1, z2) -> c2(IFPLUS(le(z0, z2), z0, z1, z2), LE(z0, z2)) IFPLUS(true, z0, z1, z2) -> c3 IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2))) LE(s(z0), 0') -> c5 LE(0', z0) -> c6 LE(s(z0), s(z1)) -> c7(LE(z0, z1)) SUM(z0) -> c8(SUMITER(z0, 0')) SUMITER(z0, z1) -> c9(IFSUM(isempty(z0), z0, z1, plus(z1, head(z0))), ISEMPTY(z0)) SUMITER(z0, z1) -> c10(IFSUM(isempty(z0), z0, z1, plus(z1, head(z0))), PLUS(z1, head(z0)), HEAD(z0)) IFSUM(true, z0, z1, z2) -> c11 IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2), TAIL(z0)) ISEMPTY(nil) -> c13 ISEMPTY(cons(z0, z1)) -> c14 HEAD(nil) -> c15 HEAD(cons(z0, z1)) -> c16 TAIL(nil) -> c17 TAIL(cons(z0, z1)) -> c18 A -> c19 A -> c20 plus(z0, z1) -> plusIter(z0, z1, 0') plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2) ifPlus(true, z0, z1, z2) -> z1 ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2)) le(s(z0), 0') -> false le(0', z0) -> true le(s(z0), s(z1)) -> le(z0, z1) sum(z0) -> sumIter(z0, 0') sumIter(z0, z1) -> ifSum(isempty(z0), z0, z1, plus(z1, head(z0))) ifSum(true, z0, z1, z2) -> z1 ifSum(false, z0, z1, z2) -> sumIter(tail(z0), z2) isempty(nil) -> true isempty(cons(z0, z1)) -> false head(nil) -> error head(cons(z0, z1)) -> z0 tail(nil) -> nil tail(cons(z0, z1)) -> z1 a -> b a -> c Types: PLUS :: 0':s:error -> 0':s:error -> c1 c1 :: c2 -> c1 PLUSITER :: 0':s:error -> 0':s:error -> 0':s:error -> c2 0' :: 0':s:error c2 :: c3:c4 -> c5:c6:c7 -> c2 IFPLUS :: true:false -> 0':s:error -> 0':s:error -> 0':s:error -> c3:c4 le :: 0':s:error -> 0':s:error -> true:false LE :: 0':s:error -> 0':s:error -> c5:c6:c7 true :: true:false c3 :: c3:c4 false :: true:false c4 :: c2 -> c3:c4 s :: 0':s:error -> 0':s:error c5 :: c5:c6:c7 c6 :: c5:c6:c7 c7 :: c5:c6:c7 -> c5:c6:c7 SUM :: nil:cons -> c8 c8 :: c9:c10 -> c8 SUMITER :: nil:cons -> 0':s:error -> c9:c10 c9 :: c11:c12 -> c13:c14 -> c9:c10 IFSUM :: true:false -> nil:cons -> 0':s:error -> 0':s:error -> c11:c12 isempty :: nil:cons -> true:false plus :: 0':s:error -> 0':s:error -> 0':s:error head :: nil:cons -> 0':s:error ISEMPTY :: nil:cons -> c13:c14 c10 :: c11:c12 -> c1 -> c15:c16 -> c9:c10 HEAD :: nil:cons -> c15:c16 c11 :: c11:c12 c12 :: c9:c10 -> c17:c18 -> c11:c12 tail :: nil:cons -> nil:cons TAIL :: nil:cons -> c17:c18 nil :: nil:cons c13 :: c13:c14 cons :: 0':s:error -> nil:cons -> nil:cons c14 :: c13:c14 c15 :: c15:c16 c16 :: c15:c16 c17 :: c17:c18 c18 :: c17:c18 A :: c19:c20 c19 :: c19:c20 c20 :: c19:c20 plusIter :: 0':s:error -> 0':s:error -> 0':s:error -> 0':s:error ifPlus :: true:false -> 0':s:error -> 0':s:error -> 0':s:error -> 0':s:error sum :: nil:cons -> 0':s:error sumIter :: nil:cons -> 0':s:error -> 0':s:error ifSum :: true:false -> nil:cons -> 0':s:error -> 0':s:error -> 0':s:error error :: 0':s:error a :: b:c b :: b:c c :: b:c hole_c11_21 :: c1 hole_0':s:error2_21 :: 0':s:error hole_c23_21 :: c2 hole_c3:c44_21 :: c3:c4 hole_c5:c6:c75_21 :: c5:c6:c7 hole_true:false6_21 :: true:false hole_c87_21 :: c8 hole_nil:cons8_21 :: nil:cons hole_c9:c109_21 :: c9:c10 hole_c11:c1210_21 :: c11:c12 hole_c13:c1411_21 :: c13:c14 hole_c15:c1612_21 :: c15:c16 hole_c17:c1813_21 :: c17:c18 hole_c19:c2014_21 :: c19:c20 hole_b:c15_21 :: b:c gen_0':s:error16_21 :: Nat -> 0':s:error gen_c5:c6:c717_21 :: Nat -> c5:c6:c7 gen_nil:cons18_21 :: Nat -> nil:cons Lemmas: le(gen_0':s:error16_21(+(1, n20_21)), gen_0':s:error16_21(n20_21)) -> false, rt in Omega(0) Generator Equations: gen_0':s:error16_21(0) <=> 0' gen_0':s:error16_21(+(x, 1)) <=> s(gen_0':s:error16_21(x)) gen_c5:c6:c717_21(0) <=> c5 gen_c5:c6:c717_21(+(x, 1)) <=> c7(gen_c5:c6:c717_21(x)) gen_nil:cons18_21(0) <=> nil gen_nil:cons18_21(+(x, 1)) <=> cons(0', gen_nil:cons18_21(x)) The following defined symbols remain to be analysed: LE, PLUSITER, SUMITER, plusIter, sumIter They will be analysed ascendingly in the following order: LE < PLUSITER ---------------------------------------- (17) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: LE(gen_0':s:error16_21(+(1, n505_21)), gen_0':s:error16_21(n505_21)) -> gen_c5:c6:c717_21(n505_21), rt in Omega(1 + n505_21) Induction Base: LE(gen_0':s:error16_21(+(1, 0)), gen_0':s:error16_21(0)) ->_R^Omega(1) c5 Induction Step: LE(gen_0':s:error16_21(+(1, +(n505_21, 1))), gen_0':s:error16_21(+(n505_21, 1))) ->_R^Omega(1) c7(LE(gen_0':s:error16_21(+(1, n505_21)), gen_0':s:error16_21(n505_21))) ->_IH c7(gen_c5:c6:c717_21(c506_21)) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (18) Complex Obligation (BEST) ---------------------------------------- (19) Obligation: Proved the lower bound n^1 for the following obligation: Innermost TRS: Rules: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0')) PLUSITER(z0, z1, z2) -> c2(IFPLUS(le(z0, z2), z0, z1, z2), LE(z0, z2)) IFPLUS(true, z0, z1, z2) -> c3 IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2))) LE(s(z0), 0') -> c5 LE(0', z0) -> c6 LE(s(z0), s(z1)) -> c7(LE(z0, z1)) SUM(z0) -> c8(SUMITER(z0, 0')) SUMITER(z0, z1) -> c9(IFSUM(isempty(z0), z0, z1, plus(z1, head(z0))), ISEMPTY(z0)) SUMITER(z0, z1) -> c10(IFSUM(isempty(z0), z0, z1, plus(z1, head(z0))), PLUS(z1, head(z0)), HEAD(z0)) IFSUM(true, z0, z1, z2) -> c11 IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2), TAIL(z0)) ISEMPTY(nil) -> c13 ISEMPTY(cons(z0, z1)) -> c14 HEAD(nil) -> c15 HEAD(cons(z0, z1)) -> c16 TAIL(nil) -> c17 TAIL(cons(z0, z1)) -> c18 A -> c19 A -> c20 plus(z0, z1) -> plusIter(z0, z1, 0') plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2) ifPlus(true, z0, z1, z2) -> z1 ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2)) le(s(z0), 0') -> false le(0', z0) -> true le(s(z0), s(z1)) -> le(z0, z1) sum(z0) -> sumIter(z0, 0') sumIter(z0, z1) -> ifSum(isempty(z0), z0, z1, plus(z1, head(z0))) ifSum(true, z0, z1, z2) -> z1 ifSum(false, z0, z1, z2) -> sumIter(tail(z0), z2) isempty(nil) -> true isempty(cons(z0, z1)) -> false head(nil) -> error head(cons(z0, z1)) -> z0 tail(nil) -> nil tail(cons(z0, z1)) -> z1 a -> b a -> c Types: PLUS :: 0':s:error -> 0':s:error -> c1 c1 :: c2 -> c1 PLUSITER :: 0':s:error -> 0':s:error -> 0':s:error -> c2 0' :: 0':s:error c2 :: c3:c4 -> c5:c6:c7 -> c2 IFPLUS :: true:false -> 0':s:error -> 0':s:error -> 0':s:error -> c3:c4 le :: 0':s:error -> 0':s:error -> true:false LE :: 0':s:error -> 0':s:error -> c5:c6:c7 true :: true:false c3 :: c3:c4 false :: true:false c4 :: c2 -> c3:c4 s :: 0':s:error -> 0':s:error c5 :: c5:c6:c7 c6 :: c5:c6:c7 c7 :: c5:c6:c7 -> c5:c6:c7 SUM :: nil:cons -> c8 c8 :: c9:c10 -> c8 SUMITER :: nil:cons -> 0':s:error -> c9:c10 c9 :: c11:c12 -> c13:c14 -> c9:c10 IFSUM :: true:false -> nil:cons -> 0':s:error -> 0':s:error -> c11:c12 isempty :: nil:cons -> true:false plus :: 0':s:error -> 0':s:error -> 0':s:error head :: nil:cons -> 0':s:error ISEMPTY :: nil:cons -> c13:c14 c10 :: c11:c12 -> c1 -> c15:c16 -> c9:c10 HEAD :: nil:cons -> c15:c16 c11 :: c11:c12 c12 :: c9:c10 -> c17:c18 -> c11:c12 tail :: nil:cons -> nil:cons TAIL :: nil:cons -> c17:c18 nil :: nil:cons c13 :: c13:c14 cons :: 0':s:error -> nil:cons -> nil:cons c14 :: c13:c14 c15 :: c15:c16 c16 :: c15:c16 c17 :: c17:c18 c18 :: c17:c18 A :: c19:c20 c19 :: c19:c20 c20 :: c19:c20 plusIter :: 0':s:error -> 0':s:error -> 0':s:error -> 0':s:error ifPlus :: true:false -> 0':s:error -> 0':s:error -> 0':s:error -> 0':s:error sum :: nil:cons -> 0':s:error sumIter :: nil:cons -> 0':s:error -> 0':s:error ifSum :: true:false -> nil:cons -> 0':s:error -> 0':s:error -> 0':s:error error :: 0':s:error a :: b:c b :: b:c c :: b:c hole_c11_21 :: c1 hole_0':s:error2_21 :: 0':s:error hole_c23_21 :: c2 hole_c3:c44_21 :: c3:c4 hole_c5:c6:c75_21 :: c5:c6:c7 hole_true:false6_21 :: true:false hole_c87_21 :: c8 hole_nil:cons8_21 :: nil:cons hole_c9:c109_21 :: c9:c10 hole_c11:c1210_21 :: c11:c12 hole_c13:c1411_21 :: c13:c14 hole_c15:c1612_21 :: c15:c16 hole_c17:c1813_21 :: c17:c18 hole_c19:c2014_21 :: c19:c20 hole_b:c15_21 :: b:c gen_0':s:error16_21 :: Nat -> 0':s:error gen_c5:c6:c717_21 :: Nat -> c5:c6:c7 gen_nil:cons18_21 :: Nat -> nil:cons Lemmas: le(gen_0':s:error16_21(+(1, n20_21)), gen_0':s:error16_21(n20_21)) -> false, rt in Omega(0) Generator Equations: gen_0':s:error16_21(0) <=> 0' gen_0':s:error16_21(+(x, 1)) <=> s(gen_0':s:error16_21(x)) gen_c5:c6:c717_21(0) <=> c5 gen_c5:c6:c717_21(+(x, 1)) <=> c7(gen_c5:c6:c717_21(x)) gen_nil:cons18_21(0) <=> nil gen_nil:cons18_21(+(x, 1)) <=> cons(0', gen_nil:cons18_21(x)) The following defined symbols remain to be analysed: LE, PLUSITER, SUMITER, plusIter, sumIter They will be analysed ascendingly in the following order: LE < PLUSITER ---------------------------------------- (20) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (21) BOUNDS(n^1, INF) ---------------------------------------- (22) Obligation: Innermost TRS: Rules: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0')) PLUSITER(z0, z1, z2) -> c2(IFPLUS(le(z0, z2), z0, z1, z2), LE(z0, z2)) IFPLUS(true, z0, z1, z2) -> c3 IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2))) LE(s(z0), 0') -> c5 LE(0', z0) -> c6 LE(s(z0), s(z1)) -> c7(LE(z0, z1)) SUM(z0) -> c8(SUMITER(z0, 0')) SUMITER(z0, z1) -> c9(IFSUM(isempty(z0), z0, z1, plus(z1, head(z0))), ISEMPTY(z0)) SUMITER(z0, z1) -> c10(IFSUM(isempty(z0), z0, z1, plus(z1, head(z0))), PLUS(z1, head(z0)), HEAD(z0)) IFSUM(true, z0, z1, z2) -> c11 IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2), TAIL(z0)) ISEMPTY(nil) -> c13 ISEMPTY(cons(z0, z1)) -> c14 HEAD(nil) -> c15 HEAD(cons(z0, z1)) -> c16 TAIL(nil) -> c17 TAIL(cons(z0, z1)) -> c18 A -> c19 A -> c20 plus(z0, z1) -> plusIter(z0, z1, 0') plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2) ifPlus(true, z0, z1, z2) -> z1 ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2)) le(s(z0), 0') -> false le(0', z0) -> true le(s(z0), s(z1)) -> le(z0, z1) sum(z0) -> sumIter(z0, 0') sumIter(z0, z1) -> ifSum(isempty(z0), z0, z1, plus(z1, head(z0))) ifSum(true, z0, z1, z2) -> z1 ifSum(false, z0, z1, z2) -> sumIter(tail(z0), z2) isempty(nil) -> true isempty(cons(z0, z1)) -> false head(nil) -> error head(cons(z0, z1)) -> z0 tail(nil) -> nil tail(cons(z0, z1)) -> z1 a -> b a -> c Types: PLUS :: 0':s:error -> 0':s:error -> c1 c1 :: c2 -> c1 PLUSITER :: 0':s:error -> 0':s:error -> 0':s:error -> c2 0' :: 0':s:error c2 :: c3:c4 -> c5:c6:c7 -> c2 IFPLUS :: true:false -> 0':s:error -> 0':s:error -> 0':s:error -> c3:c4 le :: 0':s:error -> 0':s:error -> true:false LE :: 0':s:error -> 0':s:error -> c5:c6:c7 true :: true:false c3 :: c3:c4 false :: true:false c4 :: c2 -> c3:c4 s :: 0':s:error -> 0':s:error c5 :: c5:c6:c7 c6 :: c5:c6:c7 c7 :: c5:c6:c7 -> c5:c6:c7 SUM :: nil:cons -> c8 c8 :: c9:c10 -> c8 SUMITER :: nil:cons -> 0':s:error -> c9:c10 c9 :: c11:c12 -> c13:c14 -> c9:c10 IFSUM :: true:false -> nil:cons -> 0':s:error -> 0':s:error -> c11:c12 isempty :: nil:cons -> true:false plus :: 0':s:error -> 0':s:error -> 0':s:error head :: nil:cons -> 0':s:error ISEMPTY :: nil:cons -> c13:c14 c10 :: c11:c12 -> c1 -> c15:c16 -> c9:c10 HEAD :: nil:cons -> c15:c16 c11 :: c11:c12 c12 :: c9:c10 -> c17:c18 -> c11:c12 tail :: nil:cons -> nil:cons TAIL :: nil:cons -> c17:c18 nil :: nil:cons c13 :: c13:c14 cons :: 0':s:error -> nil:cons -> nil:cons c14 :: c13:c14 c15 :: c15:c16 c16 :: c15:c16 c17 :: c17:c18 c18 :: c17:c18 A :: c19:c20 c19 :: c19:c20 c20 :: c19:c20 plusIter :: 0':s:error -> 0':s:error -> 0':s:error -> 0':s:error ifPlus :: true:false -> 0':s:error -> 0':s:error -> 0':s:error -> 0':s:error sum :: nil:cons -> 0':s:error sumIter :: nil:cons -> 0':s:error -> 0':s:error ifSum :: true:false -> nil:cons -> 0':s:error -> 0':s:error -> 0':s:error error :: 0':s:error a :: b:c b :: b:c c :: b:c hole_c11_21 :: c1 hole_0':s:error2_21 :: 0':s:error hole_c23_21 :: c2 hole_c3:c44_21 :: c3:c4 hole_c5:c6:c75_21 :: c5:c6:c7 hole_true:false6_21 :: true:false hole_c87_21 :: c8 hole_nil:cons8_21 :: nil:cons hole_c9:c109_21 :: c9:c10 hole_c11:c1210_21 :: c11:c12 hole_c13:c1411_21 :: c13:c14 hole_c15:c1612_21 :: c15:c16 hole_c17:c1813_21 :: c17:c18 hole_c19:c2014_21 :: c19:c20 hole_b:c15_21 :: b:c gen_0':s:error16_21 :: Nat -> 0':s:error gen_c5:c6:c717_21 :: Nat -> c5:c6:c7 gen_nil:cons18_21 :: Nat -> nil:cons Lemmas: le(gen_0':s:error16_21(+(1, n20_21)), gen_0':s:error16_21(n20_21)) -> false, rt in Omega(0) LE(gen_0':s:error16_21(+(1, n505_21)), gen_0':s:error16_21(n505_21)) -> gen_c5:c6:c717_21(n505_21), rt in Omega(1 + n505_21) Generator Equations: gen_0':s:error16_21(0) <=> 0' gen_0':s:error16_21(+(x, 1)) <=> s(gen_0':s:error16_21(x)) gen_c5:c6:c717_21(0) <=> c5 gen_c5:c6:c717_21(+(x, 1)) <=> c7(gen_c5:c6:c717_21(x)) gen_nil:cons18_21(0) <=> nil gen_nil:cons18_21(+(x, 1)) <=> cons(0', gen_nil:cons18_21(x)) The following defined symbols remain to be analysed: PLUSITER, SUMITER, plusIter, sumIter ---------------------------------------- (23) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: SUMITER(gen_nil:cons18_21(n6468_21), gen_0':s:error16_21(1)) -> *19_21, rt in Omega(n6468_21) Induction Base: SUMITER(gen_nil:cons18_21(0), gen_0':s:error16_21(1)) Induction Step: SUMITER(gen_nil:cons18_21(+(n6468_21, 1)), gen_0':s:error16_21(1)) ->_R^Omega(1) c9(IFSUM(isempty(gen_nil:cons18_21(+(n6468_21, 1))), gen_nil:cons18_21(+(n6468_21, 1)), gen_0':s:error16_21(1), plus(gen_0':s:error16_21(1), head(gen_nil:cons18_21(+(n6468_21, 1))))), ISEMPTY(gen_nil:cons18_21(+(n6468_21, 1)))) ->_R^Omega(0) c9(IFSUM(false, gen_nil:cons18_21(+(1, n6468_21)), gen_0':s:error16_21(1), plus(gen_0':s:error16_21(1), head(gen_nil:cons18_21(+(1, n6468_21))))), ISEMPTY(gen_nil:cons18_21(+(1, n6468_21)))) ->_R^Omega(0) c9(IFSUM(false, gen_nil:cons18_21(+(1, n6468_21)), gen_0':s:error16_21(1), plus(gen_0':s:error16_21(1), 0')), ISEMPTY(gen_nil:cons18_21(+(1, n6468_21)))) ->_R^Omega(0) c9(IFSUM(false, gen_nil:cons18_21(+(1, n6468_21)), gen_0':s:error16_21(1), plusIter(gen_0':s:error16_21(1), 0', 0')), ISEMPTY(gen_nil:cons18_21(+(1, n6468_21)))) ->_R^Omega(0) c9(IFSUM(false, gen_nil:cons18_21(+(1, n6468_21)), gen_0':s:error16_21(1), ifPlus(le(gen_0':s:error16_21(1), 0'), gen_0':s:error16_21(1), 0', 0')), ISEMPTY(gen_nil:cons18_21(+(1, n6468_21)))) ->_L^Omega(0) c9(IFSUM(false, gen_nil:cons18_21(+(1, n6468_21)), gen_0':s:error16_21(1), ifPlus(false, gen_0':s:error16_21(1), 0', 0')), ISEMPTY(gen_nil:cons18_21(+(1, n6468_21)))) ->_R^Omega(0) c9(IFSUM(false, gen_nil:cons18_21(+(1, n6468_21)), gen_0':s:error16_21(1), plusIter(gen_0':s:error16_21(1), s(0'), s(0'))), ISEMPTY(gen_nil:cons18_21(+(1, n6468_21)))) ->_R^Omega(0) c9(IFSUM(false, gen_nil:cons18_21(+(1, n6468_21)), gen_0':s:error16_21(1), ifPlus(le(gen_0':s:error16_21(1), s(0')), gen_0':s:error16_21(1), s(0'), s(0'))), ISEMPTY(gen_nil:cons18_21(+(1, n6468_21)))) ->_R^Omega(0) c9(IFSUM(false, gen_nil:cons18_21(+(1, n6468_21)), gen_0':s:error16_21(1), ifPlus(le(gen_0':s:error16_21(0), 0'), gen_0':s:error16_21(1), s(0'), s(0'))), ISEMPTY(gen_nil:cons18_21(+(1, n6468_21)))) ->_R^Omega(0) c9(IFSUM(false, gen_nil:cons18_21(+(1, n6468_21)), gen_0':s:error16_21(1), ifPlus(true, gen_0':s:error16_21(1), s(0'), s(0'))), ISEMPTY(gen_nil:cons18_21(+(1, n6468_21)))) ->_R^Omega(0) c9(IFSUM(false, gen_nil:cons18_21(+(1, n6468_21)), gen_0':s:error16_21(1), s(0')), ISEMPTY(gen_nil:cons18_21(+(1, n6468_21)))) ->_R^Omega(1) c9(c12(SUMITER(tail(gen_nil:cons18_21(+(1, n6468_21))), s(0')), TAIL(gen_nil:cons18_21(+(1, n6468_21)))), ISEMPTY(gen_nil:cons18_21(+(1, n6468_21)))) ->_R^Omega(0) c9(c12(SUMITER(gen_nil:cons18_21(n6468_21), s(0')), TAIL(gen_nil:cons18_21(+(1, n6468_21)))), ISEMPTY(gen_nil:cons18_21(+(1, n6468_21)))) ->_IH c9(c12(*19_21, TAIL(gen_nil:cons18_21(+(1, n6468_21)))), ISEMPTY(gen_nil:cons18_21(+(1, n6468_21)))) ->_R^Omega(1) c9(c12(*19_21, c18), ISEMPTY(gen_nil:cons18_21(+(1, n6468_21)))) ->_R^Omega(1) c9(c12(*19_21, c18), c14) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (24) Obligation: Innermost TRS: Rules: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0')) PLUSITER(z0, z1, z2) -> c2(IFPLUS(le(z0, z2), z0, z1, z2), LE(z0, z2)) IFPLUS(true, z0, z1, z2) -> c3 IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2))) LE(s(z0), 0') -> c5 LE(0', z0) -> c6 LE(s(z0), s(z1)) -> c7(LE(z0, z1)) SUM(z0) -> c8(SUMITER(z0, 0')) SUMITER(z0, z1) -> c9(IFSUM(isempty(z0), z0, z1, plus(z1, head(z0))), ISEMPTY(z0)) SUMITER(z0, z1) -> c10(IFSUM(isempty(z0), z0, z1, plus(z1, head(z0))), PLUS(z1, head(z0)), HEAD(z0)) IFSUM(true, z0, z1, z2) -> c11 IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2), TAIL(z0)) ISEMPTY(nil) -> c13 ISEMPTY(cons(z0, z1)) -> c14 HEAD(nil) -> c15 HEAD(cons(z0, z1)) -> c16 TAIL(nil) -> c17 TAIL(cons(z0, z1)) -> c18 A -> c19 A -> c20 plus(z0, z1) -> plusIter(z0, z1, 0') plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2) ifPlus(true, z0, z1, z2) -> z1 ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2)) le(s(z0), 0') -> false le(0', z0) -> true le(s(z0), s(z1)) -> le(z0, z1) sum(z0) -> sumIter(z0, 0') sumIter(z0, z1) -> ifSum(isempty(z0), z0, z1, plus(z1, head(z0))) ifSum(true, z0, z1, z2) -> z1 ifSum(false, z0, z1, z2) -> sumIter(tail(z0), z2) isempty(nil) -> true isempty(cons(z0, z1)) -> false head(nil) -> error head(cons(z0, z1)) -> z0 tail(nil) -> nil tail(cons(z0, z1)) -> z1 a -> b a -> c Types: PLUS :: 0':s:error -> 0':s:error -> c1 c1 :: c2 -> c1 PLUSITER :: 0':s:error -> 0':s:error -> 0':s:error -> c2 0' :: 0':s:error c2 :: c3:c4 -> c5:c6:c7 -> c2 IFPLUS :: true:false -> 0':s:error -> 0':s:error -> 0':s:error -> c3:c4 le :: 0':s:error -> 0':s:error -> true:false LE :: 0':s:error -> 0':s:error -> c5:c6:c7 true :: true:false c3 :: c3:c4 false :: true:false c4 :: c2 -> c3:c4 s :: 0':s:error -> 0':s:error c5 :: c5:c6:c7 c6 :: c5:c6:c7 c7 :: c5:c6:c7 -> c5:c6:c7 SUM :: nil:cons -> c8 c8 :: c9:c10 -> c8 SUMITER :: nil:cons -> 0':s:error -> c9:c10 c9 :: c11:c12 -> c13:c14 -> c9:c10 IFSUM :: true:false -> nil:cons -> 0':s:error -> 0':s:error -> c11:c12 isempty :: nil:cons -> true:false plus :: 0':s:error -> 0':s:error -> 0':s:error head :: nil:cons -> 0':s:error ISEMPTY :: nil:cons -> c13:c14 c10 :: c11:c12 -> c1 -> c15:c16 -> c9:c10 HEAD :: nil:cons -> c15:c16 c11 :: c11:c12 c12 :: c9:c10 -> c17:c18 -> c11:c12 tail :: nil:cons -> nil:cons TAIL :: nil:cons -> c17:c18 nil :: nil:cons c13 :: c13:c14 cons :: 0':s:error -> nil:cons -> nil:cons c14 :: c13:c14 c15 :: c15:c16 c16 :: c15:c16 c17 :: c17:c18 c18 :: c17:c18 A :: c19:c20 c19 :: c19:c20 c20 :: c19:c20 plusIter :: 0':s:error -> 0':s:error -> 0':s:error -> 0':s:error ifPlus :: true:false -> 0':s:error -> 0':s:error -> 0':s:error -> 0':s:error sum :: nil:cons -> 0':s:error sumIter :: nil:cons -> 0':s:error -> 0':s:error ifSum :: true:false -> nil:cons -> 0':s:error -> 0':s:error -> 0':s:error error :: 0':s:error a :: b:c b :: b:c c :: b:c hole_c11_21 :: c1 hole_0':s:error2_21 :: 0':s:error hole_c23_21 :: c2 hole_c3:c44_21 :: c3:c4 hole_c5:c6:c75_21 :: c5:c6:c7 hole_true:false6_21 :: true:false hole_c87_21 :: c8 hole_nil:cons8_21 :: nil:cons hole_c9:c109_21 :: c9:c10 hole_c11:c1210_21 :: c11:c12 hole_c13:c1411_21 :: c13:c14 hole_c15:c1612_21 :: c15:c16 hole_c17:c1813_21 :: c17:c18 hole_c19:c2014_21 :: c19:c20 hole_b:c15_21 :: b:c gen_0':s:error16_21 :: Nat -> 0':s:error gen_c5:c6:c717_21 :: Nat -> c5:c6:c7 gen_nil:cons18_21 :: Nat -> nil:cons Lemmas: le(gen_0':s:error16_21(+(1, n20_21)), gen_0':s:error16_21(n20_21)) -> false, rt in Omega(0) LE(gen_0':s:error16_21(+(1, n505_21)), gen_0':s:error16_21(n505_21)) -> gen_c5:c6:c717_21(n505_21), rt in Omega(1 + n505_21) SUMITER(gen_nil:cons18_21(n6468_21), gen_0':s:error16_21(1)) -> *19_21, rt in Omega(n6468_21) Generator Equations: gen_0':s:error16_21(0) <=> 0' gen_0':s:error16_21(+(x, 1)) <=> s(gen_0':s:error16_21(x)) gen_c5:c6:c717_21(0) <=> c5 gen_c5:c6:c717_21(+(x, 1)) <=> c7(gen_c5:c6:c717_21(x)) gen_nil:cons18_21(0) <=> nil gen_nil:cons18_21(+(x, 1)) <=> cons(0', gen_nil:cons18_21(x)) The following defined symbols remain to be analysed: plusIter, sumIter ---------------------------------------- (25) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: sumIter(gen_nil:cons18_21(n221484_21), gen_0':s:error16_21(1)) -> *19_21, rt in Omega(0) Induction Base: sumIter(gen_nil:cons18_21(0), gen_0':s:error16_21(1)) Induction Step: sumIter(gen_nil:cons18_21(+(n221484_21, 1)), gen_0':s:error16_21(1)) ->_R^Omega(0) ifSum(isempty(gen_nil:cons18_21(+(n221484_21, 1))), gen_nil:cons18_21(+(n221484_21, 1)), gen_0':s:error16_21(1), plus(gen_0':s:error16_21(1), head(gen_nil:cons18_21(+(n221484_21, 1))))) ->_R^Omega(0) ifSum(false, gen_nil:cons18_21(+(1, n221484_21)), gen_0':s:error16_21(1), plus(gen_0':s:error16_21(1), head(gen_nil:cons18_21(+(1, n221484_21))))) ->_R^Omega(0) ifSum(false, gen_nil:cons18_21(+(1, n221484_21)), gen_0':s:error16_21(1), plus(gen_0':s:error16_21(1), 0')) ->_R^Omega(0) ifSum(false, gen_nil:cons18_21(+(1, n221484_21)), gen_0':s:error16_21(1), plusIter(gen_0':s:error16_21(1), 0', 0')) ->_R^Omega(0) ifSum(false, gen_nil:cons18_21(+(1, n221484_21)), gen_0':s:error16_21(1), ifPlus(le(gen_0':s:error16_21(1), 0'), gen_0':s:error16_21(1), 0', 0')) ->_L^Omega(0) ifSum(false, gen_nil:cons18_21(+(1, n221484_21)), gen_0':s:error16_21(1), ifPlus(false, gen_0':s:error16_21(1), 0', 0')) ->_R^Omega(0) ifSum(false, gen_nil:cons18_21(+(1, n221484_21)), gen_0':s:error16_21(1), plusIter(gen_0':s:error16_21(1), s(0'), s(0'))) ->_R^Omega(0) ifSum(false, gen_nil:cons18_21(+(1, n221484_21)), gen_0':s:error16_21(1), ifPlus(le(gen_0':s:error16_21(1), s(0')), gen_0':s:error16_21(1), s(0'), s(0'))) ->_R^Omega(0) ifSum(false, gen_nil:cons18_21(+(1, n221484_21)), gen_0':s:error16_21(1), ifPlus(le(gen_0':s:error16_21(0), 0'), gen_0':s:error16_21(1), s(0'), s(0'))) ->_R^Omega(0) ifSum(false, gen_nil:cons18_21(+(1, n221484_21)), gen_0':s:error16_21(1), ifPlus(true, gen_0':s:error16_21(1), s(0'), s(0'))) ->_R^Omega(0) ifSum(false, gen_nil:cons18_21(+(1, n221484_21)), gen_0':s:error16_21(1), s(0')) ->_R^Omega(0) sumIter(tail(gen_nil:cons18_21(+(1, n221484_21))), s(0')) ->_R^Omega(0) sumIter(gen_nil:cons18_21(n221484_21), s(0')) ->_IH *19_21 We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0). ---------------------------------------- (26) BOUNDS(1, INF) ---------------------------------------- (27) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (28) Obligation: Complexity Dependency Tuples Problem Rules: plus(z0, z1) -> plusIter(z0, z1, 0) plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2) ifPlus(true, z0, z1, z2) -> z1 ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2)) le(s(z0), 0) -> false le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) sum(z0) -> sumIter(z0, 0) sumIter(z0, z1) -> ifSum(isempty(z0), z0, z1, plus(z1, head(z0))) ifSum(true, z0, z1, z2) -> z1 ifSum(false, z0, z1, z2) -> sumIter(tail(z0), z2) isempty(nil) -> true isempty(cons(z0, z1)) -> false head(nil) -> error head(cons(z0, z1)) -> z0 tail(nil) -> nil tail(cons(z0, z1)) -> z1 a -> b a -> c Tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) PLUSITER(z0, z1, z2) -> c2(IFPLUS(le(z0, z2), z0, z1, z2), LE(z0, z2)) IFPLUS(true, z0, z1, z2) -> c3 IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2))) LE(s(z0), 0) -> c5 LE(0, z0) -> c6 LE(s(z0), s(z1)) -> c7(LE(z0, z1)) SUM(z0) -> c8(SUMITER(z0, 0)) SUMITER(z0, z1) -> c9(IFSUM(isempty(z0), z0, z1, plus(z1, head(z0))), ISEMPTY(z0)) SUMITER(z0, z1) -> c10(IFSUM(isempty(z0), z0, z1, plus(z1, head(z0))), PLUS(z1, head(z0)), HEAD(z0)) IFSUM(true, z0, z1, z2) -> c11 IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2), TAIL(z0)) ISEMPTY(nil) -> c13 ISEMPTY(cons(z0, z1)) -> c14 HEAD(nil) -> c15 HEAD(cons(z0, z1)) -> c16 TAIL(nil) -> c17 TAIL(cons(z0, z1)) -> c18 A -> c19 A -> c20 S tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) PLUSITER(z0, z1, z2) -> c2(IFPLUS(le(z0, z2), z0, z1, z2), LE(z0, z2)) IFPLUS(true, z0, z1, z2) -> c3 IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2))) LE(s(z0), 0) -> c5 LE(0, z0) -> c6 LE(s(z0), s(z1)) -> c7(LE(z0, z1)) SUM(z0) -> c8(SUMITER(z0, 0)) SUMITER(z0, z1) -> c9(IFSUM(isempty(z0), z0, z1, plus(z1, head(z0))), ISEMPTY(z0)) SUMITER(z0, z1) -> c10(IFSUM(isempty(z0), z0, z1, plus(z1, head(z0))), PLUS(z1, head(z0)), HEAD(z0)) IFSUM(true, z0, z1, z2) -> c11 IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2), TAIL(z0)) ISEMPTY(nil) -> c13 ISEMPTY(cons(z0, z1)) -> c14 HEAD(nil) -> c15 HEAD(cons(z0, z1)) -> c16 TAIL(nil) -> c17 TAIL(cons(z0, z1)) -> c18 A -> c19 A -> c20 K tuples:none Defined Rule Symbols: plus_2, plusIter_3, ifPlus_4, le_2, sum_1, sumIter_2, ifSum_4, isempty_1, head_1, tail_1, a Defined Pair Symbols: PLUS_2, PLUSITER_3, IFPLUS_4, LE_2, SUM_1, SUMITER_2, IFSUM_4, ISEMPTY_1, HEAD_1, TAIL_1, A Compound Symbols: c1_1, c2_2, c3, c4_1, c5, c6, c7_1, c8_1, c9_2, c10_3, c11, c12_2, c13, c14, c15, c16, c17, c18, c19, c20 ---------------------------------------- (29) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 1 leading nodes: SUM(z0) -> c8(SUMITER(z0, 0)) Removed 12 trailing nodes: LE(s(z0), 0) -> c5 HEAD(cons(z0, z1)) -> c16 IFPLUS(true, z0, z1, z2) -> c3 A -> c20 A -> c19 TAIL(nil) -> c17 ISEMPTY(nil) -> c13 IFSUM(true, z0, z1, z2) -> c11 TAIL(cons(z0, z1)) -> c18 LE(0, z0) -> c6 HEAD(nil) -> c15 ISEMPTY(cons(z0, z1)) -> c14 ---------------------------------------- (30) Obligation: Complexity Dependency Tuples Problem Rules: plus(z0, z1) -> plusIter(z0, z1, 0) plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2) ifPlus(true, z0, z1, z2) -> z1 ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2)) le(s(z0), 0) -> false le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) sum(z0) -> sumIter(z0, 0) sumIter(z0, z1) -> ifSum(isempty(z0), z0, z1, plus(z1, head(z0))) ifSum(true, z0, z1, z2) -> z1 ifSum(false, z0, z1, z2) -> sumIter(tail(z0), z2) isempty(nil) -> true isempty(cons(z0, z1)) -> false head(nil) -> error head(cons(z0, z1)) -> z0 tail(nil) -> nil tail(cons(z0, z1)) -> z1 a -> b a -> c Tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) PLUSITER(z0, z1, z2) -> c2(IFPLUS(le(z0, z2), z0, z1, z2), LE(z0, z2)) IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2))) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) SUMITER(z0, z1) -> c9(IFSUM(isempty(z0), z0, z1, plus(z1, head(z0))), ISEMPTY(z0)) SUMITER(z0, z1) -> c10(IFSUM(isempty(z0), z0, z1, plus(z1, head(z0))), PLUS(z1, head(z0)), HEAD(z0)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2), TAIL(z0)) S tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) PLUSITER(z0, z1, z2) -> c2(IFPLUS(le(z0, z2), z0, z1, z2), LE(z0, z2)) IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2))) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) SUMITER(z0, z1) -> c9(IFSUM(isempty(z0), z0, z1, plus(z1, head(z0))), ISEMPTY(z0)) SUMITER(z0, z1) -> c10(IFSUM(isempty(z0), z0, z1, plus(z1, head(z0))), PLUS(z1, head(z0)), HEAD(z0)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2), TAIL(z0)) K tuples:none Defined Rule Symbols: plus_2, plusIter_3, ifPlus_4, le_2, sum_1, sumIter_2, ifSum_4, isempty_1, head_1, tail_1, a Defined Pair Symbols: PLUS_2, PLUSITER_3, IFPLUS_4, LE_2, SUMITER_2, IFSUM_4 Compound Symbols: c1_1, c2_2, c4_1, c7_1, c9_2, c10_3, c12_2 ---------------------------------------- (31) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (32) Obligation: Complexity Dependency Tuples Problem Rules: plus(z0, z1) -> plusIter(z0, z1, 0) plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2) ifPlus(true, z0, z1, z2) -> z1 ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2)) le(s(z0), 0) -> false le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) sum(z0) -> sumIter(z0, 0) sumIter(z0, z1) -> ifSum(isempty(z0), z0, z1, plus(z1, head(z0))) ifSum(true, z0, z1, z2) -> z1 ifSum(false, z0, z1, z2) -> sumIter(tail(z0), z2) isempty(nil) -> true isempty(cons(z0, z1)) -> false head(nil) -> error head(cons(z0, z1)) -> z0 tail(nil) -> nil tail(cons(z0, z1)) -> z1 a -> b a -> c Tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) PLUSITER(z0, z1, z2) -> c2(IFPLUS(le(z0, z2), z0, z1, z2), LE(z0, z2)) IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2))) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) SUMITER(z0, z1) -> c9(IFSUM(isempty(z0), z0, z1, plus(z1, head(z0)))) SUMITER(z0, z1) -> c10(IFSUM(isempty(z0), z0, z1, plus(z1, head(z0))), PLUS(z1, head(z0))) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) S tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) PLUSITER(z0, z1, z2) -> c2(IFPLUS(le(z0, z2), z0, z1, z2), LE(z0, z2)) IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2))) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) SUMITER(z0, z1) -> c9(IFSUM(isempty(z0), z0, z1, plus(z1, head(z0)))) SUMITER(z0, z1) -> c10(IFSUM(isempty(z0), z0, z1, plus(z1, head(z0))), PLUS(z1, head(z0))) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) K tuples:none Defined Rule Symbols: plus_2, plusIter_3, ifPlus_4, le_2, sum_1, sumIter_2, ifSum_4, isempty_1, head_1, tail_1, a Defined Pair Symbols: PLUS_2, PLUSITER_3, IFPLUS_4, LE_2, SUMITER_2, IFSUM_4 Compound Symbols: c1_1, c2_2, c4_1, c7_1, c9_1, c10_2, c12_1 ---------------------------------------- (33) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: sum(z0) -> sumIter(z0, 0) sumIter(z0, z1) -> ifSum(isempty(z0), z0, z1, plus(z1, head(z0))) ifSum(true, z0, z1, z2) -> z1 ifSum(false, z0, z1, z2) -> sumIter(tail(z0), z2) a -> b a -> c ---------------------------------------- (34) Obligation: Complexity Dependency Tuples Problem Rules: le(s(z0), 0) -> false le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) isempty(nil) -> true isempty(cons(z0, z1)) -> false plus(z0, z1) -> plusIter(z0, z1, 0) head(nil) -> error head(cons(z0, z1)) -> z0 plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2) ifPlus(true, z0, z1, z2) -> z1 ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2)) tail(nil) -> nil tail(cons(z0, z1)) -> z1 Tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) PLUSITER(z0, z1, z2) -> c2(IFPLUS(le(z0, z2), z0, z1, z2), LE(z0, z2)) IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2))) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) SUMITER(z0, z1) -> c9(IFSUM(isempty(z0), z0, z1, plus(z1, head(z0)))) SUMITER(z0, z1) -> c10(IFSUM(isempty(z0), z0, z1, plus(z1, head(z0))), PLUS(z1, head(z0))) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) S tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) PLUSITER(z0, z1, z2) -> c2(IFPLUS(le(z0, z2), z0, z1, z2), LE(z0, z2)) IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2))) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) SUMITER(z0, z1) -> c9(IFSUM(isempty(z0), z0, z1, plus(z1, head(z0)))) SUMITER(z0, z1) -> c10(IFSUM(isempty(z0), z0, z1, plus(z1, head(z0))), PLUS(z1, head(z0))) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) K tuples:none Defined Rule Symbols: le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1 Defined Pair Symbols: PLUS_2, PLUSITER_3, IFPLUS_4, LE_2, SUMITER_2, IFSUM_4 Compound Symbols: c1_1, c2_2, c4_1, c7_1, c9_1, c10_2, c12_1 ---------------------------------------- (35) CdtToCpxRelTrsProof (BOTH BOUNDS(ID, ID)) Converted S to standard rules, and D \ S as well as R to relative rules. ---------------------------------------- (36) 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: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) PLUSITER(z0, z1, z2) -> c2(IFPLUS(le(z0, z2), z0, z1, z2), LE(z0, z2)) IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2))) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) SUMITER(z0, z1) -> c9(IFSUM(isempty(z0), z0, z1, plus(z1, head(z0)))) SUMITER(z0, z1) -> c10(IFSUM(isempty(z0), z0, z1, plus(z1, head(z0))), PLUS(z1, head(z0))) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) The (relative) TRS S consists of the following rules: le(s(z0), 0) -> false le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) isempty(nil) -> true isempty(cons(z0, z1)) -> false plus(z0, z1) -> plusIter(z0, z1, 0) head(nil) -> error head(cons(z0, z1)) -> z0 plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2) ifPlus(true, z0, z1, z2) -> z1 ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2)) tail(nil) -> nil tail(cons(z0, z1)) -> z1 Rewrite Strategy: INNERMOST ---------------------------------------- (37) RelTrsToTrsProof (UPPER BOUND(ID)) transformed relative TRS to TRS ---------------------------------------- (38) 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: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) PLUSITER(z0, z1, z2) -> c2(IFPLUS(le(z0, z2), z0, z1, z2), LE(z0, z2)) IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2))) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) SUMITER(z0, z1) -> c9(IFSUM(isempty(z0), z0, z1, plus(z1, head(z0)))) SUMITER(z0, z1) -> c10(IFSUM(isempty(z0), z0, z1, plus(z1, head(z0))), PLUS(z1, head(z0))) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) le(s(z0), 0) -> false le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) isempty(nil) -> true isempty(cons(z0, z1)) -> false plus(z0, z1) -> plusIter(z0, z1, 0) head(nil) -> error head(cons(z0, z1)) -> z0 plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2) ifPlus(true, z0, z1, z2) -> z1 ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2)) tail(nil) -> nil tail(cons(z0, z1)) -> z1 S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (39) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID)) Transformed relative TRS to weighted TRS ---------------------------------------- (40) 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: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) [1] PLUSITER(z0, z1, z2) -> c2(IFPLUS(le(z0, z2), z0, z1, z2), LE(z0, z2)) [1] IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2))) [1] LE(s(z0), s(z1)) -> c7(LE(z0, z1)) [1] SUMITER(z0, z1) -> c9(IFSUM(isempty(z0), z0, z1, plus(z1, head(z0)))) [1] SUMITER(z0, z1) -> c10(IFSUM(isempty(z0), z0, z1, plus(z1, head(z0))), PLUS(z1, head(z0))) [1] IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) [1] le(s(z0), 0) -> false [0] le(0, z0) -> true [0] le(s(z0), s(z1)) -> le(z0, z1) [0] isempty(nil) -> true [0] isempty(cons(z0, z1)) -> false [0] plus(z0, z1) -> plusIter(z0, z1, 0) [0] head(nil) -> error [0] head(cons(z0, z1)) -> z0 [0] plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2) [0] ifPlus(true, z0, z1, z2) -> z1 [0] ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2)) [0] tail(nil) -> nil [0] tail(cons(z0, z1)) -> z1 [0] Rewrite Strategy: INNERMOST ---------------------------------------- (41) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (42) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) [1] PLUSITER(z0, z1, z2) -> c2(IFPLUS(le(z0, z2), z0, z1, z2), LE(z0, z2)) [1] IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2))) [1] LE(s(z0), s(z1)) -> c7(LE(z0, z1)) [1] SUMITER(z0, z1) -> c9(IFSUM(isempty(z0), z0, z1, plus(z1, head(z0)))) [1] SUMITER(z0, z1) -> c10(IFSUM(isempty(z0), z0, z1, plus(z1, head(z0))), PLUS(z1, head(z0))) [1] IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) [1] le(s(z0), 0) -> false [0] le(0, z0) -> true [0] le(s(z0), s(z1)) -> le(z0, z1) [0] isempty(nil) -> true [0] isempty(cons(z0, z1)) -> false [0] plus(z0, z1) -> plusIter(z0, z1, 0) [0] head(nil) -> error [0] head(cons(z0, z1)) -> z0 [0] plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2) [0] ifPlus(true, z0, z1, z2) -> z1 [0] ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2)) [0] tail(nil) -> nil [0] tail(cons(z0, z1)) -> z1 [0] The TRS has the following type information: PLUS :: 0:s:error -> 0:s:error -> c1 c1 :: c2 -> c1 PLUSITER :: 0:s:error -> 0:s:error -> 0:s:error -> c2 0 :: 0:s:error c2 :: c4 -> c7 -> c2 IFPLUS :: false:true -> 0:s:error -> 0:s:error -> 0:s:error -> c4 le :: 0:s:error -> 0:s:error -> false:true LE :: 0:s:error -> 0:s:error -> c7 false :: false:true c4 :: c2 -> c4 s :: 0:s:error -> 0:s:error c7 :: c7 -> c7 SUMITER :: nil:cons -> 0:s:error -> c9:c10 c9 :: c12 -> c9:c10 IFSUM :: false:true -> nil:cons -> 0:s:error -> 0:s:error -> c12 isempty :: nil:cons -> false:true plus :: 0:s:error -> 0:s:error -> 0:s:error head :: nil:cons -> 0:s:error c10 :: c12 -> c1 -> c9:c10 c12 :: c9:c10 -> c12 tail :: nil:cons -> nil:cons true :: false:true nil :: nil:cons cons :: 0:s:error -> nil:cons -> nil:cons plusIter :: 0:s:error -> 0:s:error -> 0:s:error -> 0:s:error error :: 0:s:error ifPlus :: false:true -> 0:s:error -> 0:s:error -> 0:s:error -> 0:s:error Rewrite Strategy: INNERMOST ---------------------------------------- (43) 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: PLUS_2 PLUSITER_3 IFPLUS_4 LE_2 SUMITER_2 IFSUM_4 (c) The following functions are completely defined: le_2 isempty_1 plus_2 head_1 plusIter_3 ifPlus_4 tail_1 Due to the following rules being added: le(v0, v1) -> null_le [0] isempty(v0) -> null_isempty [0] plus(v0, v1) -> null_plus [0] head(v0) -> null_head [0] plusIter(v0, v1, v2) -> null_plusIter [0] ifPlus(v0, v1, v2, v3) -> null_ifPlus [0] tail(v0) -> nil [0] And the following fresh constants: null_le, null_isempty, null_plus, null_head, null_plusIter, null_ifPlus, const, const1, const2, const3, const4, const5 ---------------------------------------- (44) 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: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) [1] PLUSITER(z0, z1, z2) -> c2(IFPLUS(le(z0, z2), z0, z1, z2), LE(z0, z2)) [1] IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2))) [1] LE(s(z0), s(z1)) -> c7(LE(z0, z1)) [1] SUMITER(z0, z1) -> c9(IFSUM(isempty(z0), z0, z1, plus(z1, head(z0)))) [1] SUMITER(z0, z1) -> c10(IFSUM(isempty(z0), z0, z1, plus(z1, head(z0))), PLUS(z1, head(z0))) [1] IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) [1] le(s(z0), 0) -> false [0] le(0, z0) -> true [0] le(s(z0), s(z1)) -> le(z0, z1) [0] isempty(nil) -> true [0] isempty(cons(z0, z1)) -> false [0] plus(z0, z1) -> plusIter(z0, z1, 0) [0] head(nil) -> error [0] head(cons(z0, z1)) -> z0 [0] plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2) [0] ifPlus(true, z0, z1, z2) -> z1 [0] ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2)) [0] tail(nil) -> nil [0] tail(cons(z0, z1)) -> z1 [0] le(v0, v1) -> null_le [0] isempty(v0) -> null_isempty [0] plus(v0, v1) -> null_plus [0] head(v0) -> null_head [0] plusIter(v0, v1, v2) -> null_plusIter [0] ifPlus(v0, v1, v2, v3) -> null_ifPlus [0] tail(v0) -> nil [0] The TRS has the following type information: PLUS :: 0:s:error:null_plus:null_head:null_plusIter:null_ifPlus -> 0:s:error:null_plus:null_head:null_plusIter:null_ifPlus -> c1 c1 :: c2 -> c1 PLUSITER :: 0:s:error:null_plus:null_head:null_plusIter:null_ifPlus -> 0:s:error:null_plus:null_head:null_plusIter:null_ifPlus -> 0:s:error:null_plus:null_head:null_plusIter:null_ifPlus -> c2 0 :: 0:s:error:null_plus:null_head:null_plusIter:null_ifPlus c2 :: c4 -> c7 -> c2 IFPLUS :: false:true:null_le:null_isempty -> 0:s:error:null_plus:null_head:null_plusIter:null_ifPlus -> 0:s:error:null_plus:null_head:null_plusIter:null_ifPlus -> 0:s:error:null_plus:null_head:null_plusIter:null_ifPlus -> c4 le :: 0:s:error:null_plus:null_head:null_plusIter:null_ifPlus -> 0:s:error:null_plus:null_head:null_plusIter:null_ifPlus -> false:true:null_le:null_isempty LE :: 0:s:error:null_plus:null_head:null_plusIter:null_ifPlus -> 0:s:error:null_plus:null_head:null_plusIter:null_ifPlus -> c7 false :: false:true:null_le:null_isempty c4 :: c2 -> c4 s :: 0:s:error:null_plus:null_head:null_plusIter:null_ifPlus -> 0:s:error:null_plus:null_head:null_plusIter:null_ifPlus c7 :: c7 -> c7 SUMITER :: nil:cons -> 0:s:error:null_plus:null_head:null_plusIter:null_ifPlus -> c9:c10 c9 :: c12 -> c9:c10 IFSUM :: false:true:null_le:null_isempty -> nil:cons -> 0:s:error:null_plus:null_head:null_plusIter:null_ifPlus -> 0:s:error:null_plus:null_head:null_plusIter:null_ifPlus -> c12 isempty :: nil:cons -> false:true:null_le:null_isempty plus :: 0:s:error:null_plus:null_head:null_plusIter:null_ifPlus -> 0:s:error:null_plus:null_head:null_plusIter:null_ifPlus -> 0:s:error:null_plus:null_head:null_plusIter:null_ifPlus head :: nil:cons -> 0:s:error:null_plus:null_head:null_plusIter:null_ifPlus c10 :: c12 -> c1 -> c9:c10 c12 :: c9:c10 -> c12 tail :: nil:cons -> nil:cons true :: false:true:null_le:null_isempty nil :: nil:cons cons :: 0:s:error:null_plus:null_head:null_plusIter:null_ifPlus -> nil:cons -> nil:cons plusIter :: 0:s:error:null_plus:null_head:null_plusIter:null_ifPlus -> 0:s:error:null_plus:null_head:null_plusIter:null_ifPlus -> 0:s:error:null_plus:null_head:null_plusIter:null_ifPlus -> 0:s:error:null_plus:null_head:null_plusIter:null_ifPlus error :: 0:s:error:null_plus:null_head:null_plusIter:null_ifPlus ifPlus :: false:true:null_le:null_isempty -> 0:s:error:null_plus:null_head:null_plusIter:null_ifPlus -> 0:s:error:null_plus:null_head:null_plusIter:null_ifPlus -> 0:s:error:null_plus:null_head:null_plusIter:null_ifPlus -> 0:s:error:null_plus:null_head:null_plusIter:null_ifPlus null_le :: false:true:null_le:null_isempty null_isempty :: false:true:null_le:null_isempty null_plus :: 0:s:error:null_plus:null_head:null_plusIter:null_ifPlus null_head :: 0:s:error:null_plus:null_head:null_plusIter:null_ifPlus null_plusIter :: 0:s:error:null_plus:null_head:null_plusIter:null_ifPlus null_ifPlus :: 0:s:error:null_plus:null_head:null_plusIter:null_ifPlus const :: c1 const1 :: c2 const2 :: c4 const3 :: c7 const4 :: c9:c10 const5 :: c12 Rewrite Strategy: INNERMOST ---------------------------------------- (45) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (46) 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: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) [1] PLUSITER(s(z0'), z1, 0) -> c2(IFPLUS(false, s(z0'), z1, 0), LE(s(z0'), 0)) [1] PLUSITER(0, z1, z2) -> c2(IFPLUS(true, 0, z1, z2), LE(0, z2)) [1] PLUSITER(s(z0''), z1, s(z1')) -> c2(IFPLUS(le(z0'', z1'), s(z0''), z1, s(z1')), LE(s(z0''), s(z1'))) [1] PLUSITER(z0, z1, z2) -> c2(IFPLUS(null_le, z0, z1, z2), LE(z0, z2)) [1] IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2))) [1] LE(s(z0), s(z1)) -> c7(LE(z0, z1)) [1] SUMITER(nil, z1) -> c9(IFSUM(true, nil, z1, plus(z1, error))) [1] SUMITER(nil, z1) -> c9(IFSUM(true, nil, z1, plus(z1, null_head))) [1] SUMITER(cons(z01, z1''), z1) -> c9(IFSUM(false, cons(z01, z1''), z1, plus(z1, z01))) [1] SUMITER(cons(z01, z1''), z1) -> c9(IFSUM(false, cons(z01, z1''), z1, plus(z1, null_head))) [1] SUMITER(nil, z1) -> c9(IFSUM(null_isempty, nil, z1, plus(z1, error))) [1] SUMITER(cons(z02, z11), z1) -> c9(IFSUM(null_isempty, cons(z02, z11), z1, plus(z1, z02))) [1] SUMITER(z0, z1) -> c9(IFSUM(null_isempty, z0, z1, plus(z1, null_head))) [1] SUMITER(nil, z1) -> c10(IFSUM(true, nil, z1, plus(z1, error)), PLUS(z1, error)) [1] SUMITER(nil, z1) -> c10(IFSUM(true, nil, z1, plus(z1, error)), PLUS(z1, null_head)) [1] SUMITER(nil, z1) -> c10(IFSUM(true, nil, z1, plus(z1, null_head)), PLUS(z1, error)) [1] SUMITER(nil, z1) -> c10(IFSUM(true, nil, z1, plus(z1, null_head)), PLUS(z1, null_head)) [1] SUMITER(cons(z03, z12), z1) -> c10(IFSUM(false, cons(z03, z12), z1, plus(z1, z03)), PLUS(z1, z03)) [1] SUMITER(cons(z03, z12), z1) -> c10(IFSUM(false, cons(z03, z12), z1, plus(z1, z03)), PLUS(z1, null_head)) [1] SUMITER(cons(z03, z12), z1) -> c10(IFSUM(false, cons(z03, z12), z1, plus(z1, null_head)), PLUS(z1, z03)) [1] SUMITER(cons(z03, z12), z1) -> c10(IFSUM(false, cons(z03, z12), z1, plus(z1, null_head)), PLUS(z1, null_head)) [1] SUMITER(nil, z1) -> c10(IFSUM(null_isempty, nil, z1, plus(z1, error)), PLUS(z1, error)) [1] SUMITER(nil, z1) -> c10(IFSUM(null_isempty, nil, z1, plus(z1, error)), PLUS(z1, null_head)) [1] SUMITER(cons(z04, z13), z1) -> c10(IFSUM(null_isempty, cons(z04, z13), z1, plus(z1, z04)), PLUS(z1, z04)) [1] SUMITER(cons(z04, z13), z1) -> c10(IFSUM(null_isempty, cons(z04, z13), z1, plus(z1, z04)), PLUS(z1, null_head)) [1] SUMITER(nil, z1) -> c10(IFSUM(null_isempty, nil, z1, plus(z1, null_head)), PLUS(z1, error)) [1] SUMITER(cons(z05, z14), z1) -> c10(IFSUM(null_isempty, cons(z05, z14), z1, plus(z1, null_head)), PLUS(z1, z05)) [1] SUMITER(z0, z1) -> c10(IFSUM(null_isempty, z0, z1, plus(z1, null_head)), PLUS(z1, null_head)) [1] IFSUM(false, nil, z1, z2) -> c12(SUMITER(nil, z2)) [1] IFSUM(false, cons(z06, z15), z1, z2) -> c12(SUMITER(z15, z2)) [1] IFSUM(false, z0, z1, z2) -> c12(SUMITER(nil, z2)) [1] le(s(z0), 0) -> false [0] le(0, z0) -> true [0] le(s(z0), s(z1)) -> le(z0, z1) [0] isempty(nil) -> true [0] isempty(cons(z0, z1)) -> false [0] plus(z0, z1) -> plusIter(z0, z1, 0) [0] head(nil) -> error [0] head(cons(z0, z1)) -> z0 [0] plusIter(s(z07), z1, 0) -> ifPlus(false, s(z07), z1, 0) [0] plusIter(0, z1, z2) -> ifPlus(true, 0, z1, z2) [0] plusIter(s(z08), z1, s(z16)) -> ifPlus(le(z08, z16), s(z08), z1, s(z16)) [0] plusIter(z0, z1, z2) -> ifPlus(null_le, z0, z1, z2) [0] ifPlus(true, z0, z1, z2) -> z1 [0] ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2)) [0] tail(nil) -> nil [0] tail(cons(z0, z1)) -> z1 [0] le(v0, v1) -> null_le [0] isempty(v0) -> null_isempty [0] plus(v0, v1) -> null_plus [0] head(v0) -> null_head [0] plusIter(v0, v1, v2) -> null_plusIter [0] ifPlus(v0, v1, v2, v3) -> null_ifPlus [0] tail(v0) -> nil [0] The TRS has the following type information: PLUS :: 0:s:error:null_plus:null_head:null_plusIter:null_ifPlus -> 0:s:error:null_plus:null_head:null_plusIter:null_ifPlus -> c1 c1 :: c2 -> c1 PLUSITER :: 0:s:error:null_plus:null_head:null_plusIter:null_ifPlus -> 0:s:error:null_plus:null_head:null_plusIter:null_ifPlus -> 0:s:error:null_plus:null_head:null_plusIter:null_ifPlus -> c2 0 :: 0:s:error:null_plus:null_head:null_plusIter:null_ifPlus c2 :: c4 -> c7 -> c2 IFPLUS :: false:true:null_le:null_isempty -> 0:s:error:null_plus:null_head:null_plusIter:null_ifPlus -> 0:s:error:null_plus:null_head:null_plusIter:null_ifPlus -> 0:s:error:null_plus:null_head:null_plusIter:null_ifPlus -> c4 le :: 0:s:error:null_plus:null_head:null_plusIter:null_ifPlus -> 0:s:error:null_plus:null_head:null_plusIter:null_ifPlus -> false:true:null_le:null_isempty LE :: 0:s:error:null_plus:null_head:null_plusIter:null_ifPlus -> 0:s:error:null_plus:null_head:null_plusIter:null_ifPlus -> c7 false :: false:true:null_le:null_isempty c4 :: c2 -> c4 s :: 0:s:error:null_plus:null_head:null_plusIter:null_ifPlus -> 0:s:error:null_plus:null_head:null_plusIter:null_ifPlus c7 :: c7 -> c7 SUMITER :: nil:cons -> 0:s:error:null_plus:null_head:null_plusIter:null_ifPlus -> c9:c10 c9 :: c12 -> c9:c10 IFSUM :: false:true:null_le:null_isempty -> nil:cons -> 0:s:error:null_plus:null_head:null_plusIter:null_ifPlus -> 0:s:error:null_plus:null_head:null_plusIter:null_ifPlus -> c12 isempty :: nil:cons -> false:true:null_le:null_isempty plus :: 0:s:error:null_plus:null_head:null_plusIter:null_ifPlus -> 0:s:error:null_plus:null_head:null_plusIter:null_ifPlus -> 0:s:error:null_plus:null_head:null_plusIter:null_ifPlus head :: nil:cons -> 0:s:error:null_plus:null_head:null_plusIter:null_ifPlus c10 :: c12 -> c1 -> c9:c10 c12 :: c9:c10 -> c12 tail :: nil:cons -> nil:cons true :: false:true:null_le:null_isempty nil :: nil:cons cons :: 0:s:error:null_plus:null_head:null_plusIter:null_ifPlus -> nil:cons -> nil:cons plusIter :: 0:s:error:null_plus:null_head:null_plusIter:null_ifPlus -> 0:s:error:null_plus:null_head:null_plusIter:null_ifPlus -> 0:s:error:null_plus:null_head:null_plusIter:null_ifPlus -> 0:s:error:null_plus:null_head:null_plusIter:null_ifPlus error :: 0:s:error:null_plus:null_head:null_plusIter:null_ifPlus ifPlus :: false:true:null_le:null_isempty -> 0:s:error:null_plus:null_head:null_plusIter:null_ifPlus -> 0:s:error:null_plus:null_head:null_plusIter:null_ifPlus -> 0:s:error:null_plus:null_head:null_plusIter:null_ifPlus -> 0:s:error:null_plus:null_head:null_plusIter:null_ifPlus null_le :: false:true:null_le:null_isempty null_isempty :: false:true:null_le:null_isempty null_plus :: 0:s:error:null_plus:null_head:null_plusIter:null_ifPlus null_head :: 0:s:error:null_plus:null_head:null_plusIter:null_ifPlus null_plusIter :: 0:s:error:null_plus:null_head:null_plusIter:null_ifPlus null_ifPlus :: 0:s:error:null_plus:null_head:null_plusIter:null_ifPlus const :: c1 const1 :: c2 const2 :: c4 const3 :: c7 const4 :: c9:c10 const5 :: c12 Rewrite Strategy: INNERMOST ---------------------------------------- (47) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: 0 => 0 false => 1 true => 2 nil => 0 error => 1 null_le => 0 null_isempty => 0 null_plus => 0 null_head => 0 null_plusIter => 0 null_ifPlus => 0 const => 0 const1 => 0 const2 => 0 const3 => 0 const4 => 0 const5 => 0 ---------------------------------------- (48) Obligation: Complexity RNTS consisting of the following rules: IFPLUS(z, z', z'', z3) -{ 1 }-> 1 + PLUSITER(z0, 1 + z1, 1 + z2) :|: z1 >= 0, z = 1, z0 >= 0, z3 = z2, z' = z0, z2 >= 0, z'' = z1 IFSUM(z, z', z'', z3) -{ 1 }-> 1 + SUMITER(z15, z2) :|: z15 >= 0, z1 >= 0, z = 1, z06 >= 0, z3 = z2, z2 >= 0, z' = 1 + z06 + z15, z'' = z1 IFSUM(z, z', z'', z3) -{ 1 }-> 1 + SUMITER(0, z2) :|: z1 >= 0, z = 1, z3 = z2, z2 >= 0, z' = 0, z'' = z1 IFSUM(z, z', z'', z3) -{ 1 }-> 1 + SUMITER(0, z2) :|: z1 >= 0, z = 1, z0 >= 0, z3 = z2, z' = z0, z2 >= 0, z'' = z1 LE(z, z') -{ 1 }-> 1 + LE(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1 PLUS(z, z') -{ 1 }-> 1 + PLUSITER(z0, z1, 0) :|: z = z0, z1 >= 0, z' = z1, z0 >= 0 PLUSITER(z, z', z'') -{ 1 }-> 1 + IFPLUS(le(z0'', z1'), 1 + z0'', z1, 1 + z1') + LE(1 + z0'', 1 + z1') :|: z1 >= 0, z'' = 1 + z1', z1' >= 0, z' = z1, z0'' >= 0, z = 1 + z0'' PLUSITER(z, z', z'') -{ 1 }-> 1 + IFPLUS(2, 0, z1, z2) + LE(0, z2) :|: z'' = z2, z1 >= 0, z' = z1, z = 0, z2 >= 0 PLUSITER(z, z', z'') -{ 1 }-> 1 + IFPLUS(1, 1 + z0', z1, 0) + LE(1 + z0', 0) :|: z'' = 0, z1 >= 0, z0' >= 0, z' = z1, z = 1 + z0' PLUSITER(z, z', z'') -{ 1 }-> 1 + IFPLUS(0, z0, z1, z2) + LE(z0, z2) :|: z'' = z2, z = z0, z1 >= 0, z' = z1, z0 >= 0, z2 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z1, plus(z1, 1)) :|: z1 >= 0, z' = z1, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z1, plus(z1, 0)) :|: z1 >= 0, z' = z1, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z01 + z1'', z1, plus(z1, z01)) :|: z1 >= 0, z01 >= 0, z = 1 + z01 + z1'', z' = z1, z1'' >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z01 + z1'', z1, plus(z1, 0)) :|: z1 >= 0, z01 >= 0, z = 1 + z01 + z1'', z' = z1, z1'' >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, z0, z1, plus(z1, 0)) :|: z = z0, z1 >= 0, z' = z1, z0 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 0, z1, plus(z1, 1)) :|: z1 >= 0, z' = z1, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 1 + z02 + z11, z1, plus(z1, z02)) :|: z11 >= 0, z1 >= 0, z = 1 + z02 + z11, z02 >= 0, z' = z1 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z1, plus(z1, 1)) + PLUS(z1, 1) :|: z1 >= 0, z' = z1, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z1, plus(z1, 1)) + PLUS(z1, 0) :|: z1 >= 0, z' = z1, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z1, plus(z1, 0)) + PLUS(z1, 1) :|: z1 >= 0, z' = z1, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z1, plus(z1, 0)) + PLUS(z1, 0) :|: z1 >= 0, z' = z1, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z03 + z12, z1, plus(z1, z03)) + PLUS(z1, z03) :|: z = 1 + z03 + z12, z1 >= 0, z' = z1, z12 >= 0, z03 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z03 + z12, z1, plus(z1, z03)) + PLUS(z1, 0) :|: z = 1 + z03 + z12, z1 >= 0, z' = z1, z12 >= 0, z03 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z03 + z12, z1, plus(z1, 0)) + PLUS(z1, z03) :|: z = 1 + z03 + z12, z1 >= 0, z' = z1, z12 >= 0, z03 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z03 + z12, z1, plus(z1, 0)) + PLUS(z1, 0) :|: z = 1 + z03 + z12, z1 >= 0, z' = z1, z12 >= 0, z03 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, z0, z1, plus(z1, 0)) + PLUS(z1, 0) :|: z = z0, z1 >= 0, z' = z1, z0 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 0, z1, plus(z1, 1)) + PLUS(z1, 1) :|: z1 >= 0, z' = z1, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 0, z1, plus(z1, 1)) + PLUS(z1, 0) :|: z1 >= 0, z' = z1, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 0, z1, plus(z1, 0)) + PLUS(z1, 1) :|: z1 >= 0, z' = z1, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 1 + z04 + z13, z1, plus(z1, z04)) + PLUS(z1, z04) :|: z = 1 + z04 + z13, z04 >= 0, z1 >= 0, z' = z1, z13 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 1 + z04 + z13, z1, plus(z1, z04)) + PLUS(z1, 0) :|: z = 1 + z04 + z13, z04 >= 0, z1 >= 0, z' = z1, z13 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 1 + z05 + z14, z1, plus(z1, 0)) + PLUS(z1, z05) :|: z = 1 + z05 + z14, z1 >= 0, z' = z1, z05 >= 0, z14 >= 0 head(z) -{ 0 }-> z0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 head(z) -{ 0 }-> 1 :|: z = 0 head(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 ifPlus(z, z', z'', z3) -{ 0 }-> z1 :|: z = 2, z1 >= 0, z0 >= 0, z3 = z2, z' = z0, z2 >= 0, z'' = z1 ifPlus(z, z', z'', z3) -{ 0 }-> plusIter(z0, 1 + z1, 1 + z2) :|: z1 >= 0, z = 1, z0 >= 0, z3 = z2, z' = z0, z2 >= 0, z'' = z1 ifPlus(z, z', z'', z3) -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, z3 = v3, v2 >= 0, v3 >= 0 isempty(z) -{ 0 }-> 2 :|: z = 0 isempty(z) -{ 0 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 isempty(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 le(z, z') -{ 0 }-> le(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1 le(z, z') -{ 0 }-> 2 :|: z0 >= 0, z = 0, z' = z0 le(z, z') -{ 0 }-> 1 :|: z = 1 + z0, z0 >= 0, z' = 0 le(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 plus(z, z') -{ 0 }-> plusIter(z0, z1, 0) :|: z = z0, z1 >= 0, z' = z1, z0 >= 0 plus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 plusIter(z, z', z'') -{ 0 }-> ifPlus(le(z08, z16), 1 + z08, z1, 1 + z16) :|: z08 >= 0, z1 >= 0, z = 1 + z08, z' = z1, z'' = 1 + z16, z16 >= 0 plusIter(z, z', z'') -{ 0 }-> ifPlus(2, 0, z1, z2) :|: z'' = z2, z1 >= 0, z' = z1, z = 0, z2 >= 0 plusIter(z, z', z'') -{ 0 }-> ifPlus(1, 1 + z07, z1, 0) :|: z'' = 0, z1 >= 0, z07 >= 0, z = 1 + z07, z' = z1 plusIter(z, z', z'') -{ 0 }-> ifPlus(0, z0, z1, z2) :|: z'' = z2, z = z0, z1 >= 0, z' = z1, z0 >= 0, z2 >= 0 plusIter(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 tail(z) -{ 0 }-> z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 tail(z) -{ 0 }-> 0 :|: z = 0 tail(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 ---------------------------------------- (49) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (50) Obligation: Complexity RNTS consisting of the following rules: IFPLUS(z, z', z'', z3) -{ 1 }-> 1 + PLUSITER(z', 1 + z'', 1 + z3) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0 IFSUM(z, z', z'', z3) -{ 1 }-> 1 + SUMITER(z15, z3) :|: z15 >= 0, z'' >= 0, z = 1, z06 >= 0, z3 >= 0, z' = 1 + z06 + z15 IFSUM(z, z', z'', z3) -{ 1 }-> 1 + SUMITER(0, z3) :|: z'' >= 0, z = 1, z3 >= 0, z' = 0 IFSUM(z, z', z'', z3) -{ 1 }-> 1 + SUMITER(0, z3) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0 LE(z, z') -{ 1 }-> 1 + LE(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 PLUS(z, z') -{ 1 }-> 1 + PLUSITER(z, z', 0) :|: z' >= 0, z >= 0 PLUSITER(z, z', z'') -{ 1 }-> 1 + IFPLUS(le(z - 1, z'' - 1), 1 + (z - 1), z', 1 + (z'' - 1)) + LE(1 + (z - 1), 1 + (z'' - 1)) :|: z' >= 0, z'' - 1 >= 0, z - 1 >= 0 PLUSITER(z, z', z'') -{ 1 }-> 1 + IFPLUS(2, 0, z', z'') + LE(0, z'') :|: z' >= 0, z = 0, z'' >= 0 PLUSITER(z, z', z'') -{ 1 }-> 1 + IFPLUS(1, 1 + (z - 1), z', 0) + LE(1 + (z - 1), 0) :|: z'' = 0, z' >= 0, z - 1 >= 0 PLUSITER(z, z', z'') -{ 1 }-> 1 + IFPLUS(0, z, z', z'') + LE(z, z'') :|: z' >= 0, z >= 0, z'' >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 1)) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 0)) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z01 + z1'', z', plus(z', z01)) :|: z' >= 0, z01 >= 0, z = 1 + z01 + z1'', z1'' >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z01 + z1'', z', plus(z', 0)) :|: z' >= 0, z01 >= 0, z = 1 + z01 + z1'', z1'' >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, z, z', plus(z', 0)) :|: z' >= 0, z >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 0, z', plus(z', 1)) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 1 + z02 + z11, z', plus(z', z02)) :|: z11 >= 0, z' >= 0, z = 1 + z02 + z11, z02 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 1)) + PLUS(z', 1) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 1)) + PLUS(z', 0) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 0)) + PLUS(z', 1) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 0)) + PLUS(z', 0) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z03 + z12, z', plus(z', z03)) + PLUS(z', z03) :|: z = 1 + z03 + z12, z' >= 0, z12 >= 0, z03 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z03 + z12, z', plus(z', z03)) + PLUS(z', 0) :|: z = 1 + z03 + z12, z' >= 0, z12 >= 0, z03 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z03 + z12, z', plus(z', 0)) + PLUS(z', z03) :|: z = 1 + z03 + z12, z' >= 0, z12 >= 0, z03 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z03 + z12, z', plus(z', 0)) + PLUS(z', 0) :|: z = 1 + z03 + z12, z' >= 0, z12 >= 0, z03 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, z, z', plus(z', 0)) + PLUS(z', 0) :|: z' >= 0, z >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 0, z', plus(z', 1)) + PLUS(z', 1) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 0, z', plus(z', 1)) + PLUS(z', 0) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 0, z', plus(z', 0)) + PLUS(z', 1) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 1 + z04 + z13, z', plus(z', z04)) + PLUS(z', z04) :|: z = 1 + z04 + z13, z04 >= 0, z' >= 0, z13 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 1 + z04 + z13, z', plus(z', z04)) + PLUS(z', 0) :|: z = 1 + z04 + z13, z04 >= 0, z' >= 0, z13 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 1 + z05 + z14, z', plus(z', 0)) + PLUS(z', z05) :|: z = 1 + z05 + z14, z' >= 0, z05 >= 0, z14 >= 0 head(z) -{ 0 }-> z0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 head(z) -{ 0 }-> 1 :|: z = 0 head(z) -{ 0 }-> 0 :|: z >= 0 ifPlus(z, z', z'', z3) -{ 0 }-> z'' :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0 ifPlus(z, z', z'', z3) -{ 0 }-> plusIter(z', 1 + z'', 1 + z3) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0 ifPlus(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0 isempty(z) -{ 0 }-> 2 :|: z = 0 isempty(z) -{ 0 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 isempty(z) -{ 0 }-> 0 :|: z >= 0 le(z, z') -{ 0 }-> le(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0 le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 0 }-> plusIter(z, z', 0) :|: z' >= 0, z >= 0 plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plusIter(z, z', z'') -{ 0 }-> ifPlus(le(z - 1, z'' - 1), 1 + (z - 1), z', 1 + (z'' - 1)) :|: z - 1 >= 0, z' >= 0, z'' - 1 >= 0 plusIter(z, z', z'') -{ 0 }-> ifPlus(2, 0, z', z'') :|: z' >= 0, z = 0, z'' >= 0 plusIter(z, z', z'') -{ 0 }-> ifPlus(1, 1 + (z - 1), z', 0) :|: z'' = 0, z' >= 0, z - 1 >= 0 plusIter(z, z', z'') -{ 0 }-> ifPlus(0, z, z', z'') :|: z' >= 0, z >= 0, z'' >= 0 plusIter(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 tail(z) -{ 0 }-> z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 tail(z) -{ 0 }-> 0 :|: z = 0 tail(z) -{ 0 }-> 0 :|: z >= 0 ---------------------------------------- (51) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { le } { LE } { tail } { isempty } { head } { plusIter, ifPlus } { IFPLUS, PLUSITER } { plus } { PLUS } { IFSUM, SUMITER } ---------------------------------------- (52) Obligation: Complexity RNTS consisting of the following rules: IFPLUS(z, z', z'', z3) -{ 1 }-> 1 + PLUSITER(z', 1 + z'', 1 + z3) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0 IFSUM(z, z', z'', z3) -{ 1 }-> 1 + SUMITER(z15, z3) :|: z15 >= 0, z'' >= 0, z = 1, z06 >= 0, z3 >= 0, z' = 1 + z06 + z15 IFSUM(z, z', z'', z3) -{ 1 }-> 1 + SUMITER(0, z3) :|: z'' >= 0, z = 1, z3 >= 0, z' = 0 IFSUM(z, z', z'', z3) -{ 1 }-> 1 + SUMITER(0, z3) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0 LE(z, z') -{ 1 }-> 1 + LE(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 PLUS(z, z') -{ 1 }-> 1 + PLUSITER(z, z', 0) :|: z' >= 0, z >= 0 PLUSITER(z, z', z'') -{ 1 }-> 1 + IFPLUS(le(z - 1, z'' - 1), 1 + (z - 1), z', 1 + (z'' - 1)) + LE(1 + (z - 1), 1 + (z'' - 1)) :|: z' >= 0, z'' - 1 >= 0, z - 1 >= 0 PLUSITER(z, z', z'') -{ 1 }-> 1 + IFPLUS(2, 0, z', z'') + LE(0, z'') :|: z' >= 0, z = 0, z'' >= 0 PLUSITER(z, z', z'') -{ 1 }-> 1 + IFPLUS(1, 1 + (z - 1), z', 0) + LE(1 + (z - 1), 0) :|: z'' = 0, z' >= 0, z - 1 >= 0 PLUSITER(z, z', z'') -{ 1 }-> 1 + IFPLUS(0, z, z', z'') + LE(z, z'') :|: z' >= 0, z >= 0, z'' >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 1)) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 0)) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z01 + z1'', z', plus(z', z01)) :|: z' >= 0, z01 >= 0, z = 1 + z01 + z1'', z1'' >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z01 + z1'', z', plus(z', 0)) :|: z' >= 0, z01 >= 0, z = 1 + z01 + z1'', z1'' >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, z, z', plus(z', 0)) :|: z' >= 0, z >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 0, z', plus(z', 1)) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 1 + z02 + z11, z', plus(z', z02)) :|: z11 >= 0, z' >= 0, z = 1 + z02 + z11, z02 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 1)) + PLUS(z', 1) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 1)) + PLUS(z', 0) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 0)) + PLUS(z', 1) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 0)) + PLUS(z', 0) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z03 + z12, z', plus(z', z03)) + PLUS(z', z03) :|: z = 1 + z03 + z12, z' >= 0, z12 >= 0, z03 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z03 + z12, z', plus(z', z03)) + PLUS(z', 0) :|: z = 1 + z03 + z12, z' >= 0, z12 >= 0, z03 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z03 + z12, z', plus(z', 0)) + PLUS(z', z03) :|: z = 1 + z03 + z12, z' >= 0, z12 >= 0, z03 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z03 + z12, z', plus(z', 0)) + PLUS(z', 0) :|: z = 1 + z03 + z12, z' >= 0, z12 >= 0, z03 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, z, z', plus(z', 0)) + PLUS(z', 0) :|: z' >= 0, z >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 0, z', plus(z', 1)) + PLUS(z', 1) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 0, z', plus(z', 1)) + PLUS(z', 0) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 0, z', plus(z', 0)) + PLUS(z', 1) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 1 + z04 + z13, z', plus(z', z04)) + PLUS(z', z04) :|: z = 1 + z04 + z13, z04 >= 0, z' >= 0, z13 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 1 + z04 + z13, z', plus(z', z04)) + PLUS(z', 0) :|: z = 1 + z04 + z13, z04 >= 0, z' >= 0, z13 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 1 + z05 + z14, z', plus(z', 0)) + PLUS(z', z05) :|: z = 1 + z05 + z14, z' >= 0, z05 >= 0, z14 >= 0 head(z) -{ 0 }-> z0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 head(z) -{ 0 }-> 1 :|: z = 0 head(z) -{ 0 }-> 0 :|: z >= 0 ifPlus(z, z', z'', z3) -{ 0 }-> z'' :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0 ifPlus(z, z', z'', z3) -{ 0 }-> plusIter(z', 1 + z'', 1 + z3) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0 ifPlus(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0 isempty(z) -{ 0 }-> 2 :|: z = 0 isempty(z) -{ 0 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 isempty(z) -{ 0 }-> 0 :|: z >= 0 le(z, z') -{ 0 }-> le(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0 le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 0 }-> plusIter(z, z', 0) :|: z' >= 0, z >= 0 plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plusIter(z, z', z'') -{ 0 }-> ifPlus(le(z - 1, z'' - 1), 1 + (z - 1), z', 1 + (z'' - 1)) :|: z - 1 >= 0, z' >= 0, z'' - 1 >= 0 plusIter(z, z', z'') -{ 0 }-> ifPlus(2, 0, z', z'') :|: z' >= 0, z = 0, z'' >= 0 plusIter(z, z', z'') -{ 0 }-> ifPlus(1, 1 + (z - 1), z', 0) :|: z'' = 0, z' >= 0, z - 1 >= 0 plusIter(z, z', z'') -{ 0 }-> ifPlus(0, z, z', z'') :|: z' >= 0, z >= 0, z'' >= 0 plusIter(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 tail(z) -{ 0 }-> z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 tail(z) -{ 0 }-> 0 :|: z = 0 tail(z) -{ 0 }-> 0 :|: z >= 0 Function symbols to be analyzed: {le}, {LE}, {tail}, {isempty}, {head}, {plusIter,ifPlus}, {IFPLUS,PLUSITER}, {plus}, {PLUS}, {IFSUM,SUMITER} ---------------------------------------- (53) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (54) Obligation: Complexity RNTS consisting of the following rules: IFPLUS(z, z', z'', z3) -{ 1 }-> 1 + PLUSITER(z', 1 + z'', 1 + z3) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0 IFSUM(z, z', z'', z3) -{ 1 }-> 1 + SUMITER(z15, z3) :|: z15 >= 0, z'' >= 0, z = 1, z06 >= 0, z3 >= 0, z' = 1 + z06 + z15 IFSUM(z, z', z'', z3) -{ 1 }-> 1 + SUMITER(0, z3) :|: z'' >= 0, z = 1, z3 >= 0, z' = 0 IFSUM(z, z', z'', z3) -{ 1 }-> 1 + SUMITER(0, z3) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0 LE(z, z') -{ 1 }-> 1 + LE(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 PLUS(z, z') -{ 1 }-> 1 + PLUSITER(z, z', 0) :|: z' >= 0, z >= 0 PLUSITER(z, z', z'') -{ 1 }-> 1 + IFPLUS(le(z - 1, z'' - 1), 1 + (z - 1), z', 1 + (z'' - 1)) + LE(1 + (z - 1), 1 + (z'' - 1)) :|: z' >= 0, z'' - 1 >= 0, z - 1 >= 0 PLUSITER(z, z', z'') -{ 1 }-> 1 + IFPLUS(2, 0, z', z'') + LE(0, z'') :|: z' >= 0, z = 0, z'' >= 0 PLUSITER(z, z', z'') -{ 1 }-> 1 + IFPLUS(1, 1 + (z - 1), z', 0) + LE(1 + (z - 1), 0) :|: z'' = 0, z' >= 0, z - 1 >= 0 PLUSITER(z, z', z'') -{ 1 }-> 1 + IFPLUS(0, z, z', z'') + LE(z, z'') :|: z' >= 0, z >= 0, z'' >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 1)) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 0)) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z01 + z1'', z', plus(z', z01)) :|: z' >= 0, z01 >= 0, z = 1 + z01 + z1'', z1'' >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z01 + z1'', z', plus(z', 0)) :|: z' >= 0, z01 >= 0, z = 1 + z01 + z1'', z1'' >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, z, z', plus(z', 0)) :|: z' >= 0, z >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 0, z', plus(z', 1)) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 1 + z02 + z11, z', plus(z', z02)) :|: z11 >= 0, z' >= 0, z = 1 + z02 + z11, z02 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 1)) + PLUS(z', 1) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 1)) + PLUS(z', 0) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 0)) + PLUS(z', 1) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 0)) + PLUS(z', 0) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z03 + z12, z', plus(z', z03)) + PLUS(z', z03) :|: z = 1 + z03 + z12, z' >= 0, z12 >= 0, z03 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z03 + z12, z', plus(z', z03)) + PLUS(z', 0) :|: z = 1 + z03 + z12, z' >= 0, z12 >= 0, z03 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z03 + z12, z', plus(z', 0)) + PLUS(z', z03) :|: z = 1 + z03 + z12, z' >= 0, z12 >= 0, z03 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z03 + z12, z', plus(z', 0)) + PLUS(z', 0) :|: z = 1 + z03 + z12, z' >= 0, z12 >= 0, z03 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, z, z', plus(z', 0)) + PLUS(z', 0) :|: z' >= 0, z >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 0, z', plus(z', 1)) + PLUS(z', 1) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 0, z', plus(z', 1)) + PLUS(z', 0) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 0, z', plus(z', 0)) + PLUS(z', 1) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 1 + z04 + z13, z', plus(z', z04)) + PLUS(z', z04) :|: z = 1 + z04 + z13, z04 >= 0, z' >= 0, z13 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 1 + z04 + z13, z', plus(z', z04)) + PLUS(z', 0) :|: z = 1 + z04 + z13, z04 >= 0, z' >= 0, z13 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 1 + z05 + z14, z', plus(z', 0)) + PLUS(z', z05) :|: z = 1 + z05 + z14, z' >= 0, z05 >= 0, z14 >= 0 head(z) -{ 0 }-> z0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 head(z) -{ 0 }-> 1 :|: z = 0 head(z) -{ 0 }-> 0 :|: z >= 0 ifPlus(z, z', z'', z3) -{ 0 }-> z'' :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0 ifPlus(z, z', z'', z3) -{ 0 }-> plusIter(z', 1 + z'', 1 + z3) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0 ifPlus(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0 isempty(z) -{ 0 }-> 2 :|: z = 0 isempty(z) -{ 0 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 isempty(z) -{ 0 }-> 0 :|: z >= 0 le(z, z') -{ 0 }-> le(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0 le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 0 }-> plusIter(z, z', 0) :|: z' >= 0, z >= 0 plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plusIter(z, z', z'') -{ 0 }-> ifPlus(le(z - 1, z'' - 1), 1 + (z - 1), z', 1 + (z'' - 1)) :|: z - 1 >= 0, z' >= 0, z'' - 1 >= 0 plusIter(z, z', z'') -{ 0 }-> ifPlus(2, 0, z', z'') :|: z' >= 0, z = 0, z'' >= 0 plusIter(z, z', z'') -{ 0 }-> ifPlus(1, 1 + (z - 1), z', 0) :|: z'' = 0, z' >= 0, z - 1 >= 0 plusIter(z, z', z'') -{ 0 }-> ifPlus(0, z, z', z'') :|: z' >= 0, z >= 0, z'' >= 0 plusIter(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 tail(z) -{ 0 }-> z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 tail(z) -{ 0 }-> 0 :|: z = 0 tail(z) -{ 0 }-> 0 :|: z >= 0 Function symbols to be analyzed: {le}, {LE}, {tail}, {isempty}, {head}, {plusIter,ifPlus}, {IFPLUS,PLUSITER}, {plus}, {PLUS}, {IFSUM,SUMITER} ---------------------------------------- (55) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: le after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 2 ---------------------------------------- (56) Obligation: Complexity RNTS consisting of the following rules: IFPLUS(z, z', z'', z3) -{ 1 }-> 1 + PLUSITER(z', 1 + z'', 1 + z3) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0 IFSUM(z, z', z'', z3) -{ 1 }-> 1 + SUMITER(z15, z3) :|: z15 >= 0, z'' >= 0, z = 1, z06 >= 0, z3 >= 0, z' = 1 + z06 + z15 IFSUM(z, z', z'', z3) -{ 1 }-> 1 + SUMITER(0, z3) :|: z'' >= 0, z = 1, z3 >= 0, z' = 0 IFSUM(z, z', z'', z3) -{ 1 }-> 1 + SUMITER(0, z3) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0 LE(z, z') -{ 1 }-> 1 + LE(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 PLUS(z, z') -{ 1 }-> 1 + PLUSITER(z, z', 0) :|: z' >= 0, z >= 0 PLUSITER(z, z', z'') -{ 1 }-> 1 + IFPLUS(le(z - 1, z'' - 1), 1 + (z - 1), z', 1 + (z'' - 1)) + LE(1 + (z - 1), 1 + (z'' - 1)) :|: z' >= 0, z'' - 1 >= 0, z - 1 >= 0 PLUSITER(z, z', z'') -{ 1 }-> 1 + IFPLUS(2, 0, z', z'') + LE(0, z'') :|: z' >= 0, z = 0, z'' >= 0 PLUSITER(z, z', z'') -{ 1 }-> 1 + IFPLUS(1, 1 + (z - 1), z', 0) + LE(1 + (z - 1), 0) :|: z'' = 0, z' >= 0, z - 1 >= 0 PLUSITER(z, z', z'') -{ 1 }-> 1 + IFPLUS(0, z, z', z'') + LE(z, z'') :|: z' >= 0, z >= 0, z'' >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 1)) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 0)) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z01 + z1'', z', plus(z', z01)) :|: z' >= 0, z01 >= 0, z = 1 + z01 + z1'', z1'' >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z01 + z1'', z', plus(z', 0)) :|: z' >= 0, z01 >= 0, z = 1 + z01 + z1'', z1'' >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, z, z', plus(z', 0)) :|: z' >= 0, z >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 0, z', plus(z', 1)) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 1 + z02 + z11, z', plus(z', z02)) :|: z11 >= 0, z' >= 0, z = 1 + z02 + z11, z02 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 1)) + PLUS(z', 1) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 1)) + PLUS(z', 0) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 0)) + PLUS(z', 1) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 0)) + PLUS(z', 0) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z03 + z12, z', plus(z', z03)) + PLUS(z', z03) :|: z = 1 + z03 + z12, z' >= 0, z12 >= 0, z03 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z03 + z12, z', plus(z', z03)) + PLUS(z', 0) :|: z = 1 + z03 + z12, z' >= 0, z12 >= 0, z03 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z03 + z12, z', plus(z', 0)) + PLUS(z', z03) :|: z = 1 + z03 + z12, z' >= 0, z12 >= 0, z03 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z03 + z12, z', plus(z', 0)) + PLUS(z', 0) :|: z = 1 + z03 + z12, z' >= 0, z12 >= 0, z03 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, z, z', plus(z', 0)) + PLUS(z', 0) :|: z' >= 0, z >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 0, z', plus(z', 1)) + PLUS(z', 1) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 0, z', plus(z', 1)) + PLUS(z', 0) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 0, z', plus(z', 0)) + PLUS(z', 1) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 1 + z04 + z13, z', plus(z', z04)) + PLUS(z', z04) :|: z = 1 + z04 + z13, z04 >= 0, z' >= 0, z13 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 1 + z04 + z13, z', plus(z', z04)) + PLUS(z', 0) :|: z = 1 + z04 + z13, z04 >= 0, z' >= 0, z13 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 1 + z05 + z14, z', plus(z', 0)) + PLUS(z', z05) :|: z = 1 + z05 + z14, z' >= 0, z05 >= 0, z14 >= 0 head(z) -{ 0 }-> z0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 head(z) -{ 0 }-> 1 :|: z = 0 head(z) -{ 0 }-> 0 :|: z >= 0 ifPlus(z, z', z'', z3) -{ 0 }-> z'' :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0 ifPlus(z, z', z'', z3) -{ 0 }-> plusIter(z', 1 + z'', 1 + z3) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0 ifPlus(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0 isempty(z) -{ 0 }-> 2 :|: z = 0 isempty(z) -{ 0 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 isempty(z) -{ 0 }-> 0 :|: z >= 0 le(z, z') -{ 0 }-> le(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0 le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 0 }-> plusIter(z, z', 0) :|: z' >= 0, z >= 0 plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plusIter(z, z', z'') -{ 0 }-> ifPlus(le(z - 1, z'' - 1), 1 + (z - 1), z', 1 + (z'' - 1)) :|: z - 1 >= 0, z' >= 0, z'' - 1 >= 0 plusIter(z, z', z'') -{ 0 }-> ifPlus(2, 0, z', z'') :|: z' >= 0, z = 0, z'' >= 0 plusIter(z, z', z'') -{ 0 }-> ifPlus(1, 1 + (z - 1), z', 0) :|: z'' = 0, z' >= 0, z - 1 >= 0 plusIter(z, z', z'') -{ 0 }-> ifPlus(0, z, z', z'') :|: z' >= 0, z >= 0, z'' >= 0 plusIter(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 tail(z) -{ 0 }-> z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 tail(z) -{ 0 }-> 0 :|: z = 0 tail(z) -{ 0 }-> 0 :|: z >= 0 Function symbols to be analyzed: {le}, {LE}, {tail}, {isempty}, {head}, {plusIter,ifPlus}, {IFPLUS,PLUSITER}, {plus}, {PLUS}, {IFSUM,SUMITER} Previous analysis results are: le: runtime: ?, size: O(1) [2] ---------------------------------------- (57) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: le after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (58) Obligation: Complexity RNTS consisting of the following rules: IFPLUS(z, z', z'', z3) -{ 1 }-> 1 + PLUSITER(z', 1 + z'', 1 + z3) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0 IFSUM(z, z', z'', z3) -{ 1 }-> 1 + SUMITER(z15, z3) :|: z15 >= 0, z'' >= 0, z = 1, z06 >= 0, z3 >= 0, z' = 1 + z06 + z15 IFSUM(z, z', z'', z3) -{ 1 }-> 1 + SUMITER(0, z3) :|: z'' >= 0, z = 1, z3 >= 0, z' = 0 IFSUM(z, z', z'', z3) -{ 1 }-> 1 + SUMITER(0, z3) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0 LE(z, z') -{ 1 }-> 1 + LE(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 PLUS(z, z') -{ 1 }-> 1 + PLUSITER(z, z', 0) :|: z' >= 0, z >= 0 PLUSITER(z, z', z'') -{ 1 }-> 1 + IFPLUS(le(z - 1, z'' - 1), 1 + (z - 1), z', 1 + (z'' - 1)) + LE(1 + (z - 1), 1 + (z'' - 1)) :|: z' >= 0, z'' - 1 >= 0, z - 1 >= 0 PLUSITER(z, z', z'') -{ 1 }-> 1 + IFPLUS(2, 0, z', z'') + LE(0, z'') :|: z' >= 0, z = 0, z'' >= 0 PLUSITER(z, z', z'') -{ 1 }-> 1 + IFPLUS(1, 1 + (z - 1), z', 0) + LE(1 + (z - 1), 0) :|: z'' = 0, z' >= 0, z - 1 >= 0 PLUSITER(z, z', z'') -{ 1 }-> 1 + IFPLUS(0, z, z', z'') + LE(z, z'') :|: z' >= 0, z >= 0, z'' >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 1)) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 0)) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z01 + z1'', z', plus(z', z01)) :|: z' >= 0, z01 >= 0, z = 1 + z01 + z1'', z1'' >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z01 + z1'', z', plus(z', 0)) :|: z' >= 0, z01 >= 0, z = 1 + z01 + z1'', z1'' >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, z, z', plus(z', 0)) :|: z' >= 0, z >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 0, z', plus(z', 1)) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 1 + z02 + z11, z', plus(z', z02)) :|: z11 >= 0, z' >= 0, z = 1 + z02 + z11, z02 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 1)) + PLUS(z', 1) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 1)) + PLUS(z', 0) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 0)) + PLUS(z', 1) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 0)) + PLUS(z', 0) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z03 + z12, z', plus(z', z03)) + PLUS(z', z03) :|: z = 1 + z03 + z12, z' >= 0, z12 >= 0, z03 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z03 + z12, z', plus(z', z03)) + PLUS(z', 0) :|: z = 1 + z03 + z12, z' >= 0, z12 >= 0, z03 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z03 + z12, z', plus(z', 0)) + PLUS(z', z03) :|: z = 1 + z03 + z12, z' >= 0, z12 >= 0, z03 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z03 + z12, z', plus(z', 0)) + PLUS(z', 0) :|: z = 1 + z03 + z12, z' >= 0, z12 >= 0, z03 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, z, z', plus(z', 0)) + PLUS(z', 0) :|: z' >= 0, z >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 0, z', plus(z', 1)) + PLUS(z', 1) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 0, z', plus(z', 1)) + PLUS(z', 0) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 0, z', plus(z', 0)) + PLUS(z', 1) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 1 + z04 + z13, z', plus(z', z04)) + PLUS(z', z04) :|: z = 1 + z04 + z13, z04 >= 0, z' >= 0, z13 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 1 + z04 + z13, z', plus(z', z04)) + PLUS(z', 0) :|: z = 1 + z04 + z13, z04 >= 0, z' >= 0, z13 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 1 + z05 + z14, z', plus(z', 0)) + PLUS(z', z05) :|: z = 1 + z05 + z14, z' >= 0, z05 >= 0, z14 >= 0 head(z) -{ 0 }-> z0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 head(z) -{ 0 }-> 1 :|: z = 0 head(z) -{ 0 }-> 0 :|: z >= 0 ifPlus(z, z', z'', z3) -{ 0 }-> z'' :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0 ifPlus(z, z', z'', z3) -{ 0 }-> plusIter(z', 1 + z'', 1 + z3) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0 ifPlus(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0 isempty(z) -{ 0 }-> 2 :|: z = 0 isempty(z) -{ 0 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 isempty(z) -{ 0 }-> 0 :|: z >= 0 le(z, z') -{ 0 }-> le(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0 le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 0 }-> plusIter(z, z', 0) :|: z' >= 0, z >= 0 plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plusIter(z, z', z'') -{ 0 }-> ifPlus(le(z - 1, z'' - 1), 1 + (z - 1), z', 1 + (z'' - 1)) :|: z - 1 >= 0, z' >= 0, z'' - 1 >= 0 plusIter(z, z', z'') -{ 0 }-> ifPlus(2, 0, z', z'') :|: z' >= 0, z = 0, z'' >= 0 plusIter(z, z', z'') -{ 0 }-> ifPlus(1, 1 + (z - 1), z', 0) :|: z'' = 0, z' >= 0, z - 1 >= 0 plusIter(z, z', z'') -{ 0 }-> ifPlus(0, z, z', z'') :|: z' >= 0, z >= 0, z'' >= 0 plusIter(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 tail(z) -{ 0 }-> z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 tail(z) -{ 0 }-> 0 :|: z = 0 tail(z) -{ 0 }-> 0 :|: z >= 0 Function symbols to be analyzed: {LE}, {tail}, {isempty}, {head}, {plusIter,ifPlus}, {IFPLUS,PLUSITER}, {plus}, {PLUS}, {IFSUM,SUMITER} Previous analysis results are: le: runtime: O(1) [0], size: O(1) [2] ---------------------------------------- (59) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (60) Obligation: Complexity RNTS consisting of the following rules: IFPLUS(z, z', z'', z3) -{ 1 }-> 1 + PLUSITER(z', 1 + z'', 1 + z3) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0 IFSUM(z, z', z'', z3) -{ 1 }-> 1 + SUMITER(z15, z3) :|: z15 >= 0, z'' >= 0, z = 1, z06 >= 0, z3 >= 0, z' = 1 + z06 + z15 IFSUM(z, z', z'', z3) -{ 1 }-> 1 + SUMITER(0, z3) :|: z'' >= 0, z = 1, z3 >= 0, z' = 0 IFSUM(z, z', z'', z3) -{ 1 }-> 1 + SUMITER(0, z3) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0 LE(z, z') -{ 1 }-> 1 + LE(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 PLUS(z, z') -{ 1 }-> 1 + PLUSITER(z, z', 0) :|: z' >= 0, z >= 0 PLUSITER(z, z', z'') -{ 1 }-> 1 + IFPLUS(s, 1 + (z - 1), z', 1 + (z'' - 1)) + LE(1 + (z - 1), 1 + (z'' - 1)) :|: s >= 0, s <= 2, z' >= 0, z'' - 1 >= 0, z - 1 >= 0 PLUSITER(z, z', z'') -{ 1 }-> 1 + IFPLUS(2, 0, z', z'') + LE(0, z'') :|: z' >= 0, z = 0, z'' >= 0 PLUSITER(z, z', z'') -{ 1 }-> 1 + IFPLUS(1, 1 + (z - 1), z', 0) + LE(1 + (z - 1), 0) :|: z'' = 0, z' >= 0, z - 1 >= 0 PLUSITER(z, z', z'') -{ 1 }-> 1 + IFPLUS(0, z, z', z'') + LE(z, z'') :|: z' >= 0, z >= 0, z'' >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 1)) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 0)) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z01 + z1'', z', plus(z', z01)) :|: z' >= 0, z01 >= 0, z = 1 + z01 + z1'', z1'' >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z01 + z1'', z', plus(z', 0)) :|: z' >= 0, z01 >= 0, z = 1 + z01 + z1'', z1'' >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, z, z', plus(z', 0)) :|: z' >= 0, z >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 0, z', plus(z', 1)) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 1 + z02 + z11, z', plus(z', z02)) :|: z11 >= 0, z' >= 0, z = 1 + z02 + z11, z02 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 1)) + PLUS(z', 1) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 1)) + PLUS(z', 0) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 0)) + PLUS(z', 1) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 0)) + PLUS(z', 0) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z03 + z12, z', plus(z', z03)) + PLUS(z', z03) :|: z = 1 + z03 + z12, z' >= 0, z12 >= 0, z03 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z03 + z12, z', plus(z', z03)) + PLUS(z', 0) :|: z = 1 + z03 + z12, z' >= 0, z12 >= 0, z03 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z03 + z12, z', plus(z', 0)) + PLUS(z', z03) :|: z = 1 + z03 + z12, z' >= 0, z12 >= 0, z03 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z03 + z12, z', plus(z', 0)) + PLUS(z', 0) :|: z = 1 + z03 + z12, z' >= 0, z12 >= 0, z03 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, z, z', plus(z', 0)) + PLUS(z', 0) :|: z' >= 0, z >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 0, z', plus(z', 1)) + PLUS(z', 1) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 0, z', plus(z', 1)) + PLUS(z', 0) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 0, z', plus(z', 0)) + PLUS(z', 1) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 1 + z04 + z13, z', plus(z', z04)) + PLUS(z', z04) :|: z = 1 + z04 + z13, z04 >= 0, z' >= 0, z13 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 1 + z04 + z13, z', plus(z', z04)) + PLUS(z', 0) :|: z = 1 + z04 + z13, z04 >= 0, z' >= 0, z13 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 1 + z05 + z14, z', plus(z', 0)) + PLUS(z', z05) :|: z = 1 + z05 + z14, z' >= 0, z05 >= 0, z14 >= 0 head(z) -{ 0 }-> z0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 head(z) -{ 0 }-> 1 :|: z = 0 head(z) -{ 0 }-> 0 :|: z >= 0 ifPlus(z, z', z'', z3) -{ 0 }-> z'' :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0 ifPlus(z, z', z'', z3) -{ 0 }-> plusIter(z', 1 + z'', 1 + z3) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0 ifPlus(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0 isempty(z) -{ 0 }-> 2 :|: z = 0 isempty(z) -{ 0 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 isempty(z) -{ 0 }-> 0 :|: z >= 0 le(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 2, z' - 1 >= 0, z - 1 >= 0 le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0 le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 0 }-> plusIter(z, z', 0) :|: z' >= 0, z >= 0 plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plusIter(z, z', z'') -{ 0 }-> ifPlus(s'', 1 + (z - 1), z', 1 + (z'' - 1)) :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' >= 0, z'' - 1 >= 0 plusIter(z, z', z'') -{ 0 }-> ifPlus(2, 0, z', z'') :|: z' >= 0, z = 0, z'' >= 0 plusIter(z, z', z'') -{ 0 }-> ifPlus(1, 1 + (z - 1), z', 0) :|: z'' = 0, z' >= 0, z - 1 >= 0 plusIter(z, z', z'') -{ 0 }-> ifPlus(0, z, z', z'') :|: z' >= 0, z >= 0, z'' >= 0 plusIter(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 tail(z) -{ 0 }-> z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 tail(z) -{ 0 }-> 0 :|: z = 0 tail(z) -{ 0 }-> 0 :|: z >= 0 Function symbols to be analyzed: {LE}, {tail}, {isempty}, {head}, {plusIter,ifPlus}, {IFPLUS,PLUSITER}, {plus}, {PLUS}, {IFSUM,SUMITER} Previous analysis results are: le: runtime: O(1) [0], size: O(1) [2] ---------------------------------------- (61) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: LE after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (62) Obligation: Complexity RNTS consisting of the following rules: IFPLUS(z, z', z'', z3) -{ 1 }-> 1 + PLUSITER(z', 1 + z'', 1 + z3) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0 IFSUM(z, z', z'', z3) -{ 1 }-> 1 + SUMITER(z15, z3) :|: z15 >= 0, z'' >= 0, z = 1, z06 >= 0, z3 >= 0, z' = 1 + z06 + z15 IFSUM(z, z', z'', z3) -{ 1 }-> 1 + SUMITER(0, z3) :|: z'' >= 0, z = 1, z3 >= 0, z' = 0 IFSUM(z, z', z'', z3) -{ 1 }-> 1 + SUMITER(0, z3) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0 LE(z, z') -{ 1 }-> 1 + LE(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 PLUS(z, z') -{ 1 }-> 1 + PLUSITER(z, z', 0) :|: z' >= 0, z >= 0 PLUSITER(z, z', z'') -{ 1 }-> 1 + IFPLUS(s, 1 + (z - 1), z', 1 + (z'' - 1)) + LE(1 + (z - 1), 1 + (z'' - 1)) :|: s >= 0, s <= 2, z' >= 0, z'' - 1 >= 0, z - 1 >= 0 PLUSITER(z, z', z'') -{ 1 }-> 1 + IFPLUS(2, 0, z', z'') + LE(0, z'') :|: z' >= 0, z = 0, z'' >= 0 PLUSITER(z, z', z'') -{ 1 }-> 1 + IFPLUS(1, 1 + (z - 1), z', 0) + LE(1 + (z - 1), 0) :|: z'' = 0, z' >= 0, z - 1 >= 0 PLUSITER(z, z', z'') -{ 1 }-> 1 + IFPLUS(0, z, z', z'') + LE(z, z'') :|: z' >= 0, z >= 0, z'' >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 1)) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 0)) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z01 + z1'', z', plus(z', z01)) :|: z' >= 0, z01 >= 0, z = 1 + z01 + z1'', z1'' >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z01 + z1'', z', plus(z', 0)) :|: z' >= 0, z01 >= 0, z = 1 + z01 + z1'', z1'' >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, z, z', plus(z', 0)) :|: z' >= 0, z >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 0, z', plus(z', 1)) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 1 + z02 + z11, z', plus(z', z02)) :|: z11 >= 0, z' >= 0, z = 1 + z02 + z11, z02 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 1)) + PLUS(z', 1) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 1)) + PLUS(z', 0) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 0)) + PLUS(z', 1) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 0)) + PLUS(z', 0) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z03 + z12, z', plus(z', z03)) + PLUS(z', z03) :|: z = 1 + z03 + z12, z' >= 0, z12 >= 0, z03 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z03 + z12, z', plus(z', z03)) + PLUS(z', 0) :|: z = 1 + z03 + z12, z' >= 0, z12 >= 0, z03 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z03 + z12, z', plus(z', 0)) + PLUS(z', z03) :|: z = 1 + z03 + z12, z' >= 0, z12 >= 0, z03 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z03 + z12, z', plus(z', 0)) + PLUS(z', 0) :|: z = 1 + z03 + z12, z' >= 0, z12 >= 0, z03 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, z, z', plus(z', 0)) + PLUS(z', 0) :|: z' >= 0, z >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 0, z', plus(z', 1)) + PLUS(z', 1) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 0, z', plus(z', 1)) + PLUS(z', 0) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 0, z', plus(z', 0)) + PLUS(z', 1) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 1 + z04 + z13, z', plus(z', z04)) + PLUS(z', z04) :|: z = 1 + z04 + z13, z04 >= 0, z' >= 0, z13 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 1 + z04 + z13, z', plus(z', z04)) + PLUS(z', 0) :|: z = 1 + z04 + z13, z04 >= 0, z' >= 0, z13 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 1 + z05 + z14, z', plus(z', 0)) + PLUS(z', z05) :|: z = 1 + z05 + z14, z' >= 0, z05 >= 0, z14 >= 0 head(z) -{ 0 }-> z0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 head(z) -{ 0 }-> 1 :|: z = 0 head(z) -{ 0 }-> 0 :|: z >= 0 ifPlus(z, z', z'', z3) -{ 0 }-> z'' :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0 ifPlus(z, z', z'', z3) -{ 0 }-> plusIter(z', 1 + z'', 1 + z3) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0 ifPlus(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0 isempty(z) -{ 0 }-> 2 :|: z = 0 isempty(z) -{ 0 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 isempty(z) -{ 0 }-> 0 :|: z >= 0 le(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 2, z' - 1 >= 0, z - 1 >= 0 le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0 le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 0 }-> plusIter(z, z', 0) :|: z' >= 0, z >= 0 plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plusIter(z, z', z'') -{ 0 }-> ifPlus(s'', 1 + (z - 1), z', 1 + (z'' - 1)) :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' >= 0, z'' - 1 >= 0 plusIter(z, z', z'') -{ 0 }-> ifPlus(2, 0, z', z'') :|: z' >= 0, z = 0, z'' >= 0 plusIter(z, z', z'') -{ 0 }-> ifPlus(1, 1 + (z - 1), z', 0) :|: z'' = 0, z' >= 0, z - 1 >= 0 plusIter(z, z', z'') -{ 0 }-> ifPlus(0, z, z', z'') :|: z' >= 0, z >= 0, z'' >= 0 plusIter(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 tail(z) -{ 0 }-> z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 tail(z) -{ 0 }-> 0 :|: z = 0 tail(z) -{ 0 }-> 0 :|: z >= 0 Function symbols to be analyzed: {LE}, {tail}, {isempty}, {head}, {plusIter,ifPlus}, {IFPLUS,PLUSITER}, {plus}, {PLUS}, {IFSUM,SUMITER} Previous analysis results are: le: runtime: O(1) [0], size: O(1) [2] LE: runtime: ?, size: O(1) [0] ---------------------------------------- (63) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: LE after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z' ---------------------------------------- (64) Obligation: Complexity RNTS consisting of the following rules: IFPLUS(z, z', z'', z3) -{ 1 }-> 1 + PLUSITER(z', 1 + z'', 1 + z3) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0 IFSUM(z, z', z'', z3) -{ 1 }-> 1 + SUMITER(z15, z3) :|: z15 >= 0, z'' >= 0, z = 1, z06 >= 0, z3 >= 0, z' = 1 + z06 + z15 IFSUM(z, z', z'', z3) -{ 1 }-> 1 + SUMITER(0, z3) :|: z'' >= 0, z = 1, z3 >= 0, z' = 0 IFSUM(z, z', z'', z3) -{ 1 }-> 1 + SUMITER(0, z3) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0 LE(z, z') -{ 1 }-> 1 + LE(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 PLUS(z, z') -{ 1 }-> 1 + PLUSITER(z, z', 0) :|: z' >= 0, z >= 0 PLUSITER(z, z', z'') -{ 1 }-> 1 + IFPLUS(s, 1 + (z - 1), z', 1 + (z'' - 1)) + LE(1 + (z - 1), 1 + (z'' - 1)) :|: s >= 0, s <= 2, z' >= 0, z'' - 1 >= 0, z - 1 >= 0 PLUSITER(z, z', z'') -{ 1 }-> 1 + IFPLUS(2, 0, z', z'') + LE(0, z'') :|: z' >= 0, z = 0, z'' >= 0 PLUSITER(z, z', z'') -{ 1 }-> 1 + IFPLUS(1, 1 + (z - 1), z', 0) + LE(1 + (z - 1), 0) :|: z'' = 0, z' >= 0, z - 1 >= 0 PLUSITER(z, z', z'') -{ 1 }-> 1 + IFPLUS(0, z, z', z'') + LE(z, z'') :|: z' >= 0, z >= 0, z'' >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 1)) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 0)) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z01 + z1'', z', plus(z', z01)) :|: z' >= 0, z01 >= 0, z = 1 + z01 + z1'', z1'' >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z01 + z1'', z', plus(z', 0)) :|: z' >= 0, z01 >= 0, z = 1 + z01 + z1'', z1'' >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, z, z', plus(z', 0)) :|: z' >= 0, z >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 0, z', plus(z', 1)) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 1 + z02 + z11, z', plus(z', z02)) :|: z11 >= 0, z' >= 0, z = 1 + z02 + z11, z02 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 1)) + PLUS(z', 1) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 1)) + PLUS(z', 0) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 0)) + PLUS(z', 1) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 0)) + PLUS(z', 0) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z03 + z12, z', plus(z', z03)) + PLUS(z', z03) :|: z = 1 + z03 + z12, z' >= 0, z12 >= 0, z03 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z03 + z12, z', plus(z', z03)) + PLUS(z', 0) :|: z = 1 + z03 + z12, z' >= 0, z12 >= 0, z03 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z03 + z12, z', plus(z', 0)) + PLUS(z', z03) :|: z = 1 + z03 + z12, z' >= 0, z12 >= 0, z03 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z03 + z12, z', plus(z', 0)) + PLUS(z', 0) :|: z = 1 + z03 + z12, z' >= 0, z12 >= 0, z03 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, z, z', plus(z', 0)) + PLUS(z', 0) :|: z' >= 0, z >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 0, z', plus(z', 1)) + PLUS(z', 1) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 0, z', plus(z', 1)) + PLUS(z', 0) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 0, z', plus(z', 0)) + PLUS(z', 1) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 1 + z04 + z13, z', plus(z', z04)) + PLUS(z', z04) :|: z = 1 + z04 + z13, z04 >= 0, z' >= 0, z13 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 1 + z04 + z13, z', plus(z', z04)) + PLUS(z', 0) :|: z = 1 + z04 + z13, z04 >= 0, z' >= 0, z13 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 1 + z05 + z14, z', plus(z', 0)) + PLUS(z', z05) :|: z = 1 + z05 + z14, z' >= 0, z05 >= 0, z14 >= 0 head(z) -{ 0 }-> z0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 head(z) -{ 0 }-> 1 :|: z = 0 head(z) -{ 0 }-> 0 :|: z >= 0 ifPlus(z, z', z'', z3) -{ 0 }-> z'' :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0 ifPlus(z, z', z'', z3) -{ 0 }-> plusIter(z', 1 + z'', 1 + z3) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0 ifPlus(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0 isempty(z) -{ 0 }-> 2 :|: z = 0 isempty(z) -{ 0 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 isempty(z) -{ 0 }-> 0 :|: z >= 0 le(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 2, z' - 1 >= 0, z - 1 >= 0 le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0 le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 0 }-> plusIter(z, z', 0) :|: z' >= 0, z >= 0 plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plusIter(z, z', z'') -{ 0 }-> ifPlus(s'', 1 + (z - 1), z', 1 + (z'' - 1)) :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' >= 0, z'' - 1 >= 0 plusIter(z, z', z'') -{ 0 }-> ifPlus(2, 0, z', z'') :|: z' >= 0, z = 0, z'' >= 0 plusIter(z, z', z'') -{ 0 }-> ifPlus(1, 1 + (z - 1), z', 0) :|: z'' = 0, z' >= 0, z - 1 >= 0 plusIter(z, z', z'') -{ 0 }-> ifPlus(0, z, z', z'') :|: z' >= 0, z >= 0, z'' >= 0 plusIter(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 tail(z) -{ 0 }-> z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 tail(z) -{ 0 }-> 0 :|: z = 0 tail(z) -{ 0 }-> 0 :|: z >= 0 Function symbols to be analyzed: {tail}, {isempty}, {head}, {plusIter,ifPlus}, {IFPLUS,PLUSITER}, {plus}, {PLUS}, {IFSUM,SUMITER} Previous analysis results are: le: runtime: O(1) [0], size: O(1) [2] LE: runtime: O(n^1) [z'], size: O(1) [0] ---------------------------------------- (65) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (66) Obligation: Complexity RNTS consisting of the following rules: IFPLUS(z, z', z'', z3) -{ 1 }-> 1 + PLUSITER(z', 1 + z'', 1 + z3) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0 IFSUM(z, z', z'', z3) -{ 1 }-> 1 + SUMITER(z15, z3) :|: z15 >= 0, z'' >= 0, z = 1, z06 >= 0, z3 >= 0, z' = 1 + z06 + z15 IFSUM(z, z', z'', z3) -{ 1 }-> 1 + SUMITER(0, z3) :|: z'' >= 0, z = 1, z3 >= 0, z' = 0 IFSUM(z, z', z'', z3) -{ 1 }-> 1 + SUMITER(0, z3) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0 LE(z, z') -{ z' }-> 1 + s5 :|: s5 >= 0, s5 <= 0, z' - 1 >= 0, z - 1 >= 0 PLUS(z, z') -{ 1 }-> 1 + PLUSITER(z, z', 0) :|: z' >= 0, z >= 0 PLUSITER(z, z', z'') -{ 1 + z'' }-> 1 + IFPLUS(s, 1 + (z - 1), z', 1 + (z'' - 1)) + s3 :|: s3 >= 0, s3 <= 0, s >= 0, s <= 2, z' >= 0, z'' - 1 >= 0, z - 1 >= 0 PLUSITER(z, z', z'') -{ 1 + z'' }-> 1 + IFPLUS(2, 0, z', z'') + s2 :|: s2 >= 0, s2 <= 0, z' >= 0, z = 0, z'' >= 0 PLUSITER(z, z', z'') -{ 1 }-> 1 + IFPLUS(1, 1 + (z - 1), z', 0) + s1 :|: s1 >= 0, s1 <= 0, z'' = 0, z' >= 0, z - 1 >= 0 PLUSITER(z, z', z'') -{ 1 + z'' }-> 1 + IFPLUS(0, z, z', z'') + s4 :|: s4 >= 0, s4 <= 0, z' >= 0, z >= 0, z'' >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 1)) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 0)) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z01 + z1'', z', plus(z', z01)) :|: z' >= 0, z01 >= 0, z = 1 + z01 + z1'', z1'' >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z01 + z1'', z', plus(z', 0)) :|: z' >= 0, z01 >= 0, z = 1 + z01 + z1'', z1'' >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, z, z', plus(z', 0)) :|: z' >= 0, z >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 0, z', plus(z', 1)) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 1 + z02 + z11, z', plus(z', z02)) :|: z11 >= 0, z' >= 0, z = 1 + z02 + z11, z02 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 1)) + PLUS(z', 1) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 1)) + PLUS(z', 0) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 0)) + PLUS(z', 1) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 0)) + PLUS(z', 0) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z03 + z12, z', plus(z', z03)) + PLUS(z', z03) :|: z = 1 + z03 + z12, z' >= 0, z12 >= 0, z03 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z03 + z12, z', plus(z', z03)) + PLUS(z', 0) :|: z = 1 + z03 + z12, z' >= 0, z12 >= 0, z03 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z03 + z12, z', plus(z', 0)) + PLUS(z', z03) :|: z = 1 + z03 + z12, z' >= 0, z12 >= 0, z03 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z03 + z12, z', plus(z', 0)) + PLUS(z', 0) :|: z = 1 + z03 + z12, z' >= 0, z12 >= 0, z03 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, z, z', plus(z', 0)) + PLUS(z', 0) :|: z' >= 0, z >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 0, z', plus(z', 1)) + PLUS(z', 1) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 0, z', plus(z', 1)) + PLUS(z', 0) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 0, z', plus(z', 0)) + PLUS(z', 1) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 1 + z04 + z13, z', plus(z', z04)) + PLUS(z', z04) :|: z = 1 + z04 + z13, z04 >= 0, z' >= 0, z13 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 1 + z04 + z13, z', plus(z', z04)) + PLUS(z', 0) :|: z = 1 + z04 + z13, z04 >= 0, z' >= 0, z13 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 1 + z05 + z14, z', plus(z', 0)) + PLUS(z', z05) :|: z = 1 + z05 + z14, z' >= 0, z05 >= 0, z14 >= 0 head(z) -{ 0 }-> z0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 head(z) -{ 0 }-> 1 :|: z = 0 head(z) -{ 0 }-> 0 :|: z >= 0 ifPlus(z, z', z'', z3) -{ 0 }-> z'' :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0 ifPlus(z, z', z'', z3) -{ 0 }-> plusIter(z', 1 + z'', 1 + z3) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0 ifPlus(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0 isempty(z) -{ 0 }-> 2 :|: z = 0 isempty(z) -{ 0 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 isempty(z) -{ 0 }-> 0 :|: z >= 0 le(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 2, z' - 1 >= 0, z - 1 >= 0 le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0 le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 0 }-> plusIter(z, z', 0) :|: z' >= 0, z >= 0 plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plusIter(z, z', z'') -{ 0 }-> ifPlus(s'', 1 + (z - 1), z', 1 + (z'' - 1)) :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' >= 0, z'' - 1 >= 0 plusIter(z, z', z'') -{ 0 }-> ifPlus(2, 0, z', z'') :|: z' >= 0, z = 0, z'' >= 0 plusIter(z, z', z'') -{ 0 }-> ifPlus(1, 1 + (z - 1), z', 0) :|: z'' = 0, z' >= 0, z - 1 >= 0 plusIter(z, z', z'') -{ 0 }-> ifPlus(0, z, z', z'') :|: z' >= 0, z >= 0, z'' >= 0 plusIter(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 tail(z) -{ 0 }-> z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 tail(z) -{ 0 }-> 0 :|: z = 0 tail(z) -{ 0 }-> 0 :|: z >= 0 Function symbols to be analyzed: {tail}, {isempty}, {head}, {plusIter,ifPlus}, {IFPLUS,PLUSITER}, {plus}, {PLUS}, {IFSUM,SUMITER} Previous analysis results are: le: runtime: O(1) [0], size: O(1) [2] LE: runtime: O(n^1) [z'], size: O(1) [0] ---------------------------------------- (67) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: tail after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z ---------------------------------------- (68) Obligation: Complexity RNTS consisting of the following rules: IFPLUS(z, z', z'', z3) -{ 1 }-> 1 + PLUSITER(z', 1 + z'', 1 + z3) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0 IFSUM(z, z', z'', z3) -{ 1 }-> 1 + SUMITER(z15, z3) :|: z15 >= 0, z'' >= 0, z = 1, z06 >= 0, z3 >= 0, z' = 1 + z06 + z15 IFSUM(z, z', z'', z3) -{ 1 }-> 1 + SUMITER(0, z3) :|: z'' >= 0, z = 1, z3 >= 0, z' = 0 IFSUM(z, z', z'', z3) -{ 1 }-> 1 + SUMITER(0, z3) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0 LE(z, z') -{ z' }-> 1 + s5 :|: s5 >= 0, s5 <= 0, z' - 1 >= 0, z - 1 >= 0 PLUS(z, z') -{ 1 }-> 1 + PLUSITER(z, z', 0) :|: z' >= 0, z >= 0 PLUSITER(z, z', z'') -{ 1 + z'' }-> 1 + IFPLUS(s, 1 + (z - 1), z', 1 + (z'' - 1)) + s3 :|: s3 >= 0, s3 <= 0, s >= 0, s <= 2, z' >= 0, z'' - 1 >= 0, z - 1 >= 0 PLUSITER(z, z', z'') -{ 1 + z'' }-> 1 + IFPLUS(2, 0, z', z'') + s2 :|: s2 >= 0, s2 <= 0, z' >= 0, z = 0, z'' >= 0 PLUSITER(z, z', z'') -{ 1 }-> 1 + IFPLUS(1, 1 + (z - 1), z', 0) + s1 :|: s1 >= 0, s1 <= 0, z'' = 0, z' >= 0, z - 1 >= 0 PLUSITER(z, z', z'') -{ 1 + z'' }-> 1 + IFPLUS(0, z, z', z'') + s4 :|: s4 >= 0, s4 <= 0, z' >= 0, z >= 0, z'' >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 1)) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 0)) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z01 + z1'', z', plus(z', z01)) :|: z' >= 0, z01 >= 0, z = 1 + z01 + z1'', z1'' >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z01 + z1'', z', plus(z', 0)) :|: z' >= 0, z01 >= 0, z = 1 + z01 + z1'', z1'' >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, z, z', plus(z', 0)) :|: z' >= 0, z >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 0, z', plus(z', 1)) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 1 + z02 + z11, z', plus(z', z02)) :|: z11 >= 0, z' >= 0, z = 1 + z02 + z11, z02 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 1)) + PLUS(z', 1) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 1)) + PLUS(z', 0) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 0)) + PLUS(z', 1) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 0)) + PLUS(z', 0) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z03 + z12, z', plus(z', z03)) + PLUS(z', z03) :|: z = 1 + z03 + z12, z' >= 0, z12 >= 0, z03 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z03 + z12, z', plus(z', z03)) + PLUS(z', 0) :|: z = 1 + z03 + z12, z' >= 0, z12 >= 0, z03 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z03 + z12, z', plus(z', 0)) + PLUS(z', z03) :|: z = 1 + z03 + z12, z' >= 0, z12 >= 0, z03 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z03 + z12, z', plus(z', 0)) + PLUS(z', 0) :|: z = 1 + z03 + z12, z' >= 0, z12 >= 0, z03 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, z, z', plus(z', 0)) + PLUS(z', 0) :|: z' >= 0, z >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 0, z', plus(z', 1)) + PLUS(z', 1) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 0, z', plus(z', 1)) + PLUS(z', 0) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 0, z', plus(z', 0)) + PLUS(z', 1) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 1 + z04 + z13, z', plus(z', z04)) + PLUS(z', z04) :|: z = 1 + z04 + z13, z04 >= 0, z' >= 0, z13 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 1 + z04 + z13, z', plus(z', z04)) + PLUS(z', 0) :|: z = 1 + z04 + z13, z04 >= 0, z' >= 0, z13 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 1 + z05 + z14, z', plus(z', 0)) + PLUS(z', z05) :|: z = 1 + z05 + z14, z' >= 0, z05 >= 0, z14 >= 0 head(z) -{ 0 }-> z0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 head(z) -{ 0 }-> 1 :|: z = 0 head(z) -{ 0 }-> 0 :|: z >= 0 ifPlus(z, z', z'', z3) -{ 0 }-> z'' :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0 ifPlus(z, z', z'', z3) -{ 0 }-> plusIter(z', 1 + z'', 1 + z3) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0 ifPlus(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0 isempty(z) -{ 0 }-> 2 :|: z = 0 isempty(z) -{ 0 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 isempty(z) -{ 0 }-> 0 :|: z >= 0 le(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 2, z' - 1 >= 0, z - 1 >= 0 le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0 le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 0 }-> plusIter(z, z', 0) :|: z' >= 0, z >= 0 plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plusIter(z, z', z'') -{ 0 }-> ifPlus(s'', 1 + (z - 1), z', 1 + (z'' - 1)) :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' >= 0, z'' - 1 >= 0 plusIter(z, z', z'') -{ 0 }-> ifPlus(2, 0, z', z'') :|: z' >= 0, z = 0, z'' >= 0 plusIter(z, z', z'') -{ 0 }-> ifPlus(1, 1 + (z - 1), z', 0) :|: z'' = 0, z' >= 0, z - 1 >= 0 plusIter(z, z', z'') -{ 0 }-> ifPlus(0, z, z', z'') :|: z' >= 0, z >= 0, z'' >= 0 plusIter(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 tail(z) -{ 0 }-> z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 tail(z) -{ 0 }-> 0 :|: z = 0 tail(z) -{ 0 }-> 0 :|: z >= 0 Function symbols to be analyzed: {tail}, {isempty}, {head}, {plusIter,ifPlus}, {IFPLUS,PLUSITER}, {plus}, {PLUS}, {IFSUM,SUMITER} Previous analysis results are: le: runtime: O(1) [0], size: O(1) [2] LE: runtime: O(n^1) [z'], size: O(1) [0] tail: runtime: ?, size: O(n^1) [z] ---------------------------------------- (69) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: tail after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (70) Obligation: Complexity RNTS consisting of the following rules: IFPLUS(z, z', z'', z3) -{ 1 }-> 1 + PLUSITER(z', 1 + z'', 1 + z3) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0 IFSUM(z, z', z'', z3) -{ 1 }-> 1 + SUMITER(z15, z3) :|: z15 >= 0, z'' >= 0, z = 1, z06 >= 0, z3 >= 0, z' = 1 + z06 + z15 IFSUM(z, z', z'', z3) -{ 1 }-> 1 + SUMITER(0, z3) :|: z'' >= 0, z = 1, z3 >= 0, z' = 0 IFSUM(z, z', z'', z3) -{ 1 }-> 1 + SUMITER(0, z3) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0 LE(z, z') -{ z' }-> 1 + s5 :|: s5 >= 0, s5 <= 0, z' - 1 >= 0, z - 1 >= 0 PLUS(z, z') -{ 1 }-> 1 + PLUSITER(z, z', 0) :|: z' >= 0, z >= 0 PLUSITER(z, z', z'') -{ 1 + z'' }-> 1 + IFPLUS(s, 1 + (z - 1), z', 1 + (z'' - 1)) + s3 :|: s3 >= 0, s3 <= 0, s >= 0, s <= 2, z' >= 0, z'' - 1 >= 0, z - 1 >= 0 PLUSITER(z, z', z'') -{ 1 + z'' }-> 1 + IFPLUS(2, 0, z', z'') + s2 :|: s2 >= 0, s2 <= 0, z' >= 0, z = 0, z'' >= 0 PLUSITER(z, z', z'') -{ 1 }-> 1 + IFPLUS(1, 1 + (z - 1), z', 0) + s1 :|: s1 >= 0, s1 <= 0, z'' = 0, z' >= 0, z - 1 >= 0 PLUSITER(z, z', z'') -{ 1 + z'' }-> 1 + IFPLUS(0, z, z', z'') + s4 :|: s4 >= 0, s4 <= 0, z' >= 0, z >= 0, z'' >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 1)) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 0)) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z01 + z1'', z', plus(z', z01)) :|: z' >= 0, z01 >= 0, z = 1 + z01 + z1'', z1'' >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z01 + z1'', z', plus(z', 0)) :|: z' >= 0, z01 >= 0, z = 1 + z01 + z1'', z1'' >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, z, z', plus(z', 0)) :|: z' >= 0, z >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 0, z', plus(z', 1)) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 1 + z02 + z11, z', plus(z', z02)) :|: z11 >= 0, z' >= 0, z = 1 + z02 + z11, z02 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 1)) + PLUS(z', 1) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 1)) + PLUS(z', 0) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 0)) + PLUS(z', 1) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 0)) + PLUS(z', 0) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z03 + z12, z', plus(z', z03)) + PLUS(z', z03) :|: z = 1 + z03 + z12, z' >= 0, z12 >= 0, z03 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z03 + z12, z', plus(z', z03)) + PLUS(z', 0) :|: z = 1 + z03 + z12, z' >= 0, z12 >= 0, z03 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z03 + z12, z', plus(z', 0)) + PLUS(z', z03) :|: z = 1 + z03 + z12, z' >= 0, z12 >= 0, z03 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z03 + z12, z', plus(z', 0)) + PLUS(z', 0) :|: z = 1 + z03 + z12, z' >= 0, z12 >= 0, z03 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, z, z', plus(z', 0)) + PLUS(z', 0) :|: z' >= 0, z >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 0, z', plus(z', 1)) + PLUS(z', 1) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 0, z', plus(z', 1)) + PLUS(z', 0) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 0, z', plus(z', 0)) + PLUS(z', 1) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 1 + z04 + z13, z', plus(z', z04)) + PLUS(z', z04) :|: z = 1 + z04 + z13, z04 >= 0, z' >= 0, z13 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 1 + z04 + z13, z', plus(z', z04)) + PLUS(z', 0) :|: z = 1 + z04 + z13, z04 >= 0, z' >= 0, z13 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 1 + z05 + z14, z', plus(z', 0)) + PLUS(z', z05) :|: z = 1 + z05 + z14, z' >= 0, z05 >= 0, z14 >= 0 head(z) -{ 0 }-> z0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 head(z) -{ 0 }-> 1 :|: z = 0 head(z) -{ 0 }-> 0 :|: z >= 0 ifPlus(z, z', z'', z3) -{ 0 }-> z'' :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0 ifPlus(z, z', z'', z3) -{ 0 }-> plusIter(z', 1 + z'', 1 + z3) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0 ifPlus(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0 isempty(z) -{ 0 }-> 2 :|: z = 0 isempty(z) -{ 0 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 isempty(z) -{ 0 }-> 0 :|: z >= 0 le(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 2, z' - 1 >= 0, z - 1 >= 0 le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0 le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 0 }-> plusIter(z, z', 0) :|: z' >= 0, z >= 0 plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plusIter(z, z', z'') -{ 0 }-> ifPlus(s'', 1 + (z - 1), z', 1 + (z'' - 1)) :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' >= 0, z'' - 1 >= 0 plusIter(z, z', z'') -{ 0 }-> ifPlus(2, 0, z', z'') :|: z' >= 0, z = 0, z'' >= 0 plusIter(z, z', z'') -{ 0 }-> ifPlus(1, 1 + (z - 1), z', 0) :|: z'' = 0, z' >= 0, z - 1 >= 0 plusIter(z, z', z'') -{ 0 }-> ifPlus(0, z, z', z'') :|: z' >= 0, z >= 0, z'' >= 0 plusIter(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 tail(z) -{ 0 }-> z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 tail(z) -{ 0 }-> 0 :|: z = 0 tail(z) -{ 0 }-> 0 :|: z >= 0 Function symbols to be analyzed: {isempty}, {head}, {plusIter,ifPlus}, {IFPLUS,PLUSITER}, {plus}, {PLUS}, {IFSUM,SUMITER} Previous analysis results are: le: runtime: O(1) [0], size: O(1) [2] LE: runtime: O(n^1) [z'], size: O(1) [0] tail: runtime: O(1) [0], size: O(n^1) [z] ---------------------------------------- (71) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (72) Obligation: Complexity RNTS consisting of the following rules: IFPLUS(z, z', z'', z3) -{ 1 }-> 1 + PLUSITER(z', 1 + z'', 1 + z3) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0 IFSUM(z, z', z'', z3) -{ 1 }-> 1 + SUMITER(z15, z3) :|: z15 >= 0, z'' >= 0, z = 1, z06 >= 0, z3 >= 0, z' = 1 + z06 + z15 IFSUM(z, z', z'', z3) -{ 1 }-> 1 + SUMITER(0, z3) :|: z'' >= 0, z = 1, z3 >= 0, z' = 0 IFSUM(z, z', z'', z3) -{ 1 }-> 1 + SUMITER(0, z3) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0 LE(z, z') -{ z' }-> 1 + s5 :|: s5 >= 0, s5 <= 0, z' - 1 >= 0, z - 1 >= 0 PLUS(z, z') -{ 1 }-> 1 + PLUSITER(z, z', 0) :|: z' >= 0, z >= 0 PLUSITER(z, z', z'') -{ 1 + z'' }-> 1 + IFPLUS(s, 1 + (z - 1), z', 1 + (z'' - 1)) + s3 :|: s3 >= 0, s3 <= 0, s >= 0, s <= 2, z' >= 0, z'' - 1 >= 0, z - 1 >= 0 PLUSITER(z, z', z'') -{ 1 + z'' }-> 1 + IFPLUS(2, 0, z', z'') + s2 :|: s2 >= 0, s2 <= 0, z' >= 0, z = 0, z'' >= 0 PLUSITER(z, z', z'') -{ 1 }-> 1 + IFPLUS(1, 1 + (z - 1), z', 0) + s1 :|: s1 >= 0, s1 <= 0, z'' = 0, z' >= 0, z - 1 >= 0 PLUSITER(z, z', z'') -{ 1 + z'' }-> 1 + IFPLUS(0, z, z', z'') + s4 :|: s4 >= 0, s4 <= 0, z' >= 0, z >= 0, z'' >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 1)) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 0)) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z01 + z1'', z', plus(z', z01)) :|: z' >= 0, z01 >= 0, z = 1 + z01 + z1'', z1'' >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z01 + z1'', z', plus(z', 0)) :|: z' >= 0, z01 >= 0, z = 1 + z01 + z1'', z1'' >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, z, z', plus(z', 0)) :|: z' >= 0, z >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 0, z', plus(z', 1)) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 1 + z02 + z11, z', plus(z', z02)) :|: z11 >= 0, z' >= 0, z = 1 + z02 + z11, z02 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 1)) + PLUS(z', 1) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 1)) + PLUS(z', 0) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 0)) + PLUS(z', 1) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 0)) + PLUS(z', 0) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z03 + z12, z', plus(z', z03)) + PLUS(z', z03) :|: z = 1 + z03 + z12, z' >= 0, z12 >= 0, z03 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z03 + z12, z', plus(z', z03)) + PLUS(z', 0) :|: z = 1 + z03 + z12, z' >= 0, z12 >= 0, z03 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z03 + z12, z', plus(z', 0)) + PLUS(z', z03) :|: z = 1 + z03 + z12, z' >= 0, z12 >= 0, z03 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z03 + z12, z', plus(z', 0)) + PLUS(z', 0) :|: z = 1 + z03 + z12, z' >= 0, z12 >= 0, z03 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, z, z', plus(z', 0)) + PLUS(z', 0) :|: z' >= 0, z >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 0, z', plus(z', 1)) + PLUS(z', 1) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 0, z', plus(z', 1)) + PLUS(z', 0) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 0, z', plus(z', 0)) + PLUS(z', 1) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 1 + z04 + z13, z', plus(z', z04)) + PLUS(z', z04) :|: z = 1 + z04 + z13, z04 >= 0, z' >= 0, z13 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 1 + z04 + z13, z', plus(z', z04)) + PLUS(z', 0) :|: z = 1 + z04 + z13, z04 >= 0, z' >= 0, z13 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 1 + z05 + z14, z', plus(z', 0)) + PLUS(z', z05) :|: z = 1 + z05 + z14, z' >= 0, z05 >= 0, z14 >= 0 head(z) -{ 0 }-> z0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 head(z) -{ 0 }-> 1 :|: z = 0 head(z) -{ 0 }-> 0 :|: z >= 0 ifPlus(z, z', z'', z3) -{ 0 }-> z'' :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0 ifPlus(z, z', z'', z3) -{ 0 }-> plusIter(z', 1 + z'', 1 + z3) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0 ifPlus(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0 isempty(z) -{ 0 }-> 2 :|: z = 0 isempty(z) -{ 0 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 isempty(z) -{ 0 }-> 0 :|: z >= 0 le(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 2, z' - 1 >= 0, z - 1 >= 0 le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0 le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 0 }-> plusIter(z, z', 0) :|: z' >= 0, z >= 0 plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plusIter(z, z', z'') -{ 0 }-> ifPlus(s'', 1 + (z - 1), z', 1 + (z'' - 1)) :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' >= 0, z'' - 1 >= 0 plusIter(z, z', z'') -{ 0 }-> ifPlus(2, 0, z', z'') :|: z' >= 0, z = 0, z'' >= 0 plusIter(z, z', z'') -{ 0 }-> ifPlus(1, 1 + (z - 1), z', 0) :|: z'' = 0, z' >= 0, z - 1 >= 0 plusIter(z, z', z'') -{ 0 }-> ifPlus(0, z, z', z'') :|: z' >= 0, z >= 0, z'' >= 0 plusIter(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 tail(z) -{ 0 }-> z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 tail(z) -{ 0 }-> 0 :|: z = 0 tail(z) -{ 0 }-> 0 :|: z >= 0 Function symbols to be analyzed: {isempty}, {head}, {plusIter,ifPlus}, {IFPLUS,PLUSITER}, {plus}, {PLUS}, {IFSUM,SUMITER} Previous analysis results are: le: runtime: O(1) [0], size: O(1) [2] LE: runtime: O(n^1) [z'], size: O(1) [0] tail: runtime: O(1) [0], size: O(n^1) [z] ---------------------------------------- (73) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: isempty after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 2 ---------------------------------------- (74) Obligation: Complexity RNTS consisting of the following rules: IFPLUS(z, z', z'', z3) -{ 1 }-> 1 + PLUSITER(z', 1 + z'', 1 + z3) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0 IFSUM(z, z', z'', z3) -{ 1 }-> 1 + SUMITER(z15, z3) :|: z15 >= 0, z'' >= 0, z = 1, z06 >= 0, z3 >= 0, z' = 1 + z06 + z15 IFSUM(z, z', z'', z3) -{ 1 }-> 1 + SUMITER(0, z3) :|: z'' >= 0, z = 1, z3 >= 0, z' = 0 IFSUM(z, z', z'', z3) -{ 1 }-> 1 + SUMITER(0, z3) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0 LE(z, z') -{ z' }-> 1 + s5 :|: s5 >= 0, s5 <= 0, z' - 1 >= 0, z - 1 >= 0 PLUS(z, z') -{ 1 }-> 1 + PLUSITER(z, z', 0) :|: z' >= 0, z >= 0 PLUSITER(z, z', z'') -{ 1 + z'' }-> 1 + IFPLUS(s, 1 + (z - 1), z', 1 + (z'' - 1)) + s3 :|: s3 >= 0, s3 <= 0, s >= 0, s <= 2, z' >= 0, z'' - 1 >= 0, z - 1 >= 0 PLUSITER(z, z', z'') -{ 1 + z'' }-> 1 + IFPLUS(2, 0, z', z'') + s2 :|: s2 >= 0, s2 <= 0, z' >= 0, z = 0, z'' >= 0 PLUSITER(z, z', z'') -{ 1 }-> 1 + IFPLUS(1, 1 + (z - 1), z', 0) + s1 :|: s1 >= 0, s1 <= 0, z'' = 0, z' >= 0, z - 1 >= 0 PLUSITER(z, z', z'') -{ 1 + z'' }-> 1 + IFPLUS(0, z, z', z'') + s4 :|: s4 >= 0, s4 <= 0, z' >= 0, z >= 0, z'' >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 1)) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 0)) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z01 + z1'', z', plus(z', z01)) :|: z' >= 0, z01 >= 0, z = 1 + z01 + z1'', z1'' >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z01 + z1'', z', plus(z', 0)) :|: z' >= 0, z01 >= 0, z = 1 + z01 + z1'', z1'' >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, z, z', plus(z', 0)) :|: z' >= 0, z >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 0, z', plus(z', 1)) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 1 + z02 + z11, z', plus(z', z02)) :|: z11 >= 0, z' >= 0, z = 1 + z02 + z11, z02 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 1)) + PLUS(z', 1) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 1)) + PLUS(z', 0) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 0)) + PLUS(z', 1) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 0)) + PLUS(z', 0) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z03 + z12, z', plus(z', z03)) + PLUS(z', z03) :|: z = 1 + z03 + z12, z' >= 0, z12 >= 0, z03 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z03 + z12, z', plus(z', z03)) + PLUS(z', 0) :|: z = 1 + z03 + z12, z' >= 0, z12 >= 0, z03 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z03 + z12, z', plus(z', 0)) + PLUS(z', z03) :|: z = 1 + z03 + z12, z' >= 0, z12 >= 0, z03 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z03 + z12, z', plus(z', 0)) + PLUS(z', 0) :|: z = 1 + z03 + z12, z' >= 0, z12 >= 0, z03 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, z, z', plus(z', 0)) + PLUS(z', 0) :|: z' >= 0, z >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 0, z', plus(z', 1)) + PLUS(z', 1) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 0, z', plus(z', 1)) + PLUS(z', 0) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 0, z', plus(z', 0)) + PLUS(z', 1) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 1 + z04 + z13, z', plus(z', z04)) + PLUS(z', z04) :|: z = 1 + z04 + z13, z04 >= 0, z' >= 0, z13 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 1 + z04 + z13, z', plus(z', z04)) + PLUS(z', 0) :|: z = 1 + z04 + z13, z04 >= 0, z' >= 0, z13 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 1 + z05 + z14, z', plus(z', 0)) + PLUS(z', z05) :|: z = 1 + z05 + z14, z' >= 0, z05 >= 0, z14 >= 0 head(z) -{ 0 }-> z0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 head(z) -{ 0 }-> 1 :|: z = 0 head(z) -{ 0 }-> 0 :|: z >= 0 ifPlus(z, z', z'', z3) -{ 0 }-> z'' :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0 ifPlus(z, z', z'', z3) -{ 0 }-> plusIter(z', 1 + z'', 1 + z3) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0 ifPlus(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0 isempty(z) -{ 0 }-> 2 :|: z = 0 isempty(z) -{ 0 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 isempty(z) -{ 0 }-> 0 :|: z >= 0 le(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 2, z' - 1 >= 0, z - 1 >= 0 le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0 le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 0 }-> plusIter(z, z', 0) :|: z' >= 0, z >= 0 plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plusIter(z, z', z'') -{ 0 }-> ifPlus(s'', 1 + (z - 1), z', 1 + (z'' - 1)) :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' >= 0, z'' - 1 >= 0 plusIter(z, z', z'') -{ 0 }-> ifPlus(2, 0, z', z'') :|: z' >= 0, z = 0, z'' >= 0 plusIter(z, z', z'') -{ 0 }-> ifPlus(1, 1 + (z - 1), z', 0) :|: z'' = 0, z' >= 0, z - 1 >= 0 plusIter(z, z', z'') -{ 0 }-> ifPlus(0, z, z', z'') :|: z' >= 0, z >= 0, z'' >= 0 plusIter(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 tail(z) -{ 0 }-> z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 tail(z) -{ 0 }-> 0 :|: z = 0 tail(z) -{ 0 }-> 0 :|: z >= 0 Function symbols to be analyzed: {isempty}, {head}, {plusIter,ifPlus}, {IFPLUS,PLUSITER}, {plus}, {PLUS}, {IFSUM,SUMITER} Previous analysis results are: le: runtime: O(1) [0], size: O(1) [2] LE: runtime: O(n^1) [z'], size: O(1) [0] tail: runtime: O(1) [0], size: O(n^1) [z] isempty: runtime: ?, size: O(1) [2] ---------------------------------------- (75) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: isempty after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (76) Obligation: Complexity RNTS consisting of the following rules: IFPLUS(z, z', z'', z3) -{ 1 }-> 1 + PLUSITER(z', 1 + z'', 1 + z3) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0 IFSUM(z, z', z'', z3) -{ 1 }-> 1 + SUMITER(z15, z3) :|: z15 >= 0, z'' >= 0, z = 1, z06 >= 0, z3 >= 0, z' = 1 + z06 + z15 IFSUM(z, z', z'', z3) -{ 1 }-> 1 + SUMITER(0, z3) :|: z'' >= 0, z = 1, z3 >= 0, z' = 0 IFSUM(z, z', z'', z3) -{ 1 }-> 1 + SUMITER(0, z3) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0 LE(z, z') -{ z' }-> 1 + s5 :|: s5 >= 0, s5 <= 0, z' - 1 >= 0, z - 1 >= 0 PLUS(z, z') -{ 1 }-> 1 + PLUSITER(z, z', 0) :|: z' >= 0, z >= 0 PLUSITER(z, z', z'') -{ 1 + z'' }-> 1 + IFPLUS(s, 1 + (z - 1), z', 1 + (z'' - 1)) + s3 :|: s3 >= 0, s3 <= 0, s >= 0, s <= 2, z' >= 0, z'' - 1 >= 0, z - 1 >= 0 PLUSITER(z, z', z'') -{ 1 + z'' }-> 1 + IFPLUS(2, 0, z', z'') + s2 :|: s2 >= 0, s2 <= 0, z' >= 0, z = 0, z'' >= 0 PLUSITER(z, z', z'') -{ 1 }-> 1 + IFPLUS(1, 1 + (z - 1), z', 0) + s1 :|: s1 >= 0, s1 <= 0, z'' = 0, z' >= 0, z - 1 >= 0 PLUSITER(z, z', z'') -{ 1 + z'' }-> 1 + IFPLUS(0, z, z', z'') + s4 :|: s4 >= 0, s4 <= 0, z' >= 0, z >= 0, z'' >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 1)) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 0)) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z01 + z1'', z', plus(z', z01)) :|: z' >= 0, z01 >= 0, z = 1 + z01 + z1'', z1'' >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z01 + z1'', z', plus(z', 0)) :|: z' >= 0, z01 >= 0, z = 1 + z01 + z1'', z1'' >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, z, z', plus(z', 0)) :|: z' >= 0, z >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 0, z', plus(z', 1)) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 1 + z02 + z11, z', plus(z', z02)) :|: z11 >= 0, z' >= 0, z = 1 + z02 + z11, z02 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 1)) + PLUS(z', 1) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 1)) + PLUS(z', 0) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 0)) + PLUS(z', 1) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 0)) + PLUS(z', 0) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z03 + z12, z', plus(z', z03)) + PLUS(z', z03) :|: z = 1 + z03 + z12, z' >= 0, z12 >= 0, z03 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z03 + z12, z', plus(z', z03)) + PLUS(z', 0) :|: z = 1 + z03 + z12, z' >= 0, z12 >= 0, z03 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z03 + z12, z', plus(z', 0)) + PLUS(z', z03) :|: z = 1 + z03 + z12, z' >= 0, z12 >= 0, z03 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z03 + z12, z', plus(z', 0)) + PLUS(z', 0) :|: z = 1 + z03 + z12, z' >= 0, z12 >= 0, z03 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, z, z', plus(z', 0)) + PLUS(z', 0) :|: z' >= 0, z >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 0, z', plus(z', 1)) + PLUS(z', 1) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 0, z', plus(z', 1)) + PLUS(z', 0) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 0, z', plus(z', 0)) + PLUS(z', 1) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 1 + z04 + z13, z', plus(z', z04)) + PLUS(z', z04) :|: z = 1 + z04 + z13, z04 >= 0, z' >= 0, z13 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 1 + z04 + z13, z', plus(z', z04)) + PLUS(z', 0) :|: z = 1 + z04 + z13, z04 >= 0, z' >= 0, z13 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 1 + z05 + z14, z', plus(z', 0)) + PLUS(z', z05) :|: z = 1 + z05 + z14, z' >= 0, z05 >= 0, z14 >= 0 head(z) -{ 0 }-> z0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 head(z) -{ 0 }-> 1 :|: z = 0 head(z) -{ 0 }-> 0 :|: z >= 0 ifPlus(z, z', z'', z3) -{ 0 }-> z'' :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0 ifPlus(z, z', z'', z3) -{ 0 }-> plusIter(z', 1 + z'', 1 + z3) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0 ifPlus(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0 isempty(z) -{ 0 }-> 2 :|: z = 0 isempty(z) -{ 0 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 isempty(z) -{ 0 }-> 0 :|: z >= 0 le(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 2, z' - 1 >= 0, z - 1 >= 0 le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0 le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 0 }-> plusIter(z, z', 0) :|: z' >= 0, z >= 0 plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plusIter(z, z', z'') -{ 0 }-> ifPlus(s'', 1 + (z - 1), z', 1 + (z'' - 1)) :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' >= 0, z'' - 1 >= 0 plusIter(z, z', z'') -{ 0 }-> ifPlus(2, 0, z', z'') :|: z' >= 0, z = 0, z'' >= 0 plusIter(z, z', z'') -{ 0 }-> ifPlus(1, 1 + (z - 1), z', 0) :|: z'' = 0, z' >= 0, z - 1 >= 0 plusIter(z, z', z'') -{ 0 }-> ifPlus(0, z, z', z'') :|: z' >= 0, z >= 0, z'' >= 0 plusIter(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 tail(z) -{ 0 }-> z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 tail(z) -{ 0 }-> 0 :|: z = 0 tail(z) -{ 0 }-> 0 :|: z >= 0 Function symbols to be analyzed: {head}, {plusIter,ifPlus}, {IFPLUS,PLUSITER}, {plus}, {PLUS}, {IFSUM,SUMITER} Previous analysis results are: le: runtime: O(1) [0], size: O(1) [2] LE: runtime: O(n^1) [z'], size: O(1) [0] tail: runtime: O(1) [0], size: O(n^1) [z] isempty: runtime: O(1) [0], size: O(1) [2] ---------------------------------------- (77) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (78) Obligation: Complexity RNTS consisting of the following rules: IFPLUS(z, z', z'', z3) -{ 1 }-> 1 + PLUSITER(z', 1 + z'', 1 + z3) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0 IFSUM(z, z', z'', z3) -{ 1 }-> 1 + SUMITER(z15, z3) :|: z15 >= 0, z'' >= 0, z = 1, z06 >= 0, z3 >= 0, z' = 1 + z06 + z15 IFSUM(z, z', z'', z3) -{ 1 }-> 1 + SUMITER(0, z3) :|: z'' >= 0, z = 1, z3 >= 0, z' = 0 IFSUM(z, z', z'', z3) -{ 1 }-> 1 + SUMITER(0, z3) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0 LE(z, z') -{ z' }-> 1 + s5 :|: s5 >= 0, s5 <= 0, z' - 1 >= 0, z - 1 >= 0 PLUS(z, z') -{ 1 }-> 1 + PLUSITER(z, z', 0) :|: z' >= 0, z >= 0 PLUSITER(z, z', z'') -{ 1 + z'' }-> 1 + IFPLUS(s, 1 + (z - 1), z', 1 + (z'' - 1)) + s3 :|: s3 >= 0, s3 <= 0, s >= 0, s <= 2, z' >= 0, z'' - 1 >= 0, z - 1 >= 0 PLUSITER(z, z', z'') -{ 1 + z'' }-> 1 + IFPLUS(2, 0, z', z'') + s2 :|: s2 >= 0, s2 <= 0, z' >= 0, z = 0, z'' >= 0 PLUSITER(z, z', z'') -{ 1 }-> 1 + IFPLUS(1, 1 + (z - 1), z', 0) + s1 :|: s1 >= 0, s1 <= 0, z'' = 0, z' >= 0, z - 1 >= 0 PLUSITER(z, z', z'') -{ 1 + z'' }-> 1 + IFPLUS(0, z, z', z'') + s4 :|: s4 >= 0, s4 <= 0, z' >= 0, z >= 0, z'' >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 1)) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 0)) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z01 + z1'', z', plus(z', z01)) :|: z' >= 0, z01 >= 0, z = 1 + z01 + z1'', z1'' >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z01 + z1'', z', plus(z', 0)) :|: z' >= 0, z01 >= 0, z = 1 + z01 + z1'', z1'' >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, z, z', plus(z', 0)) :|: z' >= 0, z >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 0, z', plus(z', 1)) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 1 + z02 + z11, z', plus(z', z02)) :|: z11 >= 0, z' >= 0, z = 1 + z02 + z11, z02 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 1)) + PLUS(z', 1) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 1)) + PLUS(z', 0) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 0)) + PLUS(z', 1) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 0)) + PLUS(z', 0) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z03 + z12, z', plus(z', z03)) + PLUS(z', z03) :|: z = 1 + z03 + z12, z' >= 0, z12 >= 0, z03 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z03 + z12, z', plus(z', z03)) + PLUS(z', 0) :|: z = 1 + z03 + z12, z' >= 0, z12 >= 0, z03 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z03 + z12, z', plus(z', 0)) + PLUS(z', z03) :|: z = 1 + z03 + z12, z' >= 0, z12 >= 0, z03 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z03 + z12, z', plus(z', 0)) + PLUS(z', 0) :|: z = 1 + z03 + z12, z' >= 0, z12 >= 0, z03 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, z, z', plus(z', 0)) + PLUS(z', 0) :|: z' >= 0, z >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 0, z', plus(z', 1)) + PLUS(z', 1) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 0, z', plus(z', 1)) + PLUS(z', 0) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 0, z', plus(z', 0)) + PLUS(z', 1) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 1 + z04 + z13, z', plus(z', z04)) + PLUS(z', z04) :|: z = 1 + z04 + z13, z04 >= 0, z' >= 0, z13 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 1 + z04 + z13, z', plus(z', z04)) + PLUS(z', 0) :|: z = 1 + z04 + z13, z04 >= 0, z' >= 0, z13 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 1 + z05 + z14, z', plus(z', 0)) + PLUS(z', z05) :|: z = 1 + z05 + z14, z' >= 0, z05 >= 0, z14 >= 0 head(z) -{ 0 }-> z0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 head(z) -{ 0 }-> 1 :|: z = 0 head(z) -{ 0 }-> 0 :|: z >= 0 ifPlus(z, z', z'', z3) -{ 0 }-> z'' :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0 ifPlus(z, z', z'', z3) -{ 0 }-> plusIter(z', 1 + z'', 1 + z3) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0 ifPlus(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0 isempty(z) -{ 0 }-> 2 :|: z = 0 isempty(z) -{ 0 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 isempty(z) -{ 0 }-> 0 :|: z >= 0 le(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 2, z' - 1 >= 0, z - 1 >= 0 le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0 le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 0 }-> plusIter(z, z', 0) :|: z' >= 0, z >= 0 plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plusIter(z, z', z'') -{ 0 }-> ifPlus(s'', 1 + (z - 1), z', 1 + (z'' - 1)) :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' >= 0, z'' - 1 >= 0 plusIter(z, z', z'') -{ 0 }-> ifPlus(2, 0, z', z'') :|: z' >= 0, z = 0, z'' >= 0 plusIter(z, z', z'') -{ 0 }-> ifPlus(1, 1 + (z - 1), z', 0) :|: z'' = 0, z' >= 0, z - 1 >= 0 plusIter(z, z', z'') -{ 0 }-> ifPlus(0, z, z', z'') :|: z' >= 0, z >= 0, z'' >= 0 plusIter(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 tail(z) -{ 0 }-> z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 tail(z) -{ 0 }-> 0 :|: z = 0 tail(z) -{ 0 }-> 0 :|: z >= 0 Function symbols to be analyzed: {head}, {plusIter,ifPlus}, {IFPLUS,PLUSITER}, {plus}, {PLUS}, {IFSUM,SUMITER} Previous analysis results are: le: runtime: O(1) [0], size: O(1) [2] LE: runtime: O(n^1) [z'], size: O(1) [0] tail: runtime: O(1) [0], size: O(n^1) [z] isempty: runtime: O(1) [0], size: O(1) [2] ---------------------------------------- (79) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: head after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z ---------------------------------------- (80) Obligation: Complexity RNTS consisting of the following rules: IFPLUS(z, z', z'', z3) -{ 1 }-> 1 + PLUSITER(z', 1 + z'', 1 + z3) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0 IFSUM(z, z', z'', z3) -{ 1 }-> 1 + SUMITER(z15, z3) :|: z15 >= 0, z'' >= 0, z = 1, z06 >= 0, z3 >= 0, z' = 1 + z06 + z15 IFSUM(z, z', z'', z3) -{ 1 }-> 1 + SUMITER(0, z3) :|: z'' >= 0, z = 1, z3 >= 0, z' = 0 IFSUM(z, z', z'', z3) -{ 1 }-> 1 + SUMITER(0, z3) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0 LE(z, z') -{ z' }-> 1 + s5 :|: s5 >= 0, s5 <= 0, z' - 1 >= 0, z - 1 >= 0 PLUS(z, z') -{ 1 }-> 1 + PLUSITER(z, z', 0) :|: z' >= 0, z >= 0 PLUSITER(z, z', z'') -{ 1 + z'' }-> 1 + IFPLUS(s, 1 + (z - 1), z', 1 + (z'' - 1)) + s3 :|: s3 >= 0, s3 <= 0, s >= 0, s <= 2, z' >= 0, z'' - 1 >= 0, z - 1 >= 0 PLUSITER(z, z', z'') -{ 1 + z'' }-> 1 + IFPLUS(2, 0, z', z'') + s2 :|: s2 >= 0, s2 <= 0, z' >= 0, z = 0, z'' >= 0 PLUSITER(z, z', z'') -{ 1 }-> 1 + IFPLUS(1, 1 + (z - 1), z', 0) + s1 :|: s1 >= 0, s1 <= 0, z'' = 0, z' >= 0, z - 1 >= 0 PLUSITER(z, z', z'') -{ 1 + z'' }-> 1 + IFPLUS(0, z, z', z'') + s4 :|: s4 >= 0, s4 <= 0, z' >= 0, z >= 0, z'' >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 1)) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 0)) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z01 + z1'', z', plus(z', z01)) :|: z' >= 0, z01 >= 0, z = 1 + z01 + z1'', z1'' >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z01 + z1'', z', plus(z', 0)) :|: z' >= 0, z01 >= 0, z = 1 + z01 + z1'', z1'' >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, z, z', plus(z', 0)) :|: z' >= 0, z >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 0, z', plus(z', 1)) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 1 + z02 + z11, z', plus(z', z02)) :|: z11 >= 0, z' >= 0, z = 1 + z02 + z11, z02 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 1)) + PLUS(z', 1) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 1)) + PLUS(z', 0) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 0)) + PLUS(z', 1) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 0)) + PLUS(z', 0) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z03 + z12, z', plus(z', z03)) + PLUS(z', z03) :|: z = 1 + z03 + z12, z' >= 0, z12 >= 0, z03 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z03 + z12, z', plus(z', z03)) + PLUS(z', 0) :|: z = 1 + z03 + z12, z' >= 0, z12 >= 0, z03 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z03 + z12, z', plus(z', 0)) + PLUS(z', z03) :|: z = 1 + z03 + z12, z' >= 0, z12 >= 0, z03 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z03 + z12, z', plus(z', 0)) + PLUS(z', 0) :|: z = 1 + z03 + z12, z' >= 0, z12 >= 0, z03 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, z, z', plus(z', 0)) + PLUS(z', 0) :|: z' >= 0, z >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 0, z', plus(z', 1)) + PLUS(z', 1) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 0, z', plus(z', 1)) + PLUS(z', 0) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 0, z', plus(z', 0)) + PLUS(z', 1) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 1 + z04 + z13, z', plus(z', z04)) + PLUS(z', z04) :|: z = 1 + z04 + z13, z04 >= 0, z' >= 0, z13 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 1 + z04 + z13, z', plus(z', z04)) + PLUS(z', 0) :|: z = 1 + z04 + z13, z04 >= 0, z' >= 0, z13 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 1 + z05 + z14, z', plus(z', 0)) + PLUS(z', z05) :|: z = 1 + z05 + z14, z' >= 0, z05 >= 0, z14 >= 0 head(z) -{ 0 }-> z0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 head(z) -{ 0 }-> 1 :|: z = 0 head(z) -{ 0 }-> 0 :|: z >= 0 ifPlus(z, z', z'', z3) -{ 0 }-> z'' :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0 ifPlus(z, z', z'', z3) -{ 0 }-> plusIter(z', 1 + z'', 1 + z3) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0 ifPlus(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0 isempty(z) -{ 0 }-> 2 :|: z = 0 isempty(z) -{ 0 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 isempty(z) -{ 0 }-> 0 :|: z >= 0 le(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 2, z' - 1 >= 0, z - 1 >= 0 le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0 le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 0 }-> plusIter(z, z', 0) :|: z' >= 0, z >= 0 plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plusIter(z, z', z'') -{ 0 }-> ifPlus(s'', 1 + (z - 1), z', 1 + (z'' - 1)) :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' >= 0, z'' - 1 >= 0 plusIter(z, z', z'') -{ 0 }-> ifPlus(2, 0, z', z'') :|: z' >= 0, z = 0, z'' >= 0 plusIter(z, z', z'') -{ 0 }-> ifPlus(1, 1 + (z - 1), z', 0) :|: z'' = 0, z' >= 0, z - 1 >= 0 plusIter(z, z', z'') -{ 0 }-> ifPlus(0, z, z', z'') :|: z' >= 0, z >= 0, z'' >= 0 plusIter(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 tail(z) -{ 0 }-> z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 tail(z) -{ 0 }-> 0 :|: z = 0 tail(z) -{ 0 }-> 0 :|: z >= 0 Function symbols to be analyzed: {head}, {plusIter,ifPlus}, {IFPLUS,PLUSITER}, {plus}, {PLUS}, {IFSUM,SUMITER} Previous analysis results are: le: runtime: O(1) [0], size: O(1) [2] LE: runtime: O(n^1) [z'], size: O(1) [0] tail: runtime: O(1) [0], size: O(n^1) [z] isempty: runtime: O(1) [0], size: O(1) [2] head: runtime: ?, size: O(n^1) [1 + z] ---------------------------------------- (81) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: head after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (82) Obligation: Complexity RNTS consisting of the following rules: IFPLUS(z, z', z'', z3) -{ 1 }-> 1 + PLUSITER(z', 1 + z'', 1 + z3) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0 IFSUM(z, z', z'', z3) -{ 1 }-> 1 + SUMITER(z15, z3) :|: z15 >= 0, z'' >= 0, z = 1, z06 >= 0, z3 >= 0, z' = 1 + z06 + z15 IFSUM(z, z', z'', z3) -{ 1 }-> 1 + SUMITER(0, z3) :|: z'' >= 0, z = 1, z3 >= 0, z' = 0 IFSUM(z, z', z'', z3) -{ 1 }-> 1 + SUMITER(0, z3) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0 LE(z, z') -{ z' }-> 1 + s5 :|: s5 >= 0, s5 <= 0, z' - 1 >= 0, z - 1 >= 0 PLUS(z, z') -{ 1 }-> 1 + PLUSITER(z, z', 0) :|: z' >= 0, z >= 0 PLUSITER(z, z', z'') -{ 1 + z'' }-> 1 + IFPLUS(s, 1 + (z - 1), z', 1 + (z'' - 1)) + s3 :|: s3 >= 0, s3 <= 0, s >= 0, s <= 2, z' >= 0, z'' - 1 >= 0, z - 1 >= 0 PLUSITER(z, z', z'') -{ 1 + z'' }-> 1 + IFPLUS(2, 0, z', z'') + s2 :|: s2 >= 0, s2 <= 0, z' >= 0, z = 0, z'' >= 0 PLUSITER(z, z', z'') -{ 1 }-> 1 + IFPLUS(1, 1 + (z - 1), z', 0) + s1 :|: s1 >= 0, s1 <= 0, z'' = 0, z' >= 0, z - 1 >= 0 PLUSITER(z, z', z'') -{ 1 + z'' }-> 1 + IFPLUS(0, z, z', z'') + s4 :|: s4 >= 0, s4 <= 0, z' >= 0, z >= 0, z'' >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 1)) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 0)) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z01 + z1'', z', plus(z', z01)) :|: z' >= 0, z01 >= 0, z = 1 + z01 + z1'', z1'' >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z01 + z1'', z', plus(z', 0)) :|: z' >= 0, z01 >= 0, z = 1 + z01 + z1'', z1'' >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, z, z', plus(z', 0)) :|: z' >= 0, z >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 0, z', plus(z', 1)) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 1 + z02 + z11, z', plus(z', z02)) :|: z11 >= 0, z' >= 0, z = 1 + z02 + z11, z02 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 1)) + PLUS(z', 1) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 1)) + PLUS(z', 0) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 0)) + PLUS(z', 1) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 0)) + PLUS(z', 0) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z03 + z12, z', plus(z', z03)) + PLUS(z', z03) :|: z = 1 + z03 + z12, z' >= 0, z12 >= 0, z03 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z03 + z12, z', plus(z', z03)) + PLUS(z', 0) :|: z = 1 + z03 + z12, z' >= 0, z12 >= 0, z03 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z03 + z12, z', plus(z', 0)) + PLUS(z', z03) :|: z = 1 + z03 + z12, z' >= 0, z12 >= 0, z03 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z03 + z12, z', plus(z', 0)) + PLUS(z', 0) :|: z = 1 + z03 + z12, z' >= 0, z12 >= 0, z03 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, z, z', plus(z', 0)) + PLUS(z', 0) :|: z' >= 0, z >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 0, z', plus(z', 1)) + PLUS(z', 1) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 0, z', plus(z', 1)) + PLUS(z', 0) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 0, z', plus(z', 0)) + PLUS(z', 1) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 1 + z04 + z13, z', plus(z', z04)) + PLUS(z', z04) :|: z = 1 + z04 + z13, z04 >= 0, z' >= 0, z13 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 1 + z04 + z13, z', plus(z', z04)) + PLUS(z', 0) :|: z = 1 + z04 + z13, z04 >= 0, z' >= 0, z13 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 1 + z05 + z14, z', plus(z', 0)) + PLUS(z', z05) :|: z = 1 + z05 + z14, z' >= 0, z05 >= 0, z14 >= 0 head(z) -{ 0 }-> z0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 head(z) -{ 0 }-> 1 :|: z = 0 head(z) -{ 0 }-> 0 :|: z >= 0 ifPlus(z, z', z'', z3) -{ 0 }-> z'' :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0 ifPlus(z, z', z'', z3) -{ 0 }-> plusIter(z', 1 + z'', 1 + z3) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0 ifPlus(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0 isempty(z) -{ 0 }-> 2 :|: z = 0 isempty(z) -{ 0 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 isempty(z) -{ 0 }-> 0 :|: z >= 0 le(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 2, z' - 1 >= 0, z - 1 >= 0 le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0 le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 0 }-> plusIter(z, z', 0) :|: z' >= 0, z >= 0 plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plusIter(z, z', z'') -{ 0 }-> ifPlus(s'', 1 + (z - 1), z', 1 + (z'' - 1)) :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' >= 0, z'' - 1 >= 0 plusIter(z, z', z'') -{ 0 }-> ifPlus(2, 0, z', z'') :|: z' >= 0, z = 0, z'' >= 0 plusIter(z, z', z'') -{ 0 }-> ifPlus(1, 1 + (z - 1), z', 0) :|: z'' = 0, z' >= 0, z - 1 >= 0 plusIter(z, z', z'') -{ 0 }-> ifPlus(0, z, z', z'') :|: z' >= 0, z >= 0, z'' >= 0 plusIter(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 tail(z) -{ 0 }-> z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 tail(z) -{ 0 }-> 0 :|: z = 0 tail(z) -{ 0 }-> 0 :|: z >= 0 Function symbols to be analyzed: {plusIter,ifPlus}, {IFPLUS,PLUSITER}, {plus}, {PLUS}, {IFSUM,SUMITER} Previous analysis results are: le: runtime: O(1) [0], size: O(1) [2] LE: runtime: O(n^1) [z'], size: O(1) [0] tail: runtime: O(1) [0], size: O(n^1) [z] isempty: runtime: O(1) [0], size: O(1) [2] head: runtime: O(1) [0], size: O(n^1) [1 + z] ---------------------------------------- (83) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (84) Obligation: Complexity RNTS consisting of the following rules: IFPLUS(z, z', z'', z3) -{ 1 }-> 1 + PLUSITER(z', 1 + z'', 1 + z3) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0 IFSUM(z, z', z'', z3) -{ 1 }-> 1 + SUMITER(z15, z3) :|: z15 >= 0, z'' >= 0, z = 1, z06 >= 0, z3 >= 0, z' = 1 + z06 + z15 IFSUM(z, z', z'', z3) -{ 1 }-> 1 + SUMITER(0, z3) :|: z'' >= 0, z = 1, z3 >= 0, z' = 0 IFSUM(z, z', z'', z3) -{ 1 }-> 1 + SUMITER(0, z3) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0 LE(z, z') -{ z' }-> 1 + s5 :|: s5 >= 0, s5 <= 0, z' - 1 >= 0, z - 1 >= 0 PLUS(z, z') -{ 1 }-> 1 + PLUSITER(z, z', 0) :|: z' >= 0, z >= 0 PLUSITER(z, z', z'') -{ 1 + z'' }-> 1 + IFPLUS(s, 1 + (z - 1), z', 1 + (z'' - 1)) + s3 :|: s3 >= 0, s3 <= 0, s >= 0, s <= 2, z' >= 0, z'' - 1 >= 0, z - 1 >= 0 PLUSITER(z, z', z'') -{ 1 + z'' }-> 1 + IFPLUS(2, 0, z', z'') + s2 :|: s2 >= 0, s2 <= 0, z' >= 0, z = 0, z'' >= 0 PLUSITER(z, z', z'') -{ 1 }-> 1 + IFPLUS(1, 1 + (z - 1), z', 0) + s1 :|: s1 >= 0, s1 <= 0, z'' = 0, z' >= 0, z - 1 >= 0 PLUSITER(z, z', z'') -{ 1 + z'' }-> 1 + IFPLUS(0, z, z', z'') + s4 :|: s4 >= 0, s4 <= 0, z' >= 0, z >= 0, z'' >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 1)) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 0)) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z01 + z1'', z', plus(z', z01)) :|: z' >= 0, z01 >= 0, z = 1 + z01 + z1'', z1'' >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z01 + z1'', z', plus(z', 0)) :|: z' >= 0, z01 >= 0, z = 1 + z01 + z1'', z1'' >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, z, z', plus(z', 0)) :|: z' >= 0, z >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 0, z', plus(z', 1)) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 1 + z02 + z11, z', plus(z', z02)) :|: z11 >= 0, z' >= 0, z = 1 + z02 + z11, z02 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 1)) + PLUS(z', 1) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 1)) + PLUS(z', 0) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 0)) + PLUS(z', 1) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 0)) + PLUS(z', 0) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z03 + z12, z', plus(z', z03)) + PLUS(z', z03) :|: z = 1 + z03 + z12, z' >= 0, z12 >= 0, z03 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z03 + z12, z', plus(z', z03)) + PLUS(z', 0) :|: z = 1 + z03 + z12, z' >= 0, z12 >= 0, z03 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z03 + z12, z', plus(z', 0)) + PLUS(z', z03) :|: z = 1 + z03 + z12, z' >= 0, z12 >= 0, z03 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z03 + z12, z', plus(z', 0)) + PLUS(z', 0) :|: z = 1 + z03 + z12, z' >= 0, z12 >= 0, z03 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, z, z', plus(z', 0)) + PLUS(z', 0) :|: z' >= 0, z >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 0, z', plus(z', 1)) + PLUS(z', 1) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 0, z', plus(z', 1)) + PLUS(z', 0) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 0, z', plus(z', 0)) + PLUS(z', 1) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 1 + z04 + z13, z', plus(z', z04)) + PLUS(z', z04) :|: z = 1 + z04 + z13, z04 >= 0, z' >= 0, z13 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 1 + z04 + z13, z', plus(z', z04)) + PLUS(z', 0) :|: z = 1 + z04 + z13, z04 >= 0, z' >= 0, z13 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 1 + z05 + z14, z', plus(z', 0)) + PLUS(z', z05) :|: z = 1 + z05 + z14, z' >= 0, z05 >= 0, z14 >= 0 head(z) -{ 0 }-> z0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 head(z) -{ 0 }-> 1 :|: z = 0 head(z) -{ 0 }-> 0 :|: z >= 0 ifPlus(z, z', z'', z3) -{ 0 }-> z'' :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0 ifPlus(z, z', z'', z3) -{ 0 }-> plusIter(z', 1 + z'', 1 + z3) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0 ifPlus(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0 isempty(z) -{ 0 }-> 2 :|: z = 0 isempty(z) -{ 0 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 isempty(z) -{ 0 }-> 0 :|: z >= 0 le(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 2, z' - 1 >= 0, z - 1 >= 0 le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0 le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 0 }-> plusIter(z, z', 0) :|: z' >= 0, z >= 0 plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plusIter(z, z', z'') -{ 0 }-> ifPlus(s'', 1 + (z - 1), z', 1 + (z'' - 1)) :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' >= 0, z'' - 1 >= 0 plusIter(z, z', z'') -{ 0 }-> ifPlus(2, 0, z', z'') :|: z' >= 0, z = 0, z'' >= 0 plusIter(z, z', z'') -{ 0 }-> ifPlus(1, 1 + (z - 1), z', 0) :|: z'' = 0, z' >= 0, z - 1 >= 0 plusIter(z, z', z'') -{ 0 }-> ifPlus(0, z, z', z'') :|: z' >= 0, z >= 0, z'' >= 0 plusIter(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 tail(z) -{ 0 }-> z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 tail(z) -{ 0 }-> 0 :|: z = 0 tail(z) -{ 0 }-> 0 :|: z >= 0 Function symbols to be analyzed: {plusIter,ifPlus}, {IFPLUS,PLUSITER}, {plus}, {PLUS}, {IFSUM,SUMITER} Previous analysis results are: le: runtime: O(1) [0], size: O(1) [2] LE: runtime: O(n^1) [z'], size: O(1) [0] tail: runtime: O(1) [0], size: O(n^1) [z] isempty: runtime: O(1) [0], size: O(1) [2] head: runtime: O(1) [0], size: O(n^1) [1 + z] ---------------------------------------- (85) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: plusIter after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? Computed SIZE bound using CoFloCo for: ifPlus after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (86) Obligation: Complexity RNTS consisting of the following rules: IFPLUS(z, z', z'', z3) -{ 1 }-> 1 + PLUSITER(z', 1 + z'', 1 + z3) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0 IFSUM(z, z', z'', z3) -{ 1 }-> 1 + SUMITER(z15, z3) :|: z15 >= 0, z'' >= 0, z = 1, z06 >= 0, z3 >= 0, z' = 1 + z06 + z15 IFSUM(z, z', z'', z3) -{ 1 }-> 1 + SUMITER(0, z3) :|: z'' >= 0, z = 1, z3 >= 0, z' = 0 IFSUM(z, z', z'', z3) -{ 1 }-> 1 + SUMITER(0, z3) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0 LE(z, z') -{ z' }-> 1 + s5 :|: s5 >= 0, s5 <= 0, z' - 1 >= 0, z - 1 >= 0 PLUS(z, z') -{ 1 }-> 1 + PLUSITER(z, z', 0) :|: z' >= 0, z >= 0 PLUSITER(z, z', z'') -{ 1 + z'' }-> 1 + IFPLUS(s, 1 + (z - 1), z', 1 + (z'' - 1)) + s3 :|: s3 >= 0, s3 <= 0, s >= 0, s <= 2, z' >= 0, z'' - 1 >= 0, z - 1 >= 0 PLUSITER(z, z', z'') -{ 1 + z'' }-> 1 + IFPLUS(2, 0, z', z'') + s2 :|: s2 >= 0, s2 <= 0, z' >= 0, z = 0, z'' >= 0 PLUSITER(z, z', z'') -{ 1 }-> 1 + IFPLUS(1, 1 + (z - 1), z', 0) + s1 :|: s1 >= 0, s1 <= 0, z'' = 0, z' >= 0, z - 1 >= 0 PLUSITER(z, z', z'') -{ 1 + z'' }-> 1 + IFPLUS(0, z, z', z'') + s4 :|: s4 >= 0, s4 <= 0, z' >= 0, z >= 0, z'' >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 1)) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 0)) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z01 + z1'', z', plus(z', z01)) :|: z' >= 0, z01 >= 0, z = 1 + z01 + z1'', z1'' >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z01 + z1'', z', plus(z', 0)) :|: z' >= 0, z01 >= 0, z = 1 + z01 + z1'', z1'' >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, z, z', plus(z', 0)) :|: z' >= 0, z >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 0, z', plus(z', 1)) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 1 + z02 + z11, z', plus(z', z02)) :|: z11 >= 0, z' >= 0, z = 1 + z02 + z11, z02 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 1)) + PLUS(z', 1) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 1)) + PLUS(z', 0) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 0)) + PLUS(z', 1) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 0)) + PLUS(z', 0) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z03 + z12, z', plus(z', z03)) + PLUS(z', z03) :|: z = 1 + z03 + z12, z' >= 0, z12 >= 0, z03 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z03 + z12, z', plus(z', z03)) + PLUS(z', 0) :|: z = 1 + z03 + z12, z' >= 0, z12 >= 0, z03 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z03 + z12, z', plus(z', 0)) + PLUS(z', z03) :|: z = 1 + z03 + z12, z' >= 0, z12 >= 0, z03 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z03 + z12, z', plus(z', 0)) + PLUS(z', 0) :|: z = 1 + z03 + z12, z' >= 0, z12 >= 0, z03 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, z, z', plus(z', 0)) + PLUS(z', 0) :|: z' >= 0, z >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 0, z', plus(z', 1)) + PLUS(z', 1) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 0, z', plus(z', 1)) + PLUS(z', 0) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 0, z', plus(z', 0)) + PLUS(z', 1) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 1 + z04 + z13, z', plus(z', z04)) + PLUS(z', z04) :|: z = 1 + z04 + z13, z04 >= 0, z' >= 0, z13 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 1 + z04 + z13, z', plus(z', z04)) + PLUS(z', 0) :|: z = 1 + z04 + z13, z04 >= 0, z' >= 0, z13 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 1 + z05 + z14, z', plus(z', 0)) + PLUS(z', z05) :|: z = 1 + z05 + z14, z' >= 0, z05 >= 0, z14 >= 0 head(z) -{ 0 }-> z0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 head(z) -{ 0 }-> 1 :|: z = 0 head(z) -{ 0 }-> 0 :|: z >= 0 ifPlus(z, z', z'', z3) -{ 0 }-> z'' :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0 ifPlus(z, z', z'', z3) -{ 0 }-> plusIter(z', 1 + z'', 1 + z3) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0 ifPlus(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0 isempty(z) -{ 0 }-> 2 :|: z = 0 isempty(z) -{ 0 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 isempty(z) -{ 0 }-> 0 :|: z >= 0 le(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 2, z' - 1 >= 0, z - 1 >= 0 le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0 le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 0 }-> plusIter(z, z', 0) :|: z' >= 0, z >= 0 plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plusIter(z, z', z'') -{ 0 }-> ifPlus(s'', 1 + (z - 1), z', 1 + (z'' - 1)) :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' >= 0, z'' - 1 >= 0 plusIter(z, z', z'') -{ 0 }-> ifPlus(2, 0, z', z'') :|: z' >= 0, z = 0, z'' >= 0 plusIter(z, z', z'') -{ 0 }-> ifPlus(1, 1 + (z - 1), z', 0) :|: z'' = 0, z' >= 0, z - 1 >= 0 plusIter(z, z', z'') -{ 0 }-> ifPlus(0, z, z', z'') :|: z' >= 0, z >= 0, z'' >= 0 plusIter(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 tail(z) -{ 0 }-> z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 tail(z) -{ 0 }-> 0 :|: z = 0 tail(z) -{ 0 }-> 0 :|: z >= 0 Function symbols to be analyzed: {plusIter,ifPlus}, {IFPLUS,PLUSITER}, {plus}, {PLUS}, {IFSUM,SUMITER} Previous analysis results are: le: runtime: O(1) [0], size: O(1) [2] LE: runtime: O(n^1) [z'], size: O(1) [0] tail: runtime: O(1) [0], size: O(n^1) [z] isempty: runtime: O(1) [0], size: O(1) [2] head: runtime: O(1) [0], size: O(n^1) [1 + z] plusIter: runtime: ?, size: INF ifPlus: runtime: ?, size: INF ---------------------------------------- (87) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: plusIter after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 Computed RUNTIME bound using CoFloCo for: ifPlus after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (88) Obligation: Complexity RNTS consisting of the following rules: IFPLUS(z, z', z'', z3) -{ 1 }-> 1 + PLUSITER(z', 1 + z'', 1 + z3) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0 IFSUM(z, z', z'', z3) -{ 1 }-> 1 + SUMITER(z15, z3) :|: z15 >= 0, z'' >= 0, z = 1, z06 >= 0, z3 >= 0, z' = 1 + z06 + z15 IFSUM(z, z', z'', z3) -{ 1 }-> 1 + SUMITER(0, z3) :|: z'' >= 0, z = 1, z3 >= 0, z' = 0 IFSUM(z, z', z'', z3) -{ 1 }-> 1 + SUMITER(0, z3) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0 LE(z, z') -{ z' }-> 1 + s5 :|: s5 >= 0, s5 <= 0, z' - 1 >= 0, z - 1 >= 0 PLUS(z, z') -{ 1 }-> 1 + PLUSITER(z, z', 0) :|: z' >= 0, z >= 0 PLUSITER(z, z', z'') -{ 1 + z'' }-> 1 + IFPLUS(s, 1 + (z - 1), z', 1 + (z'' - 1)) + s3 :|: s3 >= 0, s3 <= 0, s >= 0, s <= 2, z' >= 0, z'' - 1 >= 0, z - 1 >= 0 PLUSITER(z, z', z'') -{ 1 + z'' }-> 1 + IFPLUS(2, 0, z', z'') + s2 :|: s2 >= 0, s2 <= 0, z' >= 0, z = 0, z'' >= 0 PLUSITER(z, z', z'') -{ 1 }-> 1 + IFPLUS(1, 1 + (z - 1), z', 0) + s1 :|: s1 >= 0, s1 <= 0, z'' = 0, z' >= 0, z - 1 >= 0 PLUSITER(z, z', z'') -{ 1 + z'' }-> 1 + IFPLUS(0, z, z', z'') + s4 :|: s4 >= 0, s4 <= 0, z' >= 0, z >= 0, z'' >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 1)) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 0)) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z01 + z1'', z', plus(z', z01)) :|: z' >= 0, z01 >= 0, z = 1 + z01 + z1'', z1'' >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z01 + z1'', z', plus(z', 0)) :|: z' >= 0, z01 >= 0, z = 1 + z01 + z1'', z1'' >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, z, z', plus(z', 0)) :|: z' >= 0, z >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 0, z', plus(z', 1)) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 1 + z02 + z11, z', plus(z', z02)) :|: z11 >= 0, z' >= 0, z = 1 + z02 + z11, z02 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 1)) + PLUS(z', 1) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 1)) + PLUS(z', 0) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 0)) + PLUS(z', 1) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 0)) + PLUS(z', 0) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z03 + z12, z', plus(z', z03)) + PLUS(z', z03) :|: z = 1 + z03 + z12, z' >= 0, z12 >= 0, z03 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z03 + z12, z', plus(z', z03)) + PLUS(z', 0) :|: z = 1 + z03 + z12, z' >= 0, z12 >= 0, z03 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z03 + z12, z', plus(z', 0)) + PLUS(z', z03) :|: z = 1 + z03 + z12, z' >= 0, z12 >= 0, z03 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z03 + z12, z', plus(z', 0)) + PLUS(z', 0) :|: z = 1 + z03 + z12, z' >= 0, z12 >= 0, z03 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, z, z', plus(z', 0)) + PLUS(z', 0) :|: z' >= 0, z >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 0, z', plus(z', 1)) + PLUS(z', 1) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 0, z', plus(z', 1)) + PLUS(z', 0) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 0, z', plus(z', 0)) + PLUS(z', 1) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 1 + z04 + z13, z', plus(z', z04)) + PLUS(z', z04) :|: z = 1 + z04 + z13, z04 >= 0, z' >= 0, z13 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 1 + z04 + z13, z', plus(z', z04)) + PLUS(z', 0) :|: z = 1 + z04 + z13, z04 >= 0, z' >= 0, z13 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 1 + z05 + z14, z', plus(z', 0)) + PLUS(z', z05) :|: z = 1 + z05 + z14, z' >= 0, z05 >= 0, z14 >= 0 head(z) -{ 0 }-> z0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 head(z) -{ 0 }-> 1 :|: z = 0 head(z) -{ 0 }-> 0 :|: z >= 0 ifPlus(z, z', z'', z3) -{ 0 }-> z'' :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0 ifPlus(z, z', z'', z3) -{ 0 }-> plusIter(z', 1 + z'', 1 + z3) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0 ifPlus(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0 isempty(z) -{ 0 }-> 2 :|: z = 0 isempty(z) -{ 0 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 isempty(z) -{ 0 }-> 0 :|: z >= 0 le(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 2, z' - 1 >= 0, z - 1 >= 0 le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0 le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 0 }-> plusIter(z, z', 0) :|: z' >= 0, z >= 0 plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plusIter(z, z', z'') -{ 0 }-> ifPlus(s'', 1 + (z - 1), z', 1 + (z'' - 1)) :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' >= 0, z'' - 1 >= 0 plusIter(z, z', z'') -{ 0 }-> ifPlus(2, 0, z', z'') :|: z' >= 0, z = 0, z'' >= 0 plusIter(z, z', z'') -{ 0 }-> ifPlus(1, 1 + (z - 1), z', 0) :|: z'' = 0, z' >= 0, z - 1 >= 0 plusIter(z, z', z'') -{ 0 }-> ifPlus(0, z, z', z'') :|: z' >= 0, z >= 0, z'' >= 0 plusIter(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 tail(z) -{ 0 }-> z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 tail(z) -{ 0 }-> 0 :|: z = 0 tail(z) -{ 0 }-> 0 :|: z >= 0 Function symbols to be analyzed: {IFPLUS,PLUSITER}, {plus}, {PLUS}, {IFSUM,SUMITER} Previous analysis results are: le: runtime: O(1) [0], size: O(1) [2] LE: runtime: O(n^1) [z'], size: O(1) [0] tail: runtime: O(1) [0], size: O(n^1) [z] isempty: runtime: O(1) [0], size: O(1) [2] head: runtime: O(1) [0], size: O(n^1) [1 + z] plusIter: runtime: O(1) [0], size: INF ifPlus: runtime: O(1) [0], size: INF ---------------------------------------- (89) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (90) Obligation: Complexity RNTS consisting of the following rules: IFPLUS(z, z', z'', z3) -{ 1 }-> 1 + PLUSITER(z', 1 + z'', 1 + z3) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0 IFSUM(z, z', z'', z3) -{ 1 }-> 1 + SUMITER(z15, z3) :|: z15 >= 0, z'' >= 0, z = 1, z06 >= 0, z3 >= 0, z' = 1 + z06 + z15 IFSUM(z, z', z'', z3) -{ 1 }-> 1 + SUMITER(0, z3) :|: z'' >= 0, z = 1, z3 >= 0, z' = 0 IFSUM(z, z', z'', z3) -{ 1 }-> 1 + SUMITER(0, z3) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0 LE(z, z') -{ z' }-> 1 + s5 :|: s5 >= 0, s5 <= 0, z' - 1 >= 0, z - 1 >= 0 PLUS(z, z') -{ 1 }-> 1 + PLUSITER(z, z', 0) :|: z' >= 0, z >= 0 PLUSITER(z, z', z'') -{ 1 + z'' }-> 1 + IFPLUS(s, 1 + (z - 1), z', 1 + (z'' - 1)) + s3 :|: s3 >= 0, s3 <= 0, s >= 0, s <= 2, z' >= 0, z'' - 1 >= 0, z - 1 >= 0 PLUSITER(z, z', z'') -{ 1 + z'' }-> 1 + IFPLUS(2, 0, z', z'') + s2 :|: s2 >= 0, s2 <= 0, z' >= 0, z = 0, z'' >= 0 PLUSITER(z, z', z'') -{ 1 }-> 1 + IFPLUS(1, 1 + (z - 1), z', 0) + s1 :|: s1 >= 0, s1 <= 0, z'' = 0, z' >= 0, z - 1 >= 0 PLUSITER(z, z', z'') -{ 1 + z'' }-> 1 + IFPLUS(0, z, z', z'') + s4 :|: s4 >= 0, s4 <= 0, z' >= 0, z >= 0, z'' >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 1)) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 0)) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z01 + z1'', z', plus(z', z01)) :|: z' >= 0, z01 >= 0, z = 1 + z01 + z1'', z1'' >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z01 + z1'', z', plus(z', 0)) :|: z' >= 0, z01 >= 0, z = 1 + z01 + z1'', z1'' >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, z, z', plus(z', 0)) :|: z' >= 0, z >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 0, z', plus(z', 1)) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 1 + z02 + z11, z', plus(z', z02)) :|: z11 >= 0, z' >= 0, z = 1 + z02 + z11, z02 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 1)) + PLUS(z', 1) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 1)) + PLUS(z', 0) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 0)) + PLUS(z', 1) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 0)) + PLUS(z', 0) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z03 + z12, z', plus(z', z03)) + PLUS(z', z03) :|: z = 1 + z03 + z12, z' >= 0, z12 >= 0, z03 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z03 + z12, z', plus(z', z03)) + PLUS(z', 0) :|: z = 1 + z03 + z12, z' >= 0, z12 >= 0, z03 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z03 + z12, z', plus(z', 0)) + PLUS(z', z03) :|: z = 1 + z03 + z12, z' >= 0, z12 >= 0, z03 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z03 + z12, z', plus(z', 0)) + PLUS(z', 0) :|: z = 1 + z03 + z12, z' >= 0, z12 >= 0, z03 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, z, z', plus(z', 0)) + PLUS(z', 0) :|: z' >= 0, z >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 0, z', plus(z', 1)) + PLUS(z', 1) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 0, z', plus(z', 1)) + PLUS(z', 0) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 0, z', plus(z', 0)) + PLUS(z', 1) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 1 + z04 + z13, z', plus(z', z04)) + PLUS(z', z04) :|: z = 1 + z04 + z13, z04 >= 0, z' >= 0, z13 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 1 + z04 + z13, z', plus(z', z04)) + PLUS(z', 0) :|: z = 1 + z04 + z13, z04 >= 0, z' >= 0, z13 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 1 + z05 + z14, z', plus(z', 0)) + PLUS(z', z05) :|: z = 1 + z05 + z14, z' >= 0, z05 >= 0, z14 >= 0 head(z) -{ 0 }-> z0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 head(z) -{ 0 }-> 1 :|: z = 0 head(z) -{ 0 }-> 0 :|: z >= 0 ifPlus(z, z', z'', z3) -{ 0 }-> s11 :|: s11 >= 0, s11 <= inf3, z'' >= 0, z = 1, z' >= 0, z3 >= 0 ifPlus(z, z', z'', z3) -{ 0 }-> z'' :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0 ifPlus(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0 isempty(z) -{ 0 }-> 2 :|: z = 0 isempty(z) -{ 0 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 isempty(z) -{ 0 }-> 0 :|: z >= 0 le(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 2, z' - 1 >= 0, z - 1 >= 0 le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0 le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 0 }-> s6 :|: s6 >= 0, s6 <= inf, z' >= 0, z >= 0 plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plusIter(z, z', z'') -{ 0 }-> s10 :|: s10 >= 0, s10 <= inf2, z' >= 0, z >= 0, z'' >= 0 plusIter(z, z', z'') -{ 0 }-> s7 :|: s7 >= 0, s7 <= inf', z'' = 0, z' >= 0, z - 1 >= 0 plusIter(z, z', z'') -{ 0 }-> s8 :|: s8 >= 0, s8 <= inf'', z' >= 0, z = 0, z'' >= 0 plusIter(z, z', z'') -{ 0 }-> s9 :|: s9 >= 0, s9 <= inf1, s'' >= 0, s'' <= 2, z - 1 >= 0, z' >= 0, z'' - 1 >= 0 plusIter(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 tail(z) -{ 0 }-> z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 tail(z) -{ 0 }-> 0 :|: z = 0 tail(z) -{ 0 }-> 0 :|: z >= 0 Function symbols to be analyzed: {IFPLUS,PLUSITER}, {plus}, {PLUS}, {IFSUM,SUMITER} Previous analysis results are: le: runtime: O(1) [0], size: O(1) [2] LE: runtime: O(n^1) [z'], size: O(1) [0] tail: runtime: O(1) [0], size: O(n^1) [z] isempty: runtime: O(1) [0], size: O(1) [2] head: runtime: O(1) [0], size: O(n^1) [1 + z] plusIter: runtime: O(1) [0], size: INF ifPlus: runtime: O(1) [0], size: INF ---------------------------------------- (91) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: IFPLUS after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 Computed SIZE bound using CoFloCo for: PLUSITER after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (92) Obligation: Complexity RNTS consisting of the following rules: IFPLUS(z, z', z'', z3) -{ 1 }-> 1 + PLUSITER(z', 1 + z'', 1 + z3) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0 IFSUM(z, z', z'', z3) -{ 1 }-> 1 + SUMITER(z15, z3) :|: z15 >= 0, z'' >= 0, z = 1, z06 >= 0, z3 >= 0, z' = 1 + z06 + z15 IFSUM(z, z', z'', z3) -{ 1 }-> 1 + SUMITER(0, z3) :|: z'' >= 0, z = 1, z3 >= 0, z' = 0 IFSUM(z, z', z'', z3) -{ 1 }-> 1 + SUMITER(0, z3) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0 LE(z, z') -{ z' }-> 1 + s5 :|: s5 >= 0, s5 <= 0, z' - 1 >= 0, z - 1 >= 0 PLUS(z, z') -{ 1 }-> 1 + PLUSITER(z, z', 0) :|: z' >= 0, z >= 0 PLUSITER(z, z', z'') -{ 1 + z'' }-> 1 + IFPLUS(s, 1 + (z - 1), z', 1 + (z'' - 1)) + s3 :|: s3 >= 0, s3 <= 0, s >= 0, s <= 2, z' >= 0, z'' - 1 >= 0, z - 1 >= 0 PLUSITER(z, z', z'') -{ 1 + z'' }-> 1 + IFPLUS(2, 0, z', z'') + s2 :|: s2 >= 0, s2 <= 0, z' >= 0, z = 0, z'' >= 0 PLUSITER(z, z', z'') -{ 1 }-> 1 + IFPLUS(1, 1 + (z - 1), z', 0) + s1 :|: s1 >= 0, s1 <= 0, z'' = 0, z' >= 0, z - 1 >= 0 PLUSITER(z, z', z'') -{ 1 + z'' }-> 1 + IFPLUS(0, z, z', z'') + s4 :|: s4 >= 0, s4 <= 0, z' >= 0, z >= 0, z'' >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 1)) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 0)) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z01 + z1'', z', plus(z', z01)) :|: z' >= 0, z01 >= 0, z = 1 + z01 + z1'', z1'' >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z01 + z1'', z', plus(z', 0)) :|: z' >= 0, z01 >= 0, z = 1 + z01 + z1'', z1'' >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, z, z', plus(z', 0)) :|: z' >= 0, z >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 0, z', plus(z', 1)) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 1 + z02 + z11, z', plus(z', z02)) :|: z11 >= 0, z' >= 0, z = 1 + z02 + z11, z02 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 1)) + PLUS(z', 1) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 1)) + PLUS(z', 0) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 0)) + PLUS(z', 1) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 0)) + PLUS(z', 0) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z03 + z12, z', plus(z', z03)) + PLUS(z', z03) :|: z = 1 + z03 + z12, z' >= 0, z12 >= 0, z03 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z03 + z12, z', plus(z', z03)) + PLUS(z', 0) :|: z = 1 + z03 + z12, z' >= 0, z12 >= 0, z03 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z03 + z12, z', plus(z', 0)) + PLUS(z', z03) :|: z = 1 + z03 + z12, z' >= 0, z12 >= 0, z03 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z03 + z12, z', plus(z', 0)) + PLUS(z', 0) :|: z = 1 + z03 + z12, z' >= 0, z12 >= 0, z03 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, z, z', plus(z', 0)) + PLUS(z', 0) :|: z' >= 0, z >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 0, z', plus(z', 1)) + PLUS(z', 1) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 0, z', plus(z', 1)) + PLUS(z', 0) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 0, z', plus(z', 0)) + PLUS(z', 1) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 1 + z04 + z13, z', plus(z', z04)) + PLUS(z', z04) :|: z = 1 + z04 + z13, z04 >= 0, z' >= 0, z13 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 1 + z04 + z13, z', plus(z', z04)) + PLUS(z', 0) :|: z = 1 + z04 + z13, z04 >= 0, z' >= 0, z13 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 1 + z05 + z14, z', plus(z', 0)) + PLUS(z', z05) :|: z = 1 + z05 + z14, z' >= 0, z05 >= 0, z14 >= 0 head(z) -{ 0 }-> z0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 head(z) -{ 0 }-> 1 :|: z = 0 head(z) -{ 0 }-> 0 :|: z >= 0 ifPlus(z, z', z'', z3) -{ 0 }-> s11 :|: s11 >= 0, s11 <= inf3, z'' >= 0, z = 1, z' >= 0, z3 >= 0 ifPlus(z, z', z'', z3) -{ 0 }-> z'' :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0 ifPlus(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0 isempty(z) -{ 0 }-> 2 :|: z = 0 isempty(z) -{ 0 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 isempty(z) -{ 0 }-> 0 :|: z >= 0 le(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 2, z' - 1 >= 0, z - 1 >= 0 le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0 le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 0 }-> s6 :|: s6 >= 0, s6 <= inf, z' >= 0, z >= 0 plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plusIter(z, z', z'') -{ 0 }-> s10 :|: s10 >= 0, s10 <= inf2, z' >= 0, z >= 0, z'' >= 0 plusIter(z, z', z'') -{ 0 }-> s7 :|: s7 >= 0, s7 <= inf', z'' = 0, z' >= 0, z - 1 >= 0 plusIter(z, z', z'') -{ 0 }-> s8 :|: s8 >= 0, s8 <= inf'', z' >= 0, z = 0, z'' >= 0 plusIter(z, z', z'') -{ 0 }-> s9 :|: s9 >= 0, s9 <= inf1, s'' >= 0, s'' <= 2, z - 1 >= 0, z' >= 0, z'' - 1 >= 0 plusIter(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 tail(z) -{ 0 }-> z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 tail(z) -{ 0 }-> 0 :|: z = 0 tail(z) -{ 0 }-> 0 :|: z >= 0 Function symbols to be analyzed: {IFPLUS,PLUSITER}, {plus}, {PLUS}, {IFSUM,SUMITER} Previous analysis results are: le: runtime: O(1) [0], size: O(1) [2] LE: runtime: O(n^1) [z'], size: O(1) [0] tail: runtime: O(1) [0], size: O(n^1) [z] isempty: runtime: O(1) [0], size: O(1) [2] head: runtime: O(1) [0], size: O(n^1) [1 + z] plusIter: runtime: O(1) [0], size: INF ifPlus: runtime: O(1) [0], size: INF IFPLUS: runtime: ?, size: O(1) [0] PLUSITER: runtime: ?, size: O(1) [1] ---------------------------------------- (93) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: IFPLUS after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (94) Obligation: Complexity RNTS consisting of the following rules: IFPLUS(z, z', z'', z3) -{ 1 }-> 1 + PLUSITER(z', 1 + z'', 1 + z3) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0 IFSUM(z, z', z'', z3) -{ 1 }-> 1 + SUMITER(z15, z3) :|: z15 >= 0, z'' >= 0, z = 1, z06 >= 0, z3 >= 0, z' = 1 + z06 + z15 IFSUM(z, z', z'', z3) -{ 1 }-> 1 + SUMITER(0, z3) :|: z'' >= 0, z = 1, z3 >= 0, z' = 0 IFSUM(z, z', z'', z3) -{ 1 }-> 1 + SUMITER(0, z3) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0 LE(z, z') -{ z' }-> 1 + s5 :|: s5 >= 0, s5 <= 0, z' - 1 >= 0, z - 1 >= 0 PLUS(z, z') -{ 1 }-> 1 + PLUSITER(z, z', 0) :|: z' >= 0, z >= 0 PLUSITER(z, z', z'') -{ 1 + z'' }-> 1 + IFPLUS(s, 1 + (z - 1), z', 1 + (z'' - 1)) + s3 :|: s3 >= 0, s3 <= 0, s >= 0, s <= 2, z' >= 0, z'' - 1 >= 0, z - 1 >= 0 PLUSITER(z, z', z'') -{ 1 + z'' }-> 1 + IFPLUS(2, 0, z', z'') + s2 :|: s2 >= 0, s2 <= 0, z' >= 0, z = 0, z'' >= 0 PLUSITER(z, z', z'') -{ 1 }-> 1 + IFPLUS(1, 1 + (z - 1), z', 0) + s1 :|: s1 >= 0, s1 <= 0, z'' = 0, z' >= 0, z - 1 >= 0 PLUSITER(z, z', z'') -{ 1 + z'' }-> 1 + IFPLUS(0, z, z', z'') + s4 :|: s4 >= 0, s4 <= 0, z' >= 0, z >= 0, z'' >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 1)) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 0)) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z01 + z1'', z', plus(z', z01)) :|: z' >= 0, z01 >= 0, z = 1 + z01 + z1'', z1'' >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z01 + z1'', z', plus(z', 0)) :|: z' >= 0, z01 >= 0, z = 1 + z01 + z1'', z1'' >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, z, z', plus(z', 0)) :|: z' >= 0, z >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 0, z', plus(z', 1)) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 1 + z02 + z11, z', plus(z', z02)) :|: z11 >= 0, z' >= 0, z = 1 + z02 + z11, z02 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 1)) + PLUS(z', 1) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 1)) + PLUS(z', 0) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 0)) + PLUS(z', 1) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(2, 0, z', plus(z', 0)) + PLUS(z', 0) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z03 + z12, z', plus(z', z03)) + PLUS(z', z03) :|: z = 1 + z03 + z12, z' >= 0, z12 >= 0, z03 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z03 + z12, z', plus(z', z03)) + PLUS(z', 0) :|: z = 1 + z03 + z12, z' >= 0, z12 >= 0, z03 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z03 + z12, z', plus(z', 0)) + PLUS(z', z03) :|: z = 1 + z03 + z12, z' >= 0, z12 >= 0, z03 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(1, 1 + z03 + z12, z', plus(z', 0)) + PLUS(z', 0) :|: z = 1 + z03 + z12, z' >= 0, z12 >= 0, z03 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, z, z', plus(z', 0)) + PLUS(z', 0) :|: z' >= 0, z >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 0, z', plus(z', 1)) + PLUS(z', 1) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 0, z', plus(z', 1)) + PLUS(z', 0) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 0, z', plus(z', 0)) + PLUS(z', 1) :|: z' >= 0, z = 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 1 + z04 + z13, z', plus(z', z04)) + PLUS(z', z04) :|: z = 1 + z04 + z13, z04 >= 0, z' >= 0, z13 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 1 + z04 + z13, z', plus(z', z04)) + PLUS(z', 0) :|: z = 1 + z04 + z13, z04 >= 0, z' >= 0, z13 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(0, 1 + z05 + z14, z', plus(z', 0)) + PLUS(z', z05) :|: z = 1 + z05 + z14, z' >= 0, z05 >= 0, z14 >= 0 head(z) -{ 0 }-> z0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 head(z) -{ 0 }-> 1 :|: z = 0 head(z) -{ 0 }-> 0 :|: z >= 0 ifPlus(z, z', z'', z3) -{ 0 }-> s11 :|: s11 >= 0, s11 <= inf3, z'' >= 0, z = 1, z' >= 0, z3 >= 0 ifPlus(z, z', z'', z3) -{ 0 }-> z'' :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0 ifPlus(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0 isempty(z) -{ 0 }-> 2 :|: z = 0 isempty(z) -{ 0 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 isempty(z) -{ 0 }-> 0 :|: z >= 0 le(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 2, z' - 1 >= 0, z - 1 >= 0 le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0 le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 0 }-> s6 :|: s6 >= 0, s6 <= inf, z' >= 0, z >= 0 plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plusIter(z, z', z'') -{ 0 }-> s10 :|: s10 >= 0, s10 <= inf2, z' >= 0, z >= 0, z'' >= 0 plusIter(z, z', z'') -{ 0 }-> s7 :|: s7 >= 0, s7 <= inf', z'' = 0, z' >= 0, z - 1 >= 0 plusIter(z, z', z'') -{ 0 }-> s8 :|: s8 >= 0, s8 <= inf'', z' >= 0, z = 0, z'' >= 0 plusIter(z, z', z'') -{ 0 }-> s9 :|: s9 >= 0, s9 <= inf1, s'' >= 0, s'' <= 2, z - 1 >= 0, z' >= 0, z'' - 1 >= 0 plusIter(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 tail(z) -{ 0 }-> z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 tail(z) -{ 0 }-> 0 :|: z = 0 tail(z) -{ 0 }-> 0 :|: z >= 0 Function symbols to be analyzed: {IFPLUS,PLUSITER}, {plus}, {PLUS}, {IFSUM,SUMITER} Previous analysis results are: le: runtime: O(1) [0], size: O(1) [2] LE: runtime: O(n^1) [z'], size: O(1) [0] tail: runtime: O(1) [0], size: O(n^1) [z] isempty: runtime: O(1) [0], size: O(1) [2] head: runtime: O(1) [0], size: O(n^1) [1 + z] plusIter: runtime: O(1) [0], size: INF ifPlus: runtime: O(1) [0], size: INF IFPLUS: runtime: INF, size: O(1) [0] PLUSITER: runtime: ?, size: O(1) [1] ---------------------------------------- (95) 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: le(v0, v1) -> null_le [0] isempty(v0) -> null_isempty [0] plus(v0, v1) -> null_plus [0] head(v0) -> null_head [0] plusIter(v0, v1, v2) -> null_plusIter [0] ifPlus(v0, v1, v2, v3) -> null_ifPlus [0] tail(v0) -> null_tail [0] IFPLUS(v0, v1, v2, v3) -> null_IFPLUS [0] LE(v0, v1) -> null_LE [0] IFSUM(v0, v1, v2, v3) -> null_IFSUM [0] And the following fresh constants: null_le, null_isempty, null_plus, null_head, null_plusIter, null_ifPlus, null_tail, null_IFPLUS, null_LE, null_IFSUM, const, const1, const2 ---------------------------------------- (96) 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: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) [1] PLUSITER(z0, z1, z2) -> c2(IFPLUS(le(z0, z2), z0, z1, z2), LE(z0, z2)) [1] IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2))) [1] LE(s(z0), s(z1)) -> c7(LE(z0, z1)) [1] SUMITER(z0, z1) -> c9(IFSUM(isempty(z0), z0, z1, plus(z1, head(z0)))) [1] SUMITER(z0, z1) -> c10(IFSUM(isempty(z0), z0, z1, plus(z1, head(z0))), PLUS(z1, head(z0))) [1] IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) [1] le(s(z0), 0) -> false [0] le(0, z0) -> true [0] le(s(z0), s(z1)) -> le(z0, z1) [0] isempty(nil) -> true [0] isempty(cons(z0, z1)) -> false [0] plus(z0, z1) -> plusIter(z0, z1, 0) [0] head(nil) -> error [0] head(cons(z0, z1)) -> z0 [0] plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2) [0] ifPlus(true, z0, z1, z2) -> z1 [0] ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2)) [0] tail(nil) -> nil [0] tail(cons(z0, z1)) -> z1 [0] le(v0, v1) -> null_le [0] isempty(v0) -> null_isempty [0] plus(v0, v1) -> null_plus [0] head(v0) -> null_head [0] plusIter(v0, v1, v2) -> null_plusIter [0] ifPlus(v0, v1, v2, v3) -> null_ifPlus [0] tail(v0) -> null_tail [0] IFPLUS(v0, v1, v2, v3) -> null_IFPLUS [0] LE(v0, v1) -> null_LE [0] IFSUM(v0, v1, v2, v3) -> null_IFSUM [0] The TRS has the following type information: PLUS :: 0:s:error:null_plus:null_head:null_plusIter:null_ifPlus -> 0:s:error:null_plus:null_head:null_plusIter:null_ifPlus -> c1 c1 :: c2 -> c1 PLUSITER :: 0:s:error:null_plus:null_head:null_plusIter:null_ifPlus -> 0:s:error:null_plus:null_head:null_plusIter:null_ifPlus -> 0:s:error:null_plus:null_head:null_plusIter:null_ifPlus -> c2 0 :: 0:s:error:null_plus:null_head:null_plusIter:null_ifPlus c2 :: c4:null_IFPLUS -> c7:null_LE -> c2 IFPLUS :: false:true:null_le:null_isempty -> 0:s:error:null_plus:null_head:null_plusIter:null_ifPlus -> 0:s:error:null_plus:null_head:null_plusIter:null_ifPlus -> 0:s:error:null_plus:null_head:null_plusIter:null_ifPlus -> c4:null_IFPLUS le :: 0:s:error:null_plus:null_head:null_plusIter:null_ifPlus -> 0:s:error:null_plus:null_head:null_plusIter:null_ifPlus -> false:true:null_le:null_isempty LE :: 0:s:error:null_plus:null_head:null_plusIter:null_ifPlus -> 0:s:error:null_plus:null_head:null_plusIter:null_ifPlus -> c7:null_LE false :: false:true:null_le:null_isempty c4 :: c2 -> c4:null_IFPLUS s :: 0:s:error:null_plus:null_head:null_plusIter:null_ifPlus -> 0:s:error:null_plus:null_head:null_plusIter:null_ifPlus c7 :: c7:null_LE -> c7:null_LE SUMITER :: nil:cons:null_tail -> 0:s:error:null_plus:null_head:null_plusIter:null_ifPlus -> c9:c10 c9 :: c12:null_IFSUM -> c9:c10 IFSUM :: false:true:null_le:null_isempty -> nil:cons:null_tail -> 0:s:error:null_plus:null_head:null_plusIter:null_ifPlus -> 0:s:error:null_plus:null_head:null_plusIter:null_ifPlus -> c12:null_IFSUM isempty :: nil:cons:null_tail -> false:true:null_le:null_isempty plus :: 0:s:error:null_plus:null_head:null_plusIter:null_ifPlus -> 0:s:error:null_plus:null_head:null_plusIter:null_ifPlus -> 0:s:error:null_plus:null_head:null_plusIter:null_ifPlus head :: nil:cons:null_tail -> 0:s:error:null_plus:null_head:null_plusIter:null_ifPlus c10 :: c12:null_IFSUM -> c1 -> c9:c10 c12 :: c9:c10 -> c12:null_IFSUM tail :: nil:cons:null_tail -> nil:cons:null_tail true :: false:true:null_le:null_isempty nil :: nil:cons:null_tail cons :: 0:s:error:null_plus:null_head:null_plusIter:null_ifPlus -> nil:cons:null_tail -> nil:cons:null_tail plusIter :: 0:s:error:null_plus:null_head:null_plusIter:null_ifPlus -> 0:s:error:null_plus:null_head:null_plusIter:null_ifPlus -> 0:s:error:null_plus:null_head:null_plusIter:null_ifPlus -> 0:s:error:null_plus:null_head:null_plusIter:null_ifPlus error :: 0:s:error:null_plus:null_head:null_plusIter:null_ifPlus ifPlus :: false:true:null_le:null_isempty -> 0:s:error:null_plus:null_head:null_plusIter:null_ifPlus -> 0:s:error:null_plus:null_head:null_plusIter:null_ifPlus -> 0:s:error:null_plus:null_head:null_plusIter:null_ifPlus -> 0:s:error:null_plus:null_head:null_plusIter:null_ifPlus null_le :: false:true:null_le:null_isempty null_isempty :: false:true:null_le:null_isempty null_plus :: 0:s:error:null_plus:null_head:null_plusIter:null_ifPlus null_head :: 0:s:error:null_plus:null_head:null_plusIter:null_ifPlus null_plusIter :: 0:s:error:null_plus:null_head:null_plusIter:null_ifPlus null_ifPlus :: 0:s:error:null_plus:null_head:null_plusIter:null_ifPlus null_tail :: nil:cons:null_tail null_IFPLUS :: c4:null_IFPLUS null_LE :: c7:null_LE null_IFSUM :: c12:null_IFSUM const :: c1 const1 :: c2 const2 :: c9:c10 Rewrite Strategy: INNERMOST ---------------------------------------- (97) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: 0 => 0 false => 1 true => 2 nil => 0 error => 1 null_le => 0 null_isempty => 0 null_plus => 0 null_head => 0 null_plusIter => 0 null_ifPlus => 0 null_tail => 0 null_IFPLUS => 0 null_LE => 0 null_IFSUM => 0 const => 0 const1 => 0 const2 => 0 ---------------------------------------- (98) Obligation: Complexity RNTS consisting of the following rules: IFPLUS(z, z', z'', z3) -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, z3 = v3, v2 >= 0, v3 >= 0 IFPLUS(z, z', z'', z3) -{ 1 }-> 1 + PLUSITER(z0, 1 + z1, 1 + z2) :|: z1 >= 0, z = 1, z0 >= 0, z3 = z2, z' = z0, z2 >= 0, z'' = z1 IFSUM(z, z', z'', z3) -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, z3 = v3, v2 >= 0, v3 >= 0 IFSUM(z, z', z'', z3) -{ 1 }-> 1 + SUMITER(tail(z0), z2) :|: z1 >= 0, z = 1, z0 >= 0, z3 = z2, z' = z0, z2 >= 0, z'' = z1 LE(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 LE(z, z') -{ 1 }-> 1 + LE(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1 PLUS(z, z') -{ 1 }-> 1 + PLUSITER(z0, z1, 0) :|: z = z0, z1 >= 0, z' = z1, z0 >= 0 PLUSITER(z, z', z'') -{ 1 }-> 1 + IFPLUS(le(z0, z2), z0, z1, z2) + LE(z0, z2) :|: z'' = z2, z = z0, z1 >= 0, z' = z1, z0 >= 0, z2 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(isempty(z0), z0, z1, plus(z1, head(z0))) :|: z = z0, z1 >= 0, z' = z1, z0 >= 0 SUMITER(z, z') -{ 1 }-> 1 + IFSUM(isempty(z0), z0, z1, plus(z1, head(z0))) + PLUS(z1, head(z0)) :|: z = z0, z1 >= 0, z' = z1, z0 >= 0 head(z) -{ 0 }-> z0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 head(z) -{ 0 }-> 1 :|: z = 0 head(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 ifPlus(z, z', z'', z3) -{ 0 }-> z1 :|: z = 2, z1 >= 0, z0 >= 0, z3 = z2, z' = z0, z2 >= 0, z'' = z1 ifPlus(z, z', z'', z3) -{ 0 }-> plusIter(z0, 1 + z1, 1 + z2) :|: z1 >= 0, z = 1, z0 >= 0, z3 = z2, z' = z0, z2 >= 0, z'' = z1 ifPlus(z, z', z'', z3) -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, z3 = v3, v2 >= 0, v3 >= 0 isempty(z) -{ 0 }-> 2 :|: z = 0 isempty(z) -{ 0 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 isempty(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 le(z, z') -{ 0 }-> le(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1 le(z, z') -{ 0 }-> 2 :|: z0 >= 0, z = 0, z' = z0 le(z, z') -{ 0 }-> 1 :|: z = 1 + z0, z0 >= 0, z' = 0 le(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 plus(z, z') -{ 0 }-> plusIter(z0, z1, 0) :|: z = z0, z1 >= 0, z' = z1, z0 >= 0 plus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 plusIter(z, z', z'') -{ 0 }-> ifPlus(le(z0, z2), z0, z1, z2) :|: z'' = z2, z = z0, z1 >= 0, z' = z1, z0 >= 0, z2 >= 0 plusIter(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 tail(z) -{ 0 }-> z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 tail(z) -{ 0 }-> 0 :|: z = 0 tail(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (99) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace PLUSITER(z0, z1, z2) -> c2(IFPLUS(le(z0, z2), z0, z1, z2), LE(z0, z2)) by PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0), LE(s(z0), 0)) PLUSITER(0, x1, z0) -> c2(IFPLUS(true, 0, x1, z0), LE(0, z0)) PLUSITER(s(z0), x1, s(z1)) -> c2(IFPLUS(le(z0, z1), s(z0), x1, s(z1)), LE(s(z0), s(z1))) ---------------------------------------- (100) Obligation: Complexity Dependency Tuples Problem Rules: le(s(z0), 0) -> false le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) isempty(nil) -> true isempty(cons(z0, z1)) -> false plus(z0, z1) -> plusIter(z0, z1, 0) head(nil) -> error head(cons(z0, z1)) -> z0 plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2) ifPlus(true, z0, z1, z2) -> z1 ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2)) tail(nil) -> nil tail(cons(z0, z1)) -> z1 Tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2))) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) SUMITER(z0, z1) -> c9(IFSUM(isempty(z0), z0, z1, plus(z1, head(z0)))) SUMITER(z0, z1) -> c10(IFSUM(isempty(z0), z0, z1, plus(z1, head(z0))), PLUS(z1, head(z0))) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0), LE(s(z0), 0)) PLUSITER(0, x1, z0) -> c2(IFPLUS(true, 0, x1, z0), LE(0, z0)) PLUSITER(s(z0), x1, s(z1)) -> c2(IFPLUS(le(z0, z1), s(z0), x1, s(z1)), LE(s(z0), s(z1))) S tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2))) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) SUMITER(z0, z1) -> c9(IFSUM(isempty(z0), z0, z1, plus(z1, head(z0)))) SUMITER(z0, z1) -> c10(IFSUM(isempty(z0), z0, z1, plus(z1, head(z0))), PLUS(z1, head(z0))) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0), LE(s(z0), 0)) PLUSITER(0, x1, z0) -> c2(IFPLUS(true, 0, x1, z0), LE(0, z0)) PLUSITER(s(z0), x1, s(z1)) -> c2(IFPLUS(le(z0, z1), s(z0), x1, s(z1)), LE(s(z0), s(z1))) K tuples:none Defined Rule Symbols: le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1 Defined Pair Symbols: PLUS_2, IFPLUS_4, LE_2, SUMITER_2, IFSUM_4, PLUSITER_3 Compound Symbols: c1_1, c4_1, c7_1, c9_1, c10_2, c12_1, c2_2 ---------------------------------------- (101) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: PLUSITER(0, x1, z0) -> c2(IFPLUS(true, 0, x1, z0), LE(0, z0)) ---------------------------------------- (102) Obligation: Complexity Dependency Tuples Problem Rules: le(s(z0), 0) -> false le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) isempty(nil) -> true isempty(cons(z0, z1)) -> false plus(z0, z1) -> plusIter(z0, z1, 0) head(nil) -> error head(cons(z0, z1)) -> z0 plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2) ifPlus(true, z0, z1, z2) -> z1 ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2)) tail(nil) -> nil tail(cons(z0, z1)) -> z1 Tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2))) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) SUMITER(z0, z1) -> c9(IFSUM(isempty(z0), z0, z1, plus(z1, head(z0)))) SUMITER(z0, z1) -> c10(IFSUM(isempty(z0), z0, z1, plus(z1, head(z0))), PLUS(z1, head(z0))) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0), LE(s(z0), 0)) PLUSITER(s(z0), x1, s(z1)) -> c2(IFPLUS(le(z0, z1), s(z0), x1, s(z1)), LE(s(z0), s(z1))) S tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2))) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) SUMITER(z0, z1) -> c9(IFSUM(isempty(z0), z0, z1, plus(z1, head(z0)))) SUMITER(z0, z1) -> c10(IFSUM(isempty(z0), z0, z1, plus(z1, head(z0))), PLUS(z1, head(z0))) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0), LE(s(z0), 0)) PLUSITER(s(z0), x1, s(z1)) -> c2(IFPLUS(le(z0, z1), s(z0), x1, s(z1)), LE(s(z0), s(z1))) K tuples:none Defined Rule Symbols: le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1 Defined Pair Symbols: PLUS_2, IFPLUS_4, LE_2, SUMITER_2, IFSUM_4, PLUSITER_3 Compound Symbols: c1_1, c4_1, c7_1, c9_1, c10_2, c12_1, c2_2 ---------------------------------------- (103) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (104) Obligation: Complexity Dependency Tuples Problem Rules: le(s(z0), 0) -> false le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) isempty(nil) -> true isempty(cons(z0, z1)) -> false plus(z0, z1) -> plusIter(z0, z1, 0) head(nil) -> error head(cons(z0, z1)) -> z0 plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2) ifPlus(true, z0, z1, z2) -> z1 ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2)) tail(nil) -> nil tail(cons(z0, z1)) -> z1 Tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2))) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) SUMITER(z0, z1) -> c9(IFSUM(isempty(z0), z0, z1, plus(z1, head(z0)))) SUMITER(z0, z1) -> c10(IFSUM(isempty(z0), z0, z1, plus(z1, head(z0))), PLUS(z1, head(z0))) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, s(z1)) -> c2(IFPLUS(le(z0, z1), s(z0), x1, s(z1)), LE(s(z0), s(z1))) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) S tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2))) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) SUMITER(z0, z1) -> c9(IFSUM(isempty(z0), z0, z1, plus(z1, head(z0)))) SUMITER(z0, z1) -> c10(IFSUM(isempty(z0), z0, z1, plus(z1, head(z0))), PLUS(z1, head(z0))) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, s(z1)) -> c2(IFPLUS(le(z0, z1), s(z0), x1, s(z1)), LE(s(z0), s(z1))) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) K tuples:none Defined Rule Symbols: le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1 Defined Pair Symbols: PLUS_2, IFPLUS_4, LE_2, SUMITER_2, IFSUM_4, PLUSITER_3 Compound Symbols: c1_1, c4_1, c7_1, c9_1, c10_2, c12_1, c2_2, c2_1 ---------------------------------------- (105) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace SUMITER(z0, z1) -> c9(IFSUM(isempty(z0), z0, z1, plus(z1, head(z0)))) by SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, plusIter(z0, head(x0), 0))) SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plus(x1, error))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0))) SUMITER(nil, x1) -> c9(IFSUM(true, nil, x1, plus(x1, head(nil)))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1))))) ---------------------------------------- (106) Obligation: Complexity Dependency Tuples Problem Rules: le(s(z0), 0) -> false le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) isempty(nil) -> true isempty(cons(z0, z1)) -> false plus(z0, z1) -> plusIter(z0, z1, 0) head(nil) -> error head(cons(z0, z1)) -> z0 plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2) ifPlus(true, z0, z1, z2) -> z1 ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2)) tail(nil) -> nil tail(cons(z0, z1)) -> z1 Tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2))) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) SUMITER(z0, z1) -> c10(IFSUM(isempty(z0), z0, z1, plus(z1, head(z0))), PLUS(z1, head(z0))) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, s(z1)) -> c2(IFPLUS(le(z0, z1), s(z0), x1, s(z1)), LE(s(z0), s(z1))) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, plusIter(z0, head(x0), 0))) SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plus(x1, error))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0))) SUMITER(nil, x1) -> c9(IFSUM(true, nil, x1, plus(x1, head(nil)))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1))))) S tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2))) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) SUMITER(z0, z1) -> c10(IFSUM(isempty(z0), z0, z1, plus(z1, head(z0))), PLUS(z1, head(z0))) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, s(z1)) -> c2(IFPLUS(le(z0, z1), s(z0), x1, s(z1)), LE(s(z0), s(z1))) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, plusIter(z0, head(x0), 0))) SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plus(x1, error))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0))) SUMITER(nil, x1) -> c9(IFSUM(true, nil, x1, plus(x1, head(nil)))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1))))) K tuples:none Defined Rule Symbols: le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1 Defined Pair Symbols: PLUS_2, IFPLUS_4, LE_2, SUMITER_2, IFSUM_4, PLUSITER_3 Compound Symbols: c1_1, c4_1, c7_1, c10_2, c12_1, c2_2, c2_1, c9_1 ---------------------------------------- (107) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: SUMITER(nil, x1) -> c9(IFSUM(true, nil, x1, plus(x1, head(nil)))) ---------------------------------------- (108) Obligation: Complexity Dependency Tuples Problem Rules: le(s(z0), 0) -> false le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) isempty(nil) -> true isempty(cons(z0, z1)) -> false plus(z0, z1) -> plusIter(z0, z1, 0) head(nil) -> error head(cons(z0, z1)) -> z0 plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2) ifPlus(true, z0, z1, z2) -> z1 ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2)) tail(nil) -> nil tail(cons(z0, z1)) -> z1 Tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2))) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) SUMITER(z0, z1) -> c10(IFSUM(isempty(z0), z0, z1, plus(z1, head(z0))), PLUS(z1, head(z0))) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, s(z1)) -> c2(IFPLUS(le(z0, z1), s(z0), x1, s(z1)), LE(s(z0), s(z1))) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, plusIter(z0, head(x0), 0))) SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plus(x1, error))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1))))) S tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2))) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) SUMITER(z0, z1) -> c10(IFSUM(isempty(z0), z0, z1, plus(z1, head(z0))), PLUS(z1, head(z0))) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, s(z1)) -> c2(IFPLUS(le(z0, z1), s(z0), x1, s(z1)), LE(s(z0), s(z1))) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, plusIter(z0, head(x0), 0))) SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plus(x1, error))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1))))) K tuples:none Defined Rule Symbols: le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1 Defined Pair Symbols: PLUS_2, IFPLUS_4, LE_2, SUMITER_2, IFSUM_4, PLUSITER_3 Compound Symbols: c1_1, c4_1, c7_1, c10_2, c12_1, c2_2, c2_1, c9_1 ---------------------------------------- (109) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace SUMITER(z0, z1) -> c10(IFSUM(isempty(z0), z0, z1, plus(z1, head(z0))), PLUS(z1, head(z0))) by SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, plusIter(z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plus(x1, error)), PLUS(x1, head(nil))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(nil, x1) -> c10(IFSUM(true, nil, x1, plus(x1, head(nil))), PLUS(x1, head(nil))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1)))), PLUS(x1, head(cons(z0, z1)))) ---------------------------------------- (110) Obligation: Complexity Dependency Tuples Problem Rules: le(s(z0), 0) -> false le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) isempty(nil) -> true isempty(cons(z0, z1)) -> false plus(z0, z1) -> plusIter(z0, z1, 0) head(nil) -> error head(cons(z0, z1)) -> z0 plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2) ifPlus(true, z0, z1, z2) -> z1 ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2)) tail(nil) -> nil tail(cons(z0, z1)) -> z1 Tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2))) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, s(z1)) -> c2(IFPLUS(le(z0, z1), s(z0), x1, s(z1)), LE(s(z0), s(z1))) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, plusIter(z0, head(x0), 0))) SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plus(x1, error))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1))))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, plusIter(z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plus(x1, error)), PLUS(x1, head(nil))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(nil, x1) -> c10(IFSUM(true, nil, x1, plus(x1, head(nil))), PLUS(x1, head(nil))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1)))), PLUS(x1, head(cons(z0, z1)))) S tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2))) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, s(z1)) -> c2(IFPLUS(le(z0, z1), s(z0), x1, s(z1)), LE(s(z0), s(z1))) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, plusIter(z0, head(x0), 0))) SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plus(x1, error))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1))))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, plusIter(z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plus(x1, error)), PLUS(x1, head(nil))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(nil, x1) -> c10(IFSUM(true, nil, x1, plus(x1, head(nil))), PLUS(x1, head(nil))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1)))), PLUS(x1, head(cons(z0, z1)))) K tuples:none Defined Rule Symbols: le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1 Defined Pair Symbols: PLUS_2, IFPLUS_4, LE_2, IFSUM_4, PLUSITER_3, SUMITER_2 Compound Symbols: c1_1, c4_1, c7_1, c12_1, c2_2, c2_1, c9_1, c10_2 ---------------------------------------- (111) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (112) Obligation: Complexity Dependency Tuples Problem Rules: le(s(z0), 0) -> false le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) isempty(nil) -> true isempty(cons(z0, z1)) -> false plus(z0, z1) -> plusIter(z0, z1, 0) head(nil) -> error head(cons(z0, z1)) -> z0 plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2) ifPlus(true, z0, z1, z2) -> z1 ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2)) tail(nil) -> nil tail(cons(z0, z1)) -> z1 Tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2))) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, s(z1)) -> c2(IFPLUS(le(z0, z1), s(z0), x1, s(z1)), LE(s(z0), s(z1))) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, plusIter(z0, head(x0), 0))) SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plus(x1, error))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1))))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, plusIter(z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plus(x1, error)), PLUS(x1, head(nil))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1)))), PLUS(x1, head(cons(z0, z1)))) SUMITER(nil, x1) -> c10(PLUS(x1, head(nil))) S tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2))) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, s(z1)) -> c2(IFPLUS(le(z0, z1), s(z0), x1, s(z1)), LE(s(z0), s(z1))) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, plusIter(z0, head(x0), 0))) SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plus(x1, error))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1))))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, plusIter(z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plus(x1, error)), PLUS(x1, head(nil))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1)))), PLUS(x1, head(cons(z0, z1)))) SUMITER(nil, x1) -> c10(PLUS(x1, head(nil))) K tuples:none Defined Rule Symbols: le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1 Defined Pair Symbols: PLUS_2, IFPLUS_4, LE_2, IFSUM_4, PLUSITER_3, SUMITER_2 Compound Symbols: c1_1, c4_1, c7_1, c12_1, c2_2, c2_1, c9_1, c10_2, c10_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. SUMITER(nil, x1) -> c10(PLUS(x1, head(nil))) We considered the (Usable) Rules: tail(cons(z0, z1)) -> z1 isempty(nil) -> true tail(nil) -> nil isempty(cons(z0, z1)) -> false And the Tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2))) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, s(z1)) -> c2(IFPLUS(le(z0, z1), s(z0), x1, s(z1)), LE(s(z0), s(z1))) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, plusIter(z0, head(x0), 0))) SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plus(x1, error))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1))))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, plusIter(z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plus(x1, error)), PLUS(x1, head(nil))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1)))), PLUS(x1, head(cons(z0, z1)))) SUMITER(nil, x1) -> c10(PLUS(x1, head(nil))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(IFPLUS(x_1, x_2, x_3, x_4)) = x_2 POL(IFSUM(x_1, x_2, x_3, x_4)) = x_1 + x_2 POL(LE(x_1, x_2)) = 0 POL(PLUS(x_1, x_2)) = 0 POL(PLUSITER(x_1, x_2, x_3)) = x_3 POL(SUMITER(x_1, x_2)) = x_1 POL(c1(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c10(x_1, x_2)) = x_1 + x_2 POL(c12(x_1)) = x_1 POL(c2(x_1)) = x_1 POL(c2(x_1, x_2)) = x_1 + x_2 POL(c4(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c9(x_1)) = x_1 POL(cons(x_1, x_2)) = x_1 + x_2 POL(error) = 0 POL(false) = 0 POL(head(x_1)) = 0 POL(ifPlus(x_1, x_2, x_3, x_4)) = [1] + x_1 + x_2 + x_3 + x_4 POL(isempty(x_1)) = 0 POL(le(x_1, x_2)) = [1] + x_1 + x_2 POL(nil) = [1] POL(plus(x_1, x_2)) = [1] + x_1 + x_2 POL(plusIter(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3 POL(s(x_1)) = 0 POL(tail(x_1)) = x_1 POL(true) = 0 ---------------------------------------- (114) Obligation: Complexity Dependency Tuples Problem Rules: le(s(z0), 0) -> false le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) isempty(nil) -> true isempty(cons(z0, z1)) -> false plus(z0, z1) -> plusIter(z0, z1, 0) head(nil) -> error head(cons(z0, z1)) -> z0 plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2) ifPlus(true, z0, z1, z2) -> z1 ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2)) tail(nil) -> nil tail(cons(z0, z1)) -> z1 Tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2))) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, s(z1)) -> c2(IFPLUS(le(z0, z1), s(z0), x1, s(z1)), LE(s(z0), s(z1))) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, plusIter(z0, head(x0), 0))) SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plus(x1, error))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1))))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, plusIter(z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plus(x1, error)), PLUS(x1, head(nil))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1)))), PLUS(x1, head(cons(z0, z1)))) SUMITER(nil, x1) -> c10(PLUS(x1, head(nil))) S tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2))) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, s(z1)) -> c2(IFPLUS(le(z0, z1), s(z0), x1, s(z1)), LE(s(z0), s(z1))) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, plusIter(z0, head(x0), 0))) SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plus(x1, error))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1))))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, plusIter(z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plus(x1, error)), PLUS(x1, head(nil))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1)))), PLUS(x1, head(cons(z0, z1)))) K tuples: SUMITER(nil, x1) -> c10(PLUS(x1, head(nil))) Defined Rule Symbols: le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1 Defined Pair Symbols: PLUS_2, IFPLUS_4, LE_2, IFSUM_4, PLUSITER_3, SUMITER_2 Compound Symbols: c1_1, c4_1, c7_1, c12_1, c2_2, c2_1, c9_1, c10_2, c10_1 ---------------------------------------- (115) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace PLUSITER(s(z0), x1, s(z1)) -> c2(IFPLUS(le(z0, z1), s(z0), x1, s(z1)), LE(s(z0), s(z1))) by PLUSITER(s(s(z0)), x1, s(0)) -> c2(IFPLUS(false, s(s(z0)), x1, s(0)), LE(s(s(z0)), s(0))) PLUSITER(s(0), x1, s(z0)) -> c2(IFPLUS(true, s(0), x1, s(z0)), LE(s(0), s(z0))) PLUSITER(s(s(z0)), x1, s(s(z1))) -> c2(IFPLUS(le(z0, z1), s(s(z0)), x1, s(s(z1))), LE(s(s(z0)), s(s(z1)))) PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2))) ---------------------------------------- (116) Obligation: Complexity Dependency Tuples Problem Rules: le(s(z0), 0) -> false le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) isempty(nil) -> true isempty(cons(z0, z1)) -> false plus(z0, z1) -> plusIter(z0, z1, 0) head(nil) -> error head(cons(z0, z1)) -> z0 plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2) ifPlus(true, z0, z1, z2) -> z1 ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2)) tail(nil) -> nil tail(cons(z0, z1)) -> z1 Tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2))) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, plusIter(z0, head(x0), 0))) SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plus(x1, error))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1))))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, plusIter(z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plus(x1, error)), PLUS(x1, head(nil))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1)))), PLUS(x1, head(cons(z0, z1)))) SUMITER(nil, x1) -> c10(PLUS(x1, head(nil))) PLUSITER(s(s(z0)), x1, s(0)) -> c2(IFPLUS(false, s(s(z0)), x1, s(0)), LE(s(s(z0)), s(0))) PLUSITER(s(0), x1, s(z0)) -> c2(IFPLUS(true, s(0), x1, s(z0)), LE(s(0), s(z0))) PLUSITER(s(s(z0)), x1, s(s(z1))) -> c2(IFPLUS(le(z0, z1), s(s(z0)), x1, s(s(z1))), LE(s(s(z0)), s(s(z1)))) PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2))) S tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2))) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, plusIter(z0, head(x0), 0))) SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plus(x1, error))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1))))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, plusIter(z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plus(x1, error)), PLUS(x1, head(nil))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1)))), PLUS(x1, head(cons(z0, z1)))) PLUSITER(s(s(z0)), x1, s(0)) -> c2(IFPLUS(false, s(s(z0)), x1, s(0)), LE(s(s(z0)), s(0))) PLUSITER(s(0), x1, s(z0)) -> c2(IFPLUS(true, s(0), x1, s(z0)), LE(s(0), s(z0))) PLUSITER(s(s(z0)), x1, s(s(z1))) -> c2(IFPLUS(le(z0, z1), s(s(z0)), x1, s(s(z1))), LE(s(s(z0)), s(s(z1)))) PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2))) K tuples: SUMITER(nil, x1) -> c10(PLUS(x1, head(nil))) Defined Rule Symbols: le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1 Defined Pair Symbols: PLUS_2, IFPLUS_4, LE_2, IFSUM_4, PLUSITER_3, SUMITER_2 Compound Symbols: c1_1, c4_1, c7_1, c12_1, c2_1, c9_1, c10_2, c10_1, c2_2 ---------------------------------------- (117) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (118) Obligation: Complexity Dependency Tuples Problem Rules: le(s(z0), 0) -> false le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) isempty(nil) -> true isempty(cons(z0, z1)) -> false plus(z0, z1) -> plusIter(z0, z1, 0) head(nil) -> error head(cons(z0, z1)) -> z0 plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2) ifPlus(true, z0, z1, z2) -> z1 ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2)) tail(nil) -> nil tail(cons(z0, z1)) -> z1 Tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2))) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, plusIter(z0, head(x0), 0))) SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plus(x1, error))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1))))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, plusIter(z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plus(x1, error)), PLUS(x1, head(nil))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1)))), PLUS(x1, head(cons(z0, z1)))) SUMITER(nil, x1) -> c10(PLUS(x1, head(nil))) PLUSITER(s(s(z0)), x1, s(0)) -> c2(IFPLUS(false, s(s(z0)), x1, s(0)), LE(s(s(z0)), s(0))) PLUSITER(s(s(z0)), x1, s(s(z1))) -> c2(IFPLUS(le(z0, z1), s(s(z0)), x1, s(s(z1))), LE(s(s(z0)), s(s(z1)))) PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2))) PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0))) S tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2))) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, plusIter(z0, head(x0), 0))) SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plus(x1, error))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1))))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, plusIter(z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plus(x1, error)), PLUS(x1, head(nil))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1)))), PLUS(x1, head(cons(z0, z1)))) PLUSITER(s(s(z0)), x1, s(0)) -> c2(IFPLUS(false, s(s(z0)), x1, s(0)), LE(s(s(z0)), s(0))) PLUSITER(s(s(z0)), x1, s(s(z1))) -> c2(IFPLUS(le(z0, z1), s(s(z0)), x1, s(s(z1))), LE(s(s(z0)), s(s(z1)))) PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2))) PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0))) K tuples: SUMITER(nil, x1) -> c10(PLUS(x1, head(nil))) Defined Rule Symbols: le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1 Defined Pair Symbols: PLUS_2, IFPLUS_4, LE_2, IFSUM_4, PLUSITER_3, SUMITER_2 Compound Symbols: c1_1, c4_1, c7_1, c12_1, c2_1, c9_1, c10_2, c10_1, c2_2 ---------------------------------------- (119) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, plusIter(z0, head(x0), 0))) by SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0))) SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0))) SUMITER(nil, x1) -> c9(IFSUM(true, nil, x1, plusIter(x1, head(nil), 0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0))) ---------------------------------------- (120) Obligation: Complexity Dependency Tuples Problem Rules: le(s(z0), 0) -> false le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) isempty(nil) -> true isempty(cons(z0, z1)) -> false plus(z0, z1) -> plusIter(z0, z1, 0) head(nil) -> error head(cons(z0, z1)) -> z0 plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2) ifPlus(true, z0, z1, z2) -> z1 ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2)) tail(nil) -> nil tail(cons(z0, z1)) -> z1 Tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2))) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plus(x1, error))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1))))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, plusIter(z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plus(x1, error)), PLUS(x1, head(nil))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1)))), PLUS(x1, head(cons(z0, z1)))) SUMITER(nil, x1) -> c10(PLUS(x1, head(nil))) PLUSITER(s(s(z0)), x1, s(0)) -> c2(IFPLUS(false, s(s(z0)), x1, s(0)), LE(s(s(z0)), s(0))) PLUSITER(s(s(z0)), x1, s(s(z1))) -> c2(IFPLUS(le(z0, z1), s(s(z0)), x1, s(s(z1))), LE(s(s(z0)), s(s(z1)))) PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2))) PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0))) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0))) SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0))) SUMITER(nil, x1) -> c9(IFSUM(true, nil, x1, plusIter(x1, head(nil), 0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0))) S tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2))) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plus(x1, error))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1))))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, plusIter(z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plus(x1, error)), PLUS(x1, head(nil))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1)))), PLUS(x1, head(cons(z0, z1)))) PLUSITER(s(s(z0)), x1, s(0)) -> c2(IFPLUS(false, s(s(z0)), x1, s(0)), LE(s(s(z0)), s(0))) PLUSITER(s(s(z0)), x1, s(s(z1))) -> c2(IFPLUS(le(z0, z1), s(s(z0)), x1, s(s(z1))), LE(s(s(z0)), s(s(z1)))) PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2))) PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0))) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0))) SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0))) SUMITER(nil, x1) -> c9(IFSUM(true, nil, x1, plusIter(x1, head(nil), 0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0))) K tuples: SUMITER(nil, x1) -> c10(PLUS(x1, head(nil))) Defined Rule Symbols: le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1 Defined Pair Symbols: PLUS_2, IFPLUS_4, LE_2, IFSUM_4, PLUSITER_3, SUMITER_2 Compound Symbols: c1_1, c4_1, c7_1, c12_1, c2_1, c9_1, c10_2, c10_1, c2_2 ---------------------------------------- (121) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: SUMITER(nil, x1) -> c9(IFSUM(true, nil, x1, plusIter(x1, head(nil), 0))) ---------------------------------------- (122) Obligation: Complexity Dependency Tuples Problem Rules: le(s(z0), 0) -> false le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) isempty(nil) -> true isempty(cons(z0, z1)) -> false plus(z0, z1) -> plusIter(z0, z1, 0) head(nil) -> error head(cons(z0, z1)) -> z0 plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2) ifPlus(true, z0, z1, z2) -> z1 ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2)) tail(nil) -> nil tail(cons(z0, z1)) -> z1 Tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2))) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plus(x1, error))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1))))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, plusIter(z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plus(x1, error)), PLUS(x1, head(nil))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1)))), PLUS(x1, head(cons(z0, z1)))) SUMITER(nil, x1) -> c10(PLUS(x1, head(nil))) PLUSITER(s(s(z0)), x1, s(0)) -> c2(IFPLUS(false, s(s(z0)), x1, s(0)), LE(s(s(z0)), s(0))) PLUSITER(s(s(z0)), x1, s(s(z1))) -> c2(IFPLUS(le(z0, z1), s(s(z0)), x1, s(s(z1))), LE(s(s(z0)), s(s(z1)))) PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2))) PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0))) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0))) SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0))) S tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2))) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plus(x1, error))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1))))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, plusIter(z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plus(x1, error)), PLUS(x1, head(nil))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1)))), PLUS(x1, head(cons(z0, z1)))) PLUSITER(s(s(z0)), x1, s(0)) -> c2(IFPLUS(false, s(s(z0)), x1, s(0)), LE(s(s(z0)), s(0))) PLUSITER(s(s(z0)), x1, s(s(z1))) -> c2(IFPLUS(le(z0, z1), s(s(z0)), x1, s(s(z1))), LE(s(s(z0)), s(s(z1)))) PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2))) PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0))) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0))) SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0))) K tuples: SUMITER(nil, x1) -> c10(PLUS(x1, head(nil))) Defined Rule Symbols: le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1 Defined Pair Symbols: PLUS_2, IFPLUS_4, LE_2, IFSUM_4, PLUSITER_3, SUMITER_2 Compound Symbols: c1_1, c4_1, c7_1, c12_1, c2_1, c9_1, c10_2, c10_1, c2_2 ---------------------------------------- (123) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plus(x1, error))) by SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, plusIter(z0, error, 0))) SUMITER(nil, x0) -> c9(IFSUM(true, nil, x0, plus(x0, error))) ---------------------------------------- (124) Obligation: Complexity Dependency Tuples Problem Rules: le(s(z0), 0) -> false le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) isempty(nil) -> true isempty(cons(z0, z1)) -> false plus(z0, z1) -> plusIter(z0, z1, 0) head(nil) -> error head(cons(z0, z1)) -> z0 plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2) ifPlus(true, z0, z1, z2) -> z1 ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2)) tail(nil) -> nil tail(cons(z0, z1)) -> z1 Tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2))) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1))))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, plusIter(z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plus(x1, error)), PLUS(x1, head(nil))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1)))), PLUS(x1, head(cons(z0, z1)))) SUMITER(nil, x1) -> c10(PLUS(x1, head(nil))) PLUSITER(s(s(z0)), x1, s(0)) -> c2(IFPLUS(false, s(s(z0)), x1, s(0)), LE(s(s(z0)), s(0))) PLUSITER(s(s(z0)), x1, s(s(z1))) -> c2(IFPLUS(le(z0, z1), s(s(z0)), x1, s(s(z1))), LE(s(s(z0)), s(s(z1)))) PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2))) PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0))) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0))) SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0))) SUMITER(nil, x0) -> c9(IFSUM(true, nil, x0, plus(x0, error))) S tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2))) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1))))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, plusIter(z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plus(x1, error)), PLUS(x1, head(nil))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1)))), PLUS(x1, head(cons(z0, z1)))) PLUSITER(s(s(z0)), x1, s(0)) -> c2(IFPLUS(false, s(s(z0)), x1, s(0)), LE(s(s(z0)), s(0))) PLUSITER(s(s(z0)), x1, s(s(z1))) -> c2(IFPLUS(le(z0, z1), s(s(z0)), x1, s(s(z1))), LE(s(s(z0)), s(s(z1)))) PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2))) PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0))) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0))) SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0))) SUMITER(nil, x0) -> c9(IFSUM(true, nil, x0, plus(x0, error))) K tuples: SUMITER(nil, x1) -> c10(PLUS(x1, head(nil))) Defined Rule Symbols: le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1 Defined Pair Symbols: PLUS_2, IFPLUS_4, LE_2, IFSUM_4, PLUSITER_3, SUMITER_2 Compound Symbols: c1_1, c4_1, c7_1, c12_1, c2_1, c9_1, c10_2, c10_1, c2_2 ---------------------------------------- (125) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: SUMITER(nil, x0) -> c9(IFSUM(true, nil, x0, plus(x0, error))) ---------------------------------------- (126) Obligation: Complexity Dependency Tuples Problem Rules: le(s(z0), 0) -> false le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) isempty(nil) -> true isempty(cons(z0, z1)) -> false plus(z0, z1) -> plusIter(z0, z1, 0) head(nil) -> error head(cons(z0, z1)) -> z0 plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2) ifPlus(true, z0, z1, z2) -> z1 ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2)) tail(nil) -> nil tail(cons(z0, z1)) -> z1 Tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2))) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1))))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, plusIter(z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plus(x1, error)), PLUS(x1, head(nil))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1)))), PLUS(x1, head(cons(z0, z1)))) SUMITER(nil, x1) -> c10(PLUS(x1, head(nil))) PLUSITER(s(s(z0)), x1, s(0)) -> c2(IFPLUS(false, s(s(z0)), x1, s(0)), LE(s(s(z0)), s(0))) PLUSITER(s(s(z0)), x1, s(s(z1))) -> c2(IFPLUS(le(z0, z1), s(s(z0)), x1, s(s(z1))), LE(s(s(z0)), s(s(z1)))) PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2))) PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0))) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0))) SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0))) S tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2))) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1))))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, plusIter(z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plus(x1, error)), PLUS(x1, head(nil))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1)))), PLUS(x1, head(cons(z0, z1)))) PLUSITER(s(s(z0)), x1, s(0)) -> c2(IFPLUS(false, s(s(z0)), x1, s(0)), LE(s(s(z0)), s(0))) PLUSITER(s(s(z0)), x1, s(s(z1))) -> c2(IFPLUS(le(z0, z1), s(s(z0)), x1, s(s(z1))), LE(s(s(z0)), s(s(z1)))) PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2))) PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0))) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0))) SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0))) K tuples: SUMITER(nil, x1) -> c10(PLUS(x1, head(nil))) Defined Rule Symbols: le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1 Defined Pair Symbols: PLUS_2, IFPLUS_4, LE_2, IFSUM_4, PLUSITER_3, SUMITER_2 Compound Symbols: c1_1, c4_1, c7_1, c12_1, c2_1, c9_1, c10_2, c10_1, c2_2 ---------------------------------------- (127) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0))) by SUMITER(cons(z1, x1), z0) -> c9(IFSUM(isempty(cons(z1, x1)), cons(z1, x1), z0, plusIter(z0, z1, 0))) SUMITER(cons(z0, z1), x2) -> c9(IFSUM(false, cons(z0, z1), x2, plus(x2, z0))) ---------------------------------------- (128) Obligation: Complexity Dependency Tuples Problem Rules: le(s(z0), 0) -> false le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) isempty(nil) -> true isempty(cons(z0, z1)) -> false plus(z0, z1) -> plusIter(z0, z1, 0) head(nil) -> error head(cons(z0, z1)) -> z0 plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2) ifPlus(true, z0, z1, z2) -> z1 ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2)) tail(nil) -> nil tail(cons(z0, z1)) -> z1 Tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2))) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1))))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, plusIter(z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plus(x1, error)), PLUS(x1, head(nil))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1)))), PLUS(x1, head(cons(z0, z1)))) SUMITER(nil, x1) -> c10(PLUS(x1, head(nil))) PLUSITER(s(s(z0)), x1, s(0)) -> c2(IFPLUS(false, s(s(z0)), x1, s(0)), LE(s(s(z0)), s(0))) PLUSITER(s(s(z0)), x1, s(s(z1))) -> c2(IFPLUS(le(z0, z1), s(s(z0)), x1, s(s(z1))), LE(s(s(z0)), s(s(z1)))) PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2))) PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0))) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0))) SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0))) SUMITER(cons(z0, z1), x2) -> c9(IFSUM(false, cons(z0, z1), x2, plus(x2, z0))) S tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2))) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1))))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, plusIter(z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plus(x1, error)), PLUS(x1, head(nil))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1)))), PLUS(x1, head(cons(z0, z1)))) PLUSITER(s(s(z0)), x1, s(0)) -> c2(IFPLUS(false, s(s(z0)), x1, s(0)), LE(s(s(z0)), s(0))) PLUSITER(s(s(z0)), x1, s(s(z1))) -> c2(IFPLUS(le(z0, z1), s(s(z0)), x1, s(s(z1))), LE(s(s(z0)), s(s(z1)))) PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2))) PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0))) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0))) SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0))) SUMITER(cons(z0, z1), x2) -> c9(IFSUM(false, cons(z0, z1), x2, plus(x2, z0))) K tuples: SUMITER(nil, x1) -> c10(PLUS(x1, head(nil))) Defined Rule Symbols: le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1 Defined Pair Symbols: PLUS_2, IFPLUS_4, LE_2, IFSUM_4, PLUSITER_3, SUMITER_2 Compound Symbols: c1_1, c4_1, c7_1, c12_1, c2_1, c9_1, c10_2, c10_1, c2_2 ---------------------------------------- (129) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1))))) by SUMITER(cons(x0, x1), z0) -> c9(IFSUM(false, cons(x0, x1), z0, plusIter(z0, head(cons(x0, x1)), 0))) SUMITER(cons(z0, z1), x2) -> c9(IFSUM(false, cons(z0, z1), x2, plus(x2, z0))) ---------------------------------------- (130) Obligation: Complexity Dependency Tuples Problem Rules: le(s(z0), 0) -> false le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) isempty(nil) -> true isempty(cons(z0, z1)) -> false plus(z0, z1) -> plusIter(z0, z1, 0) head(nil) -> error head(cons(z0, z1)) -> z0 plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2) ifPlus(true, z0, z1, z2) -> z1 ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2)) tail(nil) -> nil tail(cons(z0, z1)) -> z1 Tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2))) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, plusIter(z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plus(x1, error)), PLUS(x1, head(nil))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1)))), PLUS(x1, head(cons(z0, z1)))) SUMITER(nil, x1) -> c10(PLUS(x1, head(nil))) PLUSITER(s(s(z0)), x1, s(0)) -> c2(IFPLUS(false, s(s(z0)), x1, s(0)), LE(s(s(z0)), s(0))) PLUSITER(s(s(z0)), x1, s(s(z1))) -> c2(IFPLUS(le(z0, z1), s(s(z0)), x1, s(s(z1))), LE(s(s(z0)), s(s(z1)))) PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2))) PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0))) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0))) SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0))) SUMITER(cons(z0, z1), x2) -> c9(IFSUM(false, cons(z0, z1), x2, plus(x2, z0))) S tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2))) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, plusIter(z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plus(x1, error)), PLUS(x1, head(nil))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1)))), PLUS(x1, head(cons(z0, z1)))) PLUSITER(s(s(z0)), x1, s(0)) -> c2(IFPLUS(false, s(s(z0)), x1, s(0)), LE(s(s(z0)), s(0))) PLUSITER(s(s(z0)), x1, s(s(z1))) -> c2(IFPLUS(le(z0, z1), s(s(z0)), x1, s(s(z1))), LE(s(s(z0)), s(s(z1)))) PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2))) PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0))) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0))) SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0))) SUMITER(cons(z0, z1), x2) -> c9(IFSUM(false, cons(z0, z1), x2, plus(x2, z0))) K tuples: SUMITER(nil, x1) -> c10(PLUS(x1, head(nil))) Defined Rule Symbols: le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1 Defined Pair Symbols: PLUS_2, IFPLUS_4, LE_2, IFSUM_4, PLUSITER_3, SUMITER_2 Compound Symbols: c1_1, c4_1, c7_1, c12_1, c2_1, c10_2, c10_1, c2_2, c9_1 ---------------------------------------- (131) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, plusIter(z0, head(x0), 0)), PLUS(z0, head(x0))) by SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(nil, x1) -> c10(IFSUM(true, nil, x1, plusIter(x1, head(nil), 0)), PLUS(x1, head(nil))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(x0, x1) -> c10(PLUS(x1, head(x0))) ---------------------------------------- (132) Obligation: Complexity Dependency Tuples Problem Rules: le(s(z0), 0) -> false le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) isempty(nil) -> true isempty(cons(z0, z1)) -> false plus(z0, z1) -> plusIter(z0, z1, 0) head(nil) -> error head(cons(z0, z1)) -> z0 plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2) ifPlus(true, z0, z1, z2) -> z1 ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2)) tail(nil) -> nil tail(cons(z0, z1)) -> z1 Tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2))) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plus(x1, error)), PLUS(x1, head(nil))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1)))), PLUS(x1, head(cons(z0, z1)))) SUMITER(nil, x1) -> c10(PLUS(x1, head(nil))) PLUSITER(s(s(z0)), x1, s(0)) -> c2(IFPLUS(false, s(s(z0)), x1, s(0)), LE(s(s(z0)), s(0))) PLUSITER(s(s(z0)), x1, s(s(z1))) -> c2(IFPLUS(le(z0, z1), s(s(z0)), x1, s(s(z1))), LE(s(s(z0)), s(s(z1)))) PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2))) PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0))) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0))) SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0))) SUMITER(cons(z0, z1), x2) -> c9(IFSUM(false, cons(z0, z1), x2, plus(x2, z0))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(nil, x1) -> c10(IFSUM(true, nil, x1, plusIter(x1, head(nil), 0)), PLUS(x1, head(nil))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(x0, x1) -> c10(PLUS(x1, head(x0))) S tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2))) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plus(x1, error)), PLUS(x1, head(nil))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1)))), PLUS(x1, head(cons(z0, z1)))) PLUSITER(s(s(z0)), x1, s(0)) -> c2(IFPLUS(false, s(s(z0)), x1, s(0)), LE(s(s(z0)), s(0))) PLUSITER(s(s(z0)), x1, s(s(z1))) -> c2(IFPLUS(le(z0, z1), s(s(z0)), x1, s(s(z1))), LE(s(s(z0)), s(s(z1)))) PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2))) PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0))) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0))) SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0))) SUMITER(cons(z0, z1), x2) -> c9(IFSUM(false, cons(z0, z1), x2, plus(x2, z0))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(nil, x1) -> c10(IFSUM(true, nil, x1, plusIter(x1, head(nil), 0)), PLUS(x1, head(nil))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(x0, x1) -> c10(PLUS(x1, head(x0))) K tuples: SUMITER(nil, x1) -> c10(PLUS(x1, head(nil))) Defined Rule Symbols: le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1 Defined Pair Symbols: PLUS_2, IFPLUS_4, LE_2, IFSUM_4, PLUSITER_3, SUMITER_2 Compound Symbols: c1_1, c4_1, c7_1, c12_1, c2_1, c10_2, c10_1, c2_2, c9_1 ---------------------------------------- (133) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (134) Obligation: Complexity Dependency Tuples Problem Rules: le(s(z0), 0) -> false le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) isempty(nil) -> true isempty(cons(z0, z1)) -> false plus(z0, z1) -> plusIter(z0, z1, 0) head(nil) -> error head(cons(z0, z1)) -> z0 plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2) ifPlus(true, z0, z1, z2) -> z1 ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2)) tail(nil) -> nil tail(cons(z0, z1)) -> z1 Tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2))) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plus(x1, error)), PLUS(x1, head(nil))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1)))), PLUS(x1, head(cons(z0, z1)))) SUMITER(nil, x1) -> c10(PLUS(x1, head(nil))) PLUSITER(s(s(z0)), x1, s(0)) -> c2(IFPLUS(false, s(s(z0)), x1, s(0)), LE(s(s(z0)), s(0))) PLUSITER(s(s(z0)), x1, s(s(z1))) -> c2(IFPLUS(le(z0, z1), s(s(z0)), x1, s(s(z1))), LE(s(s(z0)), s(s(z1)))) PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2))) PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0))) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0))) SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0))) SUMITER(cons(z0, z1), x2) -> c9(IFSUM(false, cons(z0, z1), x2, plus(x2, z0))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(x0, x1) -> c10(PLUS(x1, head(x0))) S tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2))) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plus(x1, error)), PLUS(x1, head(nil))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1)))), PLUS(x1, head(cons(z0, z1)))) PLUSITER(s(s(z0)), x1, s(0)) -> c2(IFPLUS(false, s(s(z0)), x1, s(0)), LE(s(s(z0)), s(0))) PLUSITER(s(s(z0)), x1, s(s(z1))) -> c2(IFPLUS(le(z0, z1), s(s(z0)), x1, s(s(z1))), LE(s(s(z0)), s(s(z1)))) PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2))) PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0))) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0))) SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0))) SUMITER(cons(z0, z1), x2) -> c9(IFSUM(false, cons(z0, z1), x2, plus(x2, z0))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(x0, x1) -> c10(PLUS(x1, head(x0))) SUMITER(nil, x1) -> c10(PLUS(x1, head(nil))) K tuples: SUMITER(nil, x1) -> c10(PLUS(x1, head(nil))) Defined Rule Symbols: le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1 Defined Pair Symbols: PLUS_2, IFPLUS_4, LE_2, IFSUM_4, PLUSITER_3, SUMITER_2 Compound Symbols: c1_1, c4_1, c7_1, c12_1, c2_1, c10_2, c10_1, c2_2, c9_1 ---------------------------------------- (135) CdtKnowledgeProof (BOTH BOUNDS(ID, ID)) The following tuples could be moved from S to K by knowledge propagation: SUMITER(nil, x1) -> c10(PLUS(x1, head(nil))) SUMITER(nil, x1) -> c10(PLUS(x1, head(nil))) ---------------------------------------- (136) Obligation: Complexity Dependency Tuples Problem Rules: le(s(z0), 0) -> false le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) isempty(nil) -> true isempty(cons(z0, z1)) -> false plus(z0, z1) -> plusIter(z0, z1, 0) head(nil) -> error head(cons(z0, z1)) -> z0 plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2) ifPlus(true, z0, z1, z2) -> z1 ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2)) tail(nil) -> nil tail(cons(z0, z1)) -> z1 Tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2))) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plus(x1, error)), PLUS(x1, head(nil))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1)))), PLUS(x1, head(cons(z0, z1)))) SUMITER(nil, x1) -> c10(PLUS(x1, head(nil))) PLUSITER(s(s(z0)), x1, s(0)) -> c2(IFPLUS(false, s(s(z0)), x1, s(0)), LE(s(s(z0)), s(0))) PLUSITER(s(s(z0)), x1, s(s(z1))) -> c2(IFPLUS(le(z0, z1), s(s(z0)), x1, s(s(z1))), LE(s(s(z0)), s(s(z1)))) PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2))) PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0))) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0))) SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0))) SUMITER(cons(z0, z1), x2) -> c9(IFSUM(false, cons(z0, z1), x2, plus(x2, z0))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(x0, x1) -> c10(PLUS(x1, head(x0))) S tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2))) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plus(x1, error)), PLUS(x1, head(nil))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1)))), PLUS(x1, head(cons(z0, z1)))) PLUSITER(s(s(z0)), x1, s(0)) -> c2(IFPLUS(false, s(s(z0)), x1, s(0)), LE(s(s(z0)), s(0))) PLUSITER(s(s(z0)), x1, s(s(z1))) -> c2(IFPLUS(le(z0, z1), s(s(z0)), x1, s(s(z1))), LE(s(s(z0)), s(s(z1)))) PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2))) PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0))) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0))) SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0))) SUMITER(cons(z0, z1), x2) -> c9(IFSUM(false, cons(z0, z1), x2, plus(x2, z0))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(x0, x1) -> c10(PLUS(x1, head(x0))) K tuples: SUMITER(nil, x1) -> c10(PLUS(x1, head(nil))) Defined Rule Symbols: le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1 Defined Pair Symbols: PLUS_2, IFPLUS_4, LE_2, IFSUM_4, PLUSITER_3, SUMITER_2 Compound Symbols: c1_1, c4_1, c7_1, c12_1, c2_1, c10_2, c10_1, c2_2, c9_1 ---------------------------------------- (137) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. SUMITER(x0, x1) -> c10(PLUS(x1, head(x0))) We considered the (Usable) Rules: isempty(nil) -> true isempty(cons(z0, z1)) -> false And the Tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2))) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plus(x1, error)), PLUS(x1, head(nil))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1)))), PLUS(x1, head(cons(z0, z1)))) SUMITER(nil, x1) -> c10(PLUS(x1, head(nil))) PLUSITER(s(s(z0)), x1, s(0)) -> c2(IFPLUS(false, s(s(z0)), x1, s(0)), LE(s(s(z0)), s(0))) PLUSITER(s(s(z0)), x1, s(s(z1))) -> c2(IFPLUS(le(z0, z1), s(s(z0)), x1, s(s(z1))), LE(s(s(z0)), s(s(z1)))) PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2))) PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0))) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0))) SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0))) SUMITER(cons(z0, z1), x2) -> c9(IFSUM(false, cons(z0, z1), x2, plus(x2, z0))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(x0, x1) -> c10(PLUS(x1, head(x0))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(IFPLUS(x_1, x_2, x_3, x_4)) = x_2 POL(IFSUM(x_1, x_2, x_3, x_4)) = x_1 POL(LE(x_1, x_2)) = 0 POL(PLUS(x_1, x_2)) = 0 POL(PLUSITER(x_1, x_2, x_3)) = x_3 POL(SUMITER(x_1, x_2)) = [1] POL(c1(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c10(x_1, x_2)) = x_1 + x_2 POL(c12(x_1)) = x_1 POL(c2(x_1)) = x_1 POL(c2(x_1, x_2)) = x_1 + x_2 POL(c4(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c9(x_1)) = x_1 POL(cons(x_1, x_2)) = x_2 POL(error) = 0 POL(false) = [1] POL(head(x_1)) = 0 POL(ifPlus(x_1, x_2, x_3, x_4)) = [1] + x_2 + x_4 POL(isempty(x_1)) = [1] POL(le(x_1, x_2)) = x_1 POL(nil) = 0 POL(plus(x_1, x_2)) = [1] + x_1 POL(plusIter(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3 POL(s(x_1)) = 0 POL(tail(x_1)) = [1] + x_1 POL(true) = [1] ---------------------------------------- (138) Obligation: Complexity Dependency Tuples Problem Rules: le(s(z0), 0) -> false le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) isempty(nil) -> true isempty(cons(z0, z1)) -> false plus(z0, z1) -> plusIter(z0, z1, 0) head(nil) -> error head(cons(z0, z1)) -> z0 plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2) ifPlus(true, z0, z1, z2) -> z1 ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2)) tail(nil) -> nil tail(cons(z0, z1)) -> z1 Tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2))) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plus(x1, error)), PLUS(x1, head(nil))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1)))), PLUS(x1, head(cons(z0, z1)))) SUMITER(nil, x1) -> c10(PLUS(x1, head(nil))) PLUSITER(s(s(z0)), x1, s(0)) -> c2(IFPLUS(false, s(s(z0)), x1, s(0)), LE(s(s(z0)), s(0))) PLUSITER(s(s(z0)), x1, s(s(z1))) -> c2(IFPLUS(le(z0, z1), s(s(z0)), x1, s(s(z1))), LE(s(s(z0)), s(s(z1)))) PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2))) PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0))) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0))) SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0))) SUMITER(cons(z0, z1), x2) -> c9(IFSUM(false, cons(z0, z1), x2, plus(x2, z0))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(x0, x1) -> c10(PLUS(x1, head(x0))) S tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2))) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plus(x1, error)), PLUS(x1, head(nil))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1)))), PLUS(x1, head(cons(z0, z1)))) PLUSITER(s(s(z0)), x1, s(0)) -> c2(IFPLUS(false, s(s(z0)), x1, s(0)), LE(s(s(z0)), s(0))) PLUSITER(s(s(z0)), x1, s(s(z1))) -> c2(IFPLUS(le(z0, z1), s(s(z0)), x1, s(s(z1))), LE(s(s(z0)), s(s(z1)))) PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2))) PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0))) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0))) SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0))) SUMITER(cons(z0, z1), x2) -> c9(IFSUM(false, cons(z0, z1), x2, plus(x2, z0))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1)))) K tuples: SUMITER(nil, x1) -> c10(PLUS(x1, head(nil))) SUMITER(x0, x1) -> c10(PLUS(x1, head(x0))) Defined Rule Symbols: le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1 Defined Pair Symbols: PLUS_2, IFPLUS_4, LE_2, IFSUM_4, PLUSITER_3, SUMITER_2 Compound Symbols: c1_1, c4_1, c7_1, c12_1, c2_1, c10_2, c10_1, c2_2, c9_1 ---------------------------------------- (139) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plus(x1, error)), PLUS(x1, head(nil))) by SUMITER(nil, z0) -> c10(IFSUM(isempty(nil), nil, z0, plusIter(z0, error, 0)), PLUS(z0, head(nil))) SUMITER(nil, x0) -> c10(IFSUM(true, nil, x0, plus(x0, error)), PLUS(x0, head(nil))) ---------------------------------------- (140) Obligation: Complexity Dependency Tuples Problem Rules: le(s(z0), 0) -> false le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) isempty(nil) -> true isempty(cons(z0, z1)) -> false plus(z0, z1) -> plusIter(z0, z1, 0) head(nil) -> error head(cons(z0, z1)) -> z0 plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2) ifPlus(true, z0, z1, z2) -> z1 ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2)) tail(nil) -> nil tail(cons(z0, z1)) -> z1 Tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2))) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1)))), PLUS(x1, head(cons(z0, z1)))) SUMITER(nil, x1) -> c10(PLUS(x1, head(nil))) PLUSITER(s(s(z0)), x1, s(0)) -> c2(IFPLUS(false, s(s(z0)), x1, s(0)), LE(s(s(z0)), s(0))) PLUSITER(s(s(z0)), x1, s(s(z1))) -> c2(IFPLUS(le(z0, z1), s(s(z0)), x1, s(s(z1))), LE(s(s(z0)), s(s(z1)))) PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2))) PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0))) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0))) SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0))) SUMITER(cons(z0, z1), x2) -> c9(IFSUM(false, cons(z0, z1), x2, plus(x2, z0))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(x0, x1) -> c10(PLUS(x1, head(x0))) SUMITER(nil, x0) -> c10(IFSUM(true, nil, x0, plus(x0, error)), PLUS(x0, head(nil))) S tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2))) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1)))), PLUS(x1, head(cons(z0, z1)))) PLUSITER(s(s(z0)), x1, s(0)) -> c2(IFPLUS(false, s(s(z0)), x1, s(0)), LE(s(s(z0)), s(0))) PLUSITER(s(s(z0)), x1, s(s(z1))) -> c2(IFPLUS(le(z0, z1), s(s(z0)), x1, s(s(z1))), LE(s(s(z0)), s(s(z1)))) PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2))) PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0))) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0))) SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0))) SUMITER(cons(z0, z1), x2) -> c9(IFSUM(false, cons(z0, z1), x2, plus(x2, z0))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(nil, x0) -> c10(IFSUM(true, nil, x0, plus(x0, error)), PLUS(x0, head(nil))) K tuples: SUMITER(nil, x1) -> c10(PLUS(x1, head(nil))) SUMITER(x0, x1) -> c10(PLUS(x1, head(x0))) Defined Rule Symbols: le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1 Defined Pair Symbols: PLUS_2, IFPLUS_4, LE_2, IFSUM_4, PLUSITER_3, SUMITER_2 Compound Symbols: c1_1, c4_1, c7_1, c12_1, c2_1, c10_2, c10_1, c2_2, c9_1 ---------------------------------------- (141) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (142) Obligation: Complexity Dependency Tuples Problem Rules: le(s(z0), 0) -> false le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) isempty(nil) -> true isempty(cons(z0, z1)) -> false plus(z0, z1) -> plusIter(z0, z1, 0) head(nil) -> error head(cons(z0, z1)) -> z0 plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2) ifPlus(true, z0, z1, z2) -> z1 ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2)) tail(nil) -> nil tail(cons(z0, z1)) -> z1 Tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2))) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1)))), PLUS(x1, head(cons(z0, z1)))) SUMITER(nil, x1) -> c10(PLUS(x1, head(nil))) PLUSITER(s(s(z0)), x1, s(0)) -> c2(IFPLUS(false, s(s(z0)), x1, s(0)), LE(s(s(z0)), s(0))) PLUSITER(s(s(z0)), x1, s(s(z1))) -> c2(IFPLUS(le(z0, z1), s(s(z0)), x1, s(s(z1))), LE(s(s(z0)), s(s(z1)))) PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2))) PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0))) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0))) SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0))) SUMITER(cons(z0, z1), x2) -> c9(IFSUM(false, cons(z0, z1), x2, plus(x2, z0))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(x0, x1) -> c10(PLUS(x1, head(x0))) S tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2))) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1)))), PLUS(x1, head(cons(z0, z1)))) PLUSITER(s(s(z0)), x1, s(0)) -> c2(IFPLUS(false, s(s(z0)), x1, s(0)), LE(s(s(z0)), s(0))) PLUSITER(s(s(z0)), x1, s(s(z1))) -> c2(IFPLUS(le(z0, z1), s(s(z0)), x1, s(s(z1))), LE(s(s(z0)), s(s(z1)))) PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2))) PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0))) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0))) SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0))) SUMITER(cons(z0, z1), x2) -> c9(IFSUM(false, cons(z0, z1), x2, plus(x2, z0))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(nil, x0) -> c10(PLUS(x0, head(nil))) K tuples: SUMITER(nil, x1) -> c10(PLUS(x1, head(nil))) SUMITER(x0, x1) -> c10(PLUS(x1, head(x0))) Defined Rule Symbols: le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1 Defined Pair Symbols: PLUS_2, IFPLUS_4, LE_2, IFSUM_4, PLUSITER_3, SUMITER_2 Compound Symbols: c1_1, c4_1, c7_1, c12_1, c2_1, c10_2, c10_1, c2_2, c9_1 ---------------------------------------- (143) CdtKnowledgeProof (BOTH BOUNDS(ID, ID)) The following tuples could be moved from S to K by knowledge propagation: SUMITER(nil, x1) -> c10(PLUS(x1, head(nil))) SUMITER(nil, x1) -> c10(PLUS(x1, head(nil))) SUMITER(nil, x0) -> c10(PLUS(x0, head(nil))) ---------------------------------------- (144) Obligation: Complexity Dependency Tuples Problem Rules: le(s(z0), 0) -> false le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) isempty(nil) -> true isempty(cons(z0, z1)) -> false plus(z0, z1) -> plusIter(z0, z1, 0) head(nil) -> error head(cons(z0, z1)) -> z0 plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2) ifPlus(true, z0, z1, z2) -> z1 ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2)) tail(nil) -> nil tail(cons(z0, z1)) -> z1 Tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2))) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1)))), PLUS(x1, head(cons(z0, z1)))) SUMITER(nil, x1) -> c10(PLUS(x1, head(nil))) PLUSITER(s(s(z0)), x1, s(0)) -> c2(IFPLUS(false, s(s(z0)), x1, s(0)), LE(s(s(z0)), s(0))) PLUSITER(s(s(z0)), x1, s(s(z1))) -> c2(IFPLUS(le(z0, z1), s(s(z0)), x1, s(s(z1))), LE(s(s(z0)), s(s(z1)))) PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2))) PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0))) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0))) SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0))) SUMITER(cons(z0, z1), x2) -> c9(IFSUM(false, cons(z0, z1), x2, plus(x2, z0))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(x0, x1) -> c10(PLUS(x1, head(x0))) S tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2))) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1)))), PLUS(x1, head(cons(z0, z1)))) PLUSITER(s(s(z0)), x1, s(0)) -> c2(IFPLUS(false, s(s(z0)), x1, s(0)), LE(s(s(z0)), s(0))) PLUSITER(s(s(z0)), x1, s(s(z1))) -> c2(IFPLUS(le(z0, z1), s(s(z0)), x1, s(s(z1))), LE(s(s(z0)), s(s(z1)))) PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2))) PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0))) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0))) SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0))) SUMITER(cons(z0, z1), x2) -> c9(IFSUM(false, cons(z0, z1), x2, plus(x2, z0))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1)))) K tuples: SUMITER(nil, x1) -> c10(PLUS(x1, head(nil))) SUMITER(x0, x1) -> c10(PLUS(x1, head(x0))) Defined Rule Symbols: le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1 Defined Pair Symbols: PLUS_2, IFPLUS_4, LE_2, IFSUM_4, PLUSITER_3, SUMITER_2 Compound Symbols: c1_1, c4_1, c7_1, c12_1, c2_1, c10_2, c10_1, c2_2, c9_1 ---------------------------------------- (145) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0)), PLUS(x1, head(cons(z0, z1)))) by SUMITER(cons(z1, x1), z0) -> c10(IFSUM(isempty(cons(z1, x1)), cons(z1, x1), z0, plusIter(z0, z1, 0)), PLUS(z0, head(cons(z1, x1)))) SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1)))) SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1)))) ---------------------------------------- (146) Obligation: Complexity Dependency Tuples Problem Rules: le(s(z0), 0) -> false le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) isempty(nil) -> true isempty(cons(z0, z1)) -> false plus(z0, z1) -> plusIter(z0, z1, 0) head(nil) -> error head(cons(z0, z1)) -> z0 plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2) ifPlus(true, z0, z1, z2) -> z1 ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2)) tail(nil) -> nil tail(cons(z0, z1)) -> z1 Tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2))) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1)))), PLUS(x1, head(cons(z0, z1)))) SUMITER(nil, x1) -> c10(PLUS(x1, head(nil))) PLUSITER(s(s(z0)), x1, s(0)) -> c2(IFPLUS(false, s(s(z0)), x1, s(0)), LE(s(s(z0)), s(0))) PLUSITER(s(s(z0)), x1, s(s(z1))) -> c2(IFPLUS(le(z0, z1), s(s(z0)), x1, s(s(z1))), LE(s(s(z0)), s(s(z1)))) PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2))) PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0))) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0))) SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0))) SUMITER(cons(z0, z1), x2) -> c9(IFSUM(false, cons(z0, z1), x2, plus(x2, z0))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(x0, x1) -> c10(PLUS(x1, head(x0))) SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1)))) SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1)))) S tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2))) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1)))), PLUS(x1, head(cons(z0, z1)))) PLUSITER(s(s(z0)), x1, s(0)) -> c2(IFPLUS(false, s(s(z0)), x1, s(0)), LE(s(s(z0)), s(0))) PLUSITER(s(s(z0)), x1, s(s(z1))) -> c2(IFPLUS(le(z0, z1), s(s(z0)), x1, s(s(z1))), LE(s(s(z0)), s(s(z1)))) PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2))) PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0))) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0))) SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0))) SUMITER(cons(z0, z1), x2) -> c9(IFSUM(false, cons(z0, z1), x2, plus(x2, z0))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1)))) SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1)))) K tuples: SUMITER(nil, x1) -> c10(PLUS(x1, head(nil))) SUMITER(x0, x1) -> c10(PLUS(x1, head(x0))) Defined Rule Symbols: le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1 Defined Pair Symbols: PLUS_2, IFPLUS_4, LE_2, IFSUM_4, PLUSITER_3, SUMITER_2 Compound Symbols: c1_1, c4_1, c7_1, c12_1, c2_1, c10_2, c10_1, c2_2, c9_1 ---------------------------------------- (147) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1)))) We considered the (Usable) Rules: tail(cons(z0, z1)) -> z1 tail(nil) -> nil And the Tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2))) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1)))), PLUS(x1, head(cons(z0, z1)))) SUMITER(nil, x1) -> c10(PLUS(x1, head(nil))) PLUSITER(s(s(z0)), x1, s(0)) -> c2(IFPLUS(false, s(s(z0)), x1, s(0)), LE(s(s(z0)), s(0))) PLUSITER(s(s(z0)), x1, s(s(z1))) -> c2(IFPLUS(le(z0, z1), s(s(z0)), x1, s(s(z1))), LE(s(s(z0)), s(s(z1)))) PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2))) PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0))) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0))) SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0))) SUMITER(cons(z0, z1), x2) -> c9(IFSUM(false, cons(z0, z1), x2, plus(x2, z0))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(x0, x1) -> c10(PLUS(x1, head(x0))) SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1)))) SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1)))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(IFPLUS(x_1, x_2, x_3, x_4)) = x_2 POL(IFSUM(x_1, x_2, x_3, x_4)) = x_2 POL(LE(x_1, x_2)) = 0 POL(PLUS(x_1, x_2)) = 0 POL(PLUSITER(x_1, x_2, x_3)) = x_3 POL(SUMITER(x_1, x_2)) = x_1 POL(c1(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c10(x_1, x_2)) = x_1 + x_2 POL(c12(x_1)) = x_1 POL(c2(x_1)) = x_1 POL(c2(x_1, x_2)) = x_1 + x_2 POL(c4(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c9(x_1)) = x_1 POL(cons(x_1, x_2)) = [1] + x_2 POL(error) = 0 POL(false) = 0 POL(head(x_1)) = 0 POL(ifPlus(x_1, x_2, x_3, x_4)) = [1] + x_2 + x_3 + x_4 POL(isempty(x_1)) = [1] + x_1 POL(le(x_1, x_2)) = x_1 POL(nil) = 0 POL(plus(x_1, x_2)) = [1] + x_1 + x_2 POL(plusIter(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3 POL(s(x_1)) = 0 POL(tail(x_1)) = x_1 POL(true) = [1] ---------------------------------------- (148) Obligation: Complexity Dependency Tuples Problem Rules: le(s(z0), 0) -> false le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) isempty(nil) -> true isempty(cons(z0, z1)) -> false plus(z0, z1) -> plusIter(z0, z1, 0) head(nil) -> error head(cons(z0, z1)) -> z0 plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2) ifPlus(true, z0, z1, z2) -> z1 ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2)) tail(nil) -> nil tail(cons(z0, z1)) -> z1 Tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2))) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1)))), PLUS(x1, head(cons(z0, z1)))) SUMITER(nil, x1) -> c10(PLUS(x1, head(nil))) PLUSITER(s(s(z0)), x1, s(0)) -> c2(IFPLUS(false, s(s(z0)), x1, s(0)), LE(s(s(z0)), s(0))) PLUSITER(s(s(z0)), x1, s(s(z1))) -> c2(IFPLUS(le(z0, z1), s(s(z0)), x1, s(s(z1))), LE(s(s(z0)), s(s(z1)))) PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2))) PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0))) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0))) SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0))) SUMITER(cons(z0, z1), x2) -> c9(IFSUM(false, cons(z0, z1), x2, plus(x2, z0))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(x0, x1) -> c10(PLUS(x1, head(x0))) SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1)))) SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1)))) S tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2))) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1)))), PLUS(x1, head(cons(z0, z1)))) PLUSITER(s(s(z0)), x1, s(0)) -> c2(IFPLUS(false, s(s(z0)), x1, s(0)), LE(s(s(z0)), s(0))) PLUSITER(s(s(z0)), x1, s(s(z1))) -> c2(IFPLUS(le(z0, z1), s(s(z0)), x1, s(s(z1))), LE(s(s(z0)), s(s(z1)))) PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2))) PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0))) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0))) SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0))) SUMITER(cons(z0, z1), x2) -> c9(IFSUM(false, cons(z0, z1), x2, plus(x2, z0))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1)))) K tuples: SUMITER(nil, x1) -> c10(PLUS(x1, head(nil))) SUMITER(x0, x1) -> c10(PLUS(x1, head(x0))) SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1)))) Defined Rule Symbols: le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1 Defined Pair Symbols: PLUS_2, IFPLUS_4, LE_2, IFSUM_4, PLUSITER_3, SUMITER_2 Compound Symbols: c1_1, c4_1, c7_1, c12_1, c2_1, c10_2, c10_1, c2_2, c9_1 ---------------------------------------- (149) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1)))), PLUS(x1, head(cons(z0, z1)))) by SUMITER(cons(x0, x1), z0) -> c10(IFSUM(false, cons(x0, x1), z0, plusIter(z0, head(cons(x0, x1)), 0)), PLUS(z0, head(cons(x0, x1)))) SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1)))) ---------------------------------------- (150) Obligation: Complexity Dependency Tuples Problem Rules: le(s(z0), 0) -> false le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) isempty(nil) -> true isempty(cons(z0, z1)) -> false plus(z0, z1) -> plusIter(z0, z1, 0) head(nil) -> error head(cons(z0, z1)) -> z0 plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2) ifPlus(true, z0, z1, z2) -> z1 ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2)) tail(nil) -> nil tail(cons(z0, z1)) -> z1 Tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2))) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) SUMITER(nil, x1) -> c10(PLUS(x1, head(nil))) PLUSITER(s(s(z0)), x1, s(0)) -> c2(IFPLUS(false, s(s(z0)), x1, s(0)), LE(s(s(z0)), s(0))) PLUSITER(s(s(z0)), x1, s(s(z1))) -> c2(IFPLUS(le(z0, z1), s(s(z0)), x1, s(s(z1))), LE(s(s(z0)), s(s(z1)))) PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2))) PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0))) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0))) SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0))) SUMITER(cons(z0, z1), x2) -> c9(IFSUM(false, cons(z0, z1), x2, plus(x2, z0))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(x0, x1) -> c10(PLUS(x1, head(x0))) SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1)))) SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1)))) S tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2))) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) PLUSITER(s(s(z0)), x1, s(0)) -> c2(IFPLUS(false, s(s(z0)), x1, s(0)), LE(s(s(z0)), s(0))) PLUSITER(s(s(z0)), x1, s(s(z1))) -> c2(IFPLUS(le(z0, z1), s(s(z0)), x1, s(s(z1))), LE(s(s(z0)), s(s(z1)))) PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2))) PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0))) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0))) SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0))) SUMITER(cons(z0, z1), x2) -> c9(IFSUM(false, cons(z0, z1), x2, plus(x2, z0))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1)))) K tuples: SUMITER(nil, x1) -> c10(PLUS(x1, head(nil))) SUMITER(x0, x1) -> c10(PLUS(x1, head(x0))) SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1)))) Defined Rule Symbols: le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1 Defined Pair Symbols: PLUS_2, IFPLUS_4, LE_2, IFSUM_4, PLUSITER_3, SUMITER_2 Compound Symbols: c1_1, c4_1, c7_1, c12_1, c2_1, c10_1, c2_2, c9_1, c10_2 ---------------------------------------- (151) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace SUMITER(nil, x1) -> c10(PLUS(x1, head(nil))) by SUMITER(nil, x0) -> c10(PLUS(x0, error)) ---------------------------------------- (152) Obligation: Complexity Dependency Tuples Problem Rules: le(s(z0), 0) -> false le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) isempty(nil) -> true isempty(cons(z0, z1)) -> false plus(z0, z1) -> plusIter(z0, z1, 0) head(nil) -> error head(cons(z0, z1)) -> z0 plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2) ifPlus(true, z0, z1, z2) -> z1 ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2)) tail(nil) -> nil tail(cons(z0, z1)) -> z1 Tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2))) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) SUMITER(nil, x1) -> c10(PLUS(x1, head(nil))) PLUSITER(s(s(z0)), x1, s(0)) -> c2(IFPLUS(false, s(s(z0)), x1, s(0)), LE(s(s(z0)), s(0))) PLUSITER(s(s(z0)), x1, s(s(z1))) -> c2(IFPLUS(le(z0, z1), s(s(z0)), x1, s(s(z1))), LE(s(s(z0)), s(s(z1)))) PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2))) PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0))) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0))) SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0))) SUMITER(cons(z0, z1), x2) -> c9(IFSUM(false, cons(z0, z1), x2, plus(x2, z0))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(x0, x1) -> c10(PLUS(x1, head(x0))) SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1)))) SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1)))) SUMITER(nil, x0) -> c10(PLUS(x0, error)) S tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2))) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) PLUSITER(s(s(z0)), x1, s(0)) -> c2(IFPLUS(false, s(s(z0)), x1, s(0)), LE(s(s(z0)), s(0))) PLUSITER(s(s(z0)), x1, s(s(z1))) -> c2(IFPLUS(le(z0, z1), s(s(z0)), x1, s(s(z1))), LE(s(s(z0)), s(s(z1)))) PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2))) PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0))) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0))) SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0))) SUMITER(cons(z0, z1), x2) -> c9(IFSUM(false, cons(z0, z1), x2, plus(x2, z0))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1)))) K tuples: SUMITER(nil, x1) -> c10(PLUS(x1, head(nil))) SUMITER(x0, x1) -> c10(PLUS(x1, head(x0))) SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1)))) Defined Rule Symbols: le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1 Defined Pair Symbols: PLUS_2, IFPLUS_4, LE_2, IFSUM_4, PLUSITER_3, SUMITER_2 Compound Symbols: c1_1, c4_1, c7_1, c12_1, c2_1, c10_1, c2_2, c9_1, c10_2 ---------------------------------------- (153) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0))) by SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0))) ---------------------------------------- (154) Obligation: Complexity Dependency Tuples Problem Rules: le(s(z0), 0) -> false le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) isempty(nil) -> true isempty(cons(z0, z1)) -> false plus(z0, z1) -> plusIter(z0, z1, 0) head(nil) -> error head(cons(z0, z1)) -> z0 plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2) ifPlus(true, z0, z1, z2) -> z1 ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2)) tail(nil) -> nil tail(cons(z0, z1)) -> z1 Tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2))) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) SUMITER(nil, x1) -> c10(PLUS(x1, head(nil))) PLUSITER(s(s(z0)), x1, s(0)) -> c2(IFPLUS(false, s(s(z0)), x1, s(0)), LE(s(s(z0)), s(0))) PLUSITER(s(s(z0)), x1, s(s(z1))) -> c2(IFPLUS(le(z0, z1), s(s(z0)), x1, s(s(z1))), LE(s(s(z0)), s(s(z1)))) PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2))) PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0))) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0))) SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0))) SUMITER(cons(z0, z1), x2) -> c9(IFSUM(false, cons(z0, z1), x2, plus(x2, z0))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(x0, x1) -> c10(PLUS(x1, head(x0))) SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1)))) SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1)))) SUMITER(nil, x0) -> c10(PLUS(x0, error)) SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0))) S tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2))) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) PLUSITER(s(s(z0)), x1, s(0)) -> c2(IFPLUS(false, s(s(z0)), x1, s(0)), LE(s(s(z0)), s(0))) PLUSITER(s(s(z0)), x1, s(s(z1))) -> c2(IFPLUS(le(z0, z1), s(s(z0)), x1, s(s(z1))), LE(s(s(z0)), s(s(z1)))) PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2))) PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0))) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0))) SUMITER(cons(z0, z1), x2) -> c9(IFSUM(false, cons(z0, z1), x2, plus(x2, z0))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1)))) SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0))) K tuples: SUMITER(nil, x1) -> c10(PLUS(x1, head(nil))) SUMITER(x0, x1) -> c10(PLUS(x1, head(x0))) SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1)))) Defined Rule Symbols: le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1 Defined Pair Symbols: PLUS_2, IFPLUS_4, LE_2, IFSUM_4, PLUSITER_3, SUMITER_2 Compound Symbols: c1_1, c4_1, c7_1, c12_1, c2_1, c10_1, c2_2, c9_1, c10_2 ---------------------------------------- (155) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2))) by IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0))) IFPLUS(false, s(s(x0)), x1, s(0)) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(0)))) IFPLUS(false, s(s(x0)), x1, s(s(x2))) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(s(x2))))) ---------------------------------------- (156) Obligation: Complexity Dependency Tuples Problem Rules: le(s(z0), 0) -> false le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) isempty(nil) -> true isempty(cons(z0, z1)) -> false plus(z0, z1) -> plusIter(z0, z1, 0) head(nil) -> error head(cons(z0, z1)) -> z0 plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2) ifPlus(true, z0, z1, z2) -> z1 ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2)) tail(nil) -> nil tail(cons(z0, z1)) -> z1 Tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) SUMITER(nil, x1) -> c10(PLUS(x1, head(nil))) PLUSITER(s(s(z0)), x1, s(0)) -> c2(IFPLUS(false, s(s(z0)), x1, s(0)), LE(s(s(z0)), s(0))) PLUSITER(s(s(z0)), x1, s(s(z1))) -> c2(IFPLUS(le(z0, z1), s(s(z0)), x1, s(s(z1))), LE(s(s(z0)), s(s(z1)))) PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2))) PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0))) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0))) SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0))) SUMITER(cons(z0, z1), x2) -> c9(IFSUM(false, cons(z0, z1), x2, plus(x2, z0))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(x0, x1) -> c10(PLUS(x1, head(x0))) SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1)))) SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1)))) SUMITER(nil, x0) -> c10(PLUS(x0, error)) SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0))) IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0))) IFPLUS(false, s(s(x0)), x1, s(0)) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(0)))) IFPLUS(false, s(s(x0)), x1, s(s(x2))) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(s(x2))))) S tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) PLUSITER(s(s(z0)), x1, s(0)) -> c2(IFPLUS(false, s(s(z0)), x1, s(0)), LE(s(s(z0)), s(0))) PLUSITER(s(s(z0)), x1, s(s(z1))) -> c2(IFPLUS(le(z0, z1), s(s(z0)), x1, s(s(z1))), LE(s(s(z0)), s(s(z1)))) PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2))) PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0))) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0))) SUMITER(cons(z0, z1), x2) -> c9(IFSUM(false, cons(z0, z1), x2, plus(x2, z0))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1)))) SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0))) IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0))) IFPLUS(false, s(s(x0)), x1, s(0)) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(0)))) IFPLUS(false, s(s(x0)), x1, s(s(x2))) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(s(x2))))) K tuples: SUMITER(nil, x1) -> c10(PLUS(x1, head(nil))) SUMITER(x0, x1) -> c10(PLUS(x1, head(x0))) SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1)))) Defined Rule Symbols: le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1 Defined Pair Symbols: PLUS_2, LE_2, IFSUM_4, PLUSITER_3, SUMITER_2, IFPLUS_4 Compound Symbols: c1_1, c7_1, c12_1, c2_1, c10_1, c2_2, c9_1, c10_2, c4_1 ---------------------------------------- (157) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (158) Obligation: Complexity Dependency Tuples Problem Rules: le(s(z0), 0) -> false le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) isempty(nil) -> true isempty(cons(z0, z1)) -> false plus(z0, z1) -> plusIter(z0, z1, 0) head(nil) -> error head(cons(z0, z1)) -> z0 plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2) ifPlus(true, z0, z1, z2) -> z1 ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2)) tail(nil) -> nil tail(cons(z0, z1)) -> z1 Tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) SUMITER(nil, x1) -> c10(PLUS(x1, head(nil))) PLUSITER(s(s(z0)), x1, s(s(z1))) -> c2(IFPLUS(le(z0, z1), s(s(z0)), x1, s(s(z1))), LE(s(s(z0)), s(s(z1)))) PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2))) PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0))) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0))) SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0))) SUMITER(cons(z0, z1), x2) -> c9(IFSUM(false, cons(z0, z1), x2, plus(x2, z0))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(x0, x1) -> c10(PLUS(x1, head(x0))) SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1)))) SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1)))) SUMITER(nil, x0) -> c10(PLUS(x0, error)) SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0))) IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0))) IFPLUS(false, s(s(x0)), x1, s(0)) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(0)))) IFPLUS(false, s(s(x0)), x1, s(s(x2))) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(s(x2))))) PLUSITER(s(s(z0)), x1, s(0)) -> c(IFPLUS(false, s(s(z0)), x1, s(0))) PLUSITER(s(s(z0)), x1, s(0)) -> c(LE(s(s(z0)), s(0))) S tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) PLUSITER(s(s(z0)), x1, s(s(z1))) -> c2(IFPLUS(le(z0, z1), s(s(z0)), x1, s(s(z1))), LE(s(s(z0)), s(s(z1)))) PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2))) PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0))) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0))) SUMITER(cons(z0, z1), x2) -> c9(IFSUM(false, cons(z0, z1), x2, plus(x2, z0))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1)))) SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0))) IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0))) IFPLUS(false, s(s(x0)), x1, s(0)) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(0)))) IFPLUS(false, s(s(x0)), x1, s(s(x2))) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(s(x2))))) PLUSITER(s(s(z0)), x1, s(0)) -> c(IFPLUS(false, s(s(z0)), x1, s(0))) PLUSITER(s(s(z0)), x1, s(0)) -> c(LE(s(s(z0)), s(0))) K tuples: SUMITER(nil, x1) -> c10(PLUS(x1, head(nil))) SUMITER(x0, x1) -> c10(PLUS(x1, head(x0))) SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1)))) Defined Rule Symbols: le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1 Defined Pair Symbols: PLUS_2, LE_2, IFSUM_4, PLUSITER_3, SUMITER_2, IFPLUS_4 Compound Symbols: c1_1, c7_1, c12_1, c2_1, c10_1, c2_2, c9_1, c10_2, c4_1, c_1 ---------------------------------------- (159) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0))) by SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0))) ---------------------------------------- (160) Obligation: Complexity Dependency Tuples Problem Rules: le(s(z0), 0) -> false le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) isempty(nil) -> true isempty(cons(z0, z1)) -> false plus(z0, z1) -> plusIter(z0, z1, 0) head(nil) -> error head(cons(z0, z1)) -> z0 plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2) ifPlus(true, z0, z1, z2) -> z1 ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2)) tail(nil) -> nil tail(cons(z0, z1)) -> z1 Tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) SUMITER(nil, x1) -> c10(PLUS(x1, head(nil))) PLUSITER(s(s(z0)), x1, s(s(z1))) -> c2(IFPLUS(le(z0, z1), s(s(z0)), x1, s(s(z1))), LE(s(s(z0)), s(s(z1)))) PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2))) PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0))) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0))) SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0))) SUMITER(cons(z0, z1), x2) -> c9(IFSUM(false, cons(z0, z1), x2, plus(x2, z0))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(x0, x1) -> c10(PLUS(x1, head(x0))) SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1)))) SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1)))) SUMITER(nil, x0) -> c10(PLUS(x0, error)) SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0))) IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0))) IFPLUS(false, s(s(x0)), x1, s(0)) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(0)))) IFPLUS(false, s(s(x0)), x1, s(s(x2))) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(s(x2))))) PLUSITER(s(s(z0)), x1, s(0)) -> c(IFPLUS(false, s(s(z0)), x1, s(0))) PLUSITER(s(s(z0)), x1, s(0)) -> c(LE(s(s(z0)), s(0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0))) S tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) PLUSITER(s(s(z0)), x1, s(s(z1))) -> c2(IFPLUS(le(z0, z1), s(s(z0)), x1, s(s(z1))), LE(s(s(z0)), s(s(z1)))) PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2))) PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0))) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0))) SUMITER(cons(z0, z1), x2) -> c9(IFSUM(false, cons(z0, z1), x2, plus(x2, z0))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1)))) SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0))) IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0))) IFPLUS(false, s(s(x0)), x1, s(0)) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(0)))) IFPLUS(false, s(s(x0)), x1, s(s(x2))) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(s(x2))))) PLUSITER(s(s(z0)), x1, s(0)) -> c(IFPLUS(false, s(s(z0)), x1, s(0))) PLUSITER(s(s(z0)), x1, s(0)) -> c(LE(s(s(z0)), s(0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0))) K tuples: SUMITER(nil, x1) -> c10(PLUS(x1, head(nil))) SUMITER(x0, x1) -> c10(PLUS(x1, head(x0))) SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1)))) Defined Rule Symbols: le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1 Defined Pair Symbols: PLUS_2, LE_2, IFSUM_4, PLUSITER_3, SUMITER_2, IFPLUS_4 Compound Symbols: c1_1, c7_1, c12_1, c2_1, c10_1, c2_2, c9_1, c10_2, c4_1, c_1 ---------------------------------------- (161) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace PLUSITER(s(s(z0)), x1, s(s(z1))) -> c2(IFPLUS(le(z0, z1), s(s(z0)), x1, s(s(z1))), LE(s(s(z0)), s(s(z1)))) by PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c2(IFPLUS(le(x0, 0), s(s(x0)), s(x1), s(s(0))), LE(s(s(x0)), s(s(0)))) PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(IFPLUS(le(x0, s(x2)), s(s(x0)), s(x1), s(s(s(x2)))), LE(s(s(x0)), s(s(s(x2))))) ---------------------------------------- (162) Obligation: Complexity Dependency Tuples Problem Rules: le(s(z0), 0) -> false le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) isempty(nil) -> true isempty(cons(z0, z1)) -> false plus(z0, z1) -> plusIter(z0, z1, 0) head(nil) -> error head(cons(z0, z1)) -> z0 plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2) ifPlus(true, z0, z1, z2) -> z1 ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2)) tail(nil) -> nil tail(cons(z0, z1)) -> z1 Tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) SUMITER(nil, x1) -> c10(PLUS(x1, head(nil))) PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2))) PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0))) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0))) SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0))) SUMITER(cons(z0, z1), x2) -> c9(IFSUM(false, cons(z0, z1), x2, plus(x2, z0))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(x0, x1) -> c10(PLUS(x1, head(x0))) SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1)))) SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1)))) SUMITER(nil, x0) -> c10(PLUS(x0, error)) SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0))) IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0))) IFPLUS(false, s(s(x0)), x1, s(0)) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(0)))) IFPLUS(false, s(s(x0)), x1, s(s(x2))) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(s(x2))))) PLUSITER(s(s(z0)), x1, s(0)) -> c(IFPLUS(false, s(s(z0)), x1, s(0))) PLUSITER(s(s(z0)), x1, s(0)) -> c(LE(s(s(z0)), s(0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c2(IFPLUS(le(x0, 0), s(s(x0)), s(x1), s(s(0))), LE(s(s(x0)), s(s(0)))) PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(IFPLUS(le(x0, s(x2)), s(s(x0)), s(x1), s(s(s(x2)))), LE(s(s(x0)), s(s(s(x2))))) S tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2))) PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0))) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0))) SUMITER(cons(z0, z1), x2) -> c9(IFSUM(false, cons(z0, z1), x2, plus(x2, z0))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1)))) SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0))) IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0))) IFPLUS(false, s(s(x0)), x1, s(0)) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(0)))) IFPLUS(false, s(s(x0)), x1, s(s(x2))) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(s(x2))))) PLUSITER(s(s(z0)), x1, s(0)) -> c(IFPLUS(false, s(s(z0)), x1, s(0))) PLUSITER(s(s(z0)), x1, s(0)) -> c(LE(s(s(z0)), s(0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c2(IFPLUS(le(x0, 0), s(s(x0)), s(x1), s(s(0))), LE(s(s(x0)), s(s(0)))) PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(IFPLUS(le(x0, s(x2)), s(s(x0)), s(x1), s(s(s(x2)))), LE(s(s(x0)), s(s(s(x2))))) K tuples: SUMITER(nil, x1) -> c10(PLUS(x1, head(nil))) SUMITER(x0, x1) -> c10(PLUS(x1, head(x0))) SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1)))) Defined Rule Symbols: le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1 Defined Pair Symbols: PLUS_2, LE_2, IFSUM_4, PLUSITER_3, SUMITER_2, IFPLUS_4 Compound Symbols: c1_1, c7_1, c12_1, c2_1, c10_1, c9_1, c10_2, c4_1, c_1, c2_2 ---------------------------------------- (163) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (164) Obligation: Complexity Dependency Tuples Problem Rules: le(s(z0), 0) -> false le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) isempty(nil) -> true isempty(cons(z0, z1)) -> false plus(z0, z1) -> plusIter(z0, z1, 0) head(nil) -> error head(cons(z0, z1)) -> z0 plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2) ifPlus(true, z0, z1, z2) -> z1 ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2)) tail(nil) -> nil tail(cons(z0, z1)) -> z1 Tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) SUMITER(nil, x1) -> c10(PLUS(x1, head(nil))) PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2))) PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0))) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0))) SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0))) SUMITER(cons(z0, z1), x2) -> c9(IFSUM(false, cons(z0, z1), x2, plus(x2, z0))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(x0, x1) -> c10(PLUS(x1, head(x0))) SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1)))) SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1)))) SUMITER(nil, x0) -> c10(PLUS(x0, error)) SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0))) IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0))) IFPLUS(false, s(s(x0)), x1, s(0)) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(0)))) IFPLUS(false, s(s(x0)), x1, s(s(x2))) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(s(x2))))) PLUSITER(s(s(z0)), x1, s(0)) -> c(IFPLUS(false, s(s(z0)), x1, s(0))) PLUSITER(s(s(z0)), x1, s(0)) -> c(LE(s(s(z0)), s(0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0))) PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(IFPLUS(le(x0, s(x2)), s(s(x0)), s(x1), s(s(s(x2)))), LE(s(s(x0)), s(s(s(x2))))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(x1), s(s(0)))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(LE(s(s(x0)), s(s(0)))) S tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2))) PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0))) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0))) SUMITER(cons(z0, z1), x2) -> c9(IFSUM(false, cons(z0, z1), x2, plus(x2, z0))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1)))) SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0))) IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0))) IFPLUS(false, s(s(x0)), x1, s(0)) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(0)))) IFPLUS(false, s(s(x0)), x1, s(s(x2))) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(s(x2))))) PLUSITER(s(s(z0)), x1, s(0)) -> c(IFPLUS(false, s(s(z0)), x1, s(0))) PLUSITER(s(s(z0)), x1, s(0)) -> c(LE(s(s(z0)), s(0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0))) PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(IFPLUS(le(x0, s(x2)), s(s(x0)), s(x1), s(s(s(x2)))), LE(s(s(x0)), s(s(s(x2))))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(x1), s(s(0)))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(LE(s(s(x0)), s(s(0)))) K tuples: SUMITER(nil, x1) -> c10(PLUS(x1, head(nil))) SUMITER(x0, x1) -> c10(PLUS(x1, head(x0))) SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1)))) Defined Rule Symbols: le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1 Defined Pair Symbols: PLUS_2, LE_2, IFSUM_4, PLUSITER_3, SUMITER_2, IFPLUS_4 Compound Symbols: c1_1, c7_1, c12_1, c2_1, c10_1, c9_1, c10_2, c4_1, c_1, c2_2, c3_1 ---------------------------------------- (165) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0))) by SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0))) ---------------------------------------- (166) Obligation: Complexity Dependency Tuples Problem Rules: le(s(z0), 0) -> false le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) isempty(nil) -> true isempty(cons(z0, z1)) -> false plus(z0, z1) -> plusIter(z0, z1, 0) head(nil) -> error head(cons(z0, z1)) -> z0 plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2) ifPlus(true, z0, z1, z2) -> z1 ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2)) tail(nil) -> nil tail(cons(z0, z1)) -> z1 Tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) SUMITER(nil, x1) -> c10(PLUS(x1, head(nil))) PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2))) PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0))) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0))) SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0))) SUMITER(cons(z0, z1), x2) -> c9(IFSUM(false, cons(z0, z1), x2, plus(x2, z0))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(x0, x1) -> c10(PLUS(x1, head(x0))) SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1)))) SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1)))) SUMITER(nil, x0) -> c10(PLUS(x0, error)) SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0))) IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0))) IFPLUS(false, s(s(x0)), x1, s(0)) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(0)))) IFPLUS(false, s(s(x0)), x1, s(s(x2))) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(s(x2))))) PLUSITER(s(s(z0)), x1, s(0)) -> c(IFPLUS(false, s(s(z0)), x1, s(0))) PLUSITER(s(s(z0)), x1, s(0)) -> c(LE(s(s(z0)), s(0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0))) PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(IFPLUS(le(x0, s(x2)), s(s(x0)), s(x1), s(s(s(x2)))), LE(s(s(x0)), s(s(s(x2))))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(x1), s(s(0)))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(LE(s(s(x0)), s(s(0)))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0))) S tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2))) PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0))) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0))) SUMITER(cons(z0, z1), x2) -> c9(IFSUM(false, cons(z0, z1), x2, plus(x2, z0))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1)))) SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0))) IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0))) IFPLUS(false, s(s(x0)), x1, s(0)) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(0)))) IFPLUS(false, s(s(x0)), x1, s(s(x2))) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(s(x2))))) PLUSITER(s(s(z0)), x1, s(0)) -> c(IFPLUS(false, s(s(z0)), x1, s(0))) PLUSITER(s(s(z0)), x1, s(0)) -> c(LE(s(s(z0)), s(0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0))) PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(IFPLUS(le(x0, s(x2)), s(s(x0)), s(x1), s(s(s(x2)))), LE(s(s(x0)), s(s(s(x2))))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(x1), s(s(0)))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(LE(s(s(x0)), s(s(0)))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0))) K tuples: SUMITER(nil, x1) -> c10(PLUS(x1, head(nil))) SUMITER(x0, x1) -> c10(PLUS(x1, head(x0))) SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1)))) Defined Rule Symbols: le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1 Defined Pair Symbols: PLUS_2, LE_2, IFSUM_4, PLUSITER_3, SUMITER_2, IFPLUS_4 Compound Symbols: c1_1, c7_1, c12_1, c2_1, c10_1, c9_1, c10_2, c4_1, c_1, c2_2, c3_1 ---------------------------------------- (167) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, plusIter(z0, error, 0))) by SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0))) ---------------------------------------- (168) Obligation: Complexity Dependency Tuples Problem Rules: le(s(z0), 0) -> false le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) isempty(nil) -> true isempty(cons(z0, z1)) -> false plus(z0, z1) -> plusIter(z0, z1, 0) head(nil) -> error head(cons(z0, z1)) -> z0 plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2) ifPlus(true, z0, z1, z2) -> z1 ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2)) tail(nil) -> nil tail(cons(z0, z1)) -> z1 Tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) SUMITER(nil, x1) -> c10(PLUS(x1, head(nil))) PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2))) PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0))) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0))) SUMITER(cons(z0, z1), x2) -> c9(IFSUM(false, cons(z0, z1), x2, plus(x2, z0))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(x0, x1) -> c10(PLUS(x1, head(x0))) SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1)))) SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1)))) SUMITER(nil, x0) -> c10(PLUS(x0, error)) SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0))) IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0))) IFPLUS(false, s(s(x0)), x1, s(0)) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(0)))) IFPLUS(false, s(s(x0)), x1, s(s(x2))) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(s(x2))))) PLUSITER(s(s(z0)), x1, s(0)) -> c(IFPLUS(false, s(s(z0)), x1, s(0))) PLUSITER(s(s(z0)), x1, s(0)) -> c(LE(s(s(z0)), s(0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0))) PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(IFPLUS(le(x0, s(x2)), s(s(x0)), s(x1), s(s(s(x2)))), LE(s(s(x0)), s(s(s(x2))))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(x1), s(s(0)))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(LE(s(s(x0)), s(s(0)))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0))) S tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2))) PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0))) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0))) SUMITER(cons(z0, z1), x2) -> c9(IFSUM(false, cons(z0, z1), x2, plus(x2, z0))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1)))) SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0))) IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0))) IFPLUS(false, s(s(x0)), x1, s(0)) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(0)))) IFPLUS(false, s(s(x0)), x1, s(s(x2))) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(s(x2))))) PLUSITER(s(s(z0)), x1, s(0)) -> c(IFPLUS(false, s(s(z0)), x1, s(0))) PLUSITER(s(s(z0)), x1, s(0)) -> c(LE(s(s(z0)), s(0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0))) PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(IFPLUS(le(x0, s(x2)), s(s(x0)), s(x1), s(s(s(x2)))), LE(s(s(x0)), s(s(s(x2))))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(x1), s(s(0)))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(LE(s(s(x0)), s(s(0)))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0))) K tuples: SUMITER(nil, x1) -> c10(PLUS(x1, head(nil))) SUMITER(x0, x1) -> c10(PLUS(x1, head(x0))) SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1)))) Defined Rule Symbols: le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1 Defined Pair Symbols: PLUS_2, LE_2, IFSUM_4, PLUSITER_3, SUMITER_2, IFPLUS_4 Compound Symbols: c1_1, c7_1, c12_1, c2_1, c10_1, c9_1, c10_2, c4_1, c_1, c2_2, c3_1 ---------------------------------------- (169) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace SUMITER(cons(z1, x1), z0) -> c9(IFSUM(isempty(cons(z1, x1)), cons(z1, x1), z0, plusIter(z0, z1, 0))) by SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0))) ---------------------------------------- (170) Obligation: Complexity Dependency Tuples Problem Rules: le(s(z0), 0) -> false le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) isempty(nil) -> true isempty(cons(z0, z1)) -> false plus(z0, z1) -> plusIter(z0, z1, 0) head(nil) -> error head(cons(z0, z1)) -> z0 plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2) ifPlus(true, z0, z1, z2) -> z1 ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2)) tail(nil) -> nil tail(cons(z0, z1)) -> z1 Tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) SUMITER(nil, x1) -> c10(PLUS(x1, head(nil))) PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2))) PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0))) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0))) SUMITER(cons(z0, z1), x2) -> c9(IFSUM(false, cons(z0, z1), x2, plus(x2, z0))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(x0, x1) -> c10(PLUS(x1, head(x0))) SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1)))) SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1)))) SUMITER(nil, x0) -> c10(PLUS(x0, error)) SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0))) IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0))) IFPLUS(false, s(s(x0)), x1, s(0)) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(0)))) IFPLUS(false, s(s(x0)), x1, s(s(x2))) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(s(x2))))) PLUSITER(s(s(z0)), x1, s(0)) -> c(IFPLUS(false, s(s(z0)), x1, s(0))) PLUSITER(s(s(z0)), x1, s(0)) -> c(LE(s(s(z0)), s(0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0))) PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(IFPLUS(le(x0, s(x2)), s(s(x0)), s(x1), s(s(s(x2)))), LE(s(s(x0)), s(s(s(x2))))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(x1), s(s(0)))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(LE(s(s(x0)), s(s(0)))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0))) S tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2))) PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0))) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0))) SUMITER(cons(z0, z1), x2) -> c9(IFSUM(false, cons(z0, z1), x2, plus(x2, z0))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1)))) SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0))) IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0))) IFPLUS(false, s(s(x0)), x1, s(0)) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(0)))) IFPLUS(false, s(s(x0)), x1, s(s(x2))) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(s(x2))))) PLUSITER(s(s(z0)), x1, s(0)) -> c(IFPLUS(false, s(s(z0)), x1, s(0))) PLUSITER(s(s(z0)), x1, s(0)) -> c(LE(s(s(z0)), s(0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0))) PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(IFPLUS(le(x0, s(x2)), s(s(x0)), s(x1), s(s(s(x2)))), LE(s(s(x0)), s(s(s(x2))))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(x1), s(s(0)))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(LE(s(s(x0)), s(s(0)))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0))) K tuples: SUMITER(nil, x1) -> c10(PLUS(x1, head(nil))) SUMITER(x0, x1) -> c10(PLUS(x1, head(x0))) SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1)))) Defined Rule Symbols: le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1 Defined Pair Symbols: PLUS_2, LE_2, IFSUM_4, PLUSITER_3, SUMITER_2, IFPLUS_4 Compound Symbols: c1_1, c7_1, c12_1, c2_1, c10_1, c9_1, c10_2, c4_1, c_1, c2_2, c3_1 ---------------------------------------- (171) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace SUMITER(cons(z0, z1), x2) -> c9(IFSUM(false, cons(z0, z1), x2, plus(x2, z0))) by SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0))) ---------------------------------------- (172) Obligation: Complexity Dependency Tuples Problem Rules: le(s(z0), 0) -> false le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) isempty(nil) -> true isempty(cons(z0, z1)) -> false plus(z0, z1) -> plusIter(z0, z1, 0) head(nil) -> error head(cons(z0, z1)) -> z0 plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2) ifPlus(true, z0, z1, z2) -> z1 ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2)) tail(nil) -> nil tail(cons(z0, z1)) -> z1 Tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) SUMITER(nil, x1) -> c10(PLUS(x1, head(nil))) PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2))) PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0))) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0))) SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0))) SUMITER(cons(z0, z1), x2) -> c9(IFSUM(false, cons(z0, z1), x2, plus(x2, z0))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(x0, x1) -> c10(PLUS(x1, head(x0))) SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1)))) SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1)))) SUMITER(nil, x0) -> c10(PLUS(x0, error)) SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0))) IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0))) IFPLUS(false, s(s(x0)), x1, s(0)) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(0)))) IFPLUS(false, s(s(x0)), x1, s(s(x2))) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(s(x2))))) PLUSITER(s(s(z0)), x1, s(0)) -> c(IFPLUS(false, s(s(z0)), x1, s(0))) PLUSITER(s(s(z0)), x1, s(0)) -> c(LE(s(s(z0)), s(0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0))) PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(IFPLUS(le(x0, s(x2)), s(s(x0)), s(x1), s(s(s(x2)))), LE(s(s(x0)), s(s(s(x2))))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(x1), s(s(0)))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(LE(s(s(x0)), s(s(0)))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0))) S tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2))) PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0))) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1)))) SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0))) IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0))) IFPLUS(false, s(s(x0)), x1, s(0)) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(0)))) IFPLUS(false, s(s(x0)), x1, s(s(x2))) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(s(x2))))) PLUSITER(s(s(z0)), x1, s(0)) -> c(IFPLUS(false, s(s(z0)), x1, s(0))) PLUSITER(s(s(z0)), x1, s(0)) -> c(LE(s(s(z0)), s(0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0))) PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(IFPLUS(le(x0, s(x2)), s(s(x0)), s(x1), s(s(s(x2)))), LE(s(s(x0)), s(s(s(x2))))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(x1), s(s(0)))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(LE(s(s(x0)), s(s(0)))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0))) K tuples: SUMITER(nil, x1) -> c10(PLUS(x1, head(nil))) SUMITER(x0, x1) -> c10(PLUS(x1, head(x0))) SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1)))) Defined Rule Symbols: le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1 Defined Pair Symbols: PLUS_2, LE_2, IFSUM_4, PLUSITER_3, SUMITER_2, IFPLUS_4 Compound Symbols: c1_1, c7_1, c12_1, c2_1, c10_1, c9_1, c10_2, c4_1, c_1, c2_2, c3_1 ---------------------------------------- (173) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace SUMITER(cons(x0, x1), z0) -> c9(IFSUM(false, cons(x0, x1), z0, plusIter(z0, head(cons(x0, x1)), 0))) by SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0))) ---------------------------------------- (174) Obligation: Complexity Dependency Tuples Problem Rules: le(s(z0), 0) -> false le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) isempty(nil) -> true isempty(cons(z0, z1)) -> false plus(z0, z1) -> plusIter(z0, z1, 0) head(nil) -> error head(cons(z0, z1)) -> z0 plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2) ifPlus(true, z0, z1, z2) -> z1 ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2)) tail(nil) -> nil tail(cons(z0, z1)) -> z1 Tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) SUMITER(nil, x1) -> c10(PLUS(x1, head(nil))) PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2))) PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0))) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0))) SUMITER(cons(z0, z1), x2) -> c9(IFSUM(false, cons(z0, z1), x2, plus(x2, z0))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(x0, x1) -> c10(PLUS(x1, head(x0))) SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1)))) SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1)))) SUMITER(nil, x0) -> c10(PLUS(x0, error)) SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0))) IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0))) IFPLUS(false, s(s(x0)), x1, s(0)) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(0)))) IFPLUS(false, s(s(x0)), x1, s(s(x2))) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(s(x2))))) PLUSITER(s(s(z0)), x1, s(0)) -> c(IFPLUS(false, s(s(z0)), x1, s(0))) PLUSITER(s(s(z0)), x1, s(0)) -> c(LE(s(s(z0)), s(0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0))) PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(IFPLUS(le(x0, s(x2)), s(s(x0)), s(x1), s(s(s(x2)))), LE(s(s(x0)), s(s(s(x2))))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(x1), s(s(0)))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(LE(s(s(x0)), s(s(0)))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0))) S tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2))) PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0))) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1)))) SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0))) IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0))) IFPLUS(false, s(s(x0)), x1, s(0)) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(0)))) IFPLUS(false, s(s(x0)), x1, s(s(x2))) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(s(x2))))) PLUSITER(s(s(z0)), x1, s(0)) -> c(IFPLUS(false, s(s(z0)), x1, s(0))) PLUSITER(s(s(z0)), x1, s(0)) -> c(LE(s(s(z0)), s(0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0))) PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(IFPLUS(le(x0, s(x2)), s(s(x0)), s(x1), s(s(s(x2)))), LE(s(s(x0)), s(s(s(x2))))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(x1), s(s(0)))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(LE(s(s(x0)), s(s(0)))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0))) K tuples: SUMITER(nil, x1) -> c10(PLUS(x1, head(nil))) SUMITER(x0, x1) -> c10(PLUS(x1, head(x0))) SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1)))) Defined Rule Symbols: le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1 Defined Pair Symbols: PLUS_2, LE_2, IFSUM_4, PLUSITER_3, SUMITER_2, IFPLUS_4 Compound Symbols: c1_1, c7_1, c12_1, c2_1, c10_1, c9_1, c10_2, c4_1, c_1, c2_2, c3_1 ---------------------------------------- (175) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2))) by PLUSITER(s(x0), s(x1), s(0)) -> c2(LE(s(x0), s(0))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c2(LE(s(s(x0)), s(s(0)))) PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(LE(s(s(x0)), s(s(s(x2))))) ---------------------------------------- (176) Obligation: Complexity Dependency Tuples Problem Rules: le(s(z0), 0) -> false le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) isempty(nil) -> true isempty(cons(z0, z1)) -> false plus(z0, z1) -> plusIter(z0, z1, 0) head(nil) -> error head(cons(z0, z1)) -> z0 plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2) ifPlus(true, z0, z1, z2) -> z1 ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2)) tail(nil) -> nil tail(cons(z0, z1)) -> z1 Tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) SUMITER(nil, x1) -> c10(PLUS(x1, head(nil))) PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0))) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0))) SUMITER(cons(z0, z1), x2) -> c9(IFSUM(false, cons(z0, z1), x2, plus(x2, z0))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(x0, x1) -> c10(PLUS(x1, head(x0))) SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1)))) SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1)))) SUMITER(nil, x0) -> c10(PLUS(x0, error)) SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0))) IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0))) IFPLUS(false, s(s(x0)), x1, s(0)) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(0)))) IFPLUS(false, s(s(x0)), x1, s(s(x2))) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(s(x2))))) PLUSITER(s(s(z0)), x1, s(0)) -> c(IFPLUS(false, s(s(z0)), x1, s(0))) PLUSITER(s(s(z0)), x1, s(0)) -> c(LE(s(s(z0)), s(0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0))) PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(IFPLUS(le(x0, s(x2)), s(s(x0)), s(x1), s(s(s(x2)))), LE(s(s(x0)), s(s(s(x2))))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(x1), s(s(0)))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(LE(s(s(x0)), s(s(0)))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0))) PLUSITER(s(x0), s(x1), s(0)) -> c2(LE(s(x0), s(0))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c2(LE(s(s(x0)), s(s(0)))) PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(LE(s(s(x0)), s(s(s(x2))))) S tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0))) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1)))) SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0))) IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0))) IFPLUS(false, s(s(x0)), x1, s(0)) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(0)))) IFPLUS(false, s(s(x0)), x1, s(s(x2))) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(s(x2))))) PLUSITER(s(s(z0)), x1, s(0)) -> c(IFPLUS(false, s(s(z0)), x1, s(0))) PLUSITER(s(s(z0)), x1, s(0)) -> c(LE(s(s(z0)), s(0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0))) PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(IFPLUS(le(x0, s(x2)), s(s(x0)), s(x1), s(s(s(x2)))), LE(s(s(x0)), s(s(s(x2))))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(x1), s(s(0)))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(LE(s(s(x0)), s(s(0)))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0))) PLUSITER(s(x0), s(x1), s(0)) -> c2(LE(s(x0), s(0))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c2(LE(s(s(x0)), s(s(0)))) PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(LE(s(s(x0)), s(s(s(x2))))) K tuples: SUMITER(nil, x1) -> c10(PLUS(x1, head(nil))) SUMITER(x0, x1) -> c10(PLUS(x1, head(x0))) SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1)))) Defined Rule Symbols: le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1 Defined Pair Symbols: PLUS_2, LE_2, IFSUM_4, PLUSITER_3, SUMITER_2, IFPLUS_4 Compound Symbols: c1_1, c7_1, c12_1, c2_1, c10_1, c9_1, c10_2, c4_1, c_1, c2_2, c3_1 ---------------------------------------- (177) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace SUMITER(cons(z0, z1), x2) -> c9(IFSUM(false, cons(z0, z1), x2, plus(x2, z0))) by SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0))) ---------------------------------------- (178) Obligation: Complexity Dependency Tuples Problem Rules: le(s(z0), 0) -> false le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) isempty(nil) -> true isempty(cons(z0, z1)) -> false plus(z0, z1) -> plusIter(z0, z1, 0) head(nil) -> error head(cons(z0, z1)) -> z0 plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2) ifPlus(true, z0, z1, z2) -> z1 ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2)) tail(nil) -> nil tail(cons(z0, z1)) -> z1 Tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) SUMITER(nil, x1) -> c10(PLUS(x1, head(nil))) PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0))) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(x0, x1) -> c10(PLUS(x1, head(x0))) SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1)))) SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1)))) SUMITER(nil, x0) -> c10(PLUS(x0, error)) SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0))) IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0))) IFPLUS(false, s(s(x0)), x1, s(0)) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(0)))) IFPLUS(false, s(s(x0)), x1, s(s(x2))) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(s(x2))))) PLUSITER(s(s(z0)), x1, s(0)) -> c(IFPLUS(false, s(s(z0)), x1, s(0))) PLUSITER(s(s(z0)), x1, s(0)) -> c(LE(s(s(z0)), s(0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0))) PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(IFPLUS(le(x0, s(x2)), s(s(x0)), s(x1), s(s(s(x2)))), LE(s(s(x0)), s(s(s(x2))))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(x1), s(s(0)))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(LE(s(s(x0)), s(s(0)))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0))) PLUSITER(s(x0), s(x1), s(0)) -> c2(LE(s(x0), s(0))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c2(LE(s(s(x0)), s(s(0)))) PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(LE(s(s(x0)), s(s(s(x2))))) S tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0))) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1)))) SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0))) IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0))) IFPLUS(false, s(s(x0)), x1, s(0)) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(0)))) IFPLUS(false, s(s(x0)), x1, s(s(x2))) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(s(x2))))) PLUSITER(s(s(z0)), x1, s(0)) -> c(IFPLUS(false, s(s(z0)), x1, s(0))) PLUSITER(s(s(z0)), x1, s(0)) -> c(LE(s(s(z0)), s(0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0))) PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(IFPLUS(le(x0, s(x2)), s(s(x0)), s(x1), s(s(s(x2)))), LE(s(s(x0)), s(s(s(x2))))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(x1), s(s(0)))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(LE(s(s(x0)), s(s(0)))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0))) PLUSITER(s(x0), s(x1), s(0)) -> c2(LE(s(x0), s(0))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c2(LE(s(s(x0)), s(s(0)))) PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(LE(s(s(x0)), s(s(s(x2))))) K tuples: SUMITER(nil, x1) -> c10(PLUS(x1, head(nil))) SUMITER(x0, x1) -> c10(PLUS(x1, head(x0))) SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1)))) Defined Rule Symbols: le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1 Defined Pair Symbols: PLUS_2, LE_2, IFSUM_4, PLUSITER_3, SUMITER_2, IFPLUS_4 Compound Symbols: c1_1, c7_1, c12_1, c2_1, c10_1, c9_1, c10_2, c4_1, c_1, c2_2, c3_1 ---------------------------------------- (179) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil))) by SUMITER(nil, z0) -> c10(IFSUM(isempty(nil), nil, z0, plusIter(z0, error, 0)), PLUS(z0, error)) ---------------------------------------- (180) Obligation: Complexity Dependency Tuples Problem Rules: le(s(z0), 0) -> false le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) isempty(nil) -> true isempty(cons(z0, z1)) -> false plus(z0, z1) -> plusIter(z0, z1, 0) head(nil) -> error head(cons(z0, z1)) -> z0 plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2) ifPlus(true, z0, z1, z2) -> z1 ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2)) tail(nil) -> nil tail(cons(z0, z1)) -> z1 Tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) SUMITER(nil, x1) -> c10(PLUS(x1, head(nil))) PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0))) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(x0, x1) -> c10(PLUS(x1, head(x0))) SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1)))) SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1)))) SUMITER(nil, x0) -> c10(PLUS(x0, error)) SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0))) IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0))) IFPLUS(false, s(s(x0)), x1, s(0)) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(0)))) IFPLUS(false, s(s(x0)), x1, s(s(x2))) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(s(x2))))) PLUSITER(s(s(z0)), x1, s(0)) -> c(IFPLUS(false, s(s(z0)), x1, s(0))) PLUSITER(s(s(z0)), x1, s(0)) -> c(LE(s(s(z0)), s(0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0))) PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(IFPLUS(le(x0, s(x2)), s(s(x0)), s(x1), s(s(s(x2)))), LE(s(s(x0)), s(s(s(x2))))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(x1), s(s(0)))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(LE(s(s(x0)), s(s(0)))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0))) PLUSITER(s(x0), s(x1), s(0)) -> c2(LE(s(x0), s(0))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c2(LE(s(s(x0)), s(s(0)))) PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(LE(s(s(x0)), s(s(s(x2))))) SUMITER(nil, z0) -> c10(IFSUM(isempty(nil), nil, z0, plusIter(z0, error, 0)), PLUS(z0, error)) S tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0))) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1)))) SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0))) IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0))) IFPLUS(false, s(s(x0)), x1, s(0)) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(0)))) IFPLUS(false, s(s(x0)), x1, s(s(x2))) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(s(x2))))) PLUSITER(s(s(z0)), x1, s(0)) -> c(IFPLUS(false, s(s(z0)), x1, s(0))) PLUSITER(s(s(z0)), x1, s(0)) -> c(LE(s(s(z0)), s(0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0))) PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(IFPLUS(le(x0, s(x2)), s(s(x0)), s(x1), s(s(s(x2)))), LE(s(s(x0)), s(s(s(x2))))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(x1), s(s(0)))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(LE(s(s(x0)), s(s(0)))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0))) PLUSITER(s(x0), s(x1), s(0)) -> c2(LE(s(x0), s(0))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c2(LE(s(s(x0)), s(s(0)))) PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(LE(s(s(x0)), s(s(s(x2))))) SUMITER(nil, z0) -> c10(IFSUM(isempty(nil), nil, z0, plusIter(z0, error, 0)), PLUS(z0, error)) K tuples: SUMITER(nil, x1) -> c10(PLUS(x1, head(nil))) SUMITER(x0, x1) -> c10(PLUS(x1, head(x0))) SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1)))) Defined Rule Symbols: le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1 Defined Pair Symbols: PLUS_2, LE_2, IFSUM_4, PLUSITER_3, SUMITER_2, IFPLUS_4 Compound Symbols: c1_1, c7_1, c12_1, c2_1, c10_1, c9_1, c10_2, c4_1, c_1, c2_2, c3_1 ---------------------------------------- (181) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1)))) by SUMITER(cons(z0, z1), z2) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, plusIter(z2, z0, 0)), PLUS(z2, z0)) ---------------------------------------- (182) Obligation: Complexity Dependency Tuples Problem Rules: le(s(z0), 0) -> false le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) isempty(nil) -> true isempty(cons(z0, z1)) -> false plus(z0, z1) -> plusIter(z0, z1, 0) head(nil) -> error head(cons(z0, z1)) -> z0 plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2) ifPlus(true, z0, z1, z2) -> z1 ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2)) tail(nil) -> nil tail(cons(z0, z1)) -> z1 Tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) SUMITER(nil, x1) -> c10(PLUS(x1, head(nil))) PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0))) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(x0, x1) -> c10(PLUS(x1, head(x0))) SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1)))) SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1)))) SUMITER(nil, x0) -> c10(PLUS(x0, error)) SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0))) IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0))) IFPLUS(false, s(s(x0)), x1, s(0)) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(0)))) IFPLUS(false, s(s(x0)), x1, s(s(x2))) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(s(x2))))) PLUSITER(s(s(z0)), x1, s(0)) -> c(IFPLUS(false, s(s(z0)), x1, s(0))) PLUSITER(s(s(z0)), x1, s(0)) -> c(LE(s(s(z0)), s(0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0))) PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(IFPLUS(le(x0, s(x2)), s(s(x0)), s(x1), s(s(s(x2)))), LE(s(s(x0)), s(s(s(x2))))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(x1), s(s(0)))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(LE(s(s(x0)), s(s(0)))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0))) PLUSITER(s(x0), s(x1), s(0)) -> c2(LE(s(x0), s(0))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c2(LE(s(s(x0)), s(s(0)))) PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(LE(s(s(x0)), s(s(s(x2))))) SUMITER(nil, z0) -> c10(IFSUM(isempty(nil), nil, z0, plusIter(z0, error, 0)), PLUS(z0, error)) SUMITER(cons(z0, z1), z2) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, plusIter(z2, z0, 0)), PLUS(z2, z0)) S tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0))) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1)))) SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0))) IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0))) IFPLUS(false, s(s(x0)), x1, s(0)) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(0)))) IFPLUS(false, s(s(x0)), x1, s(s(x2))) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(s(x2))))) PLUSITER(s(s(z0)), x1, s(0)) -> c(IFPLUS(false, s(s(z0)), x1, s(0))) PLUSITER(s(s(z0)), x1, s(0)) -> c(LE(s(s(z0)), s(0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0))) PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(IFPLUS(le(x0, s(x2)), s(s(x0)), s(x1), s(s(s(x2)))), LE(s(s(x0)), s(s(s(x2))))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(x1), s(s(0)))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(LE(s(s(x0)), s(s(0)))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0))) PLUSITER(s(x0), s(x1), s(0)) -> c2(LE(s(x0), s(0))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c2(LE(s(s(x0)), s(s(0)))) PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(LE(s(s(x0)), s(s(s(x2))))) SUMITER(nil, z0) -> c10(IFSUM(isempty(nil), nil, z0, plusIter(z0, error, 0)), PLUS(z0, error)) SUMITER(cons(z0, z1), z2) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, plusIter(z2, z0, 0)), PLUS(z2, z0)) K tuples: SUMITER(nil, x1) -> c10(PLUS(x1, head(nil))) SUMITER(x0, x1) -> c10(PLUS(x1, head(x0))) SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1)))) Defined Rule Symbols: le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1 Defined Pair Symbols: PLUS_2, LE_2, IFSUM_4, PLUSITER_3, SUMITER_2, IFPLUS_4 Compound Symbols: c1_1, c7_1, c12_1, c2_1, c10_1, c9_1, c10_2, c4_1, c_1, c2_2, c3_1 ---------------------------------------- (183) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0))) by PLUSITER(s(0), s(x1), s(0)) -> c2(LE(s(0), s(0))) ---------------------------------------- (184) Obligation: Complexity Dependency Tuples Problem Rules: le(s(z0), 0) -> false le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) isempty(nil) -> true isempty(cons(z0, z1)) -> false plus(z0, z1) -> plusIter(z0, z1, 0) head(nil) -> error head(cons(z0, z1)) -> z0 plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2) ifPlus(true, z0, z1, z2) -> z1 ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2)) tail(nil) -> nil tail(cons(z0, z1)) -> z1 Tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) SUMITER(nil, x1) -> c10(PLUS(x1, head(nil))) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(x0, x1) -> c10(PLUS(x1, head(x0))) SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1)))) SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1)))) SUMITER(nil, x0) -> c10(PLUS(x0, error)) SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0))) IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0))) IFPLUS(false, s(s(x0)), x1, s(0)) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(0)))) IFPLUS(false, s(s(x0)), x1, s(s(x2))) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(s(x2))))) PLUSITER(s(s(z0)), x1, s(0)) -> c(IFPLUS(false, s(s(z0)), x1, s(0))) PLUSITER(s(s(z0)), x1, s(0)) -> c(LE(s(s(z0)), s(0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0))) PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(IFPLUS(le(x0, s(x2)), s(s(x0)), s(x1), s(s(s(x2)))), LE(s(s(x0)), s(s(s(x2))))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(x1), s(s(0)))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(LE(s(s(x0)), s(s(0)))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0))) PLUSITER(s(x0), s(x1), s(0)) -> c2(LE(s(x0), s(0))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c2(LE(s(s(x0)), s(s(0)))) PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(LE(s(s(x0)), s(s(s(x2))))) SUMITER(nil, z0) -> c10(IFSUM(isempty(nil), nil, z0, plusIter(z0, error, 0)), PLUS(z0, error)) SUMITER(cons(z0, z1), z2) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, plusIter(z2, z0, 0)), PLUS(z2, z0)) PLUSITER(s(0), s(x1), s(0)) -> c2(LE(s(0), s(0))) S tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1)))) SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0))) IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0))) IFPLUS(false, s(s(x0)), x1, s(0)) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(0)))) IFPLUS(false, s(s(x0)), x1, s(s(x2))) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(s(x2))))) PLUSITER(s(s(z0)), x1, s(0)) -> c(IFPLUS(false, s(s(z0)), x1, s(0))) PLUSITER(s(s(z0)), x1, s(0)) -> c(LE(s(s(z0)), s(0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0))) PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(IFPLUS(le(x0, s(x2)), s(s(x0)), s(x1), s(s(s(x2)))), LE(s(s(x0)), s(s(s(x2))))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(x1), s(s(0)))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(LE(s(s(x0)), s(s(0)))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0))) PLUSITER(s(x0), s(x1), s(0)) -> c2(LE(s(x0), s(0))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c2(LE(s(s(x0)), s(s(0)))) PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(LE(s(s(x0)), s(s(s(x2))))) SUMITER(nil, z0) -> c10(IFSUM(isempty(nil), nil, z0, plusIter(z0, error, 0)), PLUS(z0, error)) SUMITER(cons(z0, z1), z2) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, plusIter(z2, z0, 0)), PLUS(z2, z0)) PLUSITER(s(0), s(x1), s(0)) -> c2(LE(s(0), s(0))) K tuples: SUMITER(nil, x1) -> c10(PLUS(x1, head(nil))) SUMITER(x0, x1) -> c10(PLUS(x1, head(x0))) SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1)))) Defined Rule Symbols: le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1 Defined Pair Symbols: PLUS_2, LE_2, IFSUM_4, PLUSITER_3, SUMITER_2, IFPLUS_4 Compound Symbols: c1_1, c7_1, c12_1, c2_1, c10_1, c9_1, c10_2, c4_1, c_1, c2_2, c3_1 ---------------------------------------- (185) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1)))) by SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plusIter(z2, head(cons(z0, z1)), 0)), PLUS(z2, z0)) ---------------------------------------- (186) Obligation: Complexity Dependency Tuples Problem Rules: le(s(z0), 0) -> false le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) isempty(nil) -> true isempty(cons(z0, z1)) -> false plus(z0, z1) -> plusIter(z0, z1, 0) head(nil) -> error head(cons(z0, z1)) -> z0 plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2) ifPlus(true, z0, z1, z2) -> z1 ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2)) tail(nil) -> nil tail(cons(z0, z1)) -> z1 Tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) SUMITER(nil, x1) -> c10(PLUS(x1, head(nil))) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(x0, x1) -> c10(PLUS(x1, head(x0))) SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1)))) SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1)))) SUMITER(nil, x0) -> c10(PLUS(x0, error)) SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0))) IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0))) IFPLUS(false, s(s(x0)), x1, s(0)) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(0)))) IFPLUS(false, s(s(x0)), x1, s(s(x2))) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(s(x2))))) PLUSITER(s(s(z0)), x1, s(0)) -> c(IFPLUS(false, s(s(z0)), x1, s(0))) PLUSITER(s(s(z0)), x1, s(0)) -> c(LE(s(s(z0)), s(0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0))) PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(IFPLUS(le(x0, s(x2)), s(s(x0)), s(x1), s(s(s(x2)))), LE(s(s(x0)), s(s(s(x2))))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(x1), s(s(0)))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(LE(s(s(x0)), s(s(0)))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0))) PLUSITER(s(x0), s(x1), s(0)) -> c2(LE(s(x0), s(0))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c2(LE(s(s(x0)), s(s(0)))) PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(LE(s(s(x0)), s(s(s(x2))))) SUMITER(nil, z0) -> c10(IFSUM(isempty(nil), nil, z0, plusIter(z0, error, 0)), PLUS(z0, error)) SUMITER(cons(z0, z1), z2) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, plusIter(z2, z0, 0)), PLUS(z2, z0)) PLUSITER(s(0), s(x1), s(0)) -> c2(LE(s(0), s(0))) SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plusIter(z2, head(cons(z0, z1)), 0)), PLUS(z2, z0)) S tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1)))) SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0))) IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0))) IFPLUS(false, s(s(x0)), x1, s(0)) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(0)))) IFPLUS(false, s(s(x0)), x1, s(s(x2))) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(s(x2))))) PLUSITER(s(s(z0)), x1, s(0)) -> c(IFPLUS(false, s(s(z0)), x1, s(0))) PLUSITER(s(s(z0)), x1, s(0)) -> c(LE(s(s(z0)), s(0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0))) PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(IFPLUS(le(x0, s(x2)), s(s(x0)), s(x1), s(s(s(x2)))), LE(s(s(x0)), s(s(s(x2))))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(x1), s(s(0)))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(LE(s(s(x0)), s(s(0)))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0))) PLUSITER(s(x0), s(x1), s(0)) -> c2(LE(s(x0), s(0))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c2(LE(s(s(x0)), s(s(0)))) PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(LE(s(s(x0)), s(s(s(x2))))) SUMITER(nil, z0) -> c10(IFSUM(isempty(nil), nil, z0, plusIter(z0, error, 0)), PLUS(z0, error)) SUMITER(cons(z0, z1), z2) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, plusIter(z2, z0, 0)), PLUS(z2, z0)) PLUSITER(s(0), s(x1), s(0)) -> c2(LE(s(0), s(0))) SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plusIter(z2, head(cons(z0, z1)), 0)), PLUS(z2, z0)) K tuples: SUMITER(nil, x1) -> c10(PLUS(x1, head(nil))) SUMITER(x0, x1) -> c10(PLUS(x1, head(x0))) SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1)))) Defined Rule Symbols: le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1 Defined Pair Symbols: PLUS_2, LE_2, IFSUM_4, PLUSITER_3, SUMITER_2, IFPLUS_4 Compound Symbols: c1_1, c7_1, c12_1, c2_1, c10_1, c9_1, c10_2, c4_1, c_1, c2_2, c3_1 ---------------------------------------- (187) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace SUMITER(nil, x1) -> c10(PLUS(x1, head(nil))) by SUMITER(nil, z0) -> c10(PLUS(z0, error)) ---------------------------------------- (188) Obligation: Complexity Dependency Tuples Problem Rules: le(s(z0), 0) -> false le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) isempty(nil) -> true isempty(cons(z0, z1)) -> false plus(z0, z1) -> plusIter(z0, z1, 0) head(nil) -> error head(cons(z0, z1)) -> z0 plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2) ifPlus(true, z0, z1, z2) -> z1 ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2)) tail(nil) -> nil tail(cons(z0, z1)) -> z1 Tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) SUMITER(nil, x1) -> c10(PLUS(x1, head(nil))) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(x0, x1) -> c10(PLUS(x1, head(x0))) SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1)))) SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1)))) SUMITER(nil, x0) -> c10(PLUS(x0, error)) SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0))) IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0))) IFPLUS(false, s(s(x0)), x1, s(0)) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(0)))) IFPLUS(false, s(s(x0)), x1, s(s(x2))) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(s(x2))))) PLUSITER(s(s(z0)), x1, s(0)) -> c(IFPLUS(false, s(s(z0)), x1, s(0))) PLUSITER(s(s(z0)), x1, s(0)) -> c(LE(s(s(z0)), s(0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0))) PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(IFPLUS(le(x0, s(x2)), s(s(x0)), s(x1), s(s(s(x2)))), LE(s(s(x0)), s(s(s(x2))))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(x1), s(s(0)))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(LE(s(s(x0)), s(s(0)))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0))) PLUSITER(s(x0), s(x1), s(0)) -> c2(LE(s(x0), s(0))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c2(LE(s(s(x0)), s(s(0)))) PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(LE(s(s(x0)), s(s(s(x2))))) SUMITER(nil, z0) -> c10(IFSUM(isempty(nil), nil, z0, plusIter(z0, error, 0)), PLUS(z0, error)) SUMITER(cons(z0, z1), z2) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, plusIter(z2, z0, 0)), PLUS(z2, z0)) PLUSITER(s(0), s(x1), s(0)) -> c2(LE(s(0), s(0))) SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plusIter(z2, head(cons(z0, z1)), 0)), PLUS(z2, z0)) S tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1)))) SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0))) IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0))) IFPLUS(false, s(s(x0)), x1, s(0)) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(0)))) IFPLUS(false, s(s(x0)), x1, s(s(x2))) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(s(x2))))) PLUSITER(s(s(z0)), x1, s(0)) -> c(IFPLUS(false, s(s(z0)), x1, s(0))) PLUSITER(s(s(z0)), x1, s(0)) -> c(LE(s(s(z0)), s(0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0))) PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(IFPLUS(le(x0, s(x2)), s(s(x0)), s(x1), s(s(s(x2)))), LE(s(s(x0)), s(s(s(x2))))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(x1), s(s(0)))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(LE(s(s(x0)), s(s(0)))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0))) PLUSITER(s(x0), s(x1), s(0)) -> c2(LE(s(x0), s(0))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c2(LE(s(s(x0)), s(s(0)))) PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(LE(s(s(x0)), s(s(s(x2))))) SUMITER(nil, z0) -> c10(IFSUM(isempty(nil), nil, z0, plusIter(z0, error, 0)), PLUS(z0, error)) SUMITER(cons(z0, z1), z2) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, plusIter(z2, z0, 0)), PLUS(z2, z0)) PLUSITER(s(0), s(x1), s(0)) -> c2(LE(s(0), s(0))) SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plusIter(z2, head(cons(z0, z1)), 0)), PLUS(z2, z0)) K tuples: SUMITER(nil, x1) -> c10(PLUS(x1, head(nil))) SUMITER(x0, x1) -> c10(PLUS(x1, head(x0))) SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1)))) Defined Rule Symbols: le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1 Defined Pair Symbols: PLUS_2, LE_2, IFSUM_4, PLUSITER_3, SUMITER_2, IFPLUS_4 Compound Symbols: c1_1, c7_1, c12_1, c2_1, c10_1, c9_1, c10_2, c4_1, c_1, c2_2, c3_1 ---------------------------------------- (189) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IFPLUS(false, s(s(x0)), x1, s(s(x2))) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(s(x2))))) by IFPLUS(false, s(s(x0)), s(x1), s(s(s(x2)))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2)))))) IFPLUS(false, s(s(x0)), s(x1), s(s(0))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0))))) ---------------------------------------- (190) Obligation: Complexity Dependency Tuples Problem Rules: le(s(z0), 0) -> false le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) isempty(nil) -> true isempty(cons(z0, z1)) -> false plus(z0, z1) -> plusIter(z0, z1, 0) head(nil) -> error head(cons(z0, z1)) -> z0 plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2) ifPlus(true, z0, z1, z2) -> z1 ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2)) tail(nil) -> nil tail(cons(z0, z1)) -> z1 Tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) SUMITER(nil, x1) -> c10(PLUS(x1, head(nil))) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(x0, x1) -> c10(PLUS(x1, head(x0))) SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1)))) SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1)))) SUMITER(nil, x0) -> c10(PLUS(x0, error)) SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0))) IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0))) IFPLUS(false, s(s(x0)), x1, s(0)) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(0)))) PLUSITER(s(s(z0)), x1, s(0)) -> c(IFPLUS(false, s(s(z0)), x1, s(0))) PLUSITER(s(s(z0)), x1, s(0)) -> c(LE(s(s(z0)), s(0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0))) PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(IFPLUS(le(x0, s(x2)), s(s(x0)), s(x1), s(s(s(x2)))), LE(s(s(x0)), s(s(s(x2))))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(x1), s(s(0)))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(LE(s(s(x0)), s(s(0)))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0))) PLUSITER(s(x0), s(x1), s(0)) -> c2(LE(s(x0), s(0))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c2(LE(s(s(x0)), s(s(0)))) PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(LE(s(s(x0)), s(s(s(x2))))) SUMITER(nil, z0) -> c10(IFSUM(isempty(nil), nil, z0, plusIter(z0, error, 0)), PLUS(z0, error)) SUMITER(cons(z0, z1), z2) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, plusIter(z2, z0, 0)), PLUS(z2, z0)) PLUSITER(s(0), s(x1), s(0)) -> c2(LE(s(0), s(0))) SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plusIter(z2, head(cons(z0, z1)), 0)), PLUS(z2, z0)) IFPLUS(false, s(s(x0)), s(x1), s(s(s(x2)))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2)))))) IFPLUS(false, s(s(x0)), s(x1), s(s(0))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0))))) S tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1)))) SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0))) IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0))) IFPLUS(false, s(s(x0)), x1, s(0)) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(0)))) PLUSITER(s(s(z0)), x1, s(0)) -> c(IFPLUS(false, s(s(z0)), x1, s(0))) PLUSITER(s(s(z0)), x1, s(0)) -> c(LE(s(s(z0)), s(0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0))) PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(IFPLUS(le(x0, s(x2)), s(s(x0)), s(x1), s(s(s(x2)))), LE(s(s(x0)), s(s(s(x2))))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(x1), s(s(0)))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(LE(s(s(x0)), s(s(0)))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0))) PLUSITER(s(x0), s(x1), s(0)) -> c2(LE(s(x0), s(0))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c2(LE(s(s(x0)), s(s(0)))) PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(LE(s(s(x0)), s(s(s(x2))))) SUMITER(nil, z0) -> c10(IFSUM(isempty(nil), nil, z0, plusIter(z0, error, 0)), PLUS(z0, error)) SUMITER(cons(z0, z1), z2) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, plusIter(z2, z0, 0)), PLUS(z2, z0)) PLUSITER(s(0), s(x1), s(0)) -> c2(LE(s(0), s(0))) SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plusIter(z2, head(cons(z0, z1)), 0)), PLUS(z2, z0)) IFPLUS(false, s(s(x0)), s(x1), s(s(s(x2)))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2)))))) IFPLUS(false, s(s(x0)), s(x1), s(s(0))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0))))) K tuples: SUMITER(nil, x1) -> c10(PLUS(x1, head(nil))) SUMITER(x0, x1) -> c10(PLUS(x1, head(x0))) SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1)))) Defined Rule Symbols: le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1 Defined Pair Symbols: PLUS_2, LE_2, IFSUM_4, PLUSITER_3, SUMITER_2, IFPLUS_4 Compound Symbols: c1_1, c7_1, c12_1, c2_1, c10_1, c9_1, c10_2, c4_1, c_1, c2_2, c3_1 ---------------------------------------- (191) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace PLUSITER(s(s(z0)), x1, s(0)) -> c(IFPLUS(false, s(s(z0)), x1, s(0))) by PLUSITER(s(s(z0)), s(x1), s(0)) -> c(IFPLUS(false, s(s(z0)), s(x1), s(0))) ---------------------------------------- (192) Obligation: Complexity Dependency Tuples Problem Rules: le(s(z0), 0) -> false le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) isempty(nil) -> true isempty(cons(z0, z1)) -> false plus(z0, z1) -> plusIter(z0, z1, 0) head(nil) -> error head(cons(z0, z1)) -> z0 plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2) ifPlus(true, z0, z1, z2) -> z1 ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2)) tail(nil) -> nil tail(cons(z0, z1)) -> z1 Tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) SUMITER(nil, x1) -> c10(PLUS(x1, head(nil))) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(x0, x1) -> c10(PLUS(x1, head(x0))) SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1)))) SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1)))) SUMITER(nil, x0) -> c10(PLUS(x0, error)) SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0))) IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0))) IFPLUS(false, s(s(x0)), x1, s(0)) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(0)))) PLUSITER(s(s(z0)), x1, s(0)) -> c(LE(s(s(z0)), s(0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0))) PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(IFPLUS(le(x0, s(x2)), s(s(x0)), s(x1), s(s(s(x2)))), LE(s(s(x0)), s(s(s(x2))))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(x1), s(s(0)))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(LE(s(s(x0)), s(s(0)))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0))) PLUSITER(s(x0), s(x1), s(0)) -> c2(LE(s(x0), s(0))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c2(LE(s(s(x0)), s(s(0)))) PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(LE(s(s(x0)), s(s(s(x2))))) SUMITER(nil, z0) -> c10(IFSUM(isempty(nil), nil, z0, plusIter(z0, error, 0)), PLUS(z0, error)) SUMITER(cons(z0, z1), z2) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, plusIter(z2, z0, 0)), PLUS(z2, z0)) PLUSITER(s(0), s(x1), s(0)) -> c2(LE(s(0), s(0))) SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plusIter(z2, head(cons(z0, z1)), 0)), PLUS(z2, z0)) IFPLUS(false, s(s(x0)), s(x1), s(s(s(x2)))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2)))))) IFPLUS(false, s(s(x0)), s(x1), s(s(0))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0))))) PLUSITER(s(s(z0)), s(x1), s(0)) -> c(IFPLUS(false, s(s(z0)), s(x1), s(0))) S tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1)))) SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0))) IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0))) IFPLUS(false, s(s(x0)), x1, s(0)) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(0)))) PLUSITER(s(s(z0)), x1, s(0)) -> c(LE(s(s(z0)), s(0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0))) PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(IFPLUS(le(x0, s(x2)), s(s(x0)), s(x1), s(s(s(x2)))), LE(s(s(x0)), s(s(s(x2))))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(x1), s(s(0)))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(LE(s(s(x0)), s(s(0)))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0))) PLUSITER(s(x0), s(x1), s(0)) -> c2(LE(s(x0), s(0))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c2(LE(s(s(x0)), s(s(0)))) PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(LE(s(s(x0)), s(s(s(x2))))) SUMITER(nil, z0) -> c10(IFSUM(isempty(nil), nil, z0, plusIter(z0, error, 0)), PLUS(z0, error)) SUMITER(cons(z0, z1), z2) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, plusIter(z2, z0, 0)), PLUS(z2, z0)) PLUSITER(s(0), s(x1), s(0)) -> c2(LE(s(0), s(0))) SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plusIter(z2, head(cons(z0, z1)), 0)), PLUS(z2, z0)) IFPLUS(false, s(s(x0)), s(x1), s(s(s(x2)))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2)))))) IFPLUS(false, s(s(x0)), s(x1), s(s(0))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0))))) PLUSITER(s(s(z0)), s(x1), s(0)) -> c(IFPLUS(false, s(s(z0)), s(x1), s(0))) K tuples: SUMITER(nil, x1) -> c10(PLUS(x1, head(nil))) SUMITER(x0, x1) -> c10(PLUS(x1, head(x0))) SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1)))) Defined Rule Symbols: le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1 Defined Pair Symbols: PLUS_2, LE_2, IFSUM_4, PLUSITER_3, SUMITER_2, IFPLUS_4 Compound Symbols: c1_1, c7_1, c12_1, c2_1, c10_1, c9_1, c10_2, c4_1, c_1, c2_2, c3_1 ---------------------------------------- (193) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace SUMITER(nil, z0) -> c10(IFSUM(isempty(nil), nil, z0, plusIter(z0, error, 0)), PLUS(z0, head(nil))) by SUMITER(nil, z0) -> c10(IFSUM(isempty(nil), nil, z0, plusIter(z0, error, 0)), PLUS(z0, error)) ---------------------------------------- (194) Obligation: Complexity Dependency Tuples Problem Rules: le(s(z0), 0) -> false le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) isempty(nil) -> true isempty(cons(z0, z1)) -> false plus(z0, z1) -> plusIter(z0, z1, 0) head(nil) -> error head(cons(z0, z1)) -> z0 plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2) ifPlus(true, z0, z1, z2) -> z1 ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2)) tail(nil) -> nil tail(cons(z0, z1)) -> z1 Tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) SUMITER(nil, x1) -> c10(PLUS(x1, head(nil))) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(x0, x1) -> c10(PLUS(x1, head(x0))) SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1)))) SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1)))) SUMITER(nil, x0) -> c10(PLUS(x0, error)) SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0))) IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0))) IFPLUS(false, s(s(x0)), x1, s(0)) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(0)))) PLUSITER(s(s(z0)), x1, s(0)) -> c(LE(s(s(z0)), s(0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0))) PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(IFPLUS(le(x0, s(x2)), s(s(x0)), s(x1), s(s(s(x2)))), LE(s(s(x0)), s(s(s(x2))))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(x1), s(s(0)))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(LE(s(s(x0)), s(s(0)))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0))) PLUSITER(s(x0), s(x1), s(0)) -> c2(LE(s(x0), s(0))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c2(LE(s(s(x0)), s(s(0)))) PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(LE(s(s(x0)), s(s(s(x2))))) SUMITER(nil, z0) -> c10(IFSUM(isempty(nil), nil, z0, plusIter(z0, error, 0)), PLUS(z0, error)) SUMITER(cons(z0, z1), z2) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, plusIter(z2, z0, 0)), PLUS(z2, z0)) PLUSITER(s(0), s(x1), s(0)) -> c2(LE(s(0), s(0))) SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plusIter(z2, head(cons(z0, z1)), 0)), PLUS(z2, z0)) IFPLUS(false, s(s(x0)), s(x1), s(s(s(x2)))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2)))))) IFPLUS(false, s(s(x0)), s(x1), s(s(0))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0))))) PLUSITER(s(s(z0)), s(x1), s(0)) -> c(IFPLUS(false, s(s(z0)), s(x1), s(0))) S tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1)))) SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0))) IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0))) IFPLUS(false, s(s(x0)), x1, s(0)) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(0)))) PLUSITER(s(s(z0)), x1, s(0)) -> c(LE(s(s(z0)), s(0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0))) PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(IFPLUS(le(x0, s(x2)), s(s(x0)), s(x1), s(s(s(x2)))), LE(s(s(x0)), s(s(s(x2))))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(x1), s(s(0)))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(LE(s(s(x0)), s(s(0)))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0))) PLUSITER(s(x0), s(x1), s(0)) -> c2(LE(s(x0), s(0))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c2(LE(s(s(x0)), s(s(0)))) PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(LE(s(s(x0)), s(s(s(x2))))) SUMITER(nil, z0) -> c10(IFSUM(isempty(nil), nil, z0, plusIter(z0, error, 0)), PLUS(z0, error)) SUMITER(cons(z0, z1), z2) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, plusIter(z2, z0, 0)), PLUS(z2, z0)) PLUSITER(s(0), s(x1), s(0)) -> c2(LE(s(0), s(0))) SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plusIter(z2, head(cons(z0, z1)), 0)), PLUS(z2, z0)) IFPLUS(false, s(s(x0)), s(x1), s(s(s(x2)))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2)))))) IFPLUS(false, s(s(x0)), s(x1), s(s(0))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0))))) PLUSITER(s(s(z0)), s(x1), s(0)) -> c(IFPLUS(false, s(s(z0)), s(x1), s(0))) K tuples: SUMITER(nil, x1) -> c10(PLUS(x1, head(nil))) SUMITER(x0, x1) -> c10(PLUS(x1, head(x0))) SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1)))) Defined Rule Symbols: le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1 Defined Pair Symbols: PLUS_2, LE_2, IFSUM_4, PLUSITER_3, SUMITER_2, IFPLUS_4 Compound Symbols: c1_1, c7_1, c12_1, c2_1, c10_1, c9_1, c10_2, c4_1, c_1, c2_2, c3_1 ---------------------------------------- (195) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace SUMITER(nil, x0) -> c10(PLUS(x0, head(nil))) by SUMITER(nil, z0) -> c10(PLUS(z0, error)) ---------------------------------------- (196) Obligation: Complexity Dependency Tuples Problem Rules: le(s(z0), 0) -> false le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) isempty(nil) -> true isempty(cons(z0, z1)) -> false plus(z0, z1) -> plusIter(z0, z1, 0) head(nil) -> error head(cons(z0, z1)) -> z0 plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2) ifPlus(true, z0, z1, z2) -> z1 ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2)) tail(nil) -> nil tail(cons(z0, z1)) -> z1 Tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(x0, x1) -> c10(PLUS(x1, head(x0))) SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1)))) SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1)))) SUMITER(nil, x0) -> c10(PLUS(x0, error)) SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0))) IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0))) IFPLUS(false, s(s(x0)), x1, s(0)) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(0)))) PLUSITER(s(s(z0)), x1, s(0)) -> c(LE(s(s(z0)), s(0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0))) PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(IFPLUS(le(x0, s(x2)), s(s(x0)), s(x1), s(s(s(x2)))), LE(s(s(x0)), s(s(s(x2))))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(x1), s(s(0)))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(LE(s(s(x0)), s(s(0)))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0))) PLUSITER(s(x0), s(x1), s(0)) -> c2(LE(s(x0), s(0))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c2(LE(s(s(x0)), s(s(0)))) PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(LE(s(s(x0)), s(s(s(x2))))) SUMITER(nil, z0) -> c10(IFSUM(isempty(nil), nil, z0, plusIter(z0, error, 0)), PLUS(z0, error)) SUMITER(cons(z0, z1), z2) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, plusIter(z2, z0, 0)), PLUS(z2, z0)) PLUSITER(s(0), s(x1), s(0)) -> c2(LE(s(0), s(0))) SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plusIter(z2, head(cons(z0, z1)), 0)), PLUS(z2, z0)) IFPLUS(false, s(s(x0)), s(x1), s(s(s(x2)))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2)))))) IFPLUS(false, s(s(x0)), s(x1), s(s(0))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0))))) PLUSITER(s(s(z0)), s(x1), s(0)) -> c(IFPLUS(false, s(s(z0)), s(x1), s(0))) S tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1)))) SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0))) IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0))) IFPLUS(false, s(s(x0)), x1, s(0)) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(0)))) PLUSITER(s(s(z0)), x1, s(0)) -> c(LE(s(s(z0)), s(0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0))) PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(IFPLUS(le(x0, s(x2)), s(s(x0)), s(x1), s(s(s(x2)))), LE(s(s(x0)), s(s(s(x2))))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(x1), s(s(0)))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(LE(s(s(x0)), s(s(0)))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0))) PLUSITER(s(x0), s(x1), s(0)) -> c2(LE(s(x0), s(0))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c2(LE(s(s(x0)), s(s(0)))) PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(LE(s(s(x0)), s(s(s(x2))))) SUMITER(nil, z0) -> c10(IFSUM(isempty(nil), nil, z0, plusIter(z0, error, 0)), PLUS(z0, error)) SUMITER(cons(z0, z1), z2) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, plusIter(z2, z0, 0)), PLUS(z2, z0)) PLUSITER(s(0), s(x1), s(0)) -> c2(LE(s(0), s(0))) SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plusIter(z2, head(cons(z0, z1)), 0)), PLUS(z2, z0)) IFPLUS(false, s(s(x0)), s(x1), s(s(s(x2)))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2)))))) IFPLUS(false, s(s(x0)), s(x1), s(s(0))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0))))) PLUSITER(s(s(z0)), s(x1), s(0)) -> c(IFPLUS(false, s(s(z0)), s(x1), s(0))) K tuples: SUMITER(nil, x1) -> c10(PLUS(x1, head(nil))) SUMITER(x0, x1) -> c10(PLUS(x1, head(x0))) SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1)))) Defined Rule Symbols: le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1 Defined Pair Symbols: PLUS_2, LE_2, IFSUM_4, PLUSITER_3, SUMITER_2, IFPLUS_4 Compound Symbols: c1_1, c7_1, c12_1, c2_1, c9_1, c10_2, c10_1, c4_1, c_1, c2_2, c3_1 ---------------------------------------- (197) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace SUMITER(cons(z1, x1), z0) -> c10(IFSUM(isempty(cons(z1, x1)), cons(z1, x1), z0, plusIter(z0, z1, 0)), PLUS(z0, head(cons(z1, x1)))) by SUMITER(cons(z0, z1), z2) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, plusIter(z2, z0, 0)), PLUS(z2, z0)) ---------------------------------------- (198) Obligation: Complexity Dependency Tuples Problem Rules: le(s(z0), 0) -> false le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) isempty(nil) -> true isempty(cons(z0, z1)) -> false plus(z0, z1) -> plusIter(z0, z1, 0) head(nil) -> error head(cons(z0, z1)) -> z0 plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2) ifPlus(true, z0, z1, z2) -> z1 ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2)) tail(nil) -> nil tail(cons(z0, z1)) -> z1 Tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(x0, x1) -> c10(PLUS(x1, head(x0))) SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1)))) SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1)))) SUMITER(nil, x0) -> c10(PLUS(x0, error)) SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0))) IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0))) IFPLUS(false, s(s(x0)), x1, s(0)) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(0)))) PLUSITER(s(s(z0)), x1, s(0)) -> c(LE(s(s(z0)), s(0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0))) PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(IFPLUS(le(x0, s(x2)), s(s(x0)), s(x1), s(s(s(x2)))), LE(s(s(x0)), s(s(s(x2))))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(x1), s(s(0)))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(LE(s(s(x0)), s(s(0)))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0))) PLUSITER(s(x0), s(x1), s(0)) -> c2(LE(s(x0), s(0))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c2(LE(s(s(x0)), s(s(0)))) PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(LE(s(s(x0)), s(s(s(x2))))) SUMITER(nil, z0) -> c10(IFSUM(isempty(nil), nil, z0, plusIter(z0, error, 0)), PLUS(z0, error)) SUMITER(cons(z0, z1), z2) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, plusIter(z2, z0, 0)), PLUS(z2, z0)) PLUSITER(s(0), s(x1), s(0)) -> c2(LE(s(0), s(0))) SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plusIter(z2, head(cons(z0, z1)), 0)), PLUS(z2, z0)) IFPLUS(false, s(s(x0)), s(x1), s(s(s(x2)))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2)))))) IFPLUS(false, s(s(x0)), s(x1), s(s(0))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0))))) PLUSITER(s(s(z0)), s(x1), s(0)) -> c(IFPLUS(false, s(s(z0)), s(x1), s(0))) S tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1)))) SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0))) IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0))) IFPLUS(false, s(s(x0)), x1, s(0)) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(0)))) PLUSITER(s(s(z0)), x1, s(0)) -> c(LE(s(s(z0)), s(0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0))) PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(IFPLUS(le(x0, s(x2)), s(s(x0)), s(x1), s(s(s(x2)))), LE(s(s(x0)), s(s(s(x2))))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(x1), s(s(0)))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(LE(s(s(x0)), s(s(0)))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0))) PLUSITER(s(x0), s(x1), s(0)) -> c2(LE(s(x0), s(0))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c2(LE(s(s(x0)), s(s(0)))) PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(LE(s(s(x0)), s(s(s(x2))))) SUMITER(nil, z0) -> c10(IFSUM(isempty(nil), nil, z0, plusIter(z0, error, 0)), PLUS(z0, error)) SUMITER(cons(z0, z1), z2) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, plusIter(z2, z0, 0)), PLUS(z2, z0)) PLUSITER(s(0), s(x1), s(0)) -> c2(LE(s(0), s(0))) SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plusIter(z2, head(cons(z0, z1)), 0)), PLUS(z2, z0)) IFPLUS(false, s(s(x0)), s(x1), s(s(s(x2)))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2)))))) IFPLUS(false, s(s(x0)), s(x1), s(s(0))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0))))) PLUSITER(s(s(z0)), s(x1), s(0)) -> c(IFPLUS(false, s(s(z0)), s(x1), s(0))) K tuples: SUMITER(nil, x1) -> c10(PLUS(x1, head(nil))) SUMITER(x0, x1) -> c10(PLUS(x1, head(x0))) SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1)))) Defined Rule Symbols: le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1 Defined Pair Symbols: PLUS_2, LE_2, IFSUM_4, PLUSITER_3, SUMITER_2, IFPLUS_4 Compound Symbols: c1_1, c7_1, c12_1, c2_1, c9_1, c10_2, c10_1, c4_1, c_1, c2_2, c3_1 ---------------------------------------- (199) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IFPLUS(false, s(s(x0)), x1, s(0)) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(0)))) by IFPLUS(false, s(s(x0)), s(x1), s(0)) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(0)))) ---------------------------------------- (200) Obligation: Complexity Dependency Tuples Problem Rules: le(s(z0), 0) -> false le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) isempty(nil) -> true isempty(cons(z0, z1)) -> false plus(z0, z1) -> plusIter(z0, z1, 0) head(nil) -> error head(cons(z0, z1)) -> z0 plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2) ifPlus(true, z0, z1, z2) -> z1 ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2)) tail(nil) -> nil tail(cons(z0, z1)) -> z1 Tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(x0, x1) -> c10(PLUS(x1, head(x0))) SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1)))) SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1)))) SUMITER(nil, x0) -> c10(PLUS(x0, error)) SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0))) IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0))) PLUSITER(s(s(z0)), x1, s(0)) -> c(LE(s(s(z0)), s(0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0))) PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(IFPLUS(le(x0, s(x2)), s(s(x0)), s(x1), s(s(s(x2)))), LE(s(s(x0)), s(s(s(x2))))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(x1), s(s(0)))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(LE(s(s(x0)), s(s(0)))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0))) PLUSITER(s(x0), s(x1), s(0)) -> c2(LE(s(x0), s(0))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c2(LE(s(s(x0)), s(s(0)))) PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(LE(s(s(x0)), s(s(s(x2))))) SUMITER(nil, z0) -> c10(IFSUM(isempty(nil), nil, z0, plusIter(z0, error, 0)), PLUS(z0, error)) SUMITER(cons(z0, z1), z2) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, plusIter(z2, z0, 0)), PLUS(z2, z0)) PLUSITER(s(0), s(x1), s(0)) -> c2(LE(s(0), s(0))) SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plusIter(z2, head(cons(z0, z1)), 0)), PLUS(z2, z0)) IFPLUS(false, s(s(x0)), s(x1), s(s(s(x2)))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2)))))) IFPLUS(false, s(s(x0)), s(x1), s(s(0))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0))))) PLUSITER(s(s(z0)), s(x1), s(0)) -> c(IFPLUS(false, s(s(z0)), s(x1), s(0))) IFPLUS(false, s(s(x0)), s(x1), s(0)) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(0)))) S tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1)))) SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0))) IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0))) PLUSITER(s(s(z0)), x1, s(0)) -> c(LE(s(s(z0)), s(0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0))) PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(IFPLUS(le(x0, s(x2)), s(s(x0)), s(x1), s(s(s(x2)))), LE(s(s(x0)), s(s(s(x2))))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(x1), s(s(0)))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(LE(s(s(x0)), s(s(0)))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0))) PLUSITER(s(x0), s(x1), s(0)) -> c2(LE(s(x0), s(0))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c2(LE(s(s(x0)), s(s(0)))) PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(LE(s(s(x0)), s(s(s(x2))))) SUMITER(nil, z0) -> c10(IFSUM(isempty(nil), nil, z0, plusIter(z0, error, 0)), PLUS(z0, error)) SUMITER(cons(z0, z1), z2) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, plusIter(z2, z0, 0)), PLUS(z2, z0)) PLUSITER(s(0), s(x1), s(0)) -> c2(LE(s(0), s(0))) SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plusIter(z2, head(cons(z0, z1)), 0)), PLUS(z2, z0)) IFPLUS(false, s(s(x0)), s(x1), s(s(s(x2)))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2)))))) IFPLUS(false, s(s(x0)), s(x1), s(s(0))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0))))) PLUSITER(s(s(z0)), s(x1), s(0)) -> c(IFPLUS(false, s(s(z0)), s(x1), s(0))) IFPLUS(false, s(s(x0)), s(x1), s(0)) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(0)))) K tuples: SUMITER(nil, x1) -> c10(PLUS(x1, head(nil))) SUMITER(x0, x1) -> c10(PLUS(x1, head(x0))) SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1)))) Defined Rule Symbols: le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1 Defined Pair Symbols: PLUS_2, LE_2, IFSUM_4, PLUSITER_3, SUMITER_2, IFPLUS_4 Compound Symbols: c1_1, c7_1, c12_1, c2_1, c9_1, c10_2, c10_1, c4_1, c_1, c2_2, c3_1 ---------------------------------------- (201) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace PLUSITER(s(s(z0)), x1, s(0)) -> c(LE(s(s(z0)), s(0))) by PLUSITER(s(s(z0)), s(x1), s(0)) -> c(LE(s(s(z0)), s(0))) ---------------------------------------- (202) Obligation: Complexity Dependency Tuples Problem Rules: le(s(z0), 0) -> false le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) isempty(nil) -> true isempty(cons(z0, z1)) -> false plus(z0, z1) -> plusIter(z0, z1, 0) head(nil) -> error head(cons(z0, z1)) -> z0 plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2) ifPlus(true, z0, z1, z2) -> z1 ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2)) tail(nil) -> nil tail(cons(z0, z1)) -> z1 Tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(x0, x1) -> c10(PLUS(x1, head(x0))) SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1)))) SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1)))) SUMITER(nil, x0) -> c10(PLUS(x0, error)) SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0))) IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0))) PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(IFPLUS(le(x0, s(x2)), s(s(x0)), s(x1), s(s(s(x2)))), LE(s(s(x0)), s(s(s(x2))))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(x1), s(s(0)))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(LE(s(s(x0)), s(s(0)))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0))) PLUSITER(s(x0), s(x1), s(0)) -> c2(LE(s(x0), s(0))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c2(LE(s(s(x0)), s(s(0)))) PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(LE(s(s(x0)), s(s(s(x2))))) SUMITER(nil, z0) -> c10(IFSUM(isempty(nil), nil, z0, plusIter(z0, error, 0)), PLUS(z0, error)) SUMITER(cons(z0, z1), z2) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, plusIter(z2, z0, 0)), PLUS(z2, z0)) PLUSITER(s(0), s(x1), s(0)) -> c2(LE(s(0), s(0))) SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plusIter(z2, head(cons(z0, z1)), 0)), PLUS(z2, z0)) IFPLUS(false, s(s(x0)), s(x1), s(s(s(x2)))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2)))))) IFPLUS(false, s(s(x0)), s(x1), s(s(0))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0))))) PLUSITER(s(s(z0)), s(x1), s(0)) -> c(IFPLUS(false, s(s(z0)), s(x1), s(0))) IFPLUS(false, s(s(x0)), s(x1), s(0)) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(0)))) PLUSITER(s(s(z0)), s(x1), s(0)) -> c(LE(s(s(z0)), s(0))) S tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1)))) SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0))) IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0))) PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(IFPLUS(le(x0, s(x2)), s(s(x0)), s(x1), s(s(s(x2)))), LE(s(s(x0)), s(s(s(x2))))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(x1), s(s(0)))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(LE(s(s(x0)), s(s(0)))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0))) PLUSITER(s(x0), s(x1), s(0)) -> c2(LE(s(x0), s(0))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c2(LE(s(s(x0)), s(s(0)))) PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(LE(s(s(x0)), s(s(s(x2))))) SUMITER(nil, z0) -> c10(IFSUM(isempty(nil), nil, z0, plusIter(z0, error, 0)), PLUS(z0, error)) SUMITER(cons(z0, z1), z2) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, plusIter(z2, z0, 0)), PLUS(z2, z0)) PLUSITER(s(0), s(x1), s(0)) -> c2(LE(s(0), s(0))) SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plusIter(z2, head(cons(z0, z1)), 0)), PLUS(z2, z0)) IFPLUS(false, s(s(x0)), s(x1), s(s(s(x2)))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2)))))) IFPLUS(false, s(s(x0)), s(x1), s(s(0))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0))))) PLUSITER(s(s(z0)), s(x1), s(0)) -> c(IFPLUS(false, s(s(z0)), s(x1), s(0))) IFPLUS(false, s(s(x0)), s(x1), s(0)) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(0)))) PLUSITER(s(s(z0)), s(x1), s(0)) -> c(LE(s(s(z0)), s(0))) K tuples: SUMITER(nil, x1) -> c10(PLUS(x1, head(nil))) SUMITER(x0, x1) -> c10(PLUS(x1, head(x0))) SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1)))) Defined Rule Symbols: le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1 Defined Pair Symbols: PLUS_2, LE_2, IFSUM_4, PLUSITER_3, SUMITER_2, IFPLUS_4 Compound Symbols: c1_1, c7_1, c12_1, c2_1, c9_1, c10_2, c10_1, c4_1, c2_2, c3_1, c_1 ---------------------------------------- (203) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(IFPLUS(le(x0, s(x2)), s(s(x0)), s(x1), s(s(s(x2)))), LE(s(s(x0)), s(s(s(x2))))) by PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2))))) -> c2(IFPLUS(le(x0, s(s(x2))), s(s(x0)), s(s(x1)), s(s(s(s(x2))))), LE(s(s(x0)), s(s(s(s(x2)))))) PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))) -> c2(IFPLUS(le(x0, s(0)), s(s(x0)), s(s(x1)), s(s(s(0)))), LE(s(s(x0)), s(s(s(0))))) ---------------------------------------- (204) Obligation: Complexity Dependency Tuples Problem Rules: le(s(z0), 0) -> false le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) isempty(nil) -> true isempty(cons(z0, z1)) -> false plus(z0, z1) -> plusIter(z0, z1, 0) head(nil) -> error head(cons(z0, z1)) -> z0 plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2) ifPlus(true, z0, z1, z2) -> z1 ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2)) tail(nil) -> nil tail(cons(z0, z1)) -> z1 Tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(x0, x1) -> c10(PLUS(x1, head(x0))) SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1)))) SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1)))) SUMITER(nil, x0) -> c10(PLUS(x0, error)) SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0))) IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(x1), s(s(0)))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(LE(s(s(x0)), s(s(0)))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0))) PLUSITER(s(x0), s(x1), s(0)) -> c2(LE(s(x0), s(0))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c2(LE(s(s(x0)), s(s(0)))) PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(LE(s(s(x0)), s(s(s(x2))))) SUMITER(nil, z0) -> c10(IFSUM(isempty(nil), nil, z0, plusIter(z0, error, 0)), PLUS(z0, error)) SUMITER(cons(z0, z1), z2) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, plusIter(z2, z0, 0)), PLUS(z2, z0)) PLUSITER(s(0), s(x1), s(0)) -> c2(LE(s(0), s(0))) SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plusIter(z2, head(cons(z0, z1)), 0)), PLUS(z2, z0)) IFPLUS(false, s(s(x0)), s(x1), s(s(s(x2)))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2)))))) IFPLUS(false, s(s(x0)), s(x1), s(s(0))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0))))) PLUSITER(s(s(z0)), s(x1), s(0)) -> c(IFPLUS(false, s(s(z0)), s(x1), s(0))) IFPLUS(false, s(s(x0)), s(x1), s(0)) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(0)))) PLUSITER(s(s(z0)), s(x1), s(0)) -> c(LE(s(s(z0)), s(0))) PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2))))) -> c2(IFPLUS(le(x0, s(s(x2))), s(s(x0)), s(s(x1)), s(s(s(s(x2))))), LE(s(s(x0)), s(s(s(s(x2)))))) PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))) -> c2(IFPLUS(le(x0, s(0)), s(s(x0)), s(s(x1)), s(s(s(0)))), LE(s(s(x0)), s(s(s(0))))) S tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1)))) SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0))) IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(x1), s(s(0)))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(LE(s(s(x0)), s(s(0)))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0))) PLUSITER(s(x0), s(x1), s(0)) -> c2(LE(s(x0), s(0))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c2(LE(s(s(x0)), s(s(0)))) PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(LE(s(s(x0)), s(s(s(x2))))) SUMITER(nil, z0) -> c10(IFSUM(isempty(nil), nil, z0, plusIter(z0, error, 0)), PLUS(z0, error)) SUMITER(cons(z0, z1), z2) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, plusIter(z2, z0, 0)), PLUS(z2, z0)) PLUSITER(s(0), s(x1), s(0)) -> c2(LE(s(0), s(0))) SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plusIter(z2, head(cons(z0, z1)), 0)), PLUS(z2, z0)) IFPLUS(false, s(s(x0)), s(x1), s(s(s(x2)))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2)))))) IFPLUS(false, s(s(x0)), s(x1), s(s(0))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0))))) PLUSITER(s(s(z0)), s(x1), s(0)) -> c(IFPLUS(false, s(s(z0)), s(x1), s(0))) IFPLUS(false, s(s(x0)), s(x1), s(0)) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(0)))) PLUSITER(s(s(z0)), s(x1), s(0)) -> c(LE(s(s(z0)), s(0))) PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2))))) -> c2(IFPLUS(le(x0, s(s(x2))), s(s(x0)), s(s(x1)), s(s(s(s(x2))))), LE(s(s(x0)), s(s(s(s(x2)))))) PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))) -> c2(IFPLUS(le(x0, s(0)), s(s(x0)), s(s(x1)), s(s(s(0)))), LE(s(s(x0)), s(s(s(0))))) K tuples: SUMITER(nil, x1) -> c10(PLUS(x1, head(nil))) SUMITER(x0, x1) -> c10(PLUS(x1, head(x0))) SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1)))) Defined Rule Symbols: le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1 Defined Pair Symbols: PLUS_2, LE_2, IFSUM_4, PLUSITER_3, SUMITER_2, IFPLUS_4 Compound Symbols: c1_1, c7_1, c12_1, c2_1, c9_1, c10_2, c10_1, c4_1, c3_1, c_1, c2_2 ---------------------------------------- (205) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (206) Obligation: Complexity Dependency Tuples Problem Rules: le(s(z0), 0) -> false le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) isempty(nil) -> true isempty(cons(z0, z1)) -> false plus(z0, z1) -> plusIter(z0, z1, 0) head(nil) -> error head(cons(z0, z1)) -> z0 plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2) ifPlus(true, z0, z1, z2) -> z1 ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2)) tail(nil) -> nil tail(cons(z0, z1)) -> z1 Tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(x0, x1) -> c10(PLUS(x1, head(x0))) SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1)))) SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1)))) SUMITER(nil, x0) -> c10(PLUS(x0, error)) SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0))) IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(x1), s(s(0)))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(LE(s(s(x0)), s(s(0)))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0))) PLUSITER(s(x0), s(x1), s(0)) -> c2(LE(s(x0), s(0))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c2(LE(s(s(x0)), s(s(0)))) PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(LE(s(s(x0)), s(s(s(x2))))) SUMITER(nil, z0) -> c10(IFSUM(isempty(nil), nil, z0, plusIter(z0, error, 0)), PLUS(z0, error)) SUMITER(cons(z0, z1), z2) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, plusIter(z2, z0, 0)), PLUS(z2, z0)) PLUSITER(s(0), s(x1), s(0)) -> c2(LE(s(0), s(0))) SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plusIter(z2, head(cons(z0, z1)), 0)), PLUS(z2, z0)) IFPLUS(false, s(s(x0)), s(x1), s(s(s(x2)))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2)))))) IFPLUS(false, s(s(x0)), s(x1), s(s(0))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0))))) PLUSITER(s(s(z0)), s(x1), s(0)) -> c(IFPLUS(false, s(s(z0)), s(x1), s(0))) IFPLUS(false, s(s(x0)), s(x1), s(0)) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(0)))) PLUSITER(s(s(z0)), s(x1), s(0)) -> c(LE(s(s(z0)), s(0))) PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2))))) -> c2(IFPLUS(le(x0, s(s(x2))), s(s(x0)), s(s(x1)), s(s(s(s(x2))))), LE(s(s(x0)), s(s(s(s(x2)))))) PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))) -> c5(IFPLUS(le(x0, s(0)), s(s(x0)), s(s(x1)), s(s(s(0))))) PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))) -> c5(LE(s(s(x0)), s(s(s(0))))) S tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1)))) SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0))) IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(x1), s(s(0)))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(LE(s(s(x0)), s(s(0)))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0))) PLUSITER(s(x0), s(x1), s(0)) -> c2(LE(s(x0), s(0))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c2(LE(s(s(x0)), s(s(0)))) PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(LE(s(s(x0)), s(s(s(x2))))) SUMITER(nil, z0) -> c10(IFSUM(isempty(nil), nil, z0, plusIter(z0, error, 0)), PLUS(z0, error)) SUMITER(cons(z0, z1), z2) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, plusIter(z2, z0, 0)), PLUS(z2, z0)) PLUSITER(s(0), s(x1), s(0)) -> c2(LE(s(0), s(0))) SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plusIter(z2, head(cons(z0, z1)), 0)), PLUS(z2, z0)) IFPLUS(false, s(s(x0)), s(x1), s(s(s(x2)))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2)))))) IFPLUS(false, s(s(x0)), s(x1), s(s(0))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0))))) PLUSITER(s(s(z0)), s(x1), s(0)) -> c(IFPLUS(false, s(s(z0)), s(x1), s(0))) IFPLUS(false, s(s(x0)), s(x1), s(0)) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(0)))) PLUSITER(s(s(z0)), s(x1), s(0)) -> c(LE(s(s(z0)), s(0))) PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2))))) -> c2(IFPLUS(le(x0, s(s(x2))), s(s(x0)), s(s(x1)), s(s(s(s(x2))))), LE(s(s(x0)), s(s(s(s(x2)))))) PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))) -> c5(IFPLUS(le(x0, s(0)), s(s(x0)), s(s(x1)), s(s(s(0))))) PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))) -> c5(LE(s(s(x0)), s(s(s(0))))) K tuples: SUMITER(nil, x1) -> c10(PLUS(x1, head(nil))) SUMITER(x0, x1) -> c10(PLUS(x1, head(x0))) SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1)))) Defined Rule Symbols: le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1 Defined Pair Symbols: PLUS_2, LE_2, IFSUM_4, PLUSITER_3, SUMITER_2, IFPLUS_4 Compound Symbols: c1_1, c7_1, c12_1, c2_1, c9_1, c10_2, c10_1, c4_1, c3_1, c_1, c2_2, c5_1 ---------------------------------------- (207) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(x1), s(s(0)))) by PLUSITER(s(s(x0)), s(s(x1)), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(s(x1)), s(s(0)))) ---------------------------------------- (208) Obligation: Complexity Dependency Tuples Problem Rules: le(s(z0), 0) -> false le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) isempty(nil) -> true isempty(cons(z0, z1)) -> false plus(z0, z1) -> plusIter(z0, z1, 0) head(nil) -> error head(cons(z0, z1)) -> z0 plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2) ifPlus(true, z0, z1, z2) -> z1 ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2)) tail(nil) -> nil tail(cons(z0, z1)) -> z1 Tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(x0, x1) -> c10(PLUS(x1, head(x0))) SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1)))) SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1)))) SUMITER(nil, x0) -> c10(PLUS(x0, error)) SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0))) IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(LE(s(s(x0)), s(s(0)))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0))) PLUSITER(s(x0), s(x1), s(0)) -> c2(LE(s(x0), s(0))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c2(LE(s(s(x0)), s(s(0)))) PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(LE(s(s(x0)), s(s(s(x2))))) SUMITER(nil, z0) -> c10(IFSUM(isempty(nil), nil, z0, plusIter(z0, error, 0)), PLUS(z0, error)) SUMITER(cons(z0, z1), z2) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, plusIter(z2, z0, 0)), PLUS(z2, z0)) PLUSITER(s(0), s(x1), s(0)) -> c2(LE(s(0), s(0))) SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plusIter(z2, head(cons(z0, z1)), 0)), PLUS(z2, z0)) IFPLUS(false, s(s(x0)), s(x1), s(s(s(x2)))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2)))))) IFPLUS(false, s(s(x0)), s(x1), s(s(0))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0))))) PLUSITER(s(s(z0)), s(x1), s(0)) -> c(IFPLUS(false, s(s(z0)), s(x1), s(0))) IFPLUS(false, s(s(x0)), s(x1), s(0)) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(0)))) PLUSITER(s(s(z0)), s(x1), s(0)) -> c(LE(s(s(z0)), s(0))) PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2))))) -> c2(IFPLUS(le(x0, s(s(x2))), s(s(x0)), s(s(x1)), s(s(s(s(x2))))), LE(s(s(x0)), s(s(s(s(x2)))))) PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))) -> c5(IFPLUS(le(x0, s(0)), s(s(x0)), s(s(x1)), s(s(s(0))))) PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))) -> c5(LE(s(s(x0)), s(s(s(0))))) PLUSITER(s(s(x0)), s(s(x1)), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(s(x1)), s(s(0)))) S tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1)))) SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0))) IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(LE(s(s(x0)), s(s(0)))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0))) PLUSITER(s(x0), s(x1), s(0)) -> c2(LE(s(x0), s(0))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c2(LE(s(s(x0)), s(s(0)))) PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(LE(s(s(x0)), s(s(s(x2))))) SUMITER(nil, z0) -> c10(IFSUM(isempty(nil), nil, z0, plusIter(z0, error, 0)), PLUS(z0, error)) SUMITER(cons(z0, z1), z2) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, plusIter(z2, z0, 0)), PLUS(z2, z0)) PLUSITER(s(0), s(x1), s(0)) -> c2(LE(s(0), s(0))) SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plusIter(z2, head(cons(z0, z1)), 0)), PLUS(z2, z0)) IFPLUS(false, s(s(x0)), s(x1), s(s(s(x2)))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2)))))) IFPLUS(false, s(s(x0)), s(x1), s(s(0))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0))))) PLUSITER(s(s(z0)), s(x1), s(0)) -> c(IFPLUS(false, s(s(z0)), s(x1), s(0))) IFPLUS(false, s(s(x0)), s(x1), s(0)) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(0)))) PLUSITER(s(s(z0)), s(x1), s(0)) -> c(LE(s(s(z0)), s(0))) PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2))))) -> c2(IFPLUS(le(x0, s(s(x2))), s(s(x0)), s(s(x1)), s(s(s(s(x2))))), LE(s(s(x0)), s(s(s(s(x2)))))) PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))) -> c5(IFPLUS(le(x0, s(0)), s(s(x0)), s(s(x1)), s(s(s(0))))) PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))) -> c5(LE(s(s(x0)), s(s(s(0))))) PLUSITER(s(s(x0)), s(s(x1)), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(s(x1)), s(s(0)))) K tuples: SUMITER(nil, x1) -> c10(PLUS(x1, head(nil))) SUMITER(x0, x1) -> c10(PLUS(x1, head(x0))) SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1)))) Defined Rule Symbols: le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1 Defined Pair Symbols: PLUS_2, LE_2, IFSUM_4, PLUSITER_3, SUMITER_2, IFPLUS_4 Compound Symbols: c1_1, c7_1, c12_1, c2_1, c9_1, c10_2, c10_1, c4_1, c3_1, c_1, c2_2, c5_1 ---------------------------------------- (209) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(LE(s(s(x0)), s(s(0)))) by PLUSITER(s(s(x0)), s(s(x1)), s(s(0))) -> c3(LE(s(s(x0)), s(s(0)))) ---------------------------------------- (210) Obligation: Complexity Dependency Tuples Problem Rules: le(s(z0), 0) -> false le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) isempty(nil) -> true isempty(cons(z0, z1)) -> false plus(z0, z1) -> plusIter(z0, z1, 0) head(nil) -> error head(cons(z0, z1)) -> z0 plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2) ifPlus(true, z0, z1, z2) -> z1 ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2)) tail(nil) -> nil tail(cons(z0, z1)) -> z1 Tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(x0, x1) -> c10(PLUS(x1, head(x0))) SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1)))) SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1)))) SUMITER(nil, x0) -> c10(PLUS(x0, error)) SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0))) IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0))) PLUSITER(s(x0), s(x1), s(0)) -> c2(LE(s(x0), s(0))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c2(LE(s(s(x0)), s(s(0)))) PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(LE(s(s(x0)), s(s(s(x2))))) SUMITER(nil, z0) -> c10(IFSUM(isempty(nil), nil, z0, plusIter(z0, error, 0)), PLUS(z0, error)) SUMITER(cons(z0, z1), z2) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, plusIter(z2, z0, 0)), PLUS(z2, z0)) PLUSITER(s(0), s(x1), s(0)) -> c2(LE(s(0), s(0))) SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plusIter(z2, head(cons(z0, z1)), 0)), PLUS(z2, z0)) IFPLUS(false, s(s(x0)), s(x1), s(s(s(x2)))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2)))))) IFPLUS(false, s(s(x0)), s(x1), s(s(0))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0))))) PLUSITER(s(s(z0)), s(x1), s(0)) -> c(IFPLUS(false, s(s(z0)), s(x1), s(0))) IFPLUS(false, s(s(x0)), s(x1), s(0)) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(0)))) PLUSITER(s(s(z0)), s(x1), s(0)) -> c(LE(s(s(z0)), s(0))) PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2))))) -> c2(IFPLUS(le(x0, s(s(x2))), s(s(x0)), s(s(x1)), s(s(s(s(x2))))), LE(s(s(x0)), s(s(s(s(x2)))))) PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))) -> c5(IFPLUS(le(x0, s(0)), s(s(x0)), s(s(x1)), s(s(s(0))))) PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))) -> c5(LE(s(s(x0)), s(s(s(0))))) PLUSITER(s(s(x0)), s(s(x1)), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(s(x1)), s(s(0)))) PLUSITER(s(s(x0)), s(s(x1)), s(s(0))) -> c3(LE(s(s(x0)), s(s(0)))) S tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1)))) SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0))) IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0))) PLUSITER(s(x0), s(x1), s(0)) -> c2(LE(s(x0), s(0))) PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c2(LE(s(s(x0)), s(s(0)))) PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(LE(s(s(x0)), s(s(s(x2))))) SUMITER(nil, z0) -> c10(IFSUM(isempty(nil), nil, z0, plusIter(z0, error, 0)), PLUS(z0, error)) SUMITER(cons(z0, z1), z2) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, plusIter(z2, z0, 0)), PLUS(z2, z0)) PLUSITER(s(0), s(x1), s(0)) -> c2(LE(s(0), s(0))) SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plusIter(z2, head(cons(z0, z1)), 0)), PLUS(z2, z0)) IFPLUS(false, s(s(x0)), s(x1), s(s(s(x2)))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2)))))) IFPLUS(false, s(s(x0)), s(x1), s(s(0))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0))))) PLUSITER(s(s(z0)), s(x1), s(0)) -> c(IFPLUS(false, s(s(z0)), s(x1), s(0))) IFPLUS(false, s(s(x0)), s(x1), s(0)) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(0)))) PLUSITER(s(s(z0)), s(x1), s(0)) -> c(LE(s(s(z0)), s(0))) PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2))))) -> c2(IFPLUS(le(x0, s(s(x2))), s(s(x0)), s(s(x1)), s(s(s(s(x2))))), LE(s(s(x0)), s(s(s(s(x2)))))) PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))) -> c5(IFPLUS(le(x0, s(0)), s(s(x0)), s(s(x1)), s(s(s(0))))) PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))) -> c5(LE(s(s(x0)), s(s(s(0))))) PLUSITER(s(s(x0)), s(s(x1)), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(s(x1)), s(s(0)))) PLUSITER(s(s(x0)), s(s(x1)), s(s(0))) -> c3(LE(s(s(x0)), s(s(0)))) K tuples: SUMITER(nil, x1) -> c10(PLUS(x1, head(nil))) SUMITER(x0, x1) -> c10(PLUS(x1, head(x0))) SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1)))) Defined Rule Symbols: le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1 Defined Pair Symbols: PLUS_2, LE_2, IFSUM_4, PLUSITER_3, SUMITER_2, IFPLUS_4 Compound Symbols: c1_1, c7_1, c12_1, c2_1, c9_1, c10_2, c10_1, c4_1, c_1, c2_2, c5_1, c3_1 ---------------------------------------- (211) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c2(LE(s(s(x0)), s(s(0)))) by PLUSITER(s(s(x0)), s(s(x1)), s(s(0))) -> c2(LE(s(s(x0)), s(s(0)))) ---------------------------------------- (212) Obligation: Complexity Dependency Tuples Problem Rules: le(s(z0), 0) -> false le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) isempty(nil) -> true isempty(cons(z0, z1)) -> false plus(z0, z1) -> plusIter(z0, z1, 0) head(nil) -> error head(cons(z0, z1)) -> z0 plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2) ifPlus(true, z0, z1, z2) -> z1 ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2)) tail(nil) -> nil tail(cons(z0, z1)) -> z1 Tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(x0, x1) -> c10(PLUS(x1, head(x0))) SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1)))) SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1)))) SUMITER(nil, x0) -> c10(PLUS(x0, error)) SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0))) IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0))) PLUSITER(s(x0), s(x1), s(0)) -> c2(LE(s(x0), s(0))) PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(LE(s(s(x0)), s(s(s(x2))))) SUMITER(nil, z0) -> c10(IFSUM(isempty(nil), nil, z0, plusIter(z0, error, 0)), PLUS(z0, error)) SUMITER(cons(z0, z1), z2) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, plusIter(z2, z0, 0)), PLUS(z2, z0)) PLUSITER(s(0), s(x1), s(0)) -> c2(LE(s(0), s(0))) SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plusIter(z2, head(cons(z0, z1)), 0)), PLUS(z2, z0)) IFPLUS(false, s(s(x0)), s(x1), s(s(s(x2)))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2)))))) IFPLUS(false, s(s(x0)), s(x1), s(s(0))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0))))) PLUSITER(s(s(z0)), s(x1), s(0)) -> c(IFPLUS(false, s(s(z0)), s(x1), s(0))) IFPLUS(false, s(s(x0)), s(x1), s(0)) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(0)))) PLUSITER(s(s(z0)), s(x1), s(0)) -> c(LE(s(s(z0)), s(0))) PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2))))) -> c2(IFPLUS(le(x0, s(s(x2))), s(s(x0)), s(s(x1)), s(s(s(s(x2))))), LE(s(s(x0)), s(s(s(s(x2)))))) PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))) -> c5(IFPLUS(le(x0, s(0)), s(s(x0)), s(s(x1)), s(s(s(0))))) PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))) -> c5(LE(s(s(x0)), s(s(s(0))))) PLUSITER(s(s(x0)), s(s(x1)), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(s(x1)), s(s(0)))) PLUSITER(s(s(x0)), s(s(x1)), s(s(0))) -> c3(LE(s(s(x0)), s(s(0)))) PLUSITER(s(s(x0)), s(s(x1)), s(s(0))) -> c2(LE(s(s(x0)), s(s(0)))) S tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1)))) SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0))) IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0))) PLUSITER(s(x0), s(x1), s(0)) -> c2(LE(s(x0), s(0))) PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(LE(s(s(x0)), s(s(s(x2))))) SUMITER(nil, z0) -> c10(IFSUM(isempty(nil), nil, z0, plusIter(z0, error, 0)), PLUS(z0, error)) SUMITER(cons(z0, z1), z2) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, plusIter(z2, z0, 0)), PLUS(z2, z0)) PLUSITER(s(0), s(x1), s(0)) -> c2(LE(s(0), s(0))) SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plusIter(z2, head(cons(z0, z1)), 0)), PLUS(z2, z0)) IFPLUS(false, s(s(x0)), s(x1), s(s(s(x2)))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2)))))) IFPLUS(false, s(s(x0)), s(x1), s(s(0))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0))))) PLUSITER(s(s(z0)), s(x1), s(0)) -> c(IFPLUS(false, s(s(z0)), s(x1), s(0))) IFPLUS(false, s(s(x0)), s(x1), s(0)) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(0)))) PLUSITER(s(s(z0)), s(x1), s(0)) -> c(LE(s(s(z0)), s(0))) PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2))))) -> c2(IFPLUS(le(x0, s(s(x2))), s(s(x0)), s(s(x1)), s(s(s(s(x2))))), LE(s(s(x0)), s(s(s(s(x2)))))) PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))) -> c5(IFPLUS(le(x0, s(0)), s(s(x0)), s(s(x1)), s(s(s(0))))) PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))) -> c5(LE(s(s(x0)), s(s(s(0))))) PLUSITER(s(s(x0)), s(s(x1)), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(s(x1)), s(s(0)))) PLUSITER(s(s(x0)), s(s(x1)), s(s(0))) -> c3(LE(s(s(x0)), s(s(0)))) PLUSITER(s(s(x0)), s(s(x1)), s(s(0))) -> c2(LE(s(s(x0)), s(s(0)))) K tuples: SUMITER(nil, x1) -> c10(PLUS(x1, head(nil))) SUMITER(x0, x1) -> c10(PLUS(x1, head(x0))) SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1)))) Defined Rule Symbols: le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1 Defined Pair Symbols: PLUS_2, LE_2, IFSUM_4, PLUSITER_3, SUMITER_2, IFPLUS_4 Compound Symbols: c1_1, c7_1, c12_1, c2_1, c9_1, c10_2, c10_1, c4_1, c_1, c2_2, c5_1, c3_1 ---------------------------------------- (213) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(LE(s(s(x0)), s(s(s(x2))))) by PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2))))) -> c2(LE(s(s(x0)), s(s(s(s(x2)))))) PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))) -> c2(LE(s(s(x0)), s(s(s(0))))) ---------------------------------------- (214) Obligation: Complexity Dependency Tuples Problem Rules: le(s(z0), 0) -> false le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) isempty(nil) -> true isempty(cons(z0, z1)) -> false plus(z0, z1) -> plusIter(z0, z1, 0) head(nil) -> error head(cons(z0, z1)) -> z0 plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2) ifPlus(true, z0, z1, z2) -> z1 ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2)) tail(nil) -> nil tail(cons(z0, z1)) -> z1 Tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(x0, x1) -> c10(PLUS(x1, head(x0))) SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1)))) SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1)))) SUMITER(nil, x0) -> c10(PLUS(x0, error)) SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0))) IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0))) PLUSITER(s(x0), s(x1), s(0)) -> c2(LE(s(x0), s(0))) SUMITER(nil, z0) -> c10(IFSUM(isempty(nil), nil, z0, plusIter(z0, error, 0)), PLUS(z0, error)) SUMITER(cons(z0, z1), z2) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, plusIter(z2, z0, 0)), PLUS(z2, z0)) PLUSITER(s(0), s(x1), s(0)) -> c2(LE(s(0), s(0))) SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plusIter(z2, head(cons(z0, z1)), 0)), PLUS(z2, z0)) IFPLUS(false, s(s(x0)), s(x1), s(s(s(x2)))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2)))))) IFPLUS(false, s(s(x0)), s(x1), s(s(0))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0))))) PLUSITER(s(s(z0)), s(x1), s(0)) -> c(IFPLUS(false, s(s(z0)), s(x1), s(0))) IFPLUS(false, s(s(x0)), s(x1), s(0)) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(0)))) PLUSITER(s(s(z0)), s(x1), s(0)) -> c(LE(s(s(z0)), s(0))) PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2))))) -> c2(IFPLUS(le(x0, s(s(x2))), s(s(x0)), s(s(x1)), s(s(s(s(x2))))), LE(s(s(x0)), s(s(s(s(x2)))))) PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))) -> c5(IFPLUS(le(x0, s(0)), s(s(x0)), s(s(x1)), s(s(s(0))))) PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))) -> c5(LE(s(s(x0)), s(s(s(0))))) PLUSITER(s(s(x0)), s(s(x1)), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(s(x1)), s(s(0)))) PLUSITER(s(s(x0)), s(s(x1)), s(s(0))) -> c3(LE(s(s(x0)), s(s(0)))) PLUSITER(s(s(x0)), s(s(x1)), s(s(0))) -> c2(LE(s(s(x0)), s(s(0)))) PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2))))) -> c2(LE(s(s(x0)), s(s(s(s(x2)))))) PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))) -> c2(LE(s(s(x0)), s(s(s(0))))) S tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1)))) SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0))) IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0))) PLUSITER(s(x0), s(x1), s(0)) -> c2(LE(s(x0), s(0))) SUMITER(nil, z0) -> c10(IFSUM(isempty(nil), nil, z0, plusIter(z0, error, 0)), PLUS(z0, error)) SUMITER(cons(z0, z1), z2) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, plusIter(z2, z0, 0)), PLUS(z2, z0)) PLUSITER(s(0), s(x1), s(0)) -> c2(LE(s(0), s(0))) SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plusIter(z2, head(cons(z0, z1)), 0)), PLUS(z2, z0)) IFPLUS(false, s(s(x0)), s(x1), s(s(s(x2)))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2)))))) IFPLUS(false, s(s(x0)), s(x1), s(s(0))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0))))) PLUSITER(s(s(z0)), s(x1), s(0)) -> c(IFPLUS(false, s(s(z0)), s(x1), s(0))) IFPLUS(false, s(s(x0)), s(x1), s(0)) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(0)))) PLUSITER(s(s(z0)), s(x1), s(0)) -> c(LE(s(s(z0)), s(0))) PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2))))) -> c2(IFPLUS(le(x0, s(s(x2))), s(s(x0)), s(s(x1)), s(s(s(s(x2))))), LE(s(s(x0)), s(s(s(s(x2)))))) PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))) -> c5(IFPLUS(le(x0, s(0)), s(s(x0)), s(s(x1)), s(s(s(0))))) PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))) -> c5(LE(s(s(x0)), s(s(s(0))))) PLUSITER(s(s(x0)), s(s(x1)), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(s(x1)), s(s(0)))) PLUSITER(s(s(x0)), s(s(x1)), s(s(0))) -> c3(LE(s(s(x0)), s(s(0)))) PLUSITER(s(s(x0)), s(s(x1)), s(s(0))) -> c2(LE(s(s(x0)), s(s(0)))) PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2))))) -> c2(LE(s(s(x0)), s(s(s(s(x2)))))) PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))) -> c2(LE(s(s(x0)), s(s(s(0))))) K tuples: SUMITER(nil, x1) -> c10(PLUS(x1, head(nil))) SUMITER(x0, x1) -> c10(PLUS(x1, head(x0))) SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1)))) Defined Rule Symbols: le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1 Defined Pair Symbols: PLUS_2, LE_2, IFSUM_4, PLUSITER_3, SUMITER_2, IFPLUS_4 Compound Symbols: c1_1, c7_1, c12_1, c2_1, c9_1, c10_2, c10_1, c4_1, c_1, c2_2, c5_1, c3_1 ---------------------------------------- (215) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1)))) by SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plus(z2, z0)), PLUS(z2, z0)) ---------------------------------------- (216) Obligation: Complexity Dependency Tuples Problem Rules: le(s(z0), 0) -> false le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) isempty(nil) -> true isempty(cons(z0, z1)) -> false plus(z0, z1) -> plusIter(z0, z1, 0) head(nil) -> error head(cons(z0, z1)) -> z0 plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2) ifPlus(true, z0, z1, z2) -> z1 ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2)) tail(nil) -> nil tail(cons(z0, z1)) -> z1 Tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(x0, x1) -> c10(PLUS(x1, head(x0))) SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1)))) SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1)))) SUMITER(nil, x0) -> c10(PLUS(x0, error)) SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0))) IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0))) PLUSITER(s(x0), s(x1), s(0)) -> c2(LE(s(x0), s(0))) SUMITER(nil, z0) -> c10(IFSUM(isempty(nil), nil, z0, plusIter(z0, error, 0)), PLUS(z0, error)) SUMITER(cons(z0, z1), z2) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, plusIter(z2, z0, 0)), PLUS(z2, z0)) PLUSITER(s(0), s(x1), s(0)) -> c2(LE(s(0), s(0))) SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plusIter(z2, head(cons(z0, z1)), 0)), PLUS(z2, z0)) IFPLUS(false, s(s(x0)), s(x1), s(s(s(x2)))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2)))))) IFPLUS(false, s(s(x0)), s(x1), s(s(0))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0))))) PLUSITER(s(s(z0)), s(x1), s(0)) -> c(IFPLUS(false, s(s(z0)), s(x1), s(0))) IFPLUS(false, s(s(x0)), s(x1), s(0)) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(0)))) PLUSITER(s(s(z0)), s(x1), s(0)) -> c(LE(s(s(z0)), s(0))) PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2))))) -> c2(IFPLUS(le(x0, s(s(x2))), s(s(x0)), s(s(x1)), s(s(s(s(x2))))), LE(s(s(x0)), s(s(s(s(x2)))))) PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))) -> c5(IFPLUS(le(x0, s(0)), s(s(x0)), s(s(x1)), s(s(s(0))))) PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))) -> c5(LE(s(s(x0)), s(s(s(0))))) PLUSITER(s(s(x0)), s(s(x1)), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(s(x1)), s(s(0)))) PLUSITER(s(s(x0)), s(s(x1)), s(s(0))) -> c3(LE(s(s(x0)), s(s(0)))) PLUSITER(s(s(x0)), s(s(x1)), s(s(0))) -> c2(LE(s(s(x0)), s(s(0)))) PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2))))) -> c2(LE(s(s(x0)), s(s(s(s(x2)))))) PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))) -> c2(LE(s(s(x0)), s(s(s(0))))) SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plus(z2, z0)), PLUS(z2, z0)) S tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0))) IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0))) PLUSITER(s(x0), s(x1), s(0)) -> c2(LE(s(x0), s(0))) SUMITER(nil, z0) -> c10(IFSUM(isempty(nil), nil, z0, plusIter(z0, error, 0)), PLUS(z0, error)) SUMITER(cons(z0, z1), z2) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, plusIter(z2, z0, 0)), PLUS(z2, z0)) PLUSITER(s(0), s(x1), s(0)) -> c2(LE(s(0), s(0))) SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plusIter(z2, head(cons(z0, z1)), 0)), PLUS(z2, z0)) IFPLUS(false, s(s(x0)), s(x1), s(s(s(x2)))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2)))))) IFPLUS(false, s(s(x0)), s(x1), s(s(0))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0))))) PLUSITER(s(s(z0)), s(x1), s(0)) -> c(IFPLUS(false, s(s(z0)), s(x1), s(0))) IFPLUS(false, s(s(x0)), s(x1), s(0)) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(0)))) PLUSITER(s(s(z0)), s(x1), s(0)) -> c(LE(s(s(z0)), s(0))) PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2))))) -> c2(IFPLUS(le(x0, s(s(x2))), s(s(x0)), s(s(x1)), s(s(s(s(x2))))), LE(s(s(x0)), s(s(s(s(x2)))))) PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))) -> c5(IFPLUS(le(x0, s(0)), s(s(x0)), s(s(x1)), s(s(s(0))))) PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))) -> c5(LE(s(s(x0)), s(s(s(0))))) PLUSITER(s(s(x0)), s(s(x1)), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(s(x1)), s(s(0)))) PLUSITER(s(s(x0)), s(s(x1)), s(s(0))) -> c3(LE(s(s(x0)), s(s(0)))) PLUSITER(s(s(x0)), s(s(x1)), s(s(0))) -> c2(LE(s(s(x0)), s(s(0)))) PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2))))) -> c2(LE(s(s(x0)), s(s(s(s(x2)))))) PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))) -> c2(LE(s(s(x0)), s(s(s(0))))) SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plus(z2, z0)), PLUS(z2, z0)) K tuples: SUMITER(nil, x1) -> c10(PLUS(x1, head(nil))) SUMITER(x0, x1) -> c10(PLUS(x1, head(x0))) SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1)))) Defined Rule Symbols: le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1 Defined Pair Symbols: PLUS_2, LE_2, IFSUM_4, PLUSITER_3, SUMITER_2, IFPLUS_4 Compound Symbols: c1_1, c7_1, c12_1, c2_1, c9_1, c10_2, c10_1, c4_1, c_1, c2_2, c5_1, c3_1 ---------------------------------------- (217) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1)))) by SUMITER(cons(z0, z1), z2) -> c10(PLUS(z2, z0)) ---------------------------------------- (218) Obligation: Complexity Dependency Tuples Problem Rules: le(s(z0), 0) -> false le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) isempty(nil) -> true isempty(cons(z0, z1)) -> false plus(z0, z1) -> plusIter(z0, z1, 0) head(nil) -> error head(cons(z0, z1)) -> z0 plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2) ifPlus(true, z0, z1, z2) -> z1 ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2)) tail(nil) -> nil tail(cons(z0, z1)) -> z1 Tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1)))) SUMITER(x0, x1) -> c10(PLUS(x1, head(x0))) SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1)))) SUMITER(nil, x0) -> c10(PLUS(x0, error)) SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0))) IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0))) PLUSITER(s(x0), s(x1), s(0)) -> c2(LE(s(x0), s(0))) SUMITER(nil, z0) -> c10(IFSUM(isempty(nil), nil, z0, plusIter(z0, error, 0)), PLUS(z0, error)) SUMITER(cons(z0, z1), z2) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, plusIter(z2, z0, 0)), PLUS(z2, z0)) PLUSITER(s(0), s(x1), s(0)) -> c2(LE(s(0), s(0))) SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plusIter(z2, head(cons(z0, z1)), 0)), PLUS(z2, z0)) IFPLUS(false, s(s(x0)), s(x1), s(s(s(x2)))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2)))))) IFPLUS(false, s(s(x0)), s(x1), s(s(0))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0))))) PLUSITER(s(s(z0)), s(x1), s(0)) -> c(IFPLUS(false, s(s(z0)), s(x1), s(0))) IFPLUS(false, s(s(x0)), s(x1), s(0)) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(0)))) PLUSITER(s(s(z0)), s(x1), s(0)) -> c(LE(s(s(z0)), s(0))) PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2))))) -> c2(IFPLUS(le(x0, s(s(x2))), s(s(x0)), s(s(x1)), s(s(s(s(x2))))), LE(s(s(x0)), s(s(s(s(x2)))))) PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))) -> c5(IFPLUS(le(x0, s(0)), s(s(x0)), s(s(x1)), s(s(s(0))))) PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))) -> c5(LE(s(s(x0)), s(s(s(0))))) PLUSITER(s(s(x0)), s(s(x1)), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(s(x1)), s(s(0)))) PLUSITER(s(s(x0)), s(s(x1)), s(s(0))) -> c3(LE(s(s(x0)), s(s(0)))) PLUSITER(s(s(x0)), s(s(x1)), s(s(0))) -> c2(LE(s(s(x0)), s(s(0)))) PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2))))) -> c2(LE(s(s(x0)), s(s(s(s(x2)))))) PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))) -> c2(LE(s(s(x0)), s(s(s(0))))) SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plus(z2, z0)), PLUS(z2, z0)) SUMITER(cons(z0, z1), z2) -> c10(PLUS(z2, z0)) S tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0))) IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0))) PLUSITER(s(x0), s(x1), s(0)) -> c2(LE(s(x0), s(0))) SUMITER(nil, z0) -> c10(IFSUM(isempty(nil), nil, z0, plusIter(z0, error, 0)), PLUS(z0, error)) SUMITER(cons(z0, z1), z2) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, plusIter(z2, z0, 0)), PLUS(z2, z0)) PLUSITER(s(0), s(x1), s(0)) -> c2(LE(s(0), s(0))) SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plusIter(z2, head(cons(z0, z1)), 0)), PLUS(z2, z0)) IFPLUS(false, s(s(x0)), s(x1), s(s(s(x2)))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2)))))) IFPLUS(false, s(s(x0)), s(x1), s(s(0))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0))))) PLUSITER(s(s(z0)), s(x1), s(0)) -> c(IFPLUS(false, s(s(z0)), s(x1), s(0))) IFPLUS(false, s(s(x0)), s(x1), s(0)) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(0)))) PLUSITER(s(s(z0)), s(x1), s(0)) -> c(LE(s(s(z0)), s(0))) PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2))))) -> c2(IFPLUS(le(x0, s(s(x2))), s(s(x0)), s(s(x1)), s(s(s(s(x2))))), LE(s(s(x0)), s(s(s(s(x2)))))) PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))) -> c5(IFPLUS(le(x0, s(0)), s(s(x0)), s(s(x1)), s(s(s(0))))) PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))) -> c5(LE(s(s(x0)), s(s(s(0))))) PLUSITER(s(s(x0)), s(s(x1)), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(s(x1)), s(s(0)))) PLUSITER(s(s(x0)), s(s(x1)), s(s(0))) -> c3(LE(s(s(x0)), s(s(0)))) PLUSITER(s(s(x0)), s(s(x1)), s(s(0))) -> c2(LE(s(s(x0)), s(s(0)))) PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2))))) -> c2(LE(s(s(x0)), s(s(s(s(x2)))))) PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))) -> c2(LE(s(s(x0)), s(s(s(0))))) SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plus(z2, z0)), PLUS(z2, z0)) K tuples: SUMITER(nil, x1) -> c10(PLUS(x1, head(nil))) SUMITER(x0, x1) -> c10(PLUS(x1, head(x0))) SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1)))) Defined Rule Symbols: le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1 Defined Pair Symbols: PLUS_2, LE_2, IFSUM_4, PLUSITER_3, SUMITER_2, IFPLUS_4 Compound Symbols: c1_1, c7_1, c12_1, c2_1, c9_1, c10_2, c10_1, c4_1, c_1, c2_2, c5_1, c3_1 ---------------------------------------- (219) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace SUMITER(cons(x0, x1), z0) -> c10(IFSUM(false, cons(x0, x1), z0, plusIter(z0, head(cons(x0, x1)), 0)), PLUS(z0, head(cons(x0, x1)))) by SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plusIter(z2, head(cons(z0, z1)), 0)), PLUS(z2, z0)) ---------------------------------------- (220) Obligation: Complexity Dependency Tuples Problem Rules: le(s(z0), 0) -> false le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) isempty(nil) -> true isempty(cons(z0, z1)) -> false plus(z0, z1) -> plusIter(z0, z1, 0) head(nil) -> error head(cons(z0, z1)) -> z0 plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2) ifPlus(true, z0, z1, z2) -> z1 ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2)) tail(nil) -> nil tail(cons(z0, z1)) -> z1 Tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(x0, x1) -> c10(PLUS(x1, head(x0))) SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1)))) SUMITER(nil, x0) -> c10(PLUS(x0, error)) SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0))) IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0))) PLUSITER(s(x0), s(x1), s(0)) -> c2(LE(s(x0), s(0))) SUMITER(nil, z0) -> c10(IFSUM(isempty(nil), nil, z0, plusIter(z0, error, 0)), PLUS(z0, error)) SUMITER(cons(z0, z1), z2) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, plusIter(z2, z0, 0)), PLUS(z2, z0)) PLUSITER(s(0), s(x1), s(0)) -> c2(LE(s(0), s(0))) SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plusIter(z2, head(cons(z0, z1)), 0)), PLUS(z2, z0)) IFPLUS(false, s(s(x0)), s(x1), s(s(s(x2)))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2)))))) IFPLUS(false, s(s(x0)), s(x1), s(s(0))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0))))) PLUSITER(s(s(z0)), s(x1), s(0)) -> c(IFPLUS(false, s(s(z0)), s(x1), s(0))) IFPLUS(false, s(s(x0)), s(x1), s(0)) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(0)))) PLUSITER(s(s(z0)), s(x1), s(0)) -> c(LE(s(s(z0)), s(0))) PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2))))) -> c2(IFPLUS(le(x0, s(s(x2))), s(s(x0)), s(s(x1)), s(s(s(s(x2))))), LE(s(s(x0)), s(s(s(s(x2)))))) PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))) -> c5(IFPLUS(le(x0, s(0)), s(s(x0)), s(s(x1)), s(s(s(0))))) PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))) -> c5(LE(s(s(x0)), s(s(s(0))))) PLUSITER(s(s(x0)), s(s(x1)), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(s(x1)), s(s(0)))) PLUSITER(s(s(x0)), s(s(x1)), s(s(0))) -> c3(LE(s(s(x0)), s(s(0)))) PLUSITER(s(s(x0)), s(s(x1)), s(s(0))) -> c2(LE(s(s(x0)), s(s(0)))) PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2))))) -> c2(LE(s(s(x0)), s(s(s(s(x2)))))) PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))) -> c2(LE(s(s(x0)), s(s(s(0))))) SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plus(z2, z0)), PLUS(z2, z0)) SUMITER(cons(z0, z1), z2) -> c10(PLUS(z2, z0)) S tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0))) IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0))) PLUSITER(s(x0), s(x1), s(0)) -> c2(LE(s(x0), s(0))) SUMITER(nil, z0) -> c10(IFSUM(isempty(nil), nil, z0, plusIter(z0, error, 0)), PLUS(z0, error)) SUMITER(cons(z0, z1), z2) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, plusIter(z2, z0, 0)), PLUS(z2, z0)) PLUSITER(s(0), s(x1), s(0)) -> c2(LE(s(0), s(0))) SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plusIter(z2, head(cons(z0, z1)), 0)), PLUS(z2, z0)) IFPLUS(false, s(s(x0)), s(x1), s(s(s(x2)))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2)))))) IFPLUS(false, s(s(x0)), s(x1), s(s(0))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0))))) PLUSITER(s(s(z0)), s(x1), s(0)) -> c(IFPLUS(false, s(s(z0)), s(x1), s(0))) IFPLUS(false, s(s(x0)), s(x1), s(0)) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(0)))) PLUSITER(s(s(z0)), s(x1), s(0)) -> c(LE(s(s(z0)), s(0))) PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2))))) -> c2(IFPLUS(le(x0, s(s(x2))), s(s(x0)), s(s(x1)), s(s(s(s(x2))))), LE(s(s(x0)), s(s(s(s(x2)))))) PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))) -> c5(IFPLUS(le(x0, s(0)), s(s(x0)), s(s(x1)), s(s(s(0))))) PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))) -> c5(LE(s(s(x0)), s(s(s(0))))) PLUSITER(s(s(x0)), s(s(x1)), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(s(x1)), s(s(0)))) PLUSITER(s(s(x0)), s(s(x1)), s(s(0))) -> c3(LE(s(s(x0)), s(s(0)))) PLUSITER(s(s(x0)), s(s(x1)), s(s(0))) -> c2(LE(s(s(x0)), s(s(0)))) PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2))))) -> c2(LE(s(s(x0)), s(s(s(s(x2)))))) PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))) -> c2(LE(s(s(x0)), s(s(s(0))))) SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plus(z2, z0)), PLUS(z2, z0)) K tuples: SUMITER(nil, x1) -> c10(PLUS(x1, head(nil))) SUMITER(x0, x1) -> c10(PLUS(x1, head(x0))) SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1)))) Defined Rule Symbols: le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1 Defined Pair Symbols: PLUS_2, LE_2, IFSUM_4, PLUSITER_3, SUMITER_2, IFPLUS_4 Compound Symbols: c1_1, c7_1, c12_1, c2_1, c9_1, c10_2, c10_1, c4_1, c_1, c2_2, c5_1, c3_1 ---------------------------------------- (221) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1)))) by SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plus(z2, z0)), PLUS(z2, z0)) ---------------------------------------- (222) Obligation: Complexity Dependency Tuples Problem Rules: le(s(z0), 0) -> false le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) isempty(nil) -> true isempty(cons(z0, z1)) -> false plus(z0, z1) -> plusIter(z0, z1, 0) head(nil) -> error head(cons(z0, z1)) -> z0 plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2) ifPlus(true, z0, z1, z2) -> z1 ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2)) tail(nil) -> nil tail(cons(z0, z1)) -> z1 Tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(x0, x1) -> c10(PLUS(x1, head(x0))) SUMITER(nil, x0) -> c10(PLUS(x0, error)) SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0))) IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0))) PLUSITER(s(x0), s(x1), s(0)) -> c2(LE(s(x0), s(0))) SUMITER(nil, z0) -> c10(IFSUM(isempty(nil), nil, z0, plusIter(z0, error, 0)), PLUS(z0, error)) SUMITER(cons(z0, z1), z2) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, plusIter(z2, z0, 0)), PLUS(z2, z0)) PLUSITER(s(0), s(x1), s(0)) -> c2(LE(s(0), s(0))) SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plusIter(z2, head(cons(z0, z1)), 0)), PLUS(z2, z0)) IFPLUS(false, s(s(x0)), s(x1), s(s(s(x2)))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2)))))) IFPLUS(false, s(s(x0)), s(x1), s(s(0))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0))))) PLUSITER(s(s(z0)), s(x1), s(0)) -> c(IFPLUS(false, s(s(z0)), s(x1), s(0))) IFPLUS(false, s(s(x0)), s(x1), s(0)) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(0)))) PLUSITER(s(s(z0)), s(x1), s(0)) -> c(LE(s(s(z0)), s(0))) PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2))))) -> c2(IFPLUS(le(x0, s(s(x2))), s(s(x0)), s(s(x1)), s(s(s(s(x2))))), LE(s(s(x0)), s(s(s(s(x2)))))) PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))) -> c5(IFPLUS(le(x0, s(0)), s(s(x0)), s(s(x1)), s(s(s(0))))) PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))) -> c5(LE(s(s(x0)), s(s(s(0))))) PLUSITER(s(s(x0)), s(s(x1)), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(s(x1)), s(s(0)))) PLUSITER(s(s(x0)), s(s(x1)), s(s(0))) -> c3(LE(s(s(x0)), s(s(0)))) PLUSITER(s(s(x0)), s(s(x1)), s(s(0))) -> c2(LE(s(s(x0)), s(s(0)))) PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2))))) -> c2(LE(s(s(x0)), s(s(s(s(x2)))))) PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))) -> c2(LE(s(s(x0)), s(s(s(0))))) SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plus(z2, z0)), PLUS(z2, z0)) SUMITER(cons(z0, z1), z2) -> c10(PLUS(z2, z0)) S tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0))) IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0))) PLUSITER(s(x0), s(x1), s(0)) -> c2(LE(s(x0), s(0))) SUMITER(nil, z0) -> c10(IFSUM(isempty(nil), nil, z0, plusIter(z0, error, 0)), PLUS(z0, error)) SUMITER(cons(z0, z1), z2) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, plusIter(z2, z0, 0)), PLUS(z2, z0)) PLUSITER(s(0), s(x1), s(0)) -> c2(LE(s(0), s(0))) SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plusIter(z2, head(cons(z0, z1)), 0)), PLUS(z2, z0)) IFPLUS(false, s(s(x0)), s(x1), s(s(s(x2)))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2)))))) IFPLUS(false, s(s(x0)), s(x1), s(s(0))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0))))) PLUSITER(s(s(z0)), s(x1), s(0)) -> c(IFPLUS(false, s(s(z0)), s(x1), s(0))) IFPLUS(false, s(s(x0)), s(x1), s(0)) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(0)))) PLUSITER(s(s(z0)), s(x1), s(0)) -> c(LE(s(s(z0)), s(0))) PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2))))) -> c2(IFPLUS(le(x0, s(s(x2))), s(s(x0)), s(s(x1)), s(s(s(s(x2))))), LE(s(s(x0)), s(s(s(s(x2)))))) PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))) -> c5(IFPLUS(le(x0, s(0)), s(s(x0)), s(s(x1)), s(s(s(0))))) PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))) -> c5(LE(s(s(x0)), s(s(s(0))))) PLUSITER(s(s(x0)), s(s(x1)), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(s(x1)), s(s(0)))) PLUSITER(s(s(x0)), s(s(x1)), s(s(0))) -> c3(LE(s(s(x0)), s(s(0)))) PLUSITER(s(s(x0)), s(s(x1)), s(s(0))) -> c2(LE(s(s(x0)), s(s(0)))) PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2))))) -> c2(LE(s(s(x0)), s(s(s(s(x2)))))) PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))) -> c2(LE(s(s(x0)), s(s(s(0))))) SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plus(z2, z0)), PLUS(z2, z0)) K tuples: SUMITER(nil, x1) -> c10(PLUS(x1, head(nil))) SUMITER(x0, x1) -> c10(PLUS(x1, head(x0))) SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1)))) Defined Rule Symbols: le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1 Defined Pair Symbols: PLUS_2, LE_2, IFSUM_4, PLUSITER_3, SUMITER_2, IFPLUS_4 Compound Symbols: c1_1, c7_1, c12_1, c2_1, c9_1, c10_2, c10_1, c4_1, c_1, c2_2, c5_1, c3_1 ---------------------------------------- (223) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0))) by SUMITER(nil, z0) -> c9(IFSUM(true, nil, z0, ifPlus(le(z0, 0), z0, error, 0))) ---------------------------------------- (224) Obligation: Complexity Dependency Tuples Problem Rules: le(s(z0), 0) -> false le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) isempty(nil) -> true isempty(cons(z0, z1)) -> false plus(z0, z1) -> plusIter(z0, z1, 0) head(nil) -> error head(cons(z0, z1)) -> z0 plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2) ifPlus(true, z0, z1, z2) -> z1 ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2)) tail(nil) -> nil tail(cons(z0, z1)) -> z1 Tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(x0, x1) -> c10(PLUS(x1, head(x0))) SUMITER(nil, x0) -> c10(PLUS(x0, error)) SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0))) IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0))) PLUSITER(s(x0), s(x1), s(0)) -> c2(LE(s(x0), s(0))) SUMITER(nil, z0) -> c10(IFSUM(isempty(nil), nil, z0, plusIter(z0, error, 0)), PLUS(z0, error)) SUMITER(cons(z0, z1), z2) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, plusIter(z2, z0, 0)), PLUS(z2, z0)) PLUSITER(s(0), s(x1), s(0)) -> c2(LE(s(0), s(0))) SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plusIter(z2, head(cons(z0, z1)), 0)), PLUS(z2, z0)) IFPLUS(false, s(s(x0)), s(x1), s(s(s(x2)))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2)))))) IFPLUS(false, s(s(x0)), s(x1), s(s(0))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0))))) PLUSITER(s(s(z0)), s(x1), s(0)) -> c(IFPLUS(false, s(s(z0)), s(x1), s(0))) IFPLUS(false, s(s(x0)), s(x1), s(0)) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(0)))) PLUSITER(s(s(z0)), s(x1), s(0)) -> c(LE(s(s(z0)), s(0))) PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2))))) -> c2(IFPLUS(le(x0, s(s(x2))), s(s(x0)), s(s(x1)), s(s(s(s(x2))))), LE(s(s(x0)), s(s(s(s(x2)))))) PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))) -> c5(IFPLUS(le(x0, s(0)), s(s(x0)), s(s(x1)), s(s(s(0))))) PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))) -> c5(LE(s(s(x0)), s(s(s(0))))) PLUSITER(s(s(x0)), s(s(x1)), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(s(x1)), s(s(0)))) PLUSITER(s(s(x0)), s(s(x1)), s(s(0))) -> c3(LE(s(s(x0)), s(s(0)))) PLUSITER(s(s(x0)), s(s(x1)), s(s(0))) -> c2(LE(s(s(x0)), s(s(0)))) PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2))))) -> c2(LE(s(s(x0)), s(s(s(s(x2)))))) PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))) -> c2(LE(s(s(x0)), s(s(s(0))))) SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plus(z2, z0)), PLUS(z2, z0)) SUMITER(cons(z0, z1), z2) -> c10(PLUS(z2, z0)) SUMITER(nil, z0) -> c9(IFSUM(true, nil, z0, ifPlus(le(z0, 0), z0, error, 0))) S tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0))) IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0))) PLUSITER(s(x0), s(x1), s(0)) -> c2(LE(s(x0), s(0))) SUMITER(nil, z0) -> c10(IFSUM(isempty(nil), nil, z0, plusIter(z0, error, 0)), PLUS(z0, error)) SUMITER(cons(z0, z1), z2) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, plusIter(z2, z0, 0)), PLUS(z2, z0)) PLUSITER(s(0), s(x1), s(0)) -> c2(LE(s(0), s(0))) SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plusIter(z2, head(cons(z0, z1)), 0)), PLUS(z2, z0)) IFPLUS(false, s(s(x0)), s(x1), s(s(s(x2)))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2)))))) IFPLUS(false, s(s(x0)), s(x1), s(s(0))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0))))) PLUSITER(s(s(z0)), s(x1), s(0)) -> c(IFPLUS(false, s(s(z0)), s(x1), s(0))) IFPLUS(false, s(s(x0)), s(x1), s(0)) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(0)))) PLUSITER(s(s(z0)), s(x1), s(0)) -> c(LE(s(s(z0)), s(0))) PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2))))) -> c2(IFPLUS(le(x0, s(s(x2))), s(s(x0)), s(s(x1)), s(s(s(s(x2))))), LE(s(s(x0)), s(s(s(s(x2)))))) PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))) -> c5(IFPLUS(le(x0, s(0)), s(s(x0)), s(s(x1)), s(s(s(0))))) PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))) -> c5(LE(s(s(x0)), s(s(s(0))))) PLUSITER(s(s(x0)), s(s(x1)), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(s(x1)), s(s(0)))) PLUSITER(s(s(x0)), s(s(x1)), s(s(0))) -> c3(LE(s(s(x0)), s(s(0)))) PLUSITER(s(s(x0)), s(s(x1)), s(s(0))) -> c2(LE(s(s(x0)), s(s(0)))) PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2))))) -> c2(LE(s(s(x0)), s(s(s(s(x2)))))) PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))) -> c2(LE(s(s(x0)), s(s(s(0))))) SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plus(z2, z0)), PLUS(z2, z0)) SUMITER(nil, z0) -> c9(IFSUM(true, nil, z0, ifPlus(le(z0, 0), z0, error, 0))) K tuples: SUMITER(nil, x1) -> c10(PLUS(x1, head(nil))) SUMITER(x0, x1) -> c10(PLUS(x1, head(x0))) SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1)))) Defined Rule Symbols: le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1 Defined Pair Symbols: PLUS_2, LE_2, IFSUM_4, PLUSITER_3, SUMITER_2, IFPLUS_4 Compound Symbols: c1_1, c7_1, c12_1, c2_1, c9_1, c10_2, c10_1, c4_1, c_1, c2_2, c5_1, c3_1 ---------------------------------------- (225) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: SUMITER(nil, z0) -> c9(IFSUM(true, nil, z0, ifPlus(le(z0, 0), z0, error, 0))) ---------------------------------------- (226) Obligation: Complexity Dependency Tuples Problem Rules: le(s(z0), 0) -> false le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) isempty(nil) -> true isempty(cons(z0, z1)) -> false plus(z0, z1) -> plusIter(z0, z1, 0) head(nil) -> error head(cons(z0, z1)) -> z0 plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2) ifPlus(true, z0, z1, z2) -> z1 ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2)) tail(nil) -> nil tail(cons(z0, z1)) -> z1 Tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(x0, x1) -> c10(PLUS(x1, head(x0))) SUMITER(nil, x0) -> c10(PLUS(x0, error)) IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0))) SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0))) PLUSITER(s(x0), s(x1), s(0)) -> c2(LE(s(x0), s(0))) SUMITER(nil, z0) -> c10(IFSUM(isempty(nil), nil, z0, plusIter(z0, error, 0)), PLUS(z0, error)) SUMITER(cons(z0, z1), z2) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, plusIter(z2, z0, 0)), PLUS(z2, z0)) PLUSITER(s(0), s(x1), s(0)) -> c2(LE(s(0), s(0))) SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plusIter(z2, head(cons(z0, z1)), 0)), PLUS(z2, z0)) IFPLUS(false, s(s(x0)), s(x1), s(s(s(x2)))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2)))))) IFPLUS(false, s(s(x0)), s(x1), s(s(0))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0))))) PLUSITER(s(s(z0)), s(x1), s(0)) -> c(IFPLUS(false, s(s(z0)), s(x1), s(0))) IFPLUS(false, s(s(x0)), s(x1), s(0)) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(0)))) PLUSITER(s(s(z0)), s(x1), s(0)) -> c(LE(s(s(z0)), s(0))) PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2))))) -> c2(IFPLUS(le(x0, s(s(x2))), s(s(x0)), s(s(x1)), s(s(s(s(x2))))), LE(s(s(x0)), s(s(s(s(x2)))))) PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))) -> c5(IFPLUS(le(x0, s(0)), s(s(x0)), s(s(x1)), s(s(s(0))))) PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))) -> c5(LE(s(s(x0)), s(s(s(0))))) PLUSITER(s(s(x0)), s(s(x1)), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(s(x1)), s(s(0)))) PLUSITER(s(s(x0)), s(s(x1)), s(s(0))) -> c3(LE(s(s(x0)), s(s(0)))) PLUSITER(s(s(x0)), s(s(x1)), s(s(0))) -> c2(LE(s(s(x0)), s(s(0)))) PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2))))) -> c2(LE(s(s(x0)), s(s(s(s(x2)))))) PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))) -> c2(LE(s(s(x0)), s(s(s(0))))) SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plus(z2, z0)), PLUS(z2, z0)) SUMITER(cons(z0, z1), z2) -> c10(PLUS(z2, z0)) S tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0))) IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0))) PLUSITER(s(x0), s(x1), s(0)) -> c2(LE(s(x0), s(0))) SUMITER(nil, z0) -> c10(IFSUM(isempty(nil), nil, z0, plusIter(z0, error, 0)), PLUS(z0, error)) SUMITER(cons(z0, z1), z2) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, plusIter(z2, z0, 0)), PLUS(z2, z0)) PLUSITER(s(0), s(x1), s(0)) -> c2(LE(s(0), s(0))) SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plusIter(z2, head(cons(z0, z1)), 0)), PLUS(z2, z0)) IFPLUS(false, s(s(x0)), s(x1), s(s(s(x2)))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2)))))) IFPLUS(false, s(s(x0)), s(x1), s(s(0))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0))))) PLUSITER(s(s(z0)), s(x1), s(0)) -> c(IFPLUS(false, s(s(z0)), s(x1), s(0))) IFPLUS(false, s(s(x0)), s(x1), s(0)) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(0)))) PLUSITER(s(s(z0)), s(x1), s(0)) -> c(LE(s(s(z0)), s(0))) PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2))))) -> c2(IFPLUS(le(x0, s(s(x2))), s(s(x0)), s(s(x1)), s(s(s(s(x2))))), LE(s(s(x0)), s(s(s(s(x2)))))) PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))) -> c5(IFPLUS(le(x0, s(0)), s(s(x0)), s(s(x1)), s(s(s(0))))) PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))) -> c5(LE(s(s(x0)), s(s(s(0))))) PLUSITER(s(s(x0)), s(s(x1)), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(s(x1)), s(s(0)))) PLUSITER(s(s(x0)), s(s(x1)), s(s(0))) -> c3(LE(s(s(x0)), s(s(0)))) PLUSITER(s(s(x0)), s(s(x1)), s(s(0))) -> c2(LE(s(s(x0)), s(s(0)))) PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2))))) -> c2(LE(s(s(x0)), s(s(s(s(x2)))))) PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))) -> c2(LE(s(s(x0)), s(s(s(0))))) SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plus(z2, z0)), PLUS(z2, z0)) K tuples: SUMITER(x0, x1) -> c10(PLUS(x1, head(x0))) Defined Rule Symbols: le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1 Defined Pair Symbols: PLUS_2, LE_2, IFSUM_4, PLUSITER_3, SUMITER_2, IFPLUS_4 Compound Symbols: c1_1, c7_1, c12_1, c2_1, c9_1, c10_2, c10_1, c4_1, c_1, c2_2, c5_1, c3_1 ---------------------------------------- (227) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0))) by SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0))) ---------------------------------------- (228) Obligation: Complexity Dependency Tuples Problem Rules: le(s(z0), 0) -> false le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) isempty(nil) -> true isempty(cons(z0, z1)) -> false plus(z0, z1) -> plusIter(z0, z1, 0) head(nil) -> error head(cons(z0, z1)) -> z0 plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2) ifPlus(true, z0, z1, z2) -> z1 ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2)) tail(nil) -> nil tail(cons(z0, z1)) -> z1 Tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0))) SUMITER(x0, x1) -> c10(PLUS(x1, head(x0))) SUMITER(nil, x0) -> c10(PLUS(x0, error)) IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0))) SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0))) PLUSITER(s(x0), s(x1), s(0)) -> c2(LE(s(x0), s(0))) SUMITER(nil, z0) -> c10(IFSUM(isempty(nil), nil, z0, plusIter(z0, error, 0)), PLUS(z0, error)) SUMITER(cons(z0, z1), z2) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, plusIter(z2, z0, 0)), PLUS(z2, z0)) PLUSITER(s(0), s(x1), s(0)) -> c2(LE(s(0), s(0))) SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plusIter(z2, head(cons(z0, z1)), 0)), PLUS(z2, z0)) IFPLUS(false, s(s(x0)), s(x1), s(s(s(x2)))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2)))))) IFPLUS(false, s(s(x0)), s(x1), s(s(0))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0))))) PLUSITER(s(s(z0)), s(x1), s(0)) -> c(IFPLUS(false, s(s(z0)), s(x1), s(0))) IFPLUS(false, s(s(x0)), s(x1), s(0)) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(0)))) PLUSITER(s(s(z0)), s(x1), s(0)) -> c(LE(s(s(z0)), s(0))) PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2))))) -> c2(IFPLUS(le(x0, s(s(x2))), s(s(x0)), s(s(x1)), s(s(s(s(x2))))), LE(s(s(x0)), s(s(s(s(x2)))))) PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))) -> c5(IFPLUS(le(x0, s(0)), s(s(x0)), s(s(x1)), s(s(s(0))))) PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))) -> c5(LE(s(s(x0)), s(s(s(0))))) PLUSITER(s(s(x0)), s(s(x1)), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(s(x1)), s(s(0)))) PLUSITER(s(s(x0)), s(s(x1)), s(s(0))) -> c3(LE(s(s(x0)), s(s(0)))) PLUSITER(s(s(x0)), s(s(x1)), s(s(0))) -> c2(LE(s(s(x0)), s(s(0)))) PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2))))) -> c2(LE(s(s(x0)), s(s(s(s(x2)))))) PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))) -> c2(LE(s(s(x0)), s(s(s(0))))) SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plus(z2, z0)), PLUS(z2, z0)) SUMITER(cons(z0, z1), z2) -> c10(PLUS(z2, z0)) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0))) S tuples: PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) LE(s(z0), s(z1)) -> c7(LE(z0, z1)) IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2)) PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0)) SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0))) SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0))) IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0))) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0))) PLUSITER(s(x0), s(x1), s(0)) -> c2(LE(s(x0), s(0))) SUMITER(nil, z0) -> c10(IFSUM(isempty(nil), nil, z0, plusIter(z0, error, 0)), PLUS(z0, error)) SUMITER(cons(z0, z1), z2) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, plusIter(z2, z0, 0)), PLUS(z2, z0)) PLUSITER(s(0), s(x1), s(0)) -> c2(LE(s(0), s(0))) SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plusIter(z2, head(cons(z0, z1)), 0)), PLUS(z2, z0)) IFPLUS(false, s(s(x0)), s(x1), s(s(s(x2)))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2)))))) IFPLUS(false, s(s(x0)), s(x1), s(s(0))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0))))) PLUSITER(s(s(z0)), s(x1), s(0)) -> c(IFPLUS(false, s(s(z0)), s(x1), s(0))) IFPLUS(false, s(s(x0)), s(x1), s(0)) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(0)))) PLUSITER(s(s(z0)), s(x1), s(0)) -> c(LE(s(s(z0)), s(0))) PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2))))) -> c2(IFPLUS(le(x0, s(s(x2))), s(s(x0)), s(s(x1)), s(s(s(s(x2))))), LE(s(s(x0)), s(s(s(s(x2)))))) PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))) -> c5(IFPLUS(le(x0, s(0)), s(s(x0)), s(s(x1)), s(s(s(0))))) PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))) -> c5(LE(s(s(x0)), s(s(s(0))))) PLUSITER(s(s(x0)), s(s(x1)), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(s(x1)), s(s(0)))) PLUSITER(s(s(x0)), s(s(x1)), s(s(0))) -> c3(LE(s(s(x0)), s(s(0)))) PLUSITER(s(s(x0)), s(s(x1)), s(s(0))) -> c2(LE(s(s(x0)), s(s(0)))) PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2))))) -> c2(LE(s(s(x0)), s(s(s(s(x2)))))) PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))) -> c2(LE(s(s(x0)), s(s(s(0))))) SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plus(z2, z0)), PLUS(z2, z0)) SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0))) K tuples: SUMITER(x0, x1) -> c10(PLUS(x1, head(x0))) Defined Rule Symbols: le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1 Defined Pair Symbols: PLUS_2, LE_2, IFSUM_4, PLUSITER_3, SUMITER_2, IFPLUS_4 Compound Symbols: c1_1, c7_1, c12_1, c2_1, c9_1, c10_2, c10_1, c4_1, c_1, c2_2, c5_1, c3_1 ---------------------------------------- (229) RelTrsToWeightedTrsProof (UPPER BOUND(ID)) Transformed relative TRS to weighted TRS ---------------------------------------- (230) 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: plus(x, y) -> plusIter(x, y, 0) [1] plusIter(x, y, z) -> ifPlus(le(x, z), x, y, z) [1] ifPlus(true, x, y, z) -> y [1] ifPlus(false, x, y, z) -> plusIter(x, s(y), s(z)) [1] le(s(x), 0) -> false [1] le(0, y) -> true [1] le(s(x), s(y)) -> le(x, y) [1] sum(xs) -> sumIter(xs, 0) [1] sumIter(xs, x) -> ifSum(isempty(xs), xs, x, plus(x, head(xs))) [1] ifSum(true, xs, x, y) -> x [1] ifSum(false, xs, x, y) -> sumIter(tail(xs), y) [1] isempty(nil) -> true [1] isempty(cons(x, xs)) -> false [1] head(nil) -> error [1] head(cons(x, xs)) -> x [1] tail(nil) -> nil [1] tail(cons(x, xs)) -> xs [1] a -> b [1] a -> c [1] Rewrite Strategy: INNERMOST ---------------------------------------- (231) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (232) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: plus(x, y) -> plusIter(x, y, 0) [1] plusIter(x, y, z) -> ifPlus(le(x, z), x, y, z) [1] ifPlus(true, x, y, z) -> y [1] ifPlus(false, x, y, z) -> plusIter(x, s(y), s(z)) [1] le(s(x), 0) -> false [1] le(0, y) -> true [1] le(s(x), s(y)) -> le(x, y) [1] sum(xs) -> sumIter(xs, 0) [1] sumIter(xs, x) -> ifSum(isempty(xs), xs, x, plus(x, head(xs))) [1] ifSum(true, xs, x, y) -> x [1] ifSum(false, xs, x, y) -> sumIter(tail(xs), y) [1] isempty(nil) -> true [1] isempty(cons(x, xs)) -> false [1] head(nil) -> error [1] head(cons(x, xs)) -> x [1] tail(nil) -> nil [1] tail(cons(x, xs)) -> xs [1] a -> b [1] a -> c [1] The TRS has the following type information: plus :: 0:s:error -> 0:s:error -> 0:s:error plusIter :: 0:s:error -> 0:s:error -> 0:s:error -> 0:s:error 0 :: 0:s:error ifPlus :: true:false -> 0:s:error -> 0:s:error -> 0:s:error -> 0:s:error le :: 0:s:error -> 0:s:error -> true:false true :: true:false false :: true:false s :: 0:s:error -> 0:s:error sum :: nil:cons -> 0:s:error sumIter :: nil:cons -> 0:s:error -> 0:s:error ifSum :: true:false -> nil:cons -> 0:s:error -> 0:s:error -> 0:s:error isempty :: nil:cons -> true:false head :: nil:cons -> 0:s:error tail :: nil:cons -> nil:cons nil :: nil:cons cons :: 0:s:error -> nil:cons -> nil:cons error :: 0:s:error a :: b:c b :: b:c c :: b:c Rewrite Strategy: INNERMOST ---------------------------------------- (233) 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: sum_1 sumIter_2 ifSum_4 a (c) The following functions are completely defined: le_2 isempty_1 plus_2 head_1 tail_1 plusIter_3 ifPlus_4 Due to the following rules being added: le(v0, v1) -> null_le [0] ifPlus(v0, v1, v2, v3) -> null_ifPlus [0] And the following fresh constants: null_le, null_ifPlus ---------------------------------------- (234) 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: plus(x, y) -> plusIter(x, y, 0) [1] plusIter(x, y, z) -> ifPlus(le(x, z), x, y, z) [1] ifPlus(true, x, y, z) -> y [1] ifPlus(false, x, y, z) -> plusIter(x, s(y), s(z)) [1] le(s(x), 0) -> false [1] le(0, y) -> true [1] le(s(x), s(y)) -> le(x, y) [1] sum(xs) -> sumIter(xs, 0) [1] sumIter(xs, x) -> ifSum(isempty(xs), xs, x, plus(x, head(xs))) [1] ifSum(true, xs, x, y) -> x [1] ifSum(false, xs, x, y) -> sumIter(tail(xs), y) [1] isempty(nil) -> true [1] isempty(cons(x, xs)) -> false [1] head(nil) -> error [1] head(cons(x, xs)) -> x [1] tail(nil) -> nil [1] tail(cons(x, xs)) -> xs [1] a -> b [1] a -> c [1] le(v0, v1) -> null_le [0] ifPlus(v0, v1, v2, v3) -> null_ifPlus [0] The TRS has the following type information: plus :: 0:s:error:null_ifPlus -> 0:s:error:null_ifPlus -> 0:s:error:null_ifPlus plusIter :: 0:s:error:null_ifPlus -> 0:s:error:null_ifPlus -> 0:s:error:null_ifPlus -> 0:s:error:null_ifPlus 0 :: 0:s:error:null_ifPlus ifPlus :: true:false:null_le -> 0:s:error:null_ifPlus -> 0:s:error:null_ifPlus -> 0:s:error:null_ifPlus -> 0:s:error:null_ifPlus le :: 0:s:error:null_ifPlus -> 0:s:error:null_ifPlus -> true:false:null_le true :: true:false:null_le false :: true:false:null_le s :: 0:s:error:null_ifPlus -> 0:s:error:null_ifPlus sum :: nil:cons -> 0:s:error:null_ifPlus sumIter :: nil:cons -> 0:s:error:null_ifPlus -> 0:s:error:null_ifPlus ifSum :: true:false:null_le -> nil:cons -> 0:s:error:null_ifPlus -> 0:s:error:null_ifPlus -> 0:s:error:null_ifPlus isempty :: nil:cons -> true:false:null_le head :: nil:cons -> 0:s:error:null_ifPlus tail :: nil:cons -> nil:cons nil :: nil:cons cons :: 0:s:error:null_ifPlus -> nil:cons -> nil:cons error :: 0:s:error:null_ifPlus a :: b:c b :: b:c c :: b:c null_le :: true:false:null_le null_ifPlus :: 0:s:error:null_ifPlus Rewrite Strategy: INNERMOST ---------------------------------------- (235) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (236) 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: plus(x, y) -> plusIter(x, y, 0) [1] plusIter(s(x'), y, 0) -> ifPlus(false, s(x'), y, 0) [2] plusIter(0, y, z) -> ifPlus(true, 0, y, z) [2] plusIter(s(x''), y, s(y')) -> ifPlus(le(x'', y'), s(x''), y, s(y')) [2] plusIter(x, y, z) -> ifPlus(null_le, x, y, z) [1] ifPlus(true, x, y, z) -> y [1] ifPlus(false, x, y, z) -> plusIter(x, s(y), s(z)) [1] le(s(x), 0) -> false [1] le(0, y) -> true [1] le(s(x), s(y)) -> le(x, y) [1] sum(xs) -> sumIter(xs, 0) [1] sumIter(nil, x) -> ifSum(true, nil, x, plus(x, error)) [3] sumIter(cons(x1, xs'), x) -> ifSum(false, cons(x1, xs'), x, plus(x, x1)) [3] ifSum(true, xs, x, y) -> x [1] ifSum(false, nil, x, y) -> sumIter(nil, y) [2] ifSum(false, cons(x2, xs''), x, y) -> sumIter(xs'', y) [2] isempty(nil) -> true [1] isempty(cons(x, xs)) -> false [1] head(nil) -> error [1] head(cons(x, xs)) -> x [1] tail(nil) -> nil [1] tail(cons(x, xs)) -> xs [1] a -> b [1] a -> c [1] le(v0, v1) -> null_le [0] ifPlus(v0, v1, v2, v3) -> null_ifPlus [0] The TRS has the following type information: plus :: 0:s:error:null_ifPlus -> 0:s:error:null_ifPlus -> 0:s:error:null_ifPlus plusIter :: 0:s:error:null_ifPlus -> 0:s:error:null_ifPlus -> 0:s:error:null_ifPlus -> 0:s:error:null_ifPlus 0 :: 0:s:error:null_ifPlus ifPlus :: true:false:null_le -> 0:s:error:null_ifPlus -> 0:s:error:null_ifPlus -> 0:s:error:null_ifPlus -> 0:s:error:null_ifPlus le :: 0:s:error:null_ifPlus -> 0:s:error:null_ifPlus -> true:false:null_le true :: true:false:null_le false :: true:false:null_le s :: 0:s:error:null_ifPlus -> 0:s:error:null_ifPlus sum :: nil:cons -> 0:s:error:null_ifPlus sumIter :: nil:cons -> 0:s:error:null_ifPlus -> 0:s:error:null_ifPlus ifSum :: true:false:null_le -> nil:cons -> 0:s:error:null_ifPlus -> 0:s:error:null_ifPlus -> 0:s:error:null_ifPlus isempty :: nil:cons -> true:false:null_le head :: nil:cons -> 0:s:error:null_ifPlus tail :: nil:cons -> nil:cons nil :: nil:cons cons :: 0:s:error:null_ifPlus -> nil:cons -> nil:cons error :: 0:s:error:null_ifPlus a :: b:c b :: b:c c :: b:c null_le :: true:false:null_le null_ifPlus :: 0:s:error:null_ifPlus Rewrite Strategy: INNERMOST ---------------------------------------- (237) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: 0 => 0 true => 2 false => 1 nil => 0 error => 1 b => 0 c => 1 null_le => 0 null_ifPlus => 0 ---------------------------------------- (238) Obligation: Complexity RNTS consisting of the following rules: a -{ 1 }-> 1 :|: a -{ 1 }-> 0 :|: head(z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + xs, x >= 0 head(z') -{ 1 }-> 1 :|: z' = 0 ifPlus(z', z'', z1, z2) -{ 1 }-> y :|: z1 = y, z >= 0, z' = 2, z2 = z, x >= 0, y >= 0, z'' = x ifPlus(z', z'', z1, z2) -{ 1 }-> plusIter(x, 1 + y, 1 + z) :|: z1 = y, z >= 0, z2 = z, x >= 0, y >= 0, z'' = x, z' = 1 ifPlus(z', z'', z1, z2) -{ 0 }-> 0 :|: z2 = v3, v0 >= 0, z1 = v2, v1 >= 0, z'' = v1, v2 >= 0, v3 >= 0, z' = v0 ifSum(z', z'', z1, z2) -{ 1 }-> x :|: xs >= 0, z2 = y, z' = 2, x >= 0, y >= 0, z'' = xs, z1 = x ifSum(z', z'', z1, z2) -{ 2 }-> sumIter(xs'', y) :|: z2 = y, z'' = 1 + x2 + xs'', xs'' >= 0, x >= 0, y >= 0, z' = 1, z1 = x, x2 >= 0 ifSum(z', z'', z1, z2) -{ 2 }-> sumIter(0, y) :|: z'' = 0, z2 = y, x >= 0, y >= 0, z' = 1, z1 = x isempty(z') -{ 1 }-> 2 :|: z' = 0 isempty(z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x + xs, x >= 0 le(z', z'') -{ 1 }-> le(x, y) :|: z' = 1 + x, x >= 0, y >= 0, z'' = 1 + y le(z', z'') -{ 1 }-> 2 :|: z'' = y, y >= 0, z' = 0 le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' = 1 + x, x >= 0 le(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0 plus(z', z'') -{ 1 }-> plusIter(x, y, 0) :|: z' = x, z'' = y, x >= 0, y >= 0 plusIter(z', z'', z1) -{ 2 }-> ifPlus(le(x'', y'), 1 + x'', y, 1 + y') :|: z' = 1 + x'', z'' = y, y >= 0, z1 = 1 + y', y' >= 0, x'' >= 0 plusIter(z', z'', z1) -{ 2 }-> ifPlus(2, 0, y, z) :|: z1 = z, z >= 0, z'' = y, y >= 0, z' = 0 plusIter(z', z'', z1) -{ 2 }-> ifPlus(1, 1 + x', y, 0) :|: z1 = 0, z' = 1 + x', z'' = y, x' >= 0, y >= 0 plusIter(z', z'', z1) -{ 1 }-> ifPlus(0, x, y, z) :|: z1 = z, z >= 0, z' = x, z'' = y, x >= 0, y >= 0 sum(z') -{ 1 }-> sumIter(xs, 0) :|: xs >= 0, z' = xs sumIter(z', z'') -{ 3 }-> ifSum(2, 0, x, plus(x, 1)) :|: x >= 0, z'' = x, z' = 0 sumIter(z', z'') -{ 3 }-> ifSum(1, 1 + x1 + xs', x, plus(x, x1)) :|: x1 >= 0, z' = 1 + x1 + xs', xs' >= 0, x >= 0, z'' = x tail(z') -{ 1 }-> xs :|: xs >= 0, z' = 1 + x + xs, x >= 0 tail(z') -{ 1 }-> 0 :|: z' = 0 ---------------------------------------- (239) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (240) Obligation: Complexity RNTS consisting of the following rules: a -{ 1 }-> 1 :|: a -{ 1 }-> 0 :|: head(z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + xs, x >= 0 head(z') -{ 1 }-> 1 :|: z' = 0 ifPlus(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0 ifPlus(z', z'', z1, z2) -{ 1 }-> plusIter(z'', 1 + z1, 1 + z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 ifPlus(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0 ifSum(z', z'', z1, z2) -{ 1 }-> z1 :|: z'' >= 0, z' = 2, z1 >= 0, z2 >= 0 ifSum(z', z'', z1, z2) -{ 2 }-> sumIter(xs'', z2) :|: z'' = 1 + x2 + xs'', xs'' >= 0, z1 >= 0, z2 >= 0, z' = 1, x2 >= 0 ifSum(z', z'', z1, z2) -{ 2 }-> sumIter(0, z2) :|: z'' = 0, z1 >= 0, z2 >= 0, z' = 1 isempty(z') -{ 1 }-> 2 :|: z' = 0 isempty(z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x + xs, x >= 0 le(z', z'') -{ 1 }-> le(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0 le(z', z'') -{ 1 }-> 2 :|: z'' >= 0, z' = 0 le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 le(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 plus(z', z'') -{ 1 }-> plusIter(z', z'', 0) :|: z' >= 0, z'' >= 0 plusIter(z', z'', z1) -{ 2 }-> ifPlus(le(z' - 1, z1 - 1), 1 + (z' - 1), z'', 1 + (z1 - 1)) :|: z'' >= 0, z1 - 1 >= 0, z' - 1 >= 0 plusIter(z', z'', z1) -{ 2 }-> ifPlus(2, 0, z'', z1) :|: z1 >= 0, z'' >= 0, z' = 0 plusIter(z', z'', z1) -{ 2 }-> ifPlus(1, 1 + (z' - 1), z'', 0) :|: z1 = 0, z' - 1 >= 0, z'' >= 0 plusIter(z', z'', z1) -{ 1 }-> ifPlus(0, z', z'', z1) :|: z1 >= 0, z' >= 0, z'' >= 0 sum(z') -{ 1 }-> sumIter(z', 0) :|: z' >= 0 sumIter(z', z'') -{ 3 }-> ifSum(2, 0, z'', plus(z'', 1)) :|: z'' >= 0, z' = 0 sumIter(z', z'') -{ 3 }-> ifSum(1, 1 + x1 + xs', z'', plus(z'', x1)) :|: x1 >= 0, z' = 1 + x1 + xs', xs' >= 0, z'' >= 0 tail(z') -{ 1 }-> xs :|: xs >= 0, z' = 1 + x + xs, x >= 0 tail(z') -{ 1 }-> 0 :|: z' = 0 ---------------------------------------- (241) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { le } { tail } { isempty } { head } { a } { plusIter, ifPlus } { plus } { ifSum, sumIter } { sum } ---------------------------------------- (242) Obligation: Complexity RNTS consisting of the following rules: a -{ 1 }-> 1 :|: a -{ 1 }-> 0 :|: head(z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + xs, x >= 0 head(z') -{ 1 }-> 1 :|: z' = 0 ifPlus(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0 ifPlus(z', z'', z1, z2) -{ 1 }-> plusIter(z'', 1 + z1, 1 + z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 ifPlus(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0 ifSum(z', z'', z1, z2) -{ 1 }-> z1 :|: z'' >= 0, z' = 2, z1 >= 0, z2 >= 0 ifSum(z', z'', z1, z2) -{ 2 }-> sumIter(xs'', z2) :|: z'' = 1 + x2 + xs'', xs'' >= 0, z1 >= 0, z2 >= 0, z' = 1, x2 >= 0 ifSum(z', z'', z1, z2) -{ 2 }-> sumIter(0, z2) :|: z'' = 0, z1 >= 0, z2 >= 0, z' = 1 isempty(z') -{ 1 }-> 2 :|: z' = 0 isempty(z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x + xs, x >= 0 le(z', z'') -{ 1 }-> le(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0 le(z', z'') -{ 1 }-> 2 :|: z'' >= 0, z' = 0 le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 le(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 plus(z', z'') -{ 1 }-> plusIter(z', z'', 0) :|: z' >= 0, z'' >= 0 plusIter(z', z'', z1) -{ 2 }-> ifPlus(le(z' - 1, z1 - 1), 1 + (z' - 1), z'', 1 + (z1 - 1)) :|: z'' >= 0, z1 - 1 >= 0, z' - 1 >= 0 plusIter(z', z'', z1) -{ 2 }-> ifPlus(2, 0, z'', z1) :|: z1 >= 0, z'' >= 0, z' = 0 plusIter(z', z'', z1) -{ 2 }-> ifPlus(1, 1 + (z' - 1), z'', 0) :|: z1 = 0, z' - 1 >= 0, z'' >= 0 plusIter(z', z'', z1) -{ 1 }-> ifPlus(0, z', z'', z1) :|: z1 >= 0, z' >= 0, z'' >= 0 sum(z') -{ 1 }-> sumIter(z', 0) :|: z' >= 0 sumIter(z', z'') -{ 3 }-> ifSum(2, 0, z'', plus(z'', 1)) :|: z'' >= 0, z' = 0 sumIter(z', z'') -{ 3 }-> ifSum(1, 1 + x1 + xs', z'', plus(z'', x1)) :|: x1 >= 0, z' = 1 + x1 + xs', xs' >= 0, z'' >= 0 tail(z') -{ 1 }-> xs :|: xs >= 0, z' = 1 + x + xs, x >= 0 tail(z') -{ 1 }-> 0 :|: z' = 0 Function symbols to be analyzed: {le}, {tail}, {isempty}, {head}, {a}, {plusIter,ifPlus}, {plus}, {ifSum,sumIter}, {sum} ---------------------------------------- (243) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (244) Obligation: Complexity RNTS consisting of the following rules: a -{ 1 }-> 1 :|: a -{ 1 }-> 0 :|: head(z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + xs, x >= 0 head(z') -{ 1 }-> 1 :|: z' = 0 ifPlus(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0 ifPlus(z', z'', z1, z2) -{ 1 }-> plusIter(z'', 1 + z1, 1 + z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 ifPlus(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0 ifSum(z', z'', z1, z2) -{ 1 }-> z1 :|: z'' >= 0, z' = 2, z1 >= 0, z2 >= 0 ifSum(z', z'', z1, z2) -{ 2 }-> sumIter(xs'', z2) :|: z'' = 1 + x2 + xs'', xs'' >= 0, z1 >= 0, z2 >= 0, z' = 1, x2 >= 0 ifSum(z', z'', z1, z2) -{ 2 }-> sumIter(0, z2) :|: z'' = 0, z1 >= 0, z2 >= 0, z' = 1 isempty(z') -{ 1 }-> 2 :|: z' = 0 isempty(z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x + xs, x >= 0 le(z', z'') -{ 1 }-> le(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0 le(z', z'') -{ 1 }-> 2 :|: z'' >= 0, z' = 0 le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 le(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 plus(z', z'') -{ 1 }-> plusIter(z', z'', 0) :|: z' >= 0, z'' >= 0 plusIter(z', z'', z1) -{ 2 }-> ifPlus(le(z' - 1, z1 - 1), 1 + (z' - 1), z'', 1 + (z1 - 1)) :|: z'' >= 0, z1 - 1 >= 0, z' - 1 >= 0 plusIter(z', z'', z1) -{ 2 }-> ifPlus(2, 0, z'', z1) :|: z1 >= 0, z'' >= 0, z' = 0 plusIter(z', z'', z1) -{ 2 }-> ifPlus(1, 1 + (z' - 1), z'', 0) :|: z1 = 0, z' - 1 >= 0, z'' >= 0 plusIter(z', z'', z1) -{ 1 }-> ifPlus(0, z', z'', z1) :|: z1 >= 0, z' >= 0, z'' >= 0 sum(z') -{ 1 }-> sumIter(z', 0) :|: z' >= 0 sumIter(z', z'') -{ 3 }-> ifSum(2, 0, z'', plus(z'', 1)) :|: z'' >= 0, z' = 0 sumIter(z', z'') -{ 3 }-> ifSum(1, 1 + x1 + xs', z'', plus(z'', x1)) :|: x1 >= 0, z' = 1 + x1 + xs', xs' >= 0, z'' >= 0 tail(z') -{ 1 }-> xs :|: xs >= 0, z' = 1 + x + xs, x >= 0 tail(z') -{ 1 }-> 0 :|: z' = 0 Function symbols to be analyzed: {le}, {tail}, {isempty}, {head}, {a}, {plusIter,ifPlus}, {plus}, {ifSum,sumIter}, {sum} ---------------------------------------- (245) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: le after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 2 ---------------------------------------- (246) Obligation: Complexity RNTS consisting of the following rules: a -{ 1 }-> 1 :|: a -{ 1 }-> 0 :|: head(z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + xs, x >= 0 head(z') -{ 1 }-> 1 :|: z' = 0 ifPlus(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0 ifPlus(z', z'', z1, z2) -{ 1 }-> plusIter(z'', 1 + z1, 1 + z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 ifPlus(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0 ifSum(z', z'', z1, z2) -{ 1 }-> z1 :|: z'' >= 0, z' = 2, z1 >= 0, z2 >= 0 ifSum(z', z'', z1, z2) -{ 2 }-> sumIter(xs'', z2) :|: z'' = 1 + x2 + xs'', xs'' >= 0, z1 >= 0, z2 >= 0, z' = 1, x2 >= 0 ifSum(z', z'', z1, z2) -{ 2 }-> sumIter(0, z2) :|: z'' = 0, z1 >= 0, z2 >= 0, z' = 1 isempty(z') -{ 1 }-> 2 :|: z' = 0 isempty(z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x + xs, x >= 0 le(z', z'') -{ 1 }-> le(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0 le(z', z'') -{ 1 }-> 2 :|: z'' >= 0, z' = 0 le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 le(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 plus(z', z'') -{ 1 }-> plusIter(z', z'', 0) :|: z' >= 0, z'' >= 0 plusIter(z', z'', z1) -{ 2 }-> ifPlus(le(z' - 1, z1 - 1), 1 + (z' - 1), z'', 1 + (z1 - 1)) :|: z'' >= 0, z1 - 1 >= 0, z' - 1 >= 0 plusIter(z', z'', z1) -{ 2 }-> ifPlus(2, 0, z'', z1) :|: z1 >= 0, z'' >= 0, z' = 0 plusIter(z', z'', z1) -{ 2 }-> ifPlus(1, 1 + (z' - 1), z'', 0) :|: z1 = 0, z' - 1 >= 0, z'' >= 0 plusIter(z', z'', z1) -{ 1 }-> ifPlus(0, z', z'', z1) :|: z1 >= 0, z' >= 0, z'' >= 0 sum(z') -{ 1 }-> sumIter(z', 0) :|: z' >= 0 sumIter(z', z'') -{ 3 }-> ifSum(2, 0, z'', plus(z'', 1)) :|: z'' >= 0, z' = 0 sumIter(z', z'') -{ 3 }-> ifSum(1, 1 + x1 + xs', z'', plus(z'', x1)) :|: x1 >= 0, z' = 1 + x1 + xs', xs' >= 0, z'' >= 0 tail(z') -{ 1 }-> xs :|: xs >= 0, z' = 1 + x + xs, x >= 0 tail(z') -{ 1 }-> 0 :|: z' = 0 Function symbols to be analyzed: {le}, {tail}, {isempty}, {head}, {a}, {plusIter,ifPlus}, {plus}, {ifSum,sumIter}, {sum} Previous analysis results are: le: runtime: ?, size: O(1) [2] ---------------------------------------- (247) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: le after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2 + z'' ---------------------------------------- (248) Obligation: Complexity RNTS consisting of the following rules: a -{ 1 }-> 1 :|: a -{ 1 }-> 0 :|: head(z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + xs, x >= 0 head(z') -{ 1 }-> 1 :|: z' = 0 ifPlus(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0 ifPlus(z', z'', z1, z2) -{ 1 }-> plusIter(z'', 1 + z1, 1 + z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 ifPlus(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0 ifSum(z', z'', z1, z2) -{ 1 }-> z1 :|: z'' >= 0, z' = 2, z1 >= 0, z2 >= 0 ifSum(z', z'', z1, z2) -{ 2 }-> sumIter(xs'', z2) :|: z'' = 1 + x2 + xs'', xs'' >= 0, z1 >= 0, z2 >= 0, z' = 1, x2 >= 0 ifSum(z', z'', z1, z2) -{ 2 }-> sumIter(0, z2) :|: z'' = 0, z1 >= 0, z2 >= 0, z' = 1 isempty(z') -{ 1 }-> 2 :|: z' = 0 isempty(z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x + xs, x >= 0 le(z', z'') -{ 1 }-> le(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0 le(z', z'') -{ 1 }-> 2 :|: z'' >= 0, z' = 0 le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 le(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 plus(z', z'') -{ 1 }-> plusIter(z', z'', 0) :|: z' >= 0, z'' >= 0 plusIter(z', z'', z1) -{ 2 }-> ifPlus(le(z' - 1, z1 - 1), 1 + (z' - 1), z'', 1 + (z1 - 1)) :|: z'' >= 0, z1 - 1 >= 0, z' - 1 >= 0 plusIter(z', z'', z1) -{ 2 }-> ifPlus(2, 0, z'', z1) :|: z1 >= 0, z'' >= 0, z' = 0 plusIter(z', z'', z1) -{ 2 }-> ifPlus(1, 1 + (z' - 1), z'', 0) :|: z1 = 0, z' - 1 >= 0, z'' >= 0 plusIter(z', z'', z1) -{ 1 }-> ifPlus(0, z', z'', z1) :|: z1 >= 0, z' >= 0, z'' >= 0 sum(z') -{ 1 }-> sumIter(z', 0) :|: z' >= 0 sumIter(z', z'') -{ 3 }-> ifSum(2, 0, z'', plus(z'', 1)) :|: z'' >= 0, z' = 0 sumIter(z', z'') -{ 3 }-> ifSum(1, 1 + x1 + xs', z'', plus(z'', x1)) :|: x1 >= 0, z' = 1 + x1 + xs', xs' >= 0, z'' >= 0 tail(z') -{ 1 }-> xs :|: xs >= 0, z' = 1 + x + xs, x >= 0 tail(z') -{ 1 }-> 0 :|: z' = 0 Function symbols to be analyzed: {tail}, {isempty}, {head}, {a}, {plusIter,ifPlus}, {plus}, {ifSum,sumIter}, {sum} Previous analysis results are: le: runtime: O(n^1) [2 + z''], size: O(1) [2] ---------------------------------------- (249) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (250) Obligation: Complexity RNTS consisting of the following rules: a -{ 1 }-> 1 :|: a -{ 1 }-> 0 :|: head(z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + xs, x >= 0 head(z') -{ 1 }-> 1 :|: z' = 0 ifPlus(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0 ifPlus(z', z'', z1, z2) -{ 1 }-> plusIter(z'', 1 + z1, 1 + z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 ifPlus(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0 ifSum(z', z'', z1, z2) -{ 1 }-> z1 :|: z'' >= 0, z' = 2, z1 >= 0, z2 >= 0 ifSum(z', z'', z1, z2) -{ 2 }-> sumIter(xs'', z2) :|: z'' = 1 + x2 + xs'', xs'' >= 0, z1 >= 0, z2 >= 0, z' = 1, x2 >= 0 ifSum(z', z'', z1, z2) -{ 2 }-> sumIter(0, z2) :|: z'' = 0, z1 >= 0, z2 >= 0, z' = 1 isempty(z') -{ 1 }-> 2 :|: z' = 0 isempty(z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x + xs, x >= 0 le(z', z'') -{ 2 + z'' }-> s' :|: s' >= 0, s' <= 2, z' - 1 >= 0, z'' - 1 >= 0 le(z', z'') -{ 1 }-> 2 :|: z'' >= 0, z' = 0 le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 le(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 plus(z', z'') -{ 1 }-> plusIter(z', z'', 0) :|: z' >= 0, z'' >= 0 plusIter(z', z'', z1) -{ 3 + z1 }-> ifPlus(s, 1 + (z' - 1), z'', 1 + (z1 - 1)) :|: s >= 0, s <= 2, z'' >= 0, z1 - 1 >= 0, z' - 1 >= 0 plusIter(z', z'', z1) -{ 2 }-> ifPlus(2, 0, z'', z1) :|: z1 >= 0, z'' >= 0, z' = 0 plusIter(z', z'', z1) -{ 2 }-> ifPlus(1, 1 + (z' - 1), z'', 0) :|: z1 = 0, z' - 1 >= 0, z'' >= 0 plusIter(z', z'', z1) -{ 1 }-> ifPlus(0, z', z'', z1) :|: z1 >= 0, z' >= 0, z'' >= 0 sum(z') -{ 1 }-> sumIter(z', 0) :|: z' >= 0 sumIter(z', z'') -{ 3 }-> ifSum(2, 0, z'', plus(z'', 1)) :|: z'' >= 0, z' = 0 sumIter(z', z'') -{ 3 }-> ifSum(1, 1 + x1 + xs', z'', plus(z'', x1)) :|: x1 >= 0, z' = 1 + x1 + xs', xs' >= 0, z'' >= 0 tail(z') -{ 1 }-> xs :|: xs >= 0, z' = 1 + x + xs, x >= 0 tail(z') -{ 1 }-> 0 :|: z' = 0 Function symbols to be analyzed: {tail}, {isempty}, {head}, {a}, {plusIter,ifPlus}, {plus}, {ifSum,sumIter}, {sum} Previous analysis results are: le: runtime: O(n^1) [2 + z''], size: O(1) [2] ---------------------------------------- (251) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: tail after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z' ---------------------------------------- (252) Obligation: Complexity RNTS consisting of the following rules: a -{ 1 }-> 1 :|: a -{ 1 }-> 0 :|: head(z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + xs, x >= 0 head(z') -{ 1 }-> 1 :|: z' = 0 ifPlus(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0 ifPlus(z', z'', z1, z2) -{ 1 }-> plusIter(z'', 1 + z1, 1 + z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 ifPlus(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0 ifSum(z', z'', z1, z2) -{ 1 }-> z1 :|: z'' >= 0, z' = 2, z1 >= 0, z2 >= 0 ifSum(z', z'', z1, z2) -{ 2 }-> sumIter(xs'', z2) :|: z'' = 1 + x2 + xs'', xs'' >= 0, z1 >= 0, z2 >= 0, z' = 1, x2 >= 0 ifSum(z', z'', z1, z2) -{ 2 }-> sumIter(0, z2) :|: z'' = 0, z1 >= 0, z2 >= 0, z' = 1 isempty(z') -{ 1 }-> 2 :|: z' = 0 isempty(z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x + xs, x >= 0 le(z', z'') -{ 2 + z'' }-> s' :|: s' >= 0, s' <= 2, z' - 1 >= 0, z'' - 1 >= 0 le(z', z'') -{ 1 }-> 2 :|: z'' >= 0, z' = 0 le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 le(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 plus(z', z'') -{ 1 }-> plusIter(z', z'', 0) :|: z' >= 0, z'' >= 0 plusIter(z', z'', z1) -{ 3 + z1 }-> ifPlus(s, 1 + (z' - 1), z'', 1 + (z1 - 1)) :|: s >= 0, s <= 2, z'' >= 0, z1 - 1 >= 0, z' - 1 >= 0 plusIter(z', z'', z1) -{ 2 }-> ifPlus(2, 0, z'', z1) :|: z1 >= 0, z'' >= 0, z' = 0 plusIter(z', z'', z1) -{ 2 }-> ifPlus(1, 1 + (z' - 1), z'', 0) :|: z1 = 0, z' - 1 >= 0, z'' >= 0 plusIter(z', z'', z1) -{ 1 }-> ifPlus(0, z', z'', z1) :|: z1 >= 0, z' >= 0, z'' >= 0 sum(z') -{ 1 }-> sumIter(z', 0) :|: z' >= 0 sumIter(z', z'') -{ 3 }-> ifSum(2, 0, z'', plus(z'', 1)) :|: z'' >= 0, z' = 0 sumIter(z', z'') -{ 3 }-> ifSum(1, 1 + x1 + xs', z'', plus(z'', x1)) :|: x1 >= 0, z' = 1 + x1 + xs', xs' >= 0, z'' >= 0 tail(z') -{ 1 }-> xs :|: xs >= 0, z' = 1 + x + xs, x >= 0 tail(z') -{ 1 }-> 0 :|: z' = 0 Function symbols to be analyzed: {tail}, {isempty}, {head}, {a}, {plusIter,ifPlus}, {plus}, {ifSum,sumIter}, {sum} Previous analysis results are: le: runtime: O(n^1) [2 + z''], size: O(1) [2] tail: runtime: ?, size: O(n^1) [z'] ---------------------------------------- (253) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: tail after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (254) Obligation: Complexity RNTS consisting of the following rules: a -{ 1 }-> 1 :|: a -{ 1 }-> 0 :|: head(z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + xs, x >= 0 head(z') -{ 1 }-> 1 :|: z' = 0 ifPlus(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0 ifPlus(z', z'', z1, z2) -{ 1 }-> plusIter(z'', 1 + z1, 1 + z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 ifPlus(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0 ifSum(z', z'', z1, z2) -{ 1 }-> z1 :|: z'' >= 0, z' = 2, z1 >= 0, z2 >= 0 ifSum(z', z'', z1, z2) -{ 2 }-> sumIter(xs'', z2) :|: z'' = 1 + x2 + xs'', xs'' >= 0, z1 >= 0, z2 >= 0, z' = 1, x2 >= 0 ifSum(z', z'', z1, z2) -{ 2 }-> sumIter(0, z2) :|: z'' = 0, z1 >= 0, z2 >= 0, z' = 1 isempty(z') -{ 1 }-> 2 :|: z' = 0 isempty(z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x + xs, x >= 0 le(z', z'') -{ 2 + z'' }-> s' :|: s' >= 0, s' <= 2, z' - 1 >= 0, z'' - 1 >= 0 le(z', z'') -{ 1 }-> 2 :|: z'' >= 0, z' = 0 le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 le(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 plus(z', z'') -{ 1 }-> plusIter(z', z'', 0) :|: z' >= 0, z'' >= 0 plusIter(z', z'', z1) -{ 3 + z1 }-> ifPlus(s, 1 + (z' - 1), z'', 1 + (z1 - 1)) :|: s >= 0, s <= 2, z'' >= 0, z1 - 1 >= 0, z' - 1 >= 0 plusIter(z', z'', z1) -{ 2 }-> ifPlus(2, 0, z'', z1) :|: z1 >= 0, z'' >= 0, z' = 0 plusIter(z', z'', z1) -{ 2 }-> ifPlus(1, 1 + (z' - 1), z'', 0) :|: z1 = 0, z' - 1 >= 0, z'' >= 0 plusIter(z', z'', z1) -{ 1 }-> ifPlus(0, z', z'', z1) :|: z1 >= 0, z' >= 0, z'' >= 0 sum(z') -{ 1 }-> sumIter(z', 0) :|: z' >= 0 sumIter(z', z'') -{ 3 }-> ifSum(2, 0, z'', plus(z'', 1)) :|: z'' >= 0, z' = 0 sumIter(z', z'') -{ 3 }-> ifSum(1, 1 + x1 + xs', z'', plus(z'', x1)) :|: x1 >= 0, z' = 1 + x1 + xs', xs' >= 0, z'' >= 0 tail(z') -{ 1 }-> xs :|: xs >= 0, z' = 1 + x + xs, x >= 0 tail(z') -{ 1 }-> 0 :|: z' = 0 Function symbols to be analyzed: {isempty}, {head}, {a}, {plusIter,ifPlus}, {plus}, {ifSum,sumIter}, {sum} Previous analysis results are: le: runtime: O(n^1) [2 + z''], size: O(1) [2] tail: runtime: O(1) [1], size: O(n^1) [z'] ---------------------------------------- (255) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (256) Obligation: Complexity RNTS consisting of the following rules: a -{ 1 }-> 1 :|: a -{ 1 }-> 0 :|: head(z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + xs, x >= 0 head(z') -{ 1 }-> 1 :|: z' = 0 ifPlus(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0 ifPlus(z', z'', z1, z2) -{ 1 }-> plusIter(z'', 1 + z1, 1 + z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 ifPlus(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0 ifSum(z', z'', z1, z2) -{ 1 }-> z1 :|: z'' >= 0, z' = 2, z1 >= 0, z2 >= 0 ifSum(z', z'', z1, z2) -{ 2 }-> sumIter(xs'', z2) :|: z'' = 1 + x2 + xs'', xs'' >= 0, z1 >= 0, z2 >= 0, z' = 1, x2 >= 0 ifSum(z', z'', z1, z2) -{ 2 }-> sumIter(0, z2) :|: z'' = 0, z1 >= 0, z2 >= 0, z' = 1 isempty(z') -{ 1 }-> 2 :|: z' = 0 isempty(z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x + xs, x >= 0 le(z', z'') -{ 2 + z'' }-> s' :|: s' >= 0, s' <= 2, z' - 1 >= 0, z'' - 1 >= 0 le(z', z'') -{ 1 }-> 2 :|: z'' >= 0, z' = 0 le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 le(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 plus(z', z'') -{ 1 }-> plusIter(z', z'', 0) :|: z' >= 0, z'' >= 0 plusIter(z', z'', z1) -{ 3 + z1 }-> ifPlus(s, 1 + (z' - 1), z'', 1 + (z1 - 1)) :|: s >= 0, s <= 2, z'' >= 0, z1 - 1 >= 0, z' - 1 >= 0 plusIter(z', z'', z1) -{ 2 }-> ifPlus(2, 0, z'', z1) :|: z1 >= 0, z'' >= 0, z' = 0 plusIter(z', z'', z1) -{ 2 }-> ifPlus(1, 1 + (z' - 1), z'', 0) :|: z1 = 0, z' - 1 >= 0, z'' >= 0 plusIter(z', z'', z1) -{ 1 }-> ifPlus(0, z', z'', z1) :|: z1 >= 0, z' >= 0, z'' >= 0 sum(z') -{ 1 }-> sumIter(z', 0) :|: z' >= 0 sumIter(z', z'') -{ 3 }-> ifSum(2, 0, z'', plus(z'', 1)) :|: z'' >= 0, z' = 0 sumIter(z', z'') -{ 3 }-> ifSum(1, 1 + x1 + xs', z'', plus(z'', x1)) :|: x1 >= 0, z' = 1 + x1 + xs', xs' >= 0, z'' >= 0 tail(z') -{ 1 }-> xs :|: xs >= 0, z' = 1 + x + xs, x >= 0 tail(z') -{ 1 }-> 0 :|: z' = 0 Function symbols to be analyzed: {isempty}, {head}, {a}, {plusIter,ifPlus}, {plus}, {ifSum,sumIter}, {sum} Previous analysis results are: le: runtime: O(n^1) [2 + z''], size: O(1) [2] tail: runtime: O(1) [1], size: O(n^1) [z'] ---------------------------------------- (257) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: isempty after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 2 ---------------------------------------- (258) Obligation: Complexity RNTS consisting of the following rules: a -{ 1 }-> 1 :|: a -{ 1 }-> 0 :|: head(z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + xs, x >= 0 head(z') -{ 1 }-> 1 :|: z' = 0 ifPlus(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0 ifPlus(z', z'', z1, z2) -{ 1 }-> plusIter(z'', 1 + z1, 1 + z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 ifPlus(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0 ifSum(z', z'', z1, z2) -{ 1 }-> z1 :|: z'' >= 0, z' = 2, z1 >= 0, z2 >= 0 ifSum(z', z'', z1, z2) -{ 2 }-> sumIter(xs'', z2) :|: z'' = 1 + x2 + xs'', xs'' >= 0, z1 >= 0, z2 >= 0, z' = 1, x2 >= 0 ifSum(z', z'', z1, z2) -{ 2 }-> sumIter(0, z2) :|: z'' = 0, z1 >= 0, z2 >= 0, z' = 1 isempty(z') -{ 1 }-> 2 :|: z' = 0 isempty(z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x + xs, x >= 0 le(z', z'') -{ 2 + z'' }-> s' :|: s' >= 0, s' <= 2, z' - 1 >= 0, z'' - 1 >= 0 le(z', z'') -{ 1 }-> 2 :|: z'' >= 0, z' = 0 le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 le(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 plus(z', z'') -{ 1 }-> plusIter(z', z'', 0) :|: z' >= 0, z'' >= 0 plusIter(z', z'', z1) -{ 3 + z1 }-> ifPlus(s, 1 + (z' - 1), z'', 1 + (z1 - 1)) :|: s >= 0, s <= 2, z'' >= 0, z1 - 1 >= 0, z' - 1 >= 0 plusIter(z', z'', z1) -{ 2 }-> ifPlus(2, 0, z'', z1) :|: z1 >= 0, z'' >= 0, z' = 0 plusIter(z', z'', z1) -{ 2 }-> ifPlus(1, 1 + (z' - 1), z'', 0) :|: z1 = 0, z' - 1 >= 0, z'' >= 0 plusIter(z', z'', z1) -{ 1 }-> ifPlus(0, z', z'', z1) :|: z1 >= 0, z' >= 0, z'' >= 0 sum(z') -{ 1 }-> sumIter(z', 0) :|: z' >= 0 sumIter(z', z'') -{ 3 }-> ifSum(2, 0, z'', plus(z'', 1)) :|: z'' >= 0, z' = 0 sumIter(z', z'') -{ 3 }-> ifSum(1, 1 + x1 + xs', z'', plus(z'', x1)) :|: x1 >= 0, z' = 1 + x1 + xs', xs' >= 0, z'' >= 0 tail(z') -{ 1 }-> xs :|: xs >= 0, z' = 1 + x + xs, x >= 0 tail(z') -{ 1 }-> 0 :|: z' = 0 Function symbols to be analyzed: {isempty}, {head}, {a}, {plusIter,ifPlus}, {plus}, {ifSum,sumIter}, {sum} Previous analysis results are: le: runtime: O(n^1) [2 + z''], size: O(1) [2] tail: runtime: O(1) [1], size: O(n^1) [z'] isempty: runtime: ?, size: O(1) [2] ---------------------------------------- (259) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: isempty after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (260) Obligation: Complexity RNTS consisting of the following rules: a -{ 1 }-> 1 :|: a -{ 1 }-> 0 :|: head(z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + xs, x >= 0 head(z') -{ 1 }-> 1 :|: z' = 0 ifPlus(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0 ifPlus(z', z'', z1, z2) -{ 1 }-> plusIter(z'', 1 + z1, 1 + z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 ifPlus(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0 ifSum(z', z'', z1, z2) -{ 1 }-> z1 :|: z'' >= 0, z' = 2, z1 >= 0, z2 >= 0 ifSum(z', z'', z1, z2) -{ 2 }-> sumIter(xs'', z2) :|: z'' = 1 + x2 + xs'', xs'' >= 0, z1 >= 0, z2 >= 0, z' = 1, x2 >= 0 ifSum(z', z'', z1, z2) -{ 2 }-> sumIter(0, z2) :|: z'' = 0, z1 >= 0, z2 >= 0, z' = 1 isempty(z') -{ 1 }-> 2 :|: z' = 0 isempty(z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x + xs, x >= 0 le(z', z'') -{ 2 + z'' }-> s' :|: s' >= 0, s' <= 2, z' - 1 >= 0, z'' - 1 >= 0 le(z', z'') -{ 1 }-> 2 :|: z'' >= 0, z' = 0 le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 le(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 plus(z', z'') -{ 1 }-> plusIter(z', z'', 0) :|: z' >= 0, z'' >= 0 plusIter(z', z'', z1) -{ 3 + z1 }-> ifPlus(s, 1 + (z' - 1), z'', 1 + (z1 - 1)) :|: s >= 0, s <= 2, z'' >= 0, z1 - 1 >= 0, z' - 1 >= 0 plusIter(z', z'', z1) -{ 2 }-> ifPlus(2, 0, z'', z1) :|: z1 >= 0, z'' >= 0, z' = 0 plusIter(z', z'', z1) -{ 2 }-> ifPlus(1, 1 + (z' - 1), z'', 0) :|: z1 = 0, z' - 1 >= 0, z'' >= 0 plusIter(z', z'', z1) -{ 1 }-> ifPlus(0, z', z'', z1) :|: z1 >= 0, z' >= 0, z'' >= 0 sum(z') -{ 1 }-> sumIter(z', 0) :|: z' >= 0 sumIter(z', z'') -{ 3 }-> ifSum(2, 0, z'', plus(z'', 1)) :|: z'' >= 0, z' = 0 sumIter(z', z'') -{ 3 }-> ifSum(1, 1 + x1 + xs', z'', plus(z'', x1)) :|: x1 >= 0, z' = 1 + x1 + xs', xs' >= 0, z'' >= 0 tail(z') -{ 1 }-> xs :|: xs >= 0, z' = 1 + x + xs, x >= 0 tail(z') -{ 1 }-> 0 :|: z' = 0 Function symbols to be analyzed: {head}, {a}, {plusIter,ifPlus}, {plus}, {ifSum,sumIter}, {sum} Previous analysis results are: le: runtime: O(n^1) [2 + z''], size: O(1) [2] tail: runtime: O(1) [1], size: O(n^1) [z'] isempty: runtime: O(1) [1], size: O(1) [2] ---------------------------------------- (261) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (262) Obligation: Complexity RNTS consisting of the following rules: a -{ 1 }-> 1 :|: a -{ 1 }-> 0 :|: head(z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + xs, x >= 0 head(z') -{ 1 }-> 1 :|: z' = 0 ifPlus(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0 ifPlus(z', z'', z1, z2) -{ 1 }-> plusIter(z'', 1 + z1, 1 + z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 ifPlus(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0 ifSum(z', z'', z1, z2) -{ 1 }-> z1 :|: z'' >= 0, z' = 2, z1 >= 0, z2 >= 0 ifSum(z', z'', z1, z2) -{ 2 }-> sumIter(xs'', z2) :|: z'' = 1 + x2 + xs'', xs'' >= 0, z1 >= 0, z2 >= 0, z' = 1, x2 >= 0 ifSum(z', z'', z1, z2) -{ 2 }-> sumIter(0, z2) :|: z'' = 0, z1 >= 0, z2 >= 0, z' = 1 isempty(z') -{ 1 }-> 2 :|: z' = 0 isempty(z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x + xs, x >= 0 le(z', z'') -{ 2 + z'' }-> s' :|: s' >= 0, s' <= 2, z' - 1 >= 0, z'' - 1 >= 0 le(z', z'') -{ 1 }-> 2 :|: z'' >= 0, z' = 0 le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 le(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 plus(z', z'') -{ 1 }-> plusIter(z', z'', 0) :|: z' >= 0, z'' >= 0 plusIter(z', z'', z1) -{ 3 + z1 }-> ifPlus(s, 1 + (z' - 1), z'', 1 + (z1 - 1)) :|: s >= 0, s <= 2, z'' >= 0, z1 - 1 >= 0, z' - 1 >= 0 plusIter(z', z'', z1) -{ 2 }-> ifPlus(2, 0, z'', z1) :|: z1 >= 0, z'' >= 0, z' = 0 plusIter(z', z'', z1) -{ 2 }-> ifPlus(1, 1 + (z' - 1), z'', 0) :|: z1 = 0, z' - 1 >= 0, z'' >= 0 plusIter(z', z'', z1) -{ 1 }-> ifPlus(0, z', z'', z1) :|: z1 >= 0, z' >= 0, z'' >= 0 sum(z') -{ 1 }-> sumIter(z', 0) :|: z' >= 0 sumIter(z', z'') -{ 3 }-> ifSum(2, 0, z'', plus(z'', 1)) :|: z'' >= 0, z' = 0 sumIter(z', z'') -{ 3 }-> ifSum(1, 1 + x1 + xs', z'', plus(z'', x1)) :|: x1 >= 0, z' = 1 + x1 + xs', xs' >= 0, z'' >= 0 tail(z') -{ 1 }-> xs :|: xs >= 0, z' = 1 + x + xs, x >= 0 tail(z') -{ 1 }-> 0 :|: z' = 0 Function symbols to be analyzed: {head}, {a}, {plusIter,ifPlus}, {plus}, {ifSum,sumIter}, {sum} Previous analysis results are: le: runtime: O(n^1) [2 + z''], size: O(1) [2] tail: runtime: O(1) [1], size: O(n^1) [z'] isempty: runtime: O(1) [1], size: O(1) [2] ---------------------------------------- (263) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: head after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z' ---------------------------------------- (264) Obligation: Complexity RNTS consisting of the following rules: a -{ 1 }-> 1 :|: a -{ 1 }-> 0 :|: head(z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + xs, x >= 0 head(z') -{ 1 }-> 1 :|: z' = 0 ifPlus(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0 ifPlus(z', z'', z1, z2) -{ 1 }-> plusIter(z'', 1 + z1, 1 + z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 ifPlus(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0 ifSum(z', z'', z1, z2) -{ 1 }-> z1 :|: z'' >= 0, z' = 2, z1 >= 0, z2 >= 0 ifSum(z', z'', z1, z2) -{ 2 }-> sumIter(xs'', z2) :|: z'' = 1 + x2 + xs'', xs'' >= 0, z1 >= 0, z2 >= 0, z' = 1, x2 >= 0 ifSum(z', z'', z1, z2) -{ 2 }-> sumIter(0, z2) :|: z'' = 0, z1 >= 0, z2 >= 0, z' = 1 isempty(z') -{ 1 }-> 2 :|: z' = 0 isempty(z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x + xs, x >= 0 le(z', z'') -{ 2 + z'' }-> s' :|: s' >= 0, s' <= 2, z' - 1 >= 0, z'' - 1 >= 0 le(z', z'') -{ 1 }-> 2 :|: z'' >= 0, z' = 0 le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 le(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 plus(z', z'') -{ 1 }-> plusIter(z', z'', 0) :|: z' >= 0, z'' >= 0 plusIter(z', z'', z1) -{ 3 + z1 }-> ifPlus(s, 1 + (z' - 1), z'', 1 + (z1 - 1)) :|: s >= 0, s <= 2, z'' >= 0, z1 - 1 >= 0, z' - 1 >= 0 plusIter(z', z'', z1) -{ 2 }-> ifPlus(2, 0, z'', z1) :|: z1 >= 0, z'' >= 0, z' = 0 plusIter(z', z'', z1) -{ 2 }-> ifPlus(1, 1 + (z' - 1), z'', 0) :|: z1 = 0, z' - 1 >= 0, z'' >= 0 plusIter(z', z'', z1) -{ 1 }-> ifPlus(0, z', z'', z1) :|: z1 >= 0, z' >= 0, z'' >= 0 sum(z') -{ 1 }-> sumIter(z', 0) :|: z' >= 0 sumIter(z', z'') -{ 3 }-> ifSum(2, 0, z'', plus(z'', 1)) :|: z'' >= 0, z' = 0 sumIter(z', z'') -{ 3 }-> ifSum(1, 1 + x1 + xs', z'', plus(z'', x1)) :|: x1 >= 0, z' = 1 + x1 + xs', xs' >= 0, z'' >= 0 tail(z') -{ 1 }-> xs :|: xs >= 0, z' = 1 + x + xs, x >= 0 tail(z') -{ 1 }-> 0 :|: z' = 0 Function symbols to be analyzed: {head}, {a}, {plusIter,ifPlus}, {plus}, {ifSum,sumIter}, {sum} Previous analysis results are: le: runtime: O(n^1) [2 + z''], size: O(1) [2] tail: runtime: O(1) [1], size: O(n^1) [z'] isempty: runtime: O(1) [1], size: O(1) [2] head: runtime: ?, size: O(n^1) [1 + z'] ---------------------------------------- (265) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: head after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (266) Obligation: Complexity RNTS consisting of the following rules: a -{ 1 }-> 1 :|: a -{ 1 }-> 0 :|: head(z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + xs, x >= 0 head(z') -{ 1 }-> 1 :|: z' = 0 ifPlus(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0 ifPlus(z', z'', z1, z2) -{ 1 }-> plusIter(z'', 1 + z1, 1 + z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 ifPlus(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0 ifSum(z', z'', z1, z2) -{ 1 }-> z1 :|: z'' >= 0, z' = 2, z1 >= 0, z2 >= 0 ifSum(z', z'', z1, z2) -{ 2 }-> sumIter(xs'', z2) :|: z'' = 1 + x2 + xs'', xs'' >= 0, z1 >= 0, z2 >= 0, z' = 1, x2 >= 0 ifSum(z', z'', z1, z2) -{ 2 }-> sumIter(0, z2) :|: z'' = 0, z1 >= 0, z2 >= 0, z' = 1 isempty(z') -{ 1 }-> 2 :|: z' = 0 isempty(z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x + xs, x >= 0 le(z', z'') -{ 2 + z'' }-> s' :|: s' >= 0, s' <= 2, z' - 1 >= 0, z'' - 1 >= 0 le(z', z'') -{ 1 }-> 2 :|: z'' >= 0, z' = 0 le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 le(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 plus(z', z'') -{ 1 }-> plusIter(z', z'', 0) :|: z' >= 0, z'' >= 0 plusIter(z', z'', z1) -{ 3 + z1 }-> ifPlus(s, 1 + (z' - 1), z'', 1 + (z1 - 1)) :|: s >= 0, s <= 2, z'' >= 0, z1 - 1 >= 0, z' - 1 >= 0 plusIter(z', z'', z1) -{ 2 }-> ifPlus(2, 0, z'', z1) :|: z1 >= 0, z'' >= 0, z' = 0 plusIter(z', z'', z1) -{ 2 }-> ifPlus(1, 1 + (z' - 1), z'', 0) :|: z1 = 0, z' - 1 >= 0, z'' >= 0 plusIter(z', z'', z1) -{ 1 }-> ifPlus(0, z', z'', z1) :|: z1 >= 0, z' >= 0, z'' >= 0 sum(z') -{ 1 }-> sumIter(z', 0) :|: z' >= 0 sumIter(z', z'') -{ 3 }-> ifSum(2, 0, z'', plus(z'', 1)) :|: z'' >= 0, z' = 0 sumIter(z', z'') -{ 3 }-> ifSum(1, 1 + x1 + xs', z'', plus(z'', x1)) :|: x1 >= 0, z' = 1 + x1 + xs', xs' >= 0, z'' >= 0 tail(z') -{ 1 }-> xs :|: xs >= 0, z' = 1 + x + xs, x >= 0 tail(z') -{ 1 }-> 0 :|: z' = 0 Function symbols to be analyzed: {a}, {plusIter,ifPlus}, {plus}, {ifSum,sumIter}, {sum} Previous analysis results are: le: runtime: O(n^1) [2 + z''], size: O(1) [2] tail: runtime: O(1) [1], size: O(n^1) [z'] isempty: runtime: O(1) [1], size: O(1) [2] head: runtime: O(1) [1], size: O(n^1) [1 + z'] ---------------------------------------- (267) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (268) Obligation: Complexity RNTS consisting of the following rules: a -{ 1 }-> 1 :|: a -{ 1 }-> 0 :|: head(z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + xs, x >= 0 head(z') -{ 1 }-> 1 :|: z' = 0 ifPlus(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0 ifPlus(z', z'', z1, z2) -{ 1 }-> plusIter(z'', 1 + z1, 1 + z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 ifPlus(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0 ifSum(z', z'', z1, z2) -{ 1 }-> z1 :|: z'' >= 0, z' = 2, z1 >= 0, z2 >= 0 ifSum(z', z'', z1, z2) -{ 2 }-> sumIter(xs'', z2) :|: z'' = 1 + x2 + xs'', xs'' >= 0, z1 >= 0, z2 >= 0, z' = 1, x2 >= 0 ifSum(z', z'', z1, z2) -{ 2 }-> sumIter(0, z2) :|: z'' = 0, z1 >= 0, z2 >= 0, z' = 1 isempty(z') -{ 1 }-> 2 :|: z' = 0 isempty(z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x + xs, x >= 0 le(z', z'') -{ 2 + z'' }-> s' :|: s' >= 0, s' <= 2, z' - 1 >= 0, z'' - 1 >= 0 le(z', z'') -{ 1 }-> 2 :|: z'' >= 0, z' = 0 le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 le(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 plus(z', z'') -{ 1 }-> plusIter(z', z'', 0) :|: z' >= 0, z'' >= 0 plusIter(z', z'', z1) -{ 3 + z1 }-> ifPlus(s, 1 + (z' - 1), z'', 1 + (z1 - 1)) :|: s >= 0, s <= 2, z'' >= 0, z1 - 1 >= 0, z' - 1 >= 0 plusIter(z', z'', z1) -{ 2 }-> ifPlus(2, 0, z'', z1) :|: z1 >= 0, z'' >= 0, z' = 0 plusIter(z', z'', z1) -{ 2 }-> ifPlus(1, 1 + (z' - 1), z'', 0) :|: z1 = 0, z' - 1 >= 0, z'' >= 0 plusIter(z', z'', z1) -{ 1 }-> ifPlus(0, z', z'', z1) :|: z1 >= 0, z' >= 0, z'' >= 0 sum(z') -{ 1 }-> sumIter(z', 0) :|: z' >= 0 sumIter(z', z'') -{ 3 }-> ifSum(2, 0, z'', plus(z'', 1)) :|: z'' >= 0, z' = 0 sumIter(z', z'') -{ 3 }-> ifSum(1, 1 + x1 + xs', z'', plus(z'', x1)) :|: x1 >= 0, z' = 1 + x1 + xs', xs' >= 0, z'' >= 0 tail(z') -{ 1 }-> xs :|: xs >= 0, z' = 1 + x + xs, x >= 0 tail(z') -{ 1 }-> 0 :|: z' = 0 Function symbols to be analyzed: {a}, {plusIter,ifPlus}, {plus}, {ifSum,sumIter}, {sum} Previous analysis results are: le: runtime: O(n^1) [2 + z''], size: O(1) [2] tail: runtime: O(1) [1], size: O(n^1) [z'] isempty: runtime: O(1) [1], size: O(1) [2] head: runtime: O(1) [1], size: O(n^1) [1 + z'] ---------------------------------------- (269) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: a after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (270) Obligation: Complexity RNTS consisting of the following rules: a -{ 1 }-> 1 :|: a -{ 1 }-> 0 :|: head(z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + xs, x >= 0 head(z') -{ 1 }-> 1 :|: z' = 0 ifPlus(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0 ifPlus(z', z'', z1, z2) -{ 1 }-> plusIter(z'', 1 + z1, 1 + z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 ifPlus(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0 ifSum(z', z'', z1, z2) -{ 1 }-> z1 :|: z'' >= 0, z' = 2, z1 >= 0, z2 >= 0 ifSum(z', z'', z1, z2) -{ 2 }-> sumIter(xs'', z2) :|: z'' = 1 + x2 + xs'', xs'' >= 0, z1 >= 0, z2 >= 0, z' = 1, x2 >= 0 ifSum(z', z'', z1, z2) -{ 2 }-> sumIter(0, z2) :|: z'' = 0, z1 >= 0, z2 >= 0, z' = 1 isempty(z') -{ 1 }-> 2 :|: z' = 0 isempty(z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x + xs, x >= 0 le(z', z'') -{ 2 + z'' }-> s' :|: s' >= 0, s' <= 2, z' - 1 >= 0, z'' - 1 >= 0 le(z', z'') -{ 1 }-> 2 :|: z'' >= 0, z' = 0 le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 le(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 plus(z', z'') -{ 1 }-> plusIter(z', z'', 0) :|: z' >= 0, z'' >= 0 plusIter(z', z'', z1) -{ 3 + z1 }-> ifPlus(s, 1 + (z' - 1), z'', 1 + (z1 - 1)) :|: s >= 0, s <= 2, z'' >= 0, z1 - 1 >= 0, z' - 1 >= 0 plusIter(z', z'', z1) -{ 2 }-> ifPlus(2, 0, z'', z1) :|: z1 >= 0, z'' >= 0, z' = 0 plusIter(z', z'', z1) -{ 2 }-> ifPlus(1, 1 + (z' - 1), z'', 0) :|: z1 = 0, z' - 1 >= 0, z'' >= 0 plusIter(z', z'', z1) -{ 1 }-> ifPlus(0, z', z'', z1) :|: z1 >= 0, z' >= 0, z'' >= 0 sum(z') -{ 1 }-> sumIter(z', 0) :|: z' >= 0 sumIter(z', z'') -{ 3 }-> ifSum(2, 0, z'', plus(z'', 1)) :|: z'' >= 0, z' = 0 sumIter(z', z'') -{ 3 }-> ifSum(1, 1 + x1 + xs', z'', plus(z'', x1)) :|: x1 >= 0, z' = 1 + x1 + xs', xs' >= 0, z'' >= 0 tail(z') -{ 1 }-> xs :|: xs >= 0, z' = 1 + x + xs, x >= 0 tail(z') -{ 1 }-> 0 :|: z' = 0 Function symbols to be analyzed: {a}, {plusIter,ifPlus}, {plus}, {ifSum,sumIter}, {sum} Previous analysis results are: le: runtime: O(n^1) [2 + z''], size: O(1) [2] tail: runtime: O(1) [1], size: O(n^1) [z'] isempty: runtime: O(1) [1], size: O(1) [2] head: runtime: O(1) [1], size: O(n^1) [1 + z'] a: runtime: ?, size: O(1) [1] ---------------------------------------- (271) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: a after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (272) Obligation: Complexity RNTS consisting of the following rules: a -{ 1 }-> 1 :|: a -{ 1 }-> 0 :|: head(z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + xs, x >= 0 head(z') -{ 1 }-> 1 :|: z' = 0 ifPlus(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0 ifPlus(z', z'', z1, z2) -{ 1 }-> plusIter(z'', 1 + z1, 1 + z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 ifPlus(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0 ifSum(z', z'', z1, z2) -{ 1 }-> z1 :|: z'' >= 0, z' = 2, z1 >= 0, z2 >= 0 ifSum(z', z'', z1, z2) -{ 2 }-> sumIter(xs'', z2) :|: z'' = 1 + x2 + xs'', xs'' >= 0, z1 >= 0, z2 >= 0, z' = 1, x2 >= 0 ifSum(z', z'', z1, z2) -{ 2 }-> sumIter(0, z2) :|: z'' = 0, z1 >= 0, z2 >= 0, z' = 1 isempty(z') -{ 1 }-> 2 :|: z' = 0 isempty(z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x + xs, x >= 0 le(z', z'') -{ 2 + z'' }-> s' :|: s' >= 0, s' <= 2, z' - 1 >= 0, z'' - 1 >= 0 le(z', z'') -{ 1 }-> 2 :|: z'' >= 0, z' = 0 le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 le(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 plus(z', z'') -{ 1 }-> plusIter(z', z'', 0) :|: z' >= 0, z'' >= 0 plusIter(z', z'', z1) -{ 3 + z1 }-> ifPlus(s, 1 + (z' - 1), z'', 1 + (z1 - 1)) :|: s >= 0, s <= 2, z'' >= 0, z1 - 1 >= 0, z' - 1 >= 0 plusIter(z', z'', z1) -{ 2 }-> ifPlus(2, 0, z'', z1) :|: z1 >= 0, z'' >= 0, z' = 0 plusIter(z', z'', z1) -{ 2 }-> ifPlus(1, 1 + (z' - 1), z'', 0) :|: z1 = 0, z' - 1 >= 0, z'' >= 0 plusIter(z', z'', z1) -{ 1 }-> ifPlus(0, z', z'', z1) :|: z1 >= 0, z' >= 0, z'' >= 0 sum(z') -{ 1 }-> sumIter(z', 0) :|: z' >= 0 sumIter(z', z'') -{ 3 }-> ifSum(2, 0, z'', plus(z'', 1)) :|: z'' >= 0, z' = 0 sumIter(z', z'') -{ 3 }-> ifSum(1, 1 + x1 + xs', z'', plus(z'', x1)) :|: x1 >= 0, z' = 1 + x1 + xs', xs' >= 0, z'' >= 0 tail(z') -{ 1 }-> xs :|: xs >= 0, z' = 1 + x + xs, x >= 0 tail(z') -{ 1 }-> 0 :|: z' = 0 Function symbols to be analyzed: {plusIter,ifPlus}, {plus}, {ifSum,sumIter}, {sum} Previous analysis results are: le: runtime: O(n^1) [2 + z''], size: O(1) [2] tail: runtime: O(1) [1], size: O(n^1) [z'] isempty: runtime: O(1) [1], size: O(1) [2] head: runtime: O(1) [1], size: O(n^1) [1 + z'] a: runtime: O(1) [1], size: O(1) [1] ---------------------------------------- (273) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (274) Obligation: Complexity RNTS consisting of the following rules: a -{ 1 }-> 1 :|: a -{ 1 }-> 0 :|: head(z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + xs, x >= 0 head(z') -{ 1 }-> 1 :|: z' = 0 ifPlus(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0 ifPlus(z', z'', z1, z2) -{ 1 }-> plusIter(z'', 1 + z1, 1 + z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 ifPlus(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0 ifSum(z', z'', z1, z2) -{ 1 }-> z1 :|: z'' >= 0, z' = 2, z1 >= 0, z2 >= 0 ifSum(z', z'', z1, z2) -{ 2 }-> sumIter(xs'', z2) :|: z'' = 1 + x2 + xs'', xs'' >= 0, z1 >= 0, z2 >= 0, z' = 1, x2 >= 0 ifSum(z', z'', z1, z2) -{ 2 }-> sumIter(0, z2) :|: z'' = 0, z1 >= 0, z2 >= 0, z' = 1 isempty(z') -{ 1 }-> 2 :|: z' = 0 isempty(z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x + xs, x >= 0 le(z', z'') -{ 2 + z'' }-> s' :|: s' >= 0, s' <= 2, z' - 1 >= 0, z'' - 1 >= 0 le(z', z'') -{ 1 }-> 2 :|: z'' >= 0, z' = 0 le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 le(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 plus(z', z'') -{ 1 }-> plusIter(z', z'', 0) :|: z' >= 0, z'' >= 0 plusIter(z', z'', z1) -{ 3 + z1 }-> ifPlus(s, 1 + (z' - 1), z'', 1 + (z1 - 1)) :|: s >= 0, s <= 2, z'' >= 0, z1 - 1 >= 0, z' - 1 >= 0 plusIter(z', z'', z1) -{ 2 }-> ifPlus(2, 0, z'', z1) :|: z1 >= 0, z'' >= 0, z' = 0 plusIter(z', z'', z1) -{ 2 }-> ifPlus(1, 1 + (z' - 1), z'', 0) :|: z1 = 0, z' - 1 >= 0, z'' >= 0 plusIter(z', z'', z1) -{ 1 }-> ifPlus(0, z', z'', z1) :|: z1 >= 0, z' >= 0, z'' >= 0 sum(z') -{ 1 }-> sumIter(z', 0) :|: z' >= 0 sumIter(z', z'') -{ 3 }-> ifSum(2, 0, z'', plus(z'', 1)) :|: z'' >= 0, z' = 0 sumIter(z', z'') -{ 3 }-> ifSum(1, 1 + x1 + xs', z'', plus(z'', x1)) :|: x1 >= 0, z' = 1 + x1 + xs', xs' >= 0, z'' >= 0 tail(z') -{ 1 }-> xs :|: xs >= 0, z' = 1 + x + xs, x >= 0 tail(z') -{ 1 }-> 0 :|: z' = 0 Function symbols to be analyzed: {plusIter,ifPlus}, {plus}, {ifSum,sumIter}, {sum} Previous analysis results are: le: runtime: O(n^1) [2 + z''], size: O(1) [2] tail: runtime: O(1) [1], size: O(n^1) [z'] isempty: runtime: O(1) [1], size: O(1) [2] head: runtime: O(1) [1], size: O(n^1) [1 + z'] a: runtime: O(1) [1], size: O(1) [1] ---------------------------------------- (275) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: plusIter after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? Computed SIZE bound using CoFloCo for: ifPlus after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (276) Obligation: Complexity RNTS consisting of the following rules: a -{ 1 }-> 1 :|: a -{ 1 }-> 0 :|: head(z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + xs, x >= 0 head(z') -{ 1 }-> 1 :|: z' = 0 ifPlus(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0 ifPlus(z', z'', z1, z2) -{ 1 }-> plusIter(z'', 1 + z1, 1 + z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 ifPlus(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0 ifSum(z', z'', z1, z2) -{ 1 }-> z1 :|: z'' >= 0, z' = 2, z1 >= 0, z2 >= 0 ifSum(z', z'', z1, z2) -{ 2 }-> sumIter(xs'', z2) :|: z'' = 1 + x2 + xs'', xs'' >= 0, z1 >= 0, z2 >= 0, z' = 1, x2 >= 0 ifSum(z', z'', z1, z2) -{ 2 }-> sumIter(0, z2) :|: z'' = 0, z1 >= 0, z2 >= 0, z' = 1 isempty(z') -{ 1 }-> 2 :|: z' = 0 isempty(z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x + xs, x >= 0 le(z', z'') -{ 2 + z'' }-> s' :|: s' >= 0, s' <= 2, z' - 1 >= 0, z'' - 1 >= 0 le(z', z'') -{ 1 }-> 2 :|: z'' >= 0, z' = 0 le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 le(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 plus(z', z'') -{ 1 }-> plusIter(z', z'', 0) :|: z' >= 0, z'' >= 0 plusIter(z', z'', z1) -{ 3 + z1 }-> ifPlus(s, 1 + (z' - 1), z'', 1 + (z1 - 1)) :|: s >= 0, s <= 2, z'' >= 0, z1 - 1 >= 0, z' - 1 >= 0 plusIter(z', z'', z1) -{ 2 }-> ifPlus(2, 0, z'', z1) :|: z1 >= 0, z'' >= 0, z' = 0 plusIter(z', z'', z1) -{ 2 }-> ifPlus(1, 1 + (z' - 1), z'', 0) :|: z1 = 0, z' - 1 >= 0, z'' >= 0 plusIter(z', z'', z1) -{ 1 }-> ifPlus(0, z', z'', z1) :|: z1 >= 0, z' >= 0, z'' >= 0 sum(z') -{ 1 }-> sumIter(z', 0) :|: z' >= 0 sumIter(z', z'') -{ 3 }-> ifSum(2, 0, z'', plus(z'', 1)) :|: z'' >= 0, z' = 0 sumIter(z', z'') -{ 3 }-> ifSum(1, 1 + x1 + xs', z'', plus(z'', x1)) :|: x1 >= 0, z' = 1 + x1 + xs', xs' >= 0, z'' >= 0 tail(z') -{ 1 }-> xs :|: xs >= 0, z' = 1 + x + xs, x >= 0 tail(z') -{ 1 }-> 0 :|: z' = 0 Function symbols to be analyzed: {plusIter,ifPlus}, {plus}, {ifSum,sumIter}, {sum} Previous analysis results are: le: runtime: O(n^1) [2 + z''], size: O(1) [2] tail: runtime: O(1) [1], size: O(n^1) [z'] isempty: runtime: O(1) [1], size: O(1) [2] head: runtime: O(1) [1], size: O(n^1) [1 + z'] a: runtime: O(1) [1], size: O(1) [1] plusIter: runtime: ?, size: INF ifPlus: runtime: ?, size: INF ---------------------------------------- (277) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: plusIter after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (278) Obligation: Complexity RNTS consisting of the following rules: a -{ 1 }-> 1 :|: a -{ 1 }-> 0 :|: head(z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + xs, x >= 0 head(z') -{ 1 }-> 1 :|: z' = 0 ifPlus(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0 ifPlus(z', z'', z1, z2) -{ 1 }-> plusIter(z'', 1 + z1, 1 + z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 ifPlus(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0 ifSum(z', z'', z1, z2) -{ 1 }-> z1 :|: z'' >= 0, z' = 2, z1 >= 0, z2 >= 0 ifSum(z', z'', z1, z2) -{ 2 }-> sumIter(xs'', z2) :|: z'' = 1 + x2 + xs'', xs'' >= 0, z1 >= 0, z2 >= 0, z' = 1, x2 >= 0 ifSum(z', z'', z1, z2) -{ 2 }-> sumIter(0, z2) :|: z'' = 0, z1 >= 0, z2 >= 0, z' = 1 isempty(z') -{ 1 }-> 2 :|: z' = 0 isempty(z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x + xs, x >= 0 le(z', z'') -{ 2 + z'' }-> s' :|: s' >= 0, s' <= 2, z' - 1 >= 0, z'' - 1 >= 0 le(z', z'') -{ 1 }-> 2 :|: z'' >= 0, z' = 0 le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 le(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 plus(z', z'') -{ 1 }-> plusIter(z', z'', 0) :|: z' >= 0, z'' >= 0 plusIter(z', z'', z1) -{ 3 + z1 }-> ifPlus(s, 1 + (z' - 1), z'', 1 + (z1 - 1)) :|: s >= 0, s <= 2, z'' >= 0, z1 - 1 >= 0, z' - 1 >= 0 plusIter(z', z'', z1) -{ 2 }-> ifPlus(2, 0, z'', z1) :|: z1 >= 0, z'' >= 0, z' = 0 plusIter(z', z'', z1) -{ 2 }-> ifPlus(1, 1 + (z' - 1), z'', 0) :|: z1 = 0, z' - 1 >= 0, z'' >= 0 plusIter(z', z'', z1) -{ 1 }-> ifPlus(0, z', z'', z1) :|: z1 >= 0, z' >= 0, z'' >= 0 sum(z') -{ 1 }-> sumIter(z', 0) :|: z' >= 0 sumIter(z', z'') -{ 3 }-> ifSum(2, 0, z'', plus(z'', 1)) :|: z'' >= 0, z' = 0 sumIter(z', z'') -{ 3 }-> ifSum(1, 1 + x1 + xs', z'', plus(z'', x1)) :|: x1 >= 0, z' = 1 + x1 + xs', xs' >= 0, z'' >= 0 tail(z') -{ 1 }-> xs :|: xs >= 0, z' = 1 + x + xs, x >= 0 tail(z') -{ 1 }-> 0 :|: z' = 0 Function symbols to be analyzed: {plusIter,ifPlus}, {plus}, {ifSum,sumIter}, {sum} Previous analysis results are: le: runtime: O(n^1) [2 + z''], size: O(1) [2] tail: runtime: O(1) [1], size: O(n^1) [z'] isempty: runtime: O(1) [1], size: O(1) [2] head: runtime: O(1) [1], size: O(n^1) [1 + z'] a: runtime: O(1) [1], size: O(1) [1] plusIter: runtime: INF, size: INF ifPlus: runtime: ?, size: INF ---------------------------------------- (279) 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: le(v0, v1) -> null_le [0] ifPlus(v0, v1, v2, v3) -> null_ifPlus [0] ifSum(v0, v1, v2, v3) -> null_ifSum [0] And the following fresh constants: null_le, null_ifPlus, null_ifSum ---------------------------------------- (280) 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: plus(x, y) -> plusIter(x, y, 0) [1] plusIter(x, y, z) -> ifPlus(le(x, z), x, y, z) [1] ifPlus(true, x, y, z) -> y [1] ifPlus(false, x, y, z) -> plusIter(x, s(y), s(z)) [1] le(s(x), 0) -> false [1] le(0, y) -> true [1] le(s(x), s(y)) -> le(x, y) [1] sum(xs) -> sumIter(xs, 0) [1] sumIter(xs, x) -> ifSum(isempty(xs), xs, x, plus(x, head(xs))) [1] ifSum(true, xs, x, y) -> x [1] ifSum(false, xs, x, y) -> sumIter(tail(xs), y) [1] isempty(nil) -> true [1] isempty(cons(x, xs)) -> false [1] head(nil) -> error [1] head(cons(x, xs)) -> x [1] tail(nil) -> nil [1] tail(cons(x, xs)) -> xs [1] a -> b [1] a -> c [1] le(v0, v1) -> null_le [0] ifPlus(v0, v1, v2, v3) -> null_ifPlus [0] ifSum(v0, v1, v2, v3) -> null_ifSum [0] The TRS has the following type information: plus :: 0:s:error:null_ifPlus:null_ifSum -> 0:s:error:null_ifPlus:null_ifSum -> 0:s:error:null_ifPlus:null_ifSum plusIter :: 0:s:error:null_ifPlus:null_ifSum -> 0:s:error:null_ifPlus:null_ifSum -> 0:s:error:null_ifPlus:null_ifSum -> 0:s:error:null_ifPlus:null_ifSum 0 :: 0:s:error:null_ifPlus:null_ifSum ifPlus :: true:false:null_le -> 0:s:error:null_ifPlus:null_ifSum -> 0:s:error:null_ifPlus:null_ifSum -> 0:s:error:null_ifPlus:null_ifSum -> 0:s:error:null_ifPlus:null_ifSum le :: 0:s:error:null_ifPlus:null_ifSum -> 0:s:error:null_ifPlus:null_ifSum -> true:false:null_le true :: true:false:null_le false :: true:false:null_le s :: 0:s:error:null_ifPlus:null_ifSum -> 0:s:error:null_ifPlus:null_ifSum sum :: nil:cons -> 0:s:error:null_ifPlus:null_ifSum sumIter :: nil:cons -> 0:s:error:null_ifPlus:null_ifSum -> 0:s:error:null_ifPlus:null_ifSum ifSum :: true:false:null_le -> nil:cons -> 0:s:error:null_ifPlus:null_ifSum -> 0:s:error:null_ifPlus:null_ifSum -> 0:s:error:null_ifPlus:null_ifSum isempty :: nil:cons -> true:false:null_le head :: nil:cons -> 0:s:error:null_ifPlus:null_ifSum tail :: nil:cons -> nil:cons nil :: nil:cons cons :: 0:s:error:null_ifPlus:null_ifSum -> nil:cons -> nil:cons error :: 0:s:error:null_ifPlus:null_ifSum a :: b:c b :: b:c c :: b:c null_le :: true:false:null_le null_ifPlus :: 0:s:error:null_ifPlus:null_ifSum null_ifSum :: 0:s:error:null_ifPlus:null_ifSum Rewrite Strategy: INNERMOST ---------------------------------------- (281) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: 0 => 0 true => 2 false => 1 nil => 0 error => 1 b => 0 c => 1 null_le => 0 null_ifPlus => 0 null_ifSum => 0 ---------------------------------------- (282) Obligation: Complexity RNTS consisting of the following rules: a -{ 1 }-> 1 :|: a -{ 1 }-> 0 :|: head(z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + xs, x >= 0 head(z') -{ 1 }-> 1 :|: z' = 0 ifPlus(z', z'', z1, z2) -{ 1 }-> y :|: z1 = y, z >= 0, z' = 2, z2 = z, x >= 0, y >= 0, z'' = x ifPlus(z', z'', z1, z2) -{ 1 }-> plusIter(x, 1 + y, 1 + z) :|: z1 = y, z >= 0, z2 = z, x >= 0, y >= 0, z'' = x, z' = 1 ifPlus(z', z'', z1, z2) -{ 0 }-> 0 :|: z2 = v3, v0 >= 0, z1 = v2, v1 >= 0, z'' = v1, v2 >= 0, v3 >= 0, z' = v0 ifSum(z', z'', z1, z2) -{ 1 }-> x :|: xs >= 0, z2 = y, z' = 2, x >= 0, y >= 0, z'' = xs, z1 = x ifSum(z', z'', z1, z2) -{ 1 }-> sumIter(tail(xs), y) :|: xs >= 0, z2 = y, x >= 0, y >= 0, z'' = xs, z' = 1, z1 = x ifSum(z', z'', z1, z2) -{ 0 }-> 0 :|: z2 = v3, v0 >= 0, z1 = v2, v1 >= 0, z'' = v1, v2 >= 0, v3 >= 0, z' = v0 isempty(z') -{ 1 }-> 2 :|: z' = 0 isempty(z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x + xs, x >= 0 le(z', z'') -{ 1 }-> le(x, y) :|: z' = 1 + x, x >= 0, y >= 0, z'' = 1 + y le(z', z'') -{ 1 }-> 2 :|: z'' = y, y >= 0, z' = 0 le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' = 1 + x, x >= 0 le(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0 plus(z', z'') -{ 1 }-> plusIter(x, y, 0) :|: z' = x, z'' = y, x >= 0, y >= 0 plusIter(z', z'', z1) -{ 1 }-> ifPlus(le(x, z), x, y, z) :|: z1 = z, z >= 0, z' = x, z'' = y, x >= 0, y >= 0 sum(z') -{ 1 }-> sumIter(xs, 0) :|: xs >= 0, z' = xs sumIter(z', z'') -{ 1 }-> ifSum(isempty(xs), xs, x, plus(x, head(xs))) :|: xs >= 0, x >= 0, z'' = x, z' = xs tail(z') -{ 1 }-> xs :|: xs >= 0, z' = 1 + x + xs, x >= 0 tail(z') -{ 1 }-> 0 :|: z' = 0 Only complete derivations are relevant for the runtime complexity.