KILLED proof of input_pXl0xlxzL9.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). (0) CpxTRS (1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxTRS (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (4) CpxTRS (5) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxWeightedTrs (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (8) CpxTypedWeightedTrs (9) CompletionProof [UPPER BOUND(ID), 0 ms] (10) CpxTypedWeightedCompleteTrs (11) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (12) CpxTypedWeightedCompleteTrs (13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (14) CpxRNTS (15) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (16) CpxRNTS (17) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms] (18) CpxRNTS (19) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (20) CpxRNTS (21) IntTrsBoundProof [UPPER BOUND(ID), 294 ms] (22) CpxRNTS (23) IntTrsBoundProof [UPPER BOUND(ID), 41 ms] (24) CpxRNTS (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (26) CpxRNTS (27) IntTrsBoundProof [UPPER BOUND(ID), 132 ms] (28) CpxRNTS (29) IntTrsBoundProof [UPPER BOUND(ID), 44 ms] (30) CpxRNTS (31) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (32) CpxRNTS (33) IntTrsBoundProof [UPPER BOUND(ID), 155 ms] (34) CpxRNTS (35) IntTrsBoundProof [UPPER BOUND(ID), 67 ms] (36) CpxRNTS (37) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (38) CpxRNTS (39) IntTrsBoundProof [UPPER BOUND(ID), 266 ms] (40) CpxRNTS (41) IntTrsBoundProof [UPPER BOUND(ID), 77 ms] (42) CpxRNTS (43) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (44) CpxRNTS (45) IntTrsBoundProof [UPPER BOUND(ID), 5186 ms] (46) CpxRNTS (47) IntTrsBoundProof [UPPER BOUND(ID), 3525 ms] (48) CpxRNTS (49) CompletionProof [UPPER BOUND(ID), 0 ms] (50) CpxTypedWeightedCompleteTrs (51) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 10 ms] (52) CpxRNTS (53) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (54) CdtProblem (55) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (56) CdtProblem (57) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (58) CdtProblem (59) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (60) CdtProblem (61) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (62) CdtProblem (63) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (64) CdtProblem (65) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (66) CdtProblem (67) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 77 ms] (68) CdtProblem (69) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (70) CdtProblem (71) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (72) CdtProblem (73) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 82 ms] (74) CdtProblem (75) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 65 ms] (76) CdtProblem (77) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 58 ms] (78) CdtProblem (79) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (80) CdtProblem (81) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (82) CdtProblem (83) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 67 ms] (84) CdtProblem (85) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 81 ms] (86) CdtProblem (87) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (88) CdtProblem (89) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (90) CdtProblem (91) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (92) CdtProblem (93) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 90 ms] (94) CdtProblem (95) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 147 ms] (96) CdtProblem (97) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (98) CdtProblem (99) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (100) CdtProblem (101) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (102) CdtProblem (103) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 217 ms] (104) CdtProblem (105) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 83 ms] (106) CdtProblem (107) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (108) CdtProblem (109) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (110) CdtProblem (111) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 113 ms] (112) CdtProblem (113) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (114) CdtProblem (115) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 117 ms] (116) CdtProblem (117) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (118) CdtProblem (119) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (120) CdtProblem (121) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (122) CdtProblem (123) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (124) CdtProblem (125) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (126) CdtProblem (127) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 136 ms] (128) CdtProblem (129) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 155 ms] (130) CdtProblem (131) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 144 ms] (132) CdtProblem (133) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (134) CdtProblem (135) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (136) CdtProblem (137) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (138) CdtProblem (139) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (140) CdtProblem (141) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 143 ms] (142) CdtProblem (143) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (144) CdtProblem (145) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (146) CdtProblem (147) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (148) CdtProblem (149) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 143 ms] (150) CdtProblem (151) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 198 ms] (152) CdtProblem (153) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (154) CdtProblem (155) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (156) CdtProblem (157) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 164 ms] (158) CdtProblem (159) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (160) CdtProblem (161) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (162) CdtProblem (163) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 4 ms] (164) CdtProblem (165) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (166) CdtProblem (167) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (168) CdtProblem (169) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (170) CdtProblem (171) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 146 ms] (172) CdtProblem (173) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 166 ms] (174) CdtProblem (175) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (176) CdtProblem (177) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (178) CdtProblem (179) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 182 ms] (180) CdtProblem (181) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 208 ms] (182) CdtProblem (183) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (184) CdtProblem (185) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 203 ms] (186) CdtProblem (187) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (188) CdtProblem (189) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 3 ms] (190) CdtProblem (191) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (192) CdtProblem (193) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (194) CdtProblem (195) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (196) CdtProblem (197) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 175 ms] (198) CdtProblem (199) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (200) CdtProblem (201) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (202) CdtProblem (203) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 94 ms] (204) CdtProblem (205) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (206) CdtProblem (207) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 132 ms] (208) CdtProblem (209) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (210) CdtProblem (211) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 201 ms] (212) CdtProblem (213) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (214) CdtProblem (215) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 157 ms] (216) CdtProblem (217) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (218) CdtProblem (219) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 217 ms] (220) CdtProblem (221) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (222) CdtProblem (223) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (224) CdtProblem (225) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (226) CdtProblem (227) CdtRewritingProof [BOTH BOUNDS(ID, ID), 3 ms] (228) CdtProblem (229) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 207 ms] (230) CdtProblem (231) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (232) CdtProblem (233) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (234) CdtProblem (235) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (236) CdtProblem (237) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (238) CdtProblem (239) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 184 ms] (240) CdtProblem (241) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (242) CdtProblem (243) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (244) CdtProblem (245) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (246) CdtProblem (247) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (248) CdtProblem (249) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (250) CdtProblem (251) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (252) CdtProblem (253) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (254) CdtProblem ---------------------------------------- (0) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: isEmpty(nil) -> true isEmpty(cons(x, xs)) -> false last(cons(x, nil)) -> x last(cons(x, cons(y, ys))) -> last(cons(y, ys)) dropLast(nil) -> nil dropLast(cons(x, nil)) -> nil dropLast(cons(x, cons(y, ys))) -> cons(x, dropLast(cons(y, ys))) append(nil, ys) -> ys append(cons(x, xs), ys) -> cons(x, append(xs, ys)) reverse(xs) -> rev(xs, nil) rev(xs, ys) -> if(isEmpty(xs), dropLast(xs), append(ys, last(xs)), ys) if(true, xs, ys, zs) -> zs if(false, xs, ys, zs) -> rev(xs, ys) 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: isEmpty(nil) -> true isEmpty(cons(x, xs)) -> false last(cons(x, nil)) -> x last(cons(x, cons(y, ys))) -> last(cons(y, ys)) dropLast(nil) -> nil dropLast(cons(x, nil)) -> nil dropLast(cons(x, cons(y, ys))) -> cons(x, dropLast(cons(y, ys))) append(nil, ys) -> ys append(cons(x, xs), ys) -> cons(x, append(xs, ys)) reverse(xs) -> rev(xs, nil) rev(xs, ys) -> if(isEmpty(xs), dropLast(xs), append(ys, last(xs)), ys) if(true, xs, ys, zs) -> zs if(false, xs, ys, zs) -> rev(xs, ys) 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: isEmpty(nil) -> true isEmpty(cons(x, xs)) -> false last(cons(x, nil)) -> x last(cons(x, cons(y, ys))) -> last(cons(y, ys)) dropLast(nil) -> nil dropLast(cons(x, nil)) -> nil dropLast(cons(x, cons(y, ys))) -> cons(x, dropLast(cons(y, ys))) append(nil, ys) -> ys append(cons(x, xs), ys) -> cons(x, append(xs, ys)) reverse(xs) -> rev(xs, nil) rev(xs, ys) -> if(isEmpty(xs), dropLast(xs), append(ys, last(xs)), ys) if(true, xs, ys, zs) -> zs if(false, xs, ys, zs) -> rev(xs, ys) S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (5) RelTrsToWeightedTrsProof (UPPER BOUND(ID)) Transformed relative TRS to weighted TRS ---------------------------------------- (6) Obligation: The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: isEmpty(nil) -> true [1] isEmpty(cons(x, xs)) -> false [1] last(cons(x, nil)) -> x [1] last(cons(x, cons(y, ys))) -> last(cons(y, ys)) [1] dropLast(nil) -> nil [1] dropLast(cons(x, nil)) -> nil [1] dropLast(cons(x, cons(y, ys))) -> cons(x, dropLast(cons(y, ys))) [1] append(nil, ys) -> ys [1] append(cons(x, xs), ys) -> cons(x, append(xs, ys)) [1] reverse(xs) -> rev(xs, nil) [1] rev(xs, ys) -> if(isEmpty(xs), dropLast(xs), append(ys, last(xs)), ys) [1] if(true, xs, ys, zs) -> zs [1] if(false, xs, ys, zs) -> rev(xs, ys) [1] Rewrite Strategy: INNERMOST ---------------------------------------- (7) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (8) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: isEmpty(nil) -> true [1] isEmpty(cons(x, xs)) -> false [1] last(cons(x, nil)) -> x [1] last(cons(x, cons(y, ys))) -> last(cons(y, ys)) [1] dropLast(nil) -> nil [1] dropLast(cons(x, nil)) -> nil [1] dropLast(cons(x, cons(y, ys))) -> cons(x, dropLast(cons(y, ys))) [1] append(nil, ys) -> ys [1] append(cons(x, xs), ys) -> cons(x, append(xs, ys)) [1] reverse(xs) -> rev(xs, nil) [1] rev(xs, ys) -> if(isEmpty(xs), dropLast(xs), append(ys, last(xs)), ys) [1] if(true, xs, ys, zs) -> zs [1] if(false, xs, ys, zs) -> rev(xs, ys) [1] The TRS has the following type information: isEmpty :: nil:cons -> true:false nil :: nil:cons true :: true:false cons :: nil:cons -> nil:cons -> nil:cons false :: true:false last :: nil:cons -> nil:cons dropLast :: nil:cons -> nil:cons append :: nil:cons -> nil:cons -> nil:cons reverse :: nil:cons -> nil:cons rev :: nil:cons -> nil:cons -> nil:cons if :: true:false -> nil:cons -> nil:cons -> nil:cons -> nil:cons Rewrite Strategy: INNERMOST ---------------------------------------- (9) CompletionProof (UPPER BOUND(ID)) The transformation into a RNTS is sound, since: (a) The obligation is a constructor system where every type has a constant constructor, (b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols: reverse_1 rev_2 if_4 (c) The following functions are completely defined: isEmpty_1 dropLast_1 append_2 last_1 Due to the following rules being added: last(v0) -> nil [0] And the following fresh constants: none ---------------------------------------- (10) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: isEmpty(nil) -> true [1] isEmpty(cons(x, xs)) -> false [1] last(cons(x, nil)) -> x [1] last(cons(x, cons(y, ys))) -> last(cons(y, ys)) [1] dropLast(nil) -> nil [1] dropLast(cons(x, nil)) -> nil [1] dropLast(cons(x, cons(y, ys))) -> cons(x, dropLast(cons(y, ys))) [1] append(nil, ys) -> ys [1] append(cons(x, xs), ys) -> cons(x, append(xs, ys)) [1] reverse(xs) -> rev(xs, nil) [1] rev(xs, ys) -> if(isEmpty(xs), dropLast(xs), append(ys, last(xs)), ys) [1] if(true, xs, ys, zs) -> zs [1] if(false, xs, ys, zs) -> rev(xs, ys) [1] last(v0) -> nil [0] The TRS has the following type information: isEmpty :: nil:cons -> true:false nil :: nil:cons true :: true:false cons :: nil:cons -> nil:cons -> nil:cons false :: true:false last :: nil:cons -> nil:cons dropLast :: nil:cons -> nil:cons append :: nil:cons -> nil:cons -> nil:cons reverse :: nil:cons -> nil:cons rev :: nil:cons -> nil:cons -> nil:cons if :: true:false -> nil:cons -> nil:cons -> nil:cons -> nil:cons Rewrite Strategy: INNERMOST ---------------------------------------- (11) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (12) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: isEmpty(nil) -> true [1] isEmpty(cons(x, xs)) -> false [1] last(cons(x, nil)) -> x [1] last(cons(x, cons(y, ys))) -> last(cons(y, ys)) [1] dropLast(nil) -> nil [1] dropLast(cons(x, nil)) -> nil [1] dropLast(cons(x, cons(y, ys))) -> cons(x, dropLast(cons(y, ys))) [1] append(nil, ys) -> ys [1] append(cons(x, xs), ys) -> cons(x, append(xs, ys)) [1] reverse(xs) -> rev(xs, nil) [1] rev(nil, ys) -> if(true, nil, append(ys, nil), ys) [3] rev(cons(x', nil), ys) -> if(false, nil, append(ys, x'), ys) [4] rev(cons(x', nil), ys) -> if(false, nil, append(ys, nil), ys) [3] rev(cons(x', cons(y', ys')), ys) -> if(false, cons(x', dropLast(cons(y', ys'))), append(ys, last(cons(y', ys'))), ys) [4] rev(cons(x', cons(y', ys')), ys) -> if(false, cons(x', dropLast(cons(y', ys'))), append(ys, nil), ys) [3] if(true, xs, ys, zs) -> zs [1] if(false, xs, ys, zs) -> rev(xs, ys) [1] last(v0) -> nil [0] The TRS has the following type information: isEmpty :: nil:cons -> true:false nil :: nil:cons true :: true:false cons :: nil:cons -> nil:cons -> nil:cons false :: true:false last :: nil:cons -> nil:cons dropLast :: nil:cons -> nil:cons append :: nil:cons -> nil:cons -> nil:cons reverse :: nil:cons -> nil:cons rev :: nil:cons -> nil:cons -> nil:cons if :: true:false -> nil:cons -> nil:cons -> nil:cons -> nil:cons Rewrite Strategy: INNERMOST ---------------------------------------- (13) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: nil => 0 true => 1 false => 0 ---------------------------------------- (14) Obligation: Complexity RNTS consisting of the following rules: append(z, z') -{ 1 }-> ys :|: z' = ys, ys >= 0, z = 0 append(z, z') -{ 1 }-> 1 + x + append(xs, ys) :|: z = 1 + x + xs, xs >= 0, z' = ys, ys >= 0, x >= 0 dropLast(z) -{ 1 }-> 0 :|: z = 0 dropLast(z) -{ 1 }-> 0 :|: x >= 0, z = 1 + x + 0 dropLast(z) -{ 1 }-> 1 + x + dropLast(1 + y + ys) :|: ys >= 0, x >= 0, y >= 0, z = 1 + x + (1 + y + ys) if(z, z', z'', z1) -{ 1 }-> zs :|: xs >= 0, z = 1, ys >= 0, zs >= 0, z'' = ys, z' = xs, z1 = zs if(z, z', z'', z1) -{ 1 }-> rev(xs, ys) :|: xs >= 0, ys >= 0, zs >= 0, z'' = ys, z' = xs, z = 0, z1 = zs isEmpty(z) -{ 1 }-> 1 :|: z = 0 isEmpty(z) -{ 1 }-> 0 :|: z = 1 + x + xs, xs >= 0, x >= 0 last(z) -{ 1 }-> x :|: x >= 0, z = 1 + x + 0 last(z) -{ 1 }-> last(1 + y + ys) :|: ys >= 0, x >= 0, y >= 0, z = 1 + x + (1 + y + ys) last(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 rev(z, z') -{ 3 }-> if(1, 0, append(ys, 0), ys) :|: z' = ys, ys >= 0, z = 0 rev(z, z') -{ 4 }-> if(0, 0, append(ys, x'), ys) :|: z = 1 + x' + 0, z' = ys, ys >= 0, x' >= 0 rev(z, z') -{ 3 }-> if(0, 0, append(ys, 0), ys) :|: z = 1 + x' + 0, z' = ys, ys >= 0, x' >= 0 rev(z, z') -{ 4 }-> if(0, 1 + x' + dropLast(1 + y' + ys'), append(ys, last(1 + y' + ys')), ys) :|: z' = ys, ys >= 0, z = 1 + x' + (1 + y' + ys'), x' >= 0, ys' >= 0, y' >= 0 rev(z, z') -{ 3 }-> if(0, 1 + x' + dropLast(1 + y' + ys'), append(ys, 0), ys) :|: z' = ys, ys >= 0, z = 1 + x' + (1 + y' + ys'), x' >= 0, ys' >= 0, y' >= 0 reverse(z) -{ 1 }-> rev(xs, 0) :|: xs >= 0, z = xs ---------------------------------------- (15) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (16) Obligation: Complexity RNTS consisting of the following rules: append(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 append(z, z') -{ 1 }-> 1 + x + append(xs, z') :|: z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0 dropLast(z) -{ 1 }-> 0 :|: z = 0 dropLast(z) -{ 1 }-> 0 :|: z - 1 >= 0 dropLast(z) -{ 1 }-> 1 + x + dropLast(1 + y + ys) :|: ys >= 0, x >= 0, y >= 0, z = 1 + x + (1 + y + ys) if(z, z', z'', z1) -{ 1 }-> z1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 if(z, z', z'', z1) -{ 1 }-> rev(z', z'') :|: z' >= 0, z'' >= 0, z1 >= 0, z = 0 isEmpty(z) -{ 1 }-> 1 :|: z = 0 isEmpty(z) -{ 1 }-> 0 :|: z = 1 + x + xs, xs >= 0, x >= 0 last(z) -{ 1 }-> last(1 + y + ys) :|: ys >= 0, x >= 0, y >= 0, z = 1 + x + (1 + y + ys) last(z) -{ 0 }-> 0 :|: z >= 0 last(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 rev(z, z') -{ 3 }-> if(1, 0, append(z', 0), z') :|: z' >= 0, z = 0 rev(z, z') -{ 3 }-> if(0, 0, append(z', 0), z') :|: z' >= 0, z - 1 >= 0 rev(z, z') -{ 4 }-> if(0, 0, append(z', z - 1), z') :|: z' >= 0, z - 1 >= 0 rev(z, z') -{ 4 }-> if(0, 1 + x' + dropLast(1 + y' + ys'), append(z', last(1 + y' + ys')), z') :|: z' >= 0, z = 1 + x' + (1 + y' + ys'), x' >= 0, ys' >= 0, y' >= 0 rev(z, z') -{ 3 }-> if(0, 1 + x' + dropLast(1 + y' + ys'), append(z', 0), z') :|: z' >= 0, z = 1 + x' + (1 + y' + ys'), x' >= 0, ys' >= 0, y' >= 0 reverse(z) -{ 1 }-> rev(z, 0) :|: z >= 0 ---------------------------------------- (17) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { append } { isEmpty } { dropLast } { last } { if, rev } { reverse } ---------------------------------------- (18) Obligation: Complexity RNTS consisting of the following rules: append(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 append(z, z') -{ 1 }-> 1 + x + append(xs, z') :|: z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0 dropLast(z) -{ 1 }-> 0 :|: z = 0 dropLast(z) -{ 1 }-> 0 :|: z - 1 >= 0 dropLast(z) -{ 1 }-> 1 + x + dropLast(1 + y + ys) :|: ys >= 0, x >= 0, y >= 0, z = 1 + x + (1 + y + ys) if(z, z', z'', z1) -{ 1 }-> z1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 if(z, z', z'', z1) -{ 1 }-> rev(z', z'') :|: z' >= 0, z'' >= 0, z1 >= 0, z = 0 isEmpty(z) -{ 1 }-> 1 :|: z = 0 isEmpty(z) -{ 1 }-> 0 :|: z = 1 + x + xs, xs >= 0, x >= 0 last(z) -{ 1 }-> last(1 + y + ys) :|: ys >= 0, x >= 0, y >= 0, z = 1 + x + (1 + y + ys) last(z) -{ 0 }-> 0 :|: z >= 0 last(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 rev(z, z') -{ 3 }-> if(1, 0, append(z', 0), z') :|: z' >= 0, z = 0 rev(z, z') -{ 3 }-> if(0, 0, append(z', 0), z') :|: z' >= 0, z - 1 >= 0 rev(z, z') -{ 4 }-> if(0, 0, append(z', z - 1), z') :|: z' >= 0, z - 1 >= 0 rev(z, z') -{ 4 }-> if(0, 1 + x' + dropLast(1 + y' + ys'), append(z', last(1 + y' + ys')), z') :|: z' >= 0, z = 1 + x' + (1 + y' + ys'), x' >= 0, ys' >= 0, y' >= 0 rev(z, z') -{ 3 }-> if(0, 1 + x' + dropLast(1 + y' + ys'), append(z', 0), z') :|: z' >= 0, z = 1 + x' + (1 + y' + ys'), x' >= 0, ys' >= 0, y' >= 0 reverse(z) -{ 1 }-> rev(z, 0) :|: z >= 0 Function symbols to be analyzed: {append}, {isEmpty}, {dropLast}, {last}, {if,rev}, {reverse} ---------------------------------------- (19) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: append(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 append(z, z') -{ 1 }-> 1 + x + append(xs, z') :|: z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0 dropLast(z) -{ 1 }-> 0 :|: z = 0 dropLast(z) -{ 1 }-> 0 :|: z - 1 >= 0 dropLast(z) -{ 1 }-> 1 + x + dropLast(1 + y + ys) :|: ys >= 0, x >= 0, y >= 0, z = 1 + x + (1 + y + ys) if(z, z', z'', z1) -{ 1 }-> z1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 if(z, z', z'', z1) -{ 1 }-> rev(z', z'') :|: z' >= 0, z'' >= 0, z1 >= 0, z = 0 isEmpty(z) -{ 1 }-> 1 :|: z = 0 isEmpty(z) -{ 1 }-> 0 :|: z = 1 + x + xs, xs >= 0, x >= 0 last(z) -{ 1 }-> last(1 + y + ys) :|: ys >= 0, x >= 0, y >= 0, z = 1 + x + (1 + y + ys) last(z) -{ 0 }-> 0 :|: z >= 0 last(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 rev(z, z') -{ 3 }-> if(1, 0, append(z', 0), z') :|: z' >= 0, z = 0 rev(z, z') -{ 3 }-> if(0, 0, append(z', 0), z') :|: z' >= 0, z - 1 >= 0 rev(z, z') -{ 4 }-> if(0, 0, append(z', z - 1), z') :|: z' >= 0, z - 1 >= 0 rev(z, z') -{ 4 }-> if(0, 1 + x' + dropLast(1 + y' + ys'), append(z', last(1 + y' + ys')), z') :|: z' >= 0, z = 1 + x' + (1 + y' + ys'), x' >= 0, ys' >= 0, y' >= 0 rev(z, z') -{ 3 }-> if(0, 1 + x' + dropLast(1 + y' + ys'), append(z', 0), z') :|: z' >= 0, z = 1 + x' + (1 + y' + ys'), x' >= 0, ys' >= 0, y' >= 0 reverse(z) -{ 1 }-> rev(z, 0) :|: z >= 0 Function symbols to be analyzed: {append}, {isEmpty}, {dropLast}, {last}, {if,rev}, {reverse} ---------------------------------------- (21) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: append after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z + z' ---------------------------------------- (22) Obligation: Complexity RNTS consisting of the following rules: append(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 append(z, z') -{ 1 }-> 1 + x + append(xs, z') :|: z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0 dropLast(z) -{ 1 }-> 0 :|: z = 0 dropLast(z) -{ 1 }-> 0 :|: z - 1 >= 0 dropLast(z) -{ 1 }-> 1 + x + dropLast(1 + y + ys) :|: ys >= 0, x >= 0, y >= 0, z = 1 + x + (1 + y + ys) if(z, z', z'', z1) -{ 1 }-> z1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 if(z, z', z'', z1) -{ 1 }-> rev(z', z'') :|: z' >= 0, z'' >= 0, z1 >= 0, z = 0 isEmpty(z) -{ 1 }-> 1 :|: z = 0 isEmpty(z) -{ 1 }-> 0 :|: z = 1 + x + xs, xs >= 0, x >= 0 last(z) -{ 1 }-> last(1 + y + ys) :|: ys >= 0, x >= 0, y >= 0, z = 1 + x + (1 + y + ys) last(z) -{ 0 }-> 0 :|: z >= 0 last(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 rev(z, z') -{ 3 }-> if(1, 0, append(z', 0), z') :|: z' >= 0, z = 0 rev(z, z') -{ 3 }-> if(0, 0, append(z', 0), z') :|: z' >= 0, z - 1 >= 0 rev(z, z') -{ 4 }-> if(0, 0, append(z', z - 1), z') :|: z' >= 0, z - 1 >= 0 rev(z, z') -{ 4 }-> if(0, 1 + x' + dropLast(1 + y' + ys'), append(z', last(1 + y' + ys')), z') :|: z' >= 0, z = 1 + x' + (1 + y' + ys'), x' >= 0, ys' >= 0, y' >= 0 rev(z, z') -{ 3 }-> if(0, 1 + x' + dropLast(1 + y' + ys'), append(z', 0), z') :|: z' >= 0, z = 1 + x' + (1 + y' + ys'), x' >= 0, ys' >= 0, y' >= 0 reverse(z) -{ 1 }-> rev(z, 0) :|: z >= 0 Function symbols to be analyzed: {append}, {isEmpty}, {dropLast}, {last}, {if,rev}, {reverse} Previous analysis results are: append: runtime: ?, size: O(n^1) [z + z'] ---------------------------------------- (23) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: append after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z ---------------------------------------- (24) Obligation: Complexity RNTS consisting of the following rules: append(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 append(z, z') -{ 1 }-> 1 + x + append(xs, z') :|: z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0 dropLast(z) -{ 1 }-> 0 :|: z = 0 dropLast(z) -{ 1 }-> 0 :|: z - 1 >= 0 dropLast(z) -{ 1 }-> 1 + x + dropLast(1 + y + ys) :|: ys >= 0, x >= 0, y >= 0, z = 1 + x + (1 + y + ys) if(z, z', z'', z1) -{ 1 }-> z1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 if(z, z', z'', z1) -{ 1 }-> rev(z', z'') :|: z' >= 0, z'' >= 0, z1 >= 0, z = 0 isEmpty(z) -{ 1 }-> 1 :|: z = 0 isEmpty(z) -{ 1 }-> 0 :|: z = 1 + x + xs, xs >= 0, x >= 0 last(z) -{ 1 }-> last(1 + y + ys) :|: ys >= 0, x >= 0, y >= 0, z = 1 + x + (1 + y + ys) last(z) -{ 0 }-> 0 :|: z >= 0 last(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 rev(z, z') -{ 3 }-> if(1, 0, append(z', 0), z') :|: z' >= 0, z = 0 rev(z, z') -{ 3 }-> if(0, 0, append(z', 0), z') :|: z' >= 0, z - 1 >= 0 rev(z, z') -{ 4 }-> if(0, 0, append(z', z - 1), z') :|: z' >= 0, z - 1 >= 0 rev(z, z') -{ 4 }-> if(0, 1 + x' + dropLast(1 + y' + ys'), append(z', last(1 + y' + ys')), z') :|: z' >= 0, z = 1 + x' + (1 + y' + ys'), x' >= 0, ys' >= 0, y' >= 0 rev(z, z') -{ 3 }-> if(0, 1 + x' + dropLast(1 + y' + ys'), append(z', 0), z') :|: z' >= 0, z = 1 + x' + (1 + y' + ys'), x' >= 0, ys' >= 0, y' >= 0 reverse(z) -{ 1 }-> rev(z, 0) :|: z >= 0 Function symbols to be analyzed: {isEmpty}, {dropLast}, {last}, {if,rev}, {reverse} Previous analysis results are: append: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] ---------------------------------------- (25) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (26) Obligation: Complexity RNTS consisting of the following rules: append(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 append(z, z') -{ 2 + xs }-> 1 + x + s :|: s >= 0, s <= xs + z', z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0 dropLast(z) -{ 1 }-> 0 :|: z = 0 dropLast(z) -{ 1 }-> 0 :|: z - 1 >= 0 dropLast(z) -{ 1 }-> 1 + x + dropLast(1 + y + ys) :|: ys >= 0, x >= 0, y >= 0, z = 1 + x + (1 + y + ys) if(z, z', z'', z1) -{ 1 }-> z1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 if(z, z', z'', z1) -{ 1 }-> rev(z', z'') :|: z' >= 0, z'' >= 0, z1 >= 0, z = 0 isEmpty(z) -{ 1 }-> 1 :|: z = 0 isEmpty(z) -{ 1 }-> 0 :|: z = 1 + x + xs, xs >= 0, x >= 0 last(z) -{ 1 }-> last(1 + y + ys) :|: ys >= 0, x >= 0, y >= 0, z = 1 + x + (1 + y + ys) last(z) -{ 0 }-> 0 :|: z >= 0 last(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 rev(z, z') -{ 4 + z' }-> if(1, 0, s', z') :|: s' >= 0, s' <= z' + 0, z' >= 0, z = 0 rev(z, z') -{ 5 + z' }-> if(0, 0, s'', z') :|: s'' >= 0, s'' <= z' + (z - 1), z' >= 0, z - 1 >= 0 rev(z, z') -{ 4 + z' }-> if(0, 0, s1, z') :|: s1 >= 0, s1 <= z' + 0, z' >= 0, z - 1 >= 0 rev(z, z') -{ 4 + z' }-> if(0, 1 + x' + dropLast(1 + y' + ys'), s2, z') :|: s2 >= 0, s2 <= z' + 0, z' >= 0, z = 1 + x' + (1 + y' + ys'), x' >= 0, ys' >= 0, y' >= 0 rev(z, z') -{ 4 }-> if(0, 1 + x' + dropLast(1 + y' + ys'), append(z', last(1 + y' + ys')), z') :|: z' >= 0, z = 1 + x' + (1 + y' + ys'), x' >= 0, ys' >= 0, y' >= 0 reverse(z) -{ 1 }-> rev(z, 0) :|: z >= 0 Function symbols to be analyzed: {isEmpty}, {dropLast}, {last}, {if,rev}, {reverse} Previous analysis results are: append: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] ---------------------------------------- (27) 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: 1 ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: append(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 append(z, z') -{ 2 + xs }-> 1 + x + s :|: s >= 0, s <= xs + z', z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0 dropLast(z) -{ 1 }-> 0 :|: z = 0 dropLast(z) -{ 1 }-> 0 :|: z - 1 >= 0 dropLast(z) -{ 1 }-> 1 + x + dropLast(1 + y + ys) :|: ys >= 0, x >= 0, y >= 0, z = 1 + x + (1 + y + ys) if(z, z', z'', z1) -{ 1 }-> z1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 if(z, z', z'', z1) -{ 1 }-> rev(z', z'') :|: z' >= 0, z'' >= 0, z1 >= 0, z = 0 isEmpty(z) -{ 1 }-> 1 :|: z = 0 isEmpty(z) -{ 1 }-> 0 :|: z = 1 + x + xs, xs >= 0, x >= 0 last(z) -{ 1 }-> last(1 + y + ys) :|: ys >= 0, x >= 0, y >= 0, z = 1 + x + (1 + y + ys) last(z) -{ 0 }-> 0 :|: z >= 0 last(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 rev(z, z') -{ 4 + z' }-> if(1, 0, s', z') :|: s' >= 0, s' <= z' + 0, z' >= 0, z = 0 rev(z, z') -{ 5 + z' }-> if(0, 0, s'', z') :|: s'' >= 0, s'' <= z' + (z - 1), z' >= 0, z - 1 >= 0 rev(z, z') -{ 4 + z' }-> if(0, 0, s1, z') :|: s1 >= 0, s1 <= z' + 0, z' >= 0, z - 1 >= 0 rev(z, z') -{ 4 + z' }-> if(0, 1 + x' + dropLast(1 + y' + ys'), s2, z') :|: s2 >= 0, s2 <= z' + 0, z' >= 0, z = 1 + x' + (1 + y' + ys'), x' >= 0, ys' >= 0, y' >= 0 rev(z, z') -{ 4 }-> if(0, 1 + x' + dropLast(1 + y' + ys'), append(z', last(1 + y' + ys')), z') :|: z' >= 0, z = 1 + x' + (1 + y' + ys'), x' >= 0, ys' >= 0, y' >= 0 reverse(z) -{ 1 }-> rev(z, 0) :|: z >= 0 Function symbols to be analyzed: {isEmpty}, {dropLast}, {last}, {if,rev}, {reverse} Previous analysis results are: append: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] isEmpty: runtime: ?, size: O(1) [1] ---------------------------------------- (29) 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 ---------------------------------------- (30) Obligation: Complexity RNTS consisting of the following rules: append(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 append(z, z') -{ 2 + xs }-> 1 + x + s :|: s >= 0, s <= xs + z', z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0 dropLast(z) -{ 1 }-> 0 :|: z = 0 dropLast(z) -{ 1 }-> 0 :|: z - 1 >= 0 dropLast(z) -{ 1 }-> 1 + x + dropLast(1 + y + ys) :|: ys >= 0, x >= 0, y >= 0, z = 1 + x + (1 + y + ys) if(z, z', z'', z1) -{ 1 }-> z1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 if(z, z', z'', z1) -{ 1 }-> rev(z', z'') :|: z' >= 0, z'' >= 0, z1 >= 0, z = 0 isEmpty(z) -{ 1 }-> 1 :|: z = 0 isEmpty(z) -{ 1 }-> 0 :|: z = 1 + x + xs, xs >= 0, x >= 0 last(z) -{ 1 }-> last(1 + y + ys) :|: ys >= 0, x >= 0, y >= 0, z = 1 + x + (1 + y + ys) last(z) -{ 0 }-> 0 :|: z >= 0 last(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 rev(z, z') -{ 4 + z' }-> if(1, 0, s', z') :|: s' >= 0, s' <= z' + 0, z' >= 0, z = 0 rev(z, z') -{ 5 + z' }-> if(0, 0, s'', z') :|: s'' >= 0, s'' <= z' + (z - 1), z' >= 0, z - 1 >= 0 rev(z, z') -{ 4 + z' }-> if(0, 0, s1, z') :|: s1 >= 0, s1 <= z' + 0, z' >= 0, z - 1 >= 0 rev(z, z') -{ 4 + z' }-> if(0, 1 + x' + dropLast(1 + y' + ys'), s2, z') :|: s2 >= 0, s2 <= z' + 0, z' >= 0, z = 1 + x' + (1 + y' + ys'), x' >= 0, ys' >= 0, y' >= 0 rev(z, z') -{ 4 }-> if(0, 1 + x' + dropLast(1 + y' + ys'), append(z', last(1 + y' + ys')), z') :|: z' >= 0, z = 1 + x' + (1 + y' + ys'), x' >= 0, ys' >= 0, y' >= 0 reverse(z) -{ 1 }-> rev(z, 0) :|: z >= 0 Function symbols to be analyzed: {dropLast}, {last}, {if,rev}, {reverse} Previous analysis results are: append: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] isEmpty: runtime: O(1) [1], size: O(1) [1] ---------------------------------------- (31) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (32) Obligation: Complexity RNTS consisting of the following rules: append(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 append(z, z') -{ 2 + xs }-> 1 + x + s :|: s >= 0, s <= xs + z', z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0 dropLast(z) -{ 1 }-> 0 :|: z = 0 dropLast(z) -{ 1 }-> 0 :|: z - 1 >= 0 dropLast(z) -{ 1 }-> 1 + x + dropLast(1 + y + ys) :|: ys >= 0, x >= 0, y >= 0, z = 1 + x + (1 + y + ys) if(z, z', z'', z1) -{ 1 }-> z1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 if(z, z', z'', z1) -{ 1 }-> rev(z', z'') :|: z' >= 0, z'' >= 0, z1 >= 0, z = 0 isEmpty(z) -{ 1 }-> 1 :|: z = 0 isEmpty(z) -{ 1 }-> 0 :|: z = 1 + x + xs, xs >= 0, x >= 0 last(z) -{ 1 }-> last(1 + y + ys) :|: ys >= 0, x >= 0, y >= 0, z = 1 + x + (1 + y + ys) last(z) -{ 0 }-> 0 :|: z >= 0 last(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 rev(z, z') -{ 4 + z' }-> if(1, 0, s', z') :|: s' >= 0, s' <= z' + 0, z' >= 0, z = 0 rev(z, z') -{ 5 + z' }-> if(0, 0, s'', z') :|: s'' >= 0, s'' <= z' + (z - 1), z' >= 0, z - 1 >= 0 rev(z, z') -{ 4 + z' }-> if(0, 0, s1, z') :|: s1 >= 0, s1 <= z' + 0, z' >= 0, z - 1 >= 0 rev(z, z') -{ 4 + z' }-> if(0, 1 + x' + dropLast(1 + y' + ys'), s2, z') :|: s2 >= 0, s2 <= z' + 0, z' >= 0, z = 1 + x' + (1 + y' + ys'), x' >= 0, ys' >= 0, y' >= 0 rev(z, z') -{ 4 }-> if(0, 1 + x' + dropLast(1 + y' + ys'), append(z', last(1 + y' + ys')), z') :|: z' >= 0, z = 1 + x' + (1 + y' + ys'), x' >= 0, ys' >= 0, y' >= 0 reverse(z) -{ 1 }-> rev(z, 0) :|: z >= 0 Function symbols to be analyzed: {dropLast}, {last}, {if,rev}, {reverse} Previous analysis results are: append: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] isEmpty: runtime: O(1) [1], size: O(1) [1] ---------------------------------------- (33) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: dropLast after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z ---------------------------------------- (34) Obligation: Complexity RNTS consisting of the following rules: append(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 append(z, z') -{ 2 + xs }-> 1 + x + s :|: s >= 0, s <= xs + z', z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0 dropLast(z) -{ 1 }-> 0 :|: z = 0 dropLast(z) -{ 1 }-> 0 :|: z - 1 >= 0 dropLast(z) -{ 1 }-> 1 + x + dropLast(1 + y + ys) :|: ys >= 0, x >= 0, y >= 0, z = 1 + x + (1 + y + ys) if(z, z', z'', z1) -{ 1 }-> z1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 if(z, z', z'', z1) -{ 1 }-> rev(z', z'') :|: z' >= 0, z'' >= 0, z1 >= 0, z = 0 isEmpty(z) -{ 1 }-> 1 :|: z = 0 isEmpty(z) -{ 1 }-> 0 :|: z = 1 + x + xs, xs >= 0, x >= 0 last(z) -{ 1 }-> last(1 + y + ys) :|: ys >= 0, x >= 0, y >= 0, z = 1 + x + (1 + y + ys) last(z) -{ 0 }-> 0 :|: z >= 0 last(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 rev(z, z') -{ 4 + z' }-> if(1, 0, s', z') :|: s' >= 0, s' <= z' + 0, z' >= 0, z = 0 rev(z, z') -{ 5 + z' }-> if(0, 0, s'', z') :|: s'' >= 0, s'' <= z' + (z - 1), z' >= 0, z - 1 >= 0 rev(z, z') -{ 4 + z' }-> if(0, 0, s1, z') :|: s1 >= 0, s1 <= z' + 0, z' >= 0, z - 1 >= 0 rev(z, z') -{ 4 + z' }-> if(0, 1 + x' + dropLast(1 + y' + ys'), s2, z') :|: s2 >= 0, s2 <= z' + 0, z' >= 0, z = 1 + x' + (1 + y' + ys'), x' >= 0, ys' >= 0, y' >= 0 rev(z, z') -{ 4 }-> if(0, 1 + x' + dropLast(1 + y' + ys'), append(z', last(1 + y' + ys')), z') :|: z' >= 0, z = 1 + x' + (1 + y' + ys'), x' >= 0, ys' >= 0, y' >= 0 reverse(z) -{ 1 }-> rev(z, 0) :|: z >= 0 Function symbols to be analyzed: {dropLast}, {last}, {if,rev}, {reverse} Previous analysis results are: append: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] isEmpty: runtime: O(1) [1], size: O(1) [1] dropLast: runtime: ?, size: O(n^1) [z] ---------------------------------------- (35) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: dropLast after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2 + z ---------------------------------------- (36) Obligation: Complexity RNTS consisting of the following rules: append(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 append(z, z') -{ 2 + xs }-> 1 + x + s :|: s >= 0, s <= xs + z', z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0 dropLast(z) -{ 1 }-> 0 :|: z = 0 dropLast(z) -{ 1 }-> 0 :|: z - 1 >= 0 dropLast(z) -{ 1 }-> 1 + x + dropLast(1 + y + ys) :|: ys >= 0, x >= 0, y >= 0, z = 1 + x + (1 + y + ys) if(z, z', z'', z1) -{ 1 }-> z1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 if(z, z', z'', z1) -{ 1 }-> rev(z', z'') :|: z' >= 0, z'' >= 0, z1 >= 0, z = 0 isEmpty(z) -{ 1 }-> 1 :|: z = 0 isEmpty(z) -{ 1 }-> 0 :|: z = 1 + x + xs, xs >= 0, x >= 0 last(z) -{ 1 }-> last(1 + y + ys) :|: ys >= 0, x >= 0, y >= 0, z = 1 + x + (1 + y + ys) last(z) -{ 0 }-> 0 :|: z >= 0 last(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 rev(z, z') -{ 4 + z' }-> if(1, 0, s', z') :|: s' >= 0, s' <= z' + 0, z' >= 0, z = 0 rev(z, z') -{ 5 + z' }-> if(0, 0, s'', z') :|: s'' >= 0, s'' <= z' + (z - 1), z' >= 0, z - 1 >= 0 rev(z, z') -{ 4 + z' }-> if(0, 0, s1, z') :|: s1 >= 0, s1 <= z' + 0, z' >= 0, z - 1 >= 0 rev(z, z') -{ 4 + z' }-> if(0, 1 + x' + dropLast(1 + y' + ys'), s2, z') :|: s2 >= 0, s2 <= z' + 0, z' >= 0, z = 1 + x' + (1 + y' + ys'), x' >= 0, ys' >= 0, y' >= 0 rev(z, z') -{ 4 }-> if(0, 1 + x' + dropLast(1 + y' + ys'), append(z', last(1 + y' + ys')), z') :|: z' >= 0, z = 1 + x' + (1 + y' + ys'), x' >= 0, ys' >= 0, y' >= 0 reverse(z) -{ 1 }-> rev(z, 0) :|: z >= 0 Function symbols to be analyzed: {last}, {if,rev}, {reverse} Previous analysis results are: append: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] isEmpty: runtime: O(1) [1], size: O(1) [1] dropLast: runtime: O(n^1) [2 + z], size: O(n^1) [z] ---------------------------------------- (37) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (38) Obligation: Complexity RNTS consisting of the following rules: append(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 append(z, z') -{ 2 + xs }-> 1 + x + s :|: s >= 0, s <= xs + z', z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0 dropLast(z) -{ 1 }-> 0 :|: z = 0 dropLast(z) -{ 1 }-> 0 :|: z - 1 >= 0 dropLast(z) -{ 4 + y + ys }-> 1 + x + s3 :|: s3 >= 0, s3 <= 1 + y + ys, ys >= 0, x >= 0, y >= 0, z = 1 + x + (1 + y + ys) if(z, z', z'', z1) -{ 1 }-> z1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 if(z, z', z'', z1) -{ 1 }-> rev(z', z'') :|: z' >= 0, z'' >= 0, z1 >= 0, z = 0 isEmpty(z) -{ 1 }-> 1 :|: z = 0 isEmpty(z) -{ 1 }-> 0 :|: z = 1 + x + xs, xs >= 0, x >= 0 last(z) -{ 1 }-> last(1 + y + ys) :|: ys >= 0, x >= 0, y >= 0, z = 1 + x + (1 + y + ys) last(z) -{ 0 }-> 0 :|: z >= 0 last(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 rev(z, z') -{ 4 + z' }-> if(1, 0, s', z') :|: s' >= 0, s' <= z' + 0, z' >= 0, z = 0 rev(z, z') -{ 5 + z' }-> if(0, 0, s'', z') :|: s'' >= 0, s'' <= z' + (z - 1), z' >= 0, z - 1 >= 0 rev(z, z') -{ 4 + z' }-> if(0, 0, s1, z') :|: s1 >= 0, s1 <= z' + 0, z' >= 0, z - 1 >= 0 rev(z, z') -{ 7 + y' + ys' }-> if(0, 1 + x' + s4, append(z', last(1 + y' + ys')), z') :|: s4 >= 0, s4 <= 1 + y' + ys', z' >= 0, z = 1 + x' + (1 + y' + ys'), x' >= 0, ys' >= 0, y' >= 0 rev(z, z') -{ 7 + y' + ys' + z' }-> if(0, 1 + x' + s5, s2, z') :|: s5 >= 0, s5 <= 1 + y' + ys', s2 >= 0, s2 <= z' + 0, z' >= 0, z = 1 + x' + (1 + y' + ys'), x' >= 0, ys' >= 0, y' >= 0 reverse(z) -{ 1 }-> rev(z, 0) :|: z >= 0 Function symbols to be analyzed: {last}, {if,rev}, {reverse} Previous analysis results are: append: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] isEmpty: runtime: O(1) [1], size: O(1) [1] dropLast: runtime: O(n^1) [2 + z], size: O(n^1) [z] ---------------------------------------- (39) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: last after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z ---------------------------------------- (40) Obligation: Complexity RNTS consisting of the following rules: append(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 append(z, z') -{ 2 + xs }-> 1 + x + s :|: s >= 0, s <= xs + z', z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0 dropLast(z) -{ 1 }-> 0 :|: z = 0 dropLast(z) -{ 1 }-> 0 :|: z - 1 >= 0 dropLast(z) -{ 4 + y + ys }-> 1 + x + s3 :|: s3 >= 0, s3 <= 1 + y + ys, ys >= 0, x >= 0, y >= 0, z = 1 + x + (1 + y + ys) if(z, z', z'', z1) -{ 1 }-> z1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 if(z, z', z'', z1) -{ 1 }-> rev(z', z'') :|: z' >= 0, z'' >= 0, z1 >= 0, z = 0 isEmpty(z) -{ 1 }-> 1 :|: z = 0 isEmpty(z) -{ 1 }-> 0 :|: z = 1 + x + xs, xs >= 0, x >= 0 last(z) -{ 1 }-> last(1 + y + ys) :|: ys >= 0, x >= 0, y >= 0, z = 1 + x + (1 + y + ys) last(z) -{ 0 }-> 0 :|: z >= 0 last(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 rev(z, z') -{ 4 + z' }-> if(1, 0, s', z') :|: s' >= 0, s' <= z' + 0, z' >= 0, z = 0 rev(z, z') -{ 5 + z' }-> if(0, 0, s'', z') :|: s'' >= 0, s'' <= z' + (z - 1), z' >= 0, z - 1 >= 0 rev(z, z') -{ 4 + z' }-> if(0, 0, s1, z') :|: s1 >= 0, s1 <= z' + 0, z' >= 0, z - 1 >= 0 rev(z, z') -{ 7 + y' + ys' }-> if(0, 1 + x' + s4, append(z', last(1 + y' + ys')), z') :|: s4 >= 0, s4 <= 1 + y' + ys', z' >= 0, z = 1 + x' + (1 + y' + ys'), x' >= 0, ys' >= 0, y' >= 0 rev(z, z') -{ 7 + y' + ys' + z' }-> if(0, 1 + x' + s5, s2, z') :|: s5 >= 0, s5 <= 1 + y' + ys', s2 >= 0, s2 <= z' + 0, z' >= 0, z = 1 + x' + (1 + y' + ys'), x' >= 0, ys' >= 0, y' >= 0 reverse(z) -{ 1 }-> rev(z, 0) :|: z >= 0 Function symbols to be analyzed: {last}, {if,rev}, {reverse} Previous analysis results are: append: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] isEmpty: runtime: O(1) [1], size: O(1) [1] dropLast: runtime: O(n^1) [2 + z], size: O(n^1) [z] last: runtime: ?, size: O(n^1) [z] ---------------------------------------- (41) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: last after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z ---------------------------------------- (42) Obligation: Complexity RNTS consisting of the following rules: append(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 append(z, z') -{ 2 + xs }-> 1 + x + s :|: s >= 0, s <= xs + z', z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0 dropLast(z) -{ 1 }-> 0 :|: z = 0 dropLast(z) -{ 1 }-> 0 :|: z - 1 >= 0 dropLast(z) -{ 4 + y + ys }-> 1 + x + s3 :|: s3 >= 0, s3 <= 1 + y + ys, ys >= 0, x >= 0, y >= 0, z = 1 + x + (1 + y + ys) if(z, z', z'', z1) -{ 1 }-> z1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 if(z, z', z'', z1) -{ 1 }-> rev(z', z'') :|: z' >= 0, z'' >= 0, z1 >= 0, z = 0 isEmpty(z) -{ 1 }-> 1 :|: z = 0 isEmpty(z) -{ 1 }-> 0 :|: z = 1 + x + xs, xs >= 0, x >= 0 last(z) -{ 1 }-> last(1 + y + ys) :|: ys >= 0, x >= 0, y >= 0, z = 1 + x + (1 + y + ys) last(z) -{ 0 }-> 0 :|: z >= 0 last(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 rev(z, z') -{ 4 + z' }-> if(1, 0, s', z') :|: s' >= 0, s' <= z' + 0, z' >= 0, z = 0 rev(z, z') -{ 5 + z' }-> if(0, 0, s'', z') :|: s'' >= 0, s'' <= z' + (z - 1), z' >= 0, z - 1 >= 0 rev(z, z') -{ 4 + z' }-> if(0, 0, s1, z') :|: s1 >= 0, s1 <= z' + 0, z' >= 0, z - 1 >= 0 rev(z, z') -{ 7 + y' + ys' }-> if(0, 1 + x' + s4, append(z', last(1 + y' + ys')), z') :|: s4 >= 0, s4 <= 1 + y' + ys', z' >= 0, z = 1 + x' + (1 + y' + ys'), x' >= 0, ys' >= 0, y' >= 0 rev(z, z') -{ 7 + y' + ys' + z' }-> if(0, 1 + x' + s5, s2, z') :|: s5 >= 0, s5 <= 1 + y' + ys', s2 >= 0, s2 <= z' + 0, z' >= 0, z = 1 + x' + (1 + y' + ys'), x' >= 0, ys' >= 0, y' >= 0 reverse(z) -{ 1 }-> rev(z, 0) :|: z >= 0 Function symbols to be analyzed: {if,rev}, {reverse} Previous analysis results are: append: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] isEmpty: runtime: O(1) [1], size: O(1) [1] dropLast: runtime: O(n^1) [2 + z], size: O(n^1) [z] last: runtime: O(n^1) [1 + z], size: O(n^1) [z] ---------------------------------------- (43) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (44) Obligation: Complexity RNTS consisting of the following rules: append(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 append(z, z') -{ 2 + xs }-> 1 + x + s :|: s >= 0, s <= xs + z', z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0 dropLast(z) -{ 1 }-> 0 :|: z = 0 dropLast(z) -{ 1 }-> 0 :|: z - 1 >= 0 dropLast(z) -{ 4 + y + ys }-> 1 + x + s3 :|: s3 >= 0, s3 <= 1 + y + ys, ys >= 0, x >= 0, y >= 0, z = 1 + x + (1 + y + ys) if(z, z', z'', z1) -{ 1 }-> z1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 if(z, z', z'', z1) -{ 1 }-> rev(z', z'') :|: z' >= 0, z'' >= 0, z1 >= 0, z = 0 isEmpty(z) -{ 1 }-> 1 :|: z = 0 isEmpty(z) -{ 1 }-> 0 :|: z = 1 + x + xs, xs >= 0, x >= 0 last(z) -{ 3 + y + ys }-> s6 :|: s6 >= 0, s6 <= 1 + y + ys, ys >= 0, x >= 0, y >= 0, z = 1 + x + (1 + y + ys) last(z) -{ 0 }-> 0 :|: z >= 0 last(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 rev(z, z') -{ 4 + z' }-> if(1, 0, s', z') :|: s' >= 0, s' <= z' + 0, z' >= 0, z = 0 rev(z, z') -{ 5 + z' }-> if(0, 0, s'', z') :|: s'' >= 0, s'' <= z' + (z - 1), z' >= 0, z - 1 >= 0 rev(z, z') -{ 4 + z' }-> if(0, 0, s1, z') :|: s1 >= 0, s1 <= z' + 0, z' >= 0, z - 1 >= 0 rev(z, z') -{ 10 + 2*y' + 2*ys' + z' }-> if(0, 1 + x' + s4, s8, z') :|: s7 >= 0, s7 <= 1 + y' + ys', s8 >= 0, s8 <= z' + s7, s4 >= 0, s4 <= 1 + y' + ys', z' >= 0, z = 1 + x' + (1 + y' + ys'), x' >= 0, ys' >= 0, y' >= 0 rev(z, z') -{ 7 + y' + ys' + z' }-> if(0, 1 + x' + s5, s2, z') :|: s5 >= 0, s5 <= 1 + y' + ys', s2 >= 0, s2 <= z' + 0, z' >= 0, z = 1 + x' + (1 + y' + ys'), x' >= 0, ys' >= 0, y' >= 0 reverse(z) -{ 1 }-> rev(z, 0) :|: z >= 0 Function symbols to be analyzed: {if,rev}, {reverse} Previous analysis results are: append: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] isEmpty: runtime: O(1) [1], size: O(1) [1] dropLast: runtime: O(n^1) [2 + z], size: O(n^1) [z] last: runtime: O(n^1) [1 + z], size: O(n^1) [z] ---------------------------------------- (45) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: if after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? Computed SIZE bound using CoFloCo for: rev after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (46) Obligation: Complexity RNTS consisting of the following rules: append(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 append(z, z') -{ 2 + xs }-> 1 + x + s :|: s >= 0, s <= xs + z', z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0 dropLast(z) -{ 1 }-> 0 :|: z = 0 dropLast(z) -{ 1 }-> 0 :|: z - 1 >= 0 dropLast(z) -{ 4 + y + ys }-> 1 + x + s3 :|: s3 >= 0, s3 <= 1 + y + ys, ys >= 0, x >= 0, y >= 0, z = 1 + x + (1 + y + ys) if(z, z', z'', z1) -{ 1 }-> z1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 if(z, z', z'', z1) -{ 1 }-> rev(z', z'') :|: z' >= 0, z'' >= 0, z1 >= 0, z = 0 isEmpty(z) -{ 1 }-> 1 :|: z = 0 isEmpty(z) -{ 1 }-> 0 :|: z = 1 + x + xs, xs >= 0, x >= 0 last(z) -{ 3 + y + ys }-> s6 :|: s6 >= 0, s6 <= 1 + y + ys, ys >= 0, x >= 0, y >= 0, z = 1 + x + (1 + y + ys) last(z) -{ 0 }-> 0 :|: z >= 0 last(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 rev(z, z') -{ 4 + z' }-> if(1, 0, s', z') :|: s' >= 0, s' <= z' + 0, z' >= 0, z = 0 rev(z, z') -{ 5 + z' }-> if(0, 0, s'', z') :|: s'' >= 0, s'' <= z' + (z - 1), z' >= 0, z - 1 >= 0 rev(z, z') -{ 4 + z' }-> if(0, 0, s1, z') :|: s1 >= 0, s1 <= z' + 0, z' >= 0, z - 1 >= 0 rev(z, z') -{ 10 + 2*y' + 2*ys' + z' }-> if(0, 1 + x' + s4, s8, z') :|: s7 >= 0, s7 <= 1 + y' + ys', s8 >= 0, s8 <= z' + s7, s4 >= 0, s4 <= 1 + y' + ys', z' >= 0, z = 1 + x' + (1 + y' + ys'), x' >= 0, ys' >= 0, y' >= 0 rev(z, z') -{ 7 + y' + ys' + z' }-> if(0, 1 + x' + s5, s2, z') :|: s5 >= 0, s5 <= 1 + y' + ys', s2 >= 0, s2 <= z' + 0, z' >= 0, z = 1 + x' + (1 + y' + ys'), x' >= 0, ys' >= 0, y' >= 0 reverse(z) -{ 1 }-> rev(z, 0) :|: z >= 0 Function symbols to be analyzed: {if,rev}, {reverse} Previous analysis results are: append: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] isEmpty: runtime: O(1) [1], size: O(1) [1] dropLast: runtime: O(n^1) [2 + z], size: O(n^1) [z] last: runtime: O(n^1) [1 + z], size: O(n^1) [z] if: runtime: ?, size: INF rev: runtime: ?, size: INF ---------------------------------------- (47) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: if after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (48) Obligation: Complexity RNTS consisting of the following rules: append(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 append(z, z') -{ 2 + xs }-> 1 + x + s :|: s >= 0, s <= xs + z', z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0 dropLast(z) -{ 1 }-> 0 :|: z = 0 dropLast(z) -{ 1 }-> 0 :|: z - 1 >= 0 dropLast(z) -{ 4 + y + ys }-> 1 + x + s3 :|: s3 >= 0, s3 <= 1 + y + ys, ys >= 0, x >= 0, y >= 0, z = 1 + x + (1 + y + ys) if(z, z', z'', z1) -{ 1 }-> z1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 if(z, z', z'', z1) -{ 1 }-> rev(z', z'') :|: z' >= 0, z'' >= 0, z1 >= 0, z = 0 isEmpty(z) -{ 1 }-> 1 :|: z = 0 isEmpty(z) -{ 1 }-> 0 :|: z = 1 + x + xs, xs >= 0, x >= 0 last(z) -{ 3 + y + ys }-> s6 :|: s6 >= 0, s6 <= 1 + y + ys, ys >= 0, x >= 0, y >= 0, z = 1 + x + (1 + y + ys) last(z) -{ 0 }-> 0 :|: z >= 0 last(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 rev(z, z') -{ 4 + z' }-> if(1, 0, s', z') :|: s' >= 0, s' <= z' + 0, z' >= 0, z = 0 rev(z, z') -{ 5 + z' }-> if(0, 0, s'', z') :|: s'' >= 0, s'' <= z' + (z - 1), z' >= 0, z - 1 >= 0 rev(z, z') -{ 4 + z' }-> if(0, 0, s1, z') :|: s1 >= 0, s1 <= z' + 0, z' >= 0, z - 1 >= 0 rev(z, z') -{ 10 + 2*y' + 2*ys' + z' }-> if(0, 1 + x' + s4, s8, z') :|: s7 >= 0, s7 <= 1 + y' + ys', s8 >= 0, s8 <= z' + s7, s4 >= 0, s4 <= 1 + y' + ys', z' >= 0, z = 1 + x' + (1 + y' + ys'), x' >= 0, ys' >= 0, y' >= 0 rev(z, z') -{ 7 + y' + ys' + z' }-> if(0, 1 + x' + s5, s2, z') :|: s5 >= 0, s5 <= 1 + y' + ys', s2 >= 0, s2 <= z' + 0, z' >= 0, z = 1 + x' + (1 + y' + ys'), x' >= 0, ys' >= 0, y' >= 0 reverse(z) -{ 1 }-> rev(z, 0) :|: z >= 0 Function symbols to be analyzed: {if,rev}, {reverse} Previous analysis results are: append: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] isEmpty: runtime: O(1) [1], size: O(1) [1] dropLast: runtime: O(n^1) [2 + z], size: O(n^1) [z] last: runtime: O(n^1) [1 + z], size: O(n^1) [z] if: runtime: INF, size: INF rev: runtime: ?, size: INF ---------------------------------------- (49) 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: last(v0) -> null_last [0] isEmpty(v0) -> null_isEmpty [0] dropLast(v0) -> null_dropLast [0] append(v0, v1) -> null_append [0] if(v0, v1, v2, v3) -> null_if [0] And the following fresh constants: null_last, null_isEmpty, null_dropLast, null_append, null_if ---------------------------------------- (50) 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: isEmpty(nil) -> true [1] isEmpty(cons(x, xs)) -> false [1] last(cons(x, nil)) -> x [1] last(cons(x, cons(y, ys))) -> last(cons(y, ys)) [1] dropLast(nil) -> nil [1] dropLast(cons(x, nil)) -> nil [1] dropLast(cons(x, cons(y, ys))) -> cons(x, dropLast(cons(y, ys))) [1] append(nil, ys) -> ys [1] append(cons(x, xs), ys) -> cons(x, append(xs, ys)) [1] reverse(xs) -> rev(xs, nil) [1] rev(xs, ys) -> if(isEmpty(xs), dropLast(xs), append(ys, last(xs)), ys) [1] if(true, xs, ys, zs) -> zs [1] if(false, xs, ys, zs) -> rev(xs, ys) [1] last(v0) -> null_last [0] isEmpty(v0) -> null_isEmpty [0] dropLast(v0) -> null_dropLast [0] append(v0, v1) -> null_append [0] if(v0, v1, v2, v3) -> null_if [0] The TRS has the following type information: isEmpty :: nil:cons:null_last:null_dropLast:null_append:null_if -> true:false:null_isEmpty nil :: nil:cons:null_last:null_dropLast:null_append:null_if true :: true:false:null_isEmpty cons :: nil:cons:null_last:null_dropLast:null_append:null_if -> nil:cons:null_last:null_dropLast:null_append:null_if -> nil:cons:null_last:null_dropLast:null_append:null_if false :: true:false:null_isEmpty last :: nil:cons:null_last:null_dropLast:null_append:null_if -> nil:cons:null_last:null_dropLast:null_append:null_if dropLast :: nil:cons:null_last:null_dropLast:null_append:null_if -> nil:cons:null_last:null_dropLast:null_append:null_if append :: nil:cons:null_last:null_dropLast:null_append:null_if -> nil:cons:null_last:null_dropLast:null_append:null_if -> nil:cons:null_last:null_dropLast:null_append:null_if reverse :: nil:cons:null_last:null_dropLast:null_append:null_if -> nil:cons:null_last:null_dropLast:null_append:null_if rev :: nil:cons:null_last:null_dropLast:null_append:null_if -> nil:cons:null_last:null_dropLast:null_append:null_if -> nil:cons:null_last:null_dropLast:null_append:null_if if :: true:false:null_isEmpty -> nil:cons:null_last:null_dropLast:null_append:null_if -> nil:cons:null_last:null_dropLast:null_append:null_if -> nil:cons:null_last:null_dropLast:null_append:null_if -> nil:cons:null_last:null_dropLast:null_append:null_if null_last :: nil:cons:null_last:null_dropLast:null_append:null_if null_isEmpty :: true:false:null_isEmpty null_dropLast :: nil:cons:null_last:null_dropLast:null_append:null_if null_append :: nil:cons:null_last:null_dropLast:null_append:null_if null_if :: nil:cons:null_last:null_dropLast:null_append:null_if Rewrite Strategy: INNERMOST ---------------------------------------- (51) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: nil => 0 true => 2 false => 1 null_last => 0 null_isEmpty => 0 null_dropLast => 0 null_append => 0 null_if => 0 ---------------------------------------- (52) Obligation: Complexity RNTS consisting of the following rules: append(z, z') -{ 1 }-> ys :|: z' = ys, ys >= 0, z = 0 append(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 append(z, z') -{ 1 }-> 1 + x + append(xs, ys) :|: z = 1 + x + xs, xs >= 0, z' = ys, ys >= 0, x >= 0 dropLast(z) -{ 1 }-> 0 :|: z = 0 dropLast(z) -{ 1 }-> 0 :|: x >= 0, z = 1 + x + 0 dropLast(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 dropLast(z) -{ 1 }-> 1 + x + dropLast(1 + y + ys) :|: ys >= 0, x >= 0, y >= 0, z = 1 + x + (1 + y + ys) if(z, z', z'', z1) -{ 1 }-> zs :|: z = 2, xs >= 0, ys >= 0, zs >= 0, z'' = ys, z' = xs, z1 = zs if(z, z', z'', z1) -{ 1 }-> rev(xs, ys) :|: xs >= 0, z = 1, ys >= 0, zs >= 0, z'' = ys, z' = xs, z1 = zs if(z, z', z'', z1) -{ 0 }-> 0 :|: z1 = v3, v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0, v3 >= 0 isEmpty(z) -{ 1 }-> 2 :|: z = 0 isEmpty(z) -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0 isEmpty(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 last(z) -{ 1 }-> x :|: x >= 0, z = 1 + x + 0 last(z) -{ 1 }-> last(1 + y + ys) :|: ys >= 0, x >= 0, y >= 0, z = 1 + x + (1 + y + ys) last(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 rev(z, z') -{ 1 }-> if(isEmpty(xs), dropLast(xs), append(ys, last(xs)), ys) :|: xs >= 0, z = xs, z' = ys, ys >= 0 reverse(z) -{ 1 }-> rev(xs, 0) :|: xs >= 0, z = xs Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (53) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (54) Obligation: Complexity Dependency Tuples Problem Rules: isEmpty(nil) -> true isEmpty(cons(z0, z1)) -> false last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) dropLast(nil) -> nil dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) append(nil, z0) -> z0 append(cons(z0, z1), z2) -> cons(z0, append(z1, z2)) reverse(z0) -> rev(z0, nil) rev(z0, z1) -> if(isEmpty(z0), dropLast(z0), append(z1, last(z0)), z1) if(true, z0, z1, z2) -> z2 if(false, z0, z1, z2) -> rev(z0, z1) Tuples: ISEMPTY(nil) -> c ISEMPTY(cons(z0, z1)) -> c1 LAST(cons(z0, nil)) -> c2 LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(nil) -> c4 DROPLAST(cons(z0, nil)) -> c5 DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(nil, z0) -> c7 APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) REVERSE(z0) -> c9(REV(z0, nil)) REV(z0, z1) -> c10(IF(isEmpty(z0), dropLast(z0), append(z1, last(z0)), z1), ISEMPTY(z0)) REV(z0, z1) -> c11(IF(isEmpty(z0), dropLast(z0), append(z1, last(z0)), z1), DROPLAST(z0)) REV(z0, z1) -> c12(IF(isEmpty(z0), dropLast(z0), append(z1, last(z0)), z1), APPEND(z1, last(z0)), LAST(z0)) IF(true, z0, z1, z2) -> c13 IF(false, z0, z1, z2) -> c14(REV(z0, z1)) S tuples: ISEMPTY(nil) -> c ISEMPTY(cons(z0, z1)) -> c1 LAST(cons(z0, nil)) -> c2 LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(nil) -> c4 DROPLAST(cons(z0, nil)) -> c5 DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(nil, z0) -> c7 APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) REVERSE(z0) -> c9(REV(z0, nil)) REV(z0, z1) -> c10(IF(isEmpty(z0), dropLast(z0), append(z1, last(z0)), z1), ISEMPTY(z0)) REV(z0, z1) -> c11(IF(isEmpty(z0), dropLast(z0), append(z1, last(z0)), z1), DROPLAST(z0)) REV(z0, z1) -> c12(IF(isEmpty(z0), dropLast(z0), append(z1, last(z0)), z1), APPEND(z1, last(z0)), LAST(z0)) IF(true, z0, z1, z2) -> c13 IF(false, z0, z1, z2) -> c14(REV(z0, z1)) K tuples:none Defined Rule Symbols: isEmpty_1, last_1, dropLast_1, append_2, reverse_1, rev_2, if_4 Defined Pair Symbols: ISEMPTY_1, LAST_1, DROPLAST_1, APPEND_2, REVERSE_1, REV_2, IF_4 Compound Symbols: c, c1, c2, c3_1, c4, c5, c6_1, c7, c8_1, c9_1, c10_2, c11_2, c12_3, c13, c14_1 ---------------------------------------- (55) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 1 leading nodes: REVERSE(z0) -> c9(REV(z0, nil)) Removed 7 trailing nodes: APPEND(nil, z0) -> c7 LAST(cons(z0, nil)) -> c2 ISEMPTY(cons(z0, z1)) -> c1 ISEMPTY(nil) -> c DROPLAST(nil) -> c4 IF(true, z0, z1, z2) -> c13 DROPLAST(cons(z0, nil)) -> c5 ---------------------------------------- (56) Obligation: Complexity Dependency Tuples Problem Rules: isEmpty(nil) -> true isEmpty(cons(z0, z1)) -> false last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) dropLast(nil) -> nil dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) append(nil, z0) -> z0 append(cons(z0, z1), z2) -> cons(z0, append(z1, z2)) reverse(z0) -> rev(z0, nil) rev(z0, z1) -> if(isEmpty(z0), dropLast(z0), append(z1, last(z0)), z1) if(true, z0, z1, z2) -> z2 if(false, z0, z1, z2) -> rev(z0, z1) Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) REV(z0, z1) -> c10(IF(isEmpty(z0), dropLast(z0), append(z1, last(z0)), z1), ISEMPTY(z0)) REV(z0, z1) -> c11(IF(isEmpty(z0), dropLast(z0), append(z1, last(z0)), z1), DROPLAST(z0)) REV(z0, z1) -> c12(IF(isEmpty(z0), dropLast(z0), append(z1, last(z0)), z1), APPEND(z1, last(z0)), LAST(z0)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) S tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) REV(z0, z1) -> c10(IF(isEmpty(z0), dropLast(z0), append(z1, last(z0)), z1), ISEMPTY(z0)) REV(z0, z1) -> c11(IF(isEmpty(z0), dropLast(z0), append(z1, last(z0)), z1), DROPLAST(z0)) REV(z0, z1) -> c12(IF(isEmpty(z0), dropLast(z0), append(z1, last(z0)), z1), APPEND(z1, last(z0)), LAST(z0)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) K tuples:none Defined Rule Symbols: isEmpty_1, last_1, dropLast_1, append_2, reverse_1, rev_2, if_4 Defined Pair Symbols: LAST_1, DROPLAST_1, APPEND_2, REV_2, IF_4 Compound Symbols: c3_1, c6_1, c8_1, c10_2, c11_2, c12_3, c14_1 ---------------------------------------- (57) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (58) Obligation: Complexity Dependency Tuples Problem Rules: isEmpty(nil) -> true isEmpty(cons(z0, z1)) -> false last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) dropLast(nil) -> nil dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) append(nil, z0) -> z0 append(cons(z0, z1), z2) -> cons(z0, append(z1, z2)) reverse(z0) -> rev(z0, nil) rev(z0, z1) -> if(isEmpty(z0), dropLast(z0), append(z1, last(z0)), z1) if(true, z0, z1, z2) -> z2 if(false, z0, z1, z2) -> rev(z0, z1) Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) REV(z0, z1) -> c11(IF(isEmpty(z0), dropLast(z0), append(z1, last(z0)), z1), DROPLAST(z0)) REV(z0, z1) -> c12(IF(isEmpty(z0), dropLast(z0), append(z1, last(z0)), z1), APPEND(z1, last(z0)), LAST(z0)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(z0, z1) -> c10(IF(isEmpty(z0), dropLast(z0), append(z1, last(z0)), z1)) S tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) REV(z0, z1) -> c11(IF(isEmpty(z0), dropLast(z0), append(z1, last(z0)), z1), DROPLAST(z0)) REV(z0, z1) -> c12(IF(isEmpty(z0), dropLast(z0), append(z1, last(z0)), z1), APPEND(z1, last(z0)), LAST(z0)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(z0, z1) -> c10(IF(isEmpty(z0), dropLast(z0), append(z1, last(z0)), z1)) K tuples:none Defined Rule Symbols: isEmpty_1, last_1, dropLast_1, append_2, reverse_1, rev_2, if_4 Defined Pair Symbols: LAST_1, DROPLAST_1, APPEND_2, REV_2, IF_4 Compound Symbols: c3_1, c6_1, c8_1, c11_2, c12_3, c14_1, c10_1 ---------------------------------------- (59) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: reverse(z0) -> rev(z0, nil) rev(z0, z1) -> if(isEmpty(z0), dropLast(z0), append(z1, last(z0)), z1) if(true, z0, z1, z2) -> z2 if(false, z0, z1, z2) -> rev(z0, z1) ---------------------------------------- (60) Obligation: Complexity Dependency Tuples Problem Rules: isEmpty(nil) -> true isEmpty(cons(z0, z1)) -> false dropLast(nil) -> nil dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) append(nil, z0) -> z0 append(cons(z0, z1), z2) -> cons(z0, append(z1, z2)) last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) REV(z0, z1) -> c11(IF(isEmpty(z0), dropLast(z0), append(z1, last(z0)), z1), DROPLAST(z0)) REV(z0, z1) -> c12(IF(isEmpty(z0), dropLast(z0), append(z1, last(z0)), z1), APPEND(z1, last(z0)), LAST(z0)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(z0, z1) -> c10(IF(isEmpty(z0), dropLast(z0), append(z1, last(z0)), z1)) S tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) REV(z0, z1) -> c11(IF(isEmpty(z0), dropLast(z0), append(z1, last(z0)), z1), DROPLAST(z0)) REV(z0, z1) -> c12(IF(isEmpty(z0), dropLast(z0), append(z1, last(z0)), z1), APPEND(z1, last(z0)), LAST(z0)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(z0, z1) -> c10(IF(isEmpty(z0), dropLast(z0), append(z1, last(z0)), z1)) K tuples:none Defined Rule Symbols: isEmpty_1, dropLast_1, append_2, last_1 Defined Pair Symbols: LAST_1, DROPLAST_1, APPEND_2, REV_2, IF_4 Compound Symbols: c3_1, c6_1, c8_1, c11_2, c12_3, c14_1, c10_1 ---------------------------------------- (61) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace REV(z0, z1) -> c11(IF(isEmpty(z0), dropLast(z0), append(z1, last(z0)), z1), DROPLAST(z0)) by REV(x0, nil) -> c11(IF(isEmpty(x0), dropLast(x0), last(x0), nil), DROPLAST(x0)) REV(x0, cons(z0, z1)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1)), DROPLAST(x0)) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1), DROPLAST(cons(z0, nil))) REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), DROPLAST(cons(z0, cons(z1, z2)))) REV(nil, x1) -> c11(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), DROPLAST(nil)) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), DROPLAST(cons(z0, nil))) REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), DROPLAST(cons(z0, cons(z1, z2)))) REV(nil, x1) -> c11(IF(true, dropLast(nil), append(x1, last(nil)), x1), DROPLAST(nil)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) ---------------------------------------- (62) Obligation: Complexity Dependency Tuples Problem Rules: isEmpty(nil) -> true isEmpty(cons(z0, z1)) -> false dropLast(nil) -> nil dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) append(nil, z0) -> z0 append(cons(z0, z1), z2) -> cons(z0, append(z1, z2)) last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) REV(z0, z1) -> c12(IF(isEmpty(z0), dropLast(z0), append(z1, last(z0)), z1), APPEND(z1, last(z0)), LAST(z0)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(z0, z1) -> c10(IF(isEmpty(z0), dropLast(z0), append(z1, last(z0)), z1)) REV(x0, nil) -> c11(IF(isEmpty(x0), dropLast(x0), last(x0), nil), DROPLAST(x0)) REV(x0, cons(z0, z1)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1)), DROPLAST(x0)) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1), DROPLAST(cons(z0, nil))) REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), DROPLAST(cons(z0, cons(z1, z2)))) REV(nil, x1) -> c11(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), DROPLAST(nil)) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), DROPLAST(cons(z0, nil))) REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), DROPLAST(cons(z0, cons(z1, z2)))) REV(nil, x1) -> c11(IF(true, dropLast(nil), append(x1, last(nil)), x1), DROPLAST(nil)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) S tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) REV(z0, z1) -> c12(IF(isEmpty(z0), dropLast(z0), append(z1, last(z0)), z1), APPEND(z1, last(z0)), LAST(z0)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(z0, z1) -> c10(IF(isEmpty(z0), dropLast(z0), append(z1, last(z0)), z1)) REV(x0, nil) -> c11(IF(isEmpty(x0), dropLast(x0), last(x0), nil), DROPLAST(x0)) REV(x0, cons(z0, z1)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1)), DROPLAST(x0)) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1), DROPLAST(cons(z0, nil))) REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), DROPLAST(cons(z0, cons(z1, z2)))) REV(nil, x1) -> c11(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), DROPLAST(nil)) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), DROPLAST(cons(z0, nil))) REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), DROPLAST(cons(z0, cons(z1, z2)))) REV(nil, x1) -> c11(IF(true, dropLast(nil), append(x1, last(nil)), x1), DROPLAST(nil)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) K tuples:none Defined Rule Symbols: isEmpty_1, dropLast_1, append_2, last_1 Defined Pair Symbols: LAST_1, DROPLAST_1, APPEND_2, REV_2, IF_4 Compound Symbols: c3_1, c6_1, c8_1, c12_3, c14_1, c10_1, c11_2 ---------------------------------------- (63) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: REV(nil, x1) -> c11(IF(true, dropLast(nil), append(x1, last(nil)), x1), DROPLAST(nil)) ---------------------------------------- (64) Obligation: Complexity Dependency Tuples Problem Rules: isEmpty(nil) -> true isEmpty(cons(z0, z1)) -> false dropLast(nil) -> nil dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) append(nil, z0) -> z0 append(cons(z0, z1), z2) -> cons(z0, append(z1, z2)) last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) REV(z0, z1) -> c12(IF(isEmpty(z0), dropLast(z0), append(z1, last(z0)), z1), APPEND(z1, last(z0)), LAST(z0)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(z0, z1) -> c10(IF(isEmpty(z0), dropLast(z0), append(z1, last(z0)), z1)) REV(x0, nil) -> c11(IF(isEmpty(x0), dropLast(x0), last(x0), nil), DROPLAST(x0)) REV(x0, cons(z0, z1)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1)), DROPLAST(x0)) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1), DROPLAST(cons(z0, nil))) REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), DROPLAST(cons(z0, cons(z1, z2)))) REV(nil, x1) -> c11(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), DROPLAST(nil)) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), DROPLAST(cons(z0, nil))) REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) S tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) REV(z0, z1) -> c12(IF(isEmpty(z0), dropLast(z0), append(z1, last(z0)), z1), APPEND(z1, last(z0)), LAST(z0)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(z0, z1) -> c10(IF(isEmpty(z0), dropLast(z0), append(z1, last(z0)), z1)) REV(x0, nil) -> c11(IF(isEmpty(x0), dropLast(x0), last(x0), nil), DROPLAST(x0)) REV(x0, cons(z0, z1)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1)), DROPLAST(x0)) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1), DROPLAST(cons(z0, nil))) REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), DROPLAST(cons(z0, cons(z1, z2)))) REV(nil, x1) -> c11(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), DROPLAST(nil)) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), DROPLAST(cons(z0, nil))) REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) K tuples:none Defined Rule Symbols: isEmpty_1, dropLast_1, append_2, last_1 Defined Pair Symbols: LAST_1, DROPLAST_1, APPEND_2, REV_2, IF_4 Compound Symbols: c3_1, c6_1, c8_1, c12_3, c14_1, c10_1, c11_2 ---------------------------------------- (65) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (66) Obligation: Complexity Dependency Tuples Problem Rules: isEmpty(nil) -> true isEmpty(cons(z0, z1)) -> false dropLast(nil) -> nil dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) append(nil, z0) -> z0 append(cons(z0, z1), z2) -> cons(z0, append(z1, z2)) last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) REV(z0, z1) -> c12(IF(isEmpty(z0), dropLast(z0), append(z1, last(z0)), z1), APPEND(z1, last(z0)), LAST(z0)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(z0, z1) -> c10(IF(isEmpty(z0), dropLast(z0), append(z1, last(z0)), z1)) REV(x0, nil) -> c11(IF(isEmpty(x0), dropLast(x0), last(x0), nil), DROPLAST(x0)) REV(x0, cons(z0, z1)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1)), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(nil, x1) -> c11(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) S tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) REV(z0, z1) -> c12(IF(isEmpty(z0), dropLast(z0), append(z1, last(z0)), z1), APPEND(z1, last(z0)), LAST(z0)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(z0, z1) -> c10(IF(isEmpty(z0), dropLast(z0), append(z1, last(z0)), z1)) REV(x0, nil) -> c11(IF(isEmpty(x0), dropLast(x0), last(x0), nil), DROPLAST(x0)) REV(x0, cons(z0, z1)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1)), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(nil, x1) -> c11(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) K tuples:none Defined Rule Symbols: isEmpty_1, dropLast_1, append_2, last_1 Defined Pair Symbols: LAST_1, DROPLAST_1, APPEND_2, REV_2, IF_4 Compound Symbols: c3_1, c6_1, c8_1, c12_3, c14_1, c10_1, c11_2, c11_1 ---------------------------------------- (67) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. REV(nil, x1) -> c11(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) We considered the (Usable) Rules: dropLast(nil) -> nil isEmpty(nil) -> true dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) isEmpty(cons(z0, z1)) -> false And the Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) REV(z0, z1) -> c12(IF(isEmpty(z0), dropLast(z0), append(z1, last(z0)), z1), APPEND(z1, last(z0)), LAST(z0)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(z0, z1) -> c10(IF(isEmpty(z0), dropLast(z0), append(z1, last(z0)), z1)) REV(x0, nil) -> c11(IF(isEmpty(x0), dropLast(x0), last(x0), nil), DROPLAST(x0)) REV(x0, cons(z0, z1)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1)), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(nil, x1) -> c11(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) The order we found is given by the following interpretation: Polynomial interpretation : POL(APPEND(x_1, x_2)) = 0 POL(DROPLAST(x_1)) = 0 POL(IF(x_1, x_2, x_3, x_4)) = x_1 + x_2 POL(LAST(x_1)) = 0 POL(REV(x_1, x_2)) = [1] + x_1 POL(append(x_1, x_2)) = [1] + x_1 POL(c10(x_1)) = x_1 POL(c11(x_1)) = x_1 POL(c11(x_1, x_2)) = x_1 + x_2 POL(c12(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c14(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(cons(x_1, x_2)) = [1] POL(dropLast(x_1)) = [1] POL(false) = [1] POL(isEmpty(x_1)) = x_1 POL(last(x_1)) = [1] POL(nil) = 0 POL(true) = 0 ---------------------------------------- (68) Obligation: Complexity Dependency Tuples Problem Rules: isEmpty(nil) -> true isEmpty(cons(z0, z1)) -> false dropLast(nil) -> nil dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) append(nil, z0) -> z0 append(cons(z0, z1), z2) -> cons(z0, append(z1, z2)) last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) REV(z0, z1) -> c12(IF(isEmpty(z0), dropLast(z0), append(z1, last(z0)), z1), APPEND(z1, last(z0)), LAST(z0)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(z0, z1) -> c10(IF(isEmpty(z0), dropLast(z0), append(z1, last(z0)), z1)) REV(x0, nil) -> c11(IF(isEmpty(x0), dropLast(x0), last(x0), nil), DROPLAST(x0)) REV(x0, cons(z0, z1)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1)), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(nil, x1) -> c11(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) S tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) REV(z0, z1) -> c12(IF(isEmpty(z0), dropLast(z0), append(z1, last(z0)), z1), APPEND(z1, last(z0)), LAST(z0)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(z0, z1) -> c10(IF(isEmpty(z0), dropLast(z0), append(z1, last(z0)), z1)) REV(x0, nil) -> c11(IF(isEmpty(x0), dropLast(x0), last(x0), nil), DROPLAST(x0)) REV(x0, cons(z0, z1)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1)), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) K tuples: REV(nil, x1) -> c11(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) Defined Rule Symbols: isEmpty_1, dropLast_1, append_2, last_1 Defined Pair Symbols: LAST_1, DROPLAST_1, APPEND_2, REV_2, IF_4 Compound Symbols: c3_1, c6_1, c8_1, c12_3, c14_1, c10_1, c11_2, c11_1 ---------------------------------------- (69) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace REV(z0, z1) -> c12(IF(isEmpty(z0), dropLast(z0), append(z1, last(z0)), z1), APPEND(z1, last(z0)), LAST(z0)) by REV(x0, nil) -> c12(IF(isEmpty(x0), dropLast(x0), last(x0), nil), APPEND(nil, last(x0)), LAST(x0)) REV(x0, cons(z0, z1)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1)), APPEND(cons(z0, z1), last(x0)), LAST(x0)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil))), LAST(cons(z0, nil))) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(nil, x1) -> c12(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), APPEND(x1, last(nil)), LAST(nil)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil))), LAST(cons(z0, nil))) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(nil, x1) -> c12(IF(true, dropLast(nil), append(x1, last(nil)), x1), APPEND(x1, last(nil)), LAST(nil)) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) ---------------------------------------- (70) Obligation: Complexity Dependency Tuples Problem Rules: isEmpty(nil) -> true isEmpty(cons(z0, z1)) -> false dropLast(nil) -> nil dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) append(nil, z0) -> z0 append(cons(z0, z1), z2) -> cons(z0, append(z1, z2)) last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(z0, z1) -> c10(IF(isEmpty(z0), dropLast(z0), append(z1, last(z0)), z1)) REV(x0, nil) -> c11(IF(isEmpty(x0), dropLast(x0), last(x0), nil), DROPLAST(x0)) REV(x0, cons(z0, z1)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1)), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(nil, x1) -> c11(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, nil) -> c12(IF(isEmpty(x0), dropLast(x0), last(x0), nil), APPEND(nil, last(x0)), LAST(x0)) REV(x0, cons(z0, z1)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1)), APPEND(cons(z0, z1), last(x0)), LAST(x0)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil))), LAST(cons(z0, nil))) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(nil, x1) -> c12(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), APPEND(x1, last(nil)), LAST(nil)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil))), LAST(cons(z0, nil))) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(nil, x1) -> c12(IF(true, dropLast(nil), append(x1, last(nil)), x1), APPEND(x1, last(nil)), LAST(nil)) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) S tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(z0, z1) -> c10(IF(isEmpty(z0), dropLast(z0), append(z1, last(z0)), z1)) REV(x0, nil) -> c11(IF(isEmpty(x0), dropLast(x0), last(x0), nil), DROPLAST(x0)) REV(x0, cons(z0, z1)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1)), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(x0, nil) -> c12(IF(isEmpty(x0), dropLast(x0), last(x0), nil), APPEND(nil, last(x0)), LAST(x0)) REV(x0, cons(z0, z1)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1)), APPEND(cons(z0, z1), last(x0)), LAST(x0)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil))), LAST(cons(z0, nil))) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(nil, x1) -> c12(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), APPEND(x1, last(nil)), LAST(nil)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil))), LAST(cons(z0, nil))) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(nil, x1) -> c12(IF(true, dropLast(nil), append(x1, last(nil)), x1), APPEND(x1, last(nil)), LAST(nil)) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) K tuples: REV(nil, x1) -> c11(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) Defined Rule Symbols: isEmpty_1, dropLast_1, append_2, last_1 Defined Pair Symbols: LAST_1, DROPLAST_1, APPEND_2, IF_4, REV_2 Compound Symbols: c3_1, c6_1, c8_1, c14_1, c10_1, c11_2, c11_1, c12_3 ---------------------------------------- (71) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 6 trailing tuple parts ---------------------------------------- (72) Obligation: Complexity Dependency Tuples Problem Rules: isEmpty(nil) -> true isEmpty(cons(z0, z1)) -> false dropLast(nil) -> nil dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) append(nil, z0) -> z0 append(cons(z0, z1), z2) -> cons(z0, append(z1, z2)) last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(z0, z1) -> c10(IF(isEmpty(z0), dropLast(z0), append(z1, last(z0)), z1)) REV(x0, nil) -> c11(IF(isEmpty(x0), dropLast(x0), last(x0), nil), DROPLAST(x0)) REV(x0, cons(z0, z1)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1)), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(nil, x1) -> c11(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, cons(z0, z1)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1)), APPEND(cons(z0, z1), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, nil) -> c12(IF(isEmpty(x0), dropLast(x0), last(x0), nil), LAST(x0)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) S tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(z0, z1) -> c10(IF(isEmpty(z0), dropLast(z0), append(z1, last(z0)), z1)) REV(x0, nil) -> c11(IF(isEmpty(x0), dropLast(x0), last(x0), nil), DROPLAST(x0)) REV(x0, cons(z0, z1)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1)), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(x0, cons(z0, z1)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1)), APPEND(cons(z0, z1), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, nil) -> c12(IF(isEmpty(x0), dropLast(x0), last(x0), nil), LAST(x0)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) K tuples: REV(nil, x1) -> c11(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) Defined Rule Symbols: isEmpty_1, dropLast_1, append_2, last_1 Defined Pair Symbols: LAST_1, DROPLAST_1, APPEND_2, IF_4, REV_2 Compound Symbols: c3_1, c6_1, c8_1, c14_1, c10_1, c11_2, c11_1, c12_3, c12_2, c12_1 ---------------------------------------- (73) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. REV(nil, x1) -> c12(APPEND(x1, last(nil))) We considered the (Usable) Rules: dropLast(nil) -> nil isEmpty(nil) -> true dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) isEmpty(cons(z0, z1)) -> false And the Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(z0, z1) -> c10(IF(isEmpty(z0), dropLast(z0), append(z1, last(z0)), z1)) REV(x0, nil) -> c11(IF(isEmpty(x0), dropLast(x0), last(x0), nil), DROPLAST(x0)) REV(x0, cons(z0, z1)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1)), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(nil, x1) -> c11(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, cons(z0, z1)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1)), APPEND(cons(z0, z1), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, nil) -> c12(IF(isEmpty(x0), dropLast(x0), last(x0), nil), LAST(x0)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) The order we found is given by the following interpretation: Polynomial interpretation : POL(APPEND(x_1, x_2)) = 0 POL(DROPLAST(x_1)) = 0 POL(IF(x_1, x_2, x_3, x_4)) = x_1 + x_2 POL(LAST(x_1)) = 0 POL(REV(x_1, x_2)) = x_1 POL(append(x_1, x_2)) = [1] + x_1 POL(c10(x_1)) = x_1 POL(c11(x_1)) = x_1 POL(c11(x_1, x_2)) = x_1 + x_2 POL(c12(x_1)) = x_1 POL(c12(x_1, x_2)) = x_1 + x_2 POL(c12(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c14(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(cons(x_1, x_2)) = x_2 POL(dropLast(x_1)) = x_1 POL(false) = 0 POL(isEmpty(x_1)) = 0 POL(last(x_1)) = 0 POL(nil) = [1] POL(true) = 0 ---------------------------------------- (74) Obligation: Complexity Dependency Tuples Problem Rules: isEmpty(nil) -> true isEmpty(cons(z0, z1)) -> false dropLast(nil) -> nil dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) append(nil, z0) -> z0 append(cons(z0, z1), z2) -> cons(z0, append(z1, z2)) last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(z0, z1) -> c10(IF(isEmpty(z0), dropLast(z0), append(z1, last(z0)), z1)) REV(x0, nil) -> c11(IF(isEmpty(x0), dropLast(x0), last(x0), nil), DROPLAST(x0)) REV(x0, cons(z0, z1)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1)), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(nil, x1) -> c11(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, cons(z0, z1)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1)), APPEND(cons(z0, z1), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, nil) -> c12(IF(isEmpty(x0), dropLast(x0), last(x0), nil), LAST(x0)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) S tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(z0, z1) -> c10(IF(isEmpty(z0), dropLast(z0), append(z1, last(z0)), z1)) REV(x0, nil) -> c11(IF(isEmpty(x0), dropLast(x0), last(x0), nil), DROPLAST(x0)) REV(x0, cons(z0, z1)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1)), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(x0, cons(z0, z1)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1)), APPEND(cons(z0, z1), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, nil) -> c12(IF(isEmpty(x0), dropLast(x0), last(x0), nil), LAST(x0)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) K tuples: REV(nil, x1) -> c11(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(nil, x1) -> c12(APPEND(x1, last(nil))) Defined Rule Symbols: isEmpty_1, dropLast_1, append_2, last_1 Defined Pair Symbols: LAST_1, DROPLAST_1, APPEND_2, IF_4, REV_2 Compound Symbols: c3_1, c6_1, c8_1, c14_1, c10_1, c11_2, c11_1, c12_3, c12_2, c12_1 ---------------------------------------- (75) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) We considered the (Usable) Rules: dropLast(nil) -> nil isEmpty(nil) -> true dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) isEmpty(cons(z0, z1)) -> false And the Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(z0, z1) -> c10(IF(isEmpty(z0), dropLast(z0), append(z1, last(z0)), z1)) REV(x0, nil) -> c11(IF(isEmpty(x0), dropLast(x0), last(x0), nil), DROPLAST(x0)) REV(x0, cons(z0, z1)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1)), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(nil, x1) -> c11(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, cons(z0, z1)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1)), APPEND(cons(z0, z1), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, nil) -> c12(IF(isEmpty(x0), dropLast(x0), last(x0), nil), LAST(x0)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) The order we found is given by the following interpretation: Polynomial interpretation : POL(APPEND(x_1, x_2)) = 0 POL(DROPLAST(x_1)) = 0 POL(IF(x_1, x_2, x_3, x_4)) = x_1 + x_2 POL(LAST(x_1)) = 0 POL(REV(x_1, x_2)) = x_1 POL(append(x_1, x_2)) = [1] + x_1 POL(c10(x_1)) = x_1 POL(c11(x_1)) = x_1 POL(c11(x_1, x_2)) = x_1 + x_2 POL(c12(x_1)) = x_1 POL(c12(x_1, x_2)) = x_1 + x_2 POL(c12(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c14(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(cons(x_1, x_2)) = [1] + x_2 POL(dropLast(x_1)) = x_1 POL(false) = 0 POL(isEmpty(x_1)) = 0 POL(last(x_1)) = 0 POL(nil) = 0 POL(true) = 0 ---------------------------------------- (76) Obligation: Complexity Dependency Tuples Problem Rules: isEmpty(nil) -> true isEmpty(cons(z0, z1)) -> false dropLast(nil) -> nil dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) append(nil, z0) -> z0 append(cons(z0, z1), z2) -> cons(z0, append(z1, z2)) last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(z0, z1) -> c10(IF(isEmpty(z0), dropLast(z0), append(z1, last(z0)), z1)) REV(x0, nil) -> c11(IF(isEmpty(x0), dropLast(x0), last(x0), nil), DROPLAST(x0)) REV(x0, cons(z0, z1)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1)), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(nil, x1) -> c11(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, cons(z0, z1)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1)), APPEND(cons(z0, z1), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, nil) -> c12(IF(isEmpty(x0), dropLast(x0), last(x0), nil), LAST(x0)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) S tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(z0, z1) -> c10(IF(isEmpty(z0), dropLast(z0), append(z1, last(z0)), z1)) REV(x0, nil) -> c11(IF(isEmpty(x0), dropLast(x0), last(x0), nil), DROPLAST(x0)) REV(x0, cons(z0, z1)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1)), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(x0, cons(z0, z1)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1)), APPEND(cons(z0, z1), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, nil) -> c12(IF(isEmpty(x0), dropLast(x0), last(x0), nil), LAST(x0)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), APPEND(x1, last(nil))) K tuples: REV(nil, x1) -> c11(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) Defined Rule Symbols: isEmpty_1, dropLast_1, append_2, last_1 Defined Pair Symbols: LAST_1, DROPLAST_1, APPEND_2, IF_4, REV_2 Compound Symbols: c3_1, c6_1, c8_1, c14_1, c10_1, c11_2, c11_1, c12_3, c12_2, c12_1 ---------------------------------------- (77) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. REV(nil, x1) -> c12(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), APPEND(x1, last(nil))) We considered the (Usable) Rules: dropLast(nil) -> nil isEmpty(nil) -> true dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) isEmpty(cons(z0, z1)) -> false And the Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(z0, z1) -> c10(IF(isEmpty(z0), dropLast(z0), append(z1, last(z0)), z1)) REV(x0, nil) -> c11(IF(isEmpty(x0), dropLast(x0), last(x0), nil), DROPLAST(x0)) REV(x0, cons(z0, z1)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1)), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(nil, x1) -> c11(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, cons(z0, z1)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1)), APPEND(cons(z0, z1), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, nil) -> c12(IF(isEmpty(x0), dropLast(x0), last(x0), nil), LAST(x0)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) The order we found is given by the following interpretation: Polynomial interpretation : POL(APPEND(x_1, x_2)) = 0 POL(DROPLAST(x_1)) = 0 POL(IF(x_1, x_2, x_3, x_4)) = x_1 + x_2 POL(LAST(x_1)) = 0 POL(REV(x_1, x_2)) = [1] + x_1 POL(append(x_1, x_2)) = [1] + x_1 POL(c10(x_1)) = x_1 POL(c11(x_1)) = x_1 POL(c11(x_1, x_2)) = x_1 + x_2 POL(c12(x_1)) = x_1 POL(c12(x_1, x_2)) = x_1 + x_2 POL(c12(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c14(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(cons(x_1, x_2)) = [1] POL(dropLast(x_1)) = [1] POL(false) = [1] POL(isEmpty(x_1)) = x_1 POL(last(x_1)) = 0 POL(nil) = 0 POL(true) = 0 ---------------------------------------- (78) Obligation: Complexity Dependency Tuples Problem Rules: isEmpty(nil) -> true isEmpty(cons(z0, z1)) -> false dropLast(nil) -> nil dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) append(nil, z0) -> z0 append(cons(z0, z1), z2) -> cons(z0, append(z1, z2)) last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(z0, z1) -> c10(IF(isEmpty(z0), dropLast(z0), append(z1, last(z0)), z1)) REV(x0, nil) -> c11(IF(isEmpty(x0), dropLast(x0), last(x0), nil), DROPLAST(x0)) REV(x0, cons(z0, z1)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1)), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(nil, x1) -> c11(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, cons(z0, z1)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1)), APPEND(cons(z0, z1), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, nil) -> c12(IF(isEmpty(x0), dropLast(x0), last(x0), nil), LAST(x0)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) S tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(z0, z1) -> c10(IF(isEmpty(z0), dropLast(z0), append(z1, last(z0)), z1)) REV(x0, nil) -> c11(IF(isEmpty(x0), dropLast(x0), last(x0), nil), DROPLAST(x0)) REV(x0, cons(z0, z1)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1)), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(x0, cons(z0, z1)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1)), APPEND(cons(z0, z1), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, nil) -> c12(IF(isEmpty(x0), dropLast(x0), last(x0), nil), LAST(x0)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) K tuples: REV(nil, x1) -> c11(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), APPEND(x1, last(nil))) Defined Rule Symbols: isEmpty_1, dropLast_1, append_2, last_1 Defined Pair Symbols: LAST_1, DROPLAST_1, APPEND_2, IF_4, REV_2 Compound Symbols: c3_1, c6_1, c8_1, c14_1, c10_1, c11_2, c11_1, c12_3, c12_2, c12_1 ---------------------------------------- (79) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace REV(z0, z1) -> c10(IF(isEmpty(z0), dropLast(z0), append(z1, last(z0)), z1)) by REV(x0, nil) -> c10(IF(isEmpty(x0), dropLast(x0), last(x0), nil)) REV(x0, cons(z0, z1)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(nil, x1) -> c10(IF(true, dropLast(nil), append(x1, last(nil)), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) ---------------------------------------- (80) Obligation: Complexity Dependency Tuples Problem Rules: isEmpty(nil) -> true isEmpty(cons(z0, z1)) -> false dropLast(nil) -> nil dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) append(nil, z0) -> z0 append(cons(z0, z1), z2) -> cons(z0, append(z1, z2)) last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(x0, nil) -> c11(IF(isEmpty(x0), dropLast(x0), last(x0), nil), DROPLAST(x0)) REV(x0, cons(z0, z1)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1)), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(nil, x1) -> c11(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, cons(z0, z1)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1)), APPEND(cons(z0, z1), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, nil) -> c12(IF(isEmpty(x0), dropLast(x0), last(x0), nil), LAST(x0)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(x0, nil) -> c10(IF(isEmpty(x0), dropLast(x0), last(x0), nil)) REV(x0, cons(z0, z1)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(nil, x1) -> c10(IF(true, dropLast(nil), append(x1, last(nil)), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) S tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(x0, nil) -> c11(IF(isEmpty(x0), dropLast(x0), last(x0), nil), DROPLAST(x0)) REV(x0, cons(z0, z1)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1)), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(x0, cons(z0, z1)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1)), APPEND(cons(z0, z1), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, nil) -> c12(IF(isEmpty(x0), dropLast(x0), last(x0), nil), LAST(x0)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(x0, nil) -> c10(IF(isEmpty(x0), dropLast(x0), last(x0), nil)) REV(x0, cons(z0, z1)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(nil, x1) -> c10(IF(true, dropLast(nil), append(x1, last(nil)), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) K tuples: REV(nil, x1) -> c11(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), APPEND(x1, last(nil))) Defined Rule Symbols: isEmpty_1, dropLast_1, append_2, last_1 Defined Pair Symbols: LAST_1, DROPLAST_1, APPEND_2, IF_4, REV_2 Compound Symbols: c3_1, c6_1, c8_1, c14_1, c11_2, c11_1, c12_3, c12_2, c12_1, c10_1 ---------------------------------------- (81) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: REV(nil, x1) -> c10(IF(true, dropLast(nil), append(x1, last(nil)), x1)) ---------------------------------------- (82) Obligation: Complexity Dependency Tuples Problem Rules: isEmpty(nil) -> true isEmpty(cons(z0, z1)) -> false dropLast(nil) -> nil dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) append(nil, z0) -> z0 append(cons(z0, z1), z2) -> cons(z0, append(z1, z2)) last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(x0, nil) -> c11(IF(isEmpty(x0), dropLast(x0), last(x0), nil), DROPLAST(x0)) REV(x0, cons(z0, z1)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1)), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(nil, x1) -> c11(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, cons(z0, z1)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1)), APPEND(cons(z0, z1), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, nil) -> c12(IF(isEmpty(x0), dropLast(x0), last(x0), nil), LAST(x0)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(x0, nil) -> c10(IF(isEmpty(x0), dropLast(x0), last(x0), nil)) REV(x0, cons(z0, z1)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) S tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(x0, nil) -> c11(IF(isEmpty(x0), dropLast(x0), last(x0), nil), DROPLAST(x0)) REV(x0, cons(z0, z1)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1)), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(x0, cons(z0, z1)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1)), APPEND(cons(z0, z1), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, nil) -> c12(IF(isEmpty(x0), dropLast(x0), last(x0), nil), LAST(x0)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(x0, nil) -> c10(IF(isEmpty(x0), dropLast(x0), last(x0), nil)) REV(x0, cons(z0, z1)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) K tuples: REV(nil, x1) -> c11(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), APPEND(x1, last(nil))) Defined Rule Symbols: isEmpty_1, dropLast_1, append_2, last_1 Defined Pair Symbols: LAST_1, DROPLAST_1, APPEND_2, IF_4, REV_2 Compound Symbols: c3_1, c6_1, c8_1, c14_1, c11_2, c11_1, c12_3, c12_2, c12_1, c10_1 ---------------------------------------- (83) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) We considered the (Usable) Rules: dropLast(nil) -> nil isEmpty(nil) -> true dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) isEmpty(cons(z0, z1)) -> false And the Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(x0, nil) -> c11(IF(isEmpty(x0), dropLast(x0), last(x0), nil), DROPLAST(x0)) REV(x0, cons(z0, z1)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1)), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(nil, x1) -> c11(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, cons(z0, z1)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1)), APPEND(cons(z0, z1), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, nil) -> c12(IF(isEmpty(x0), dropLast(x0), last(x0), nil), LAST(x0)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(x0, nil) -> c10(IF(isEmpty(x0), dropLast(x0), last(x0), nil)) REV(x0, cons(z0, z1)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) The order we found is given by the following interpretation: Polynomial interpretation : POL(APPEND(x_1, x_2)) = 0 POL(DROPLAST(x_1)) = 0 POL(IF(x_1, x_2, x_3, x_4)) = x_1 + x_2 POL(LAST(x_1)) = 0 POL(REV(x_1, x_2)) = x_1 POL(append(x_1, x_2)) = [1] + x_1 POL(c10(x_1)) = x_1 POL(c11(x_1)) = x_1 POL(c11(x_1, x_2)) = x_1 + x_2 POL(c12(x_1)) = x_1 POL(c12(x_1, x_2)) = x_1 + x_2 POL(c12(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c14(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(cons(x_1, x_2)) = [1] + x_2 POL(dropLast(x_1)) = x_1 POL(false) = 0 POL(isEmpty(x_1)) = 0 POL(last(x_1)) = 0 POL(nil) = [1] POL(true) = 0 ---------------------------------------- (84) Obligation: Complexity Dependency Tuples Problem Rules: isEmpty(nil) -> true isEmpty(cons(z0, z1)) -> false dropLast(nil) -> nil dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) append(nil, z0) -> z0 append(cons(z0, z1), z2) -> cons(z0, append(z1, z2)) last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(x0, nil) -> c11(IF(isEmpty(x0), dropLast(x0), last(x0), nil), DROPLAST(x0)) REV(x0, cons(z0, z1)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1)), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(nil, x1) -> c11(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, cons(z0, z1)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1)), APPEND(cons(z0, z1), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, nil) -> c12(IF(isEmpty(x0), dropLast(x0), last(x0), nil), LAST(x0)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(x0, nil) -> c10(IF(isEmpty(x0), dropLast(x0), last(x0), nil)) REV(x0, cons(z0, z1)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) S tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(x0, nil) -> c11(IF(isEmpty(x0), dropLast(x0), last(x0), nil), DROPLAST(x0)) REV(x0, cons(z0, z1)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1)), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(x0, cons(z0, z1)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1)), APPEND(cons(z0, z1), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, nil) -> c12(IF(isEmpty(x0), dropLast(x0), last(x0), nil), LAST(x0)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(x0, nil) -> c10(IF(isEmpty(x0), dropLast(x0), last(x0), nil)) REV(x0, cons(z0, z1)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) K tuples: REV(nil, x1) -> c11(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) Defined Rule Symbols: isEmpty_1, dropLast_1, append_2, last_1 Defined Pair Symbols: LAST_1, DROPLAST_1, APPEND_2, IF_4, REV_2 Compound Symbols: c3_1, c6_1, c8_1, c14_1, c11_2, c11_1, c12_3, c12_2, c12_1, c10_1 ---------------------------------------- (85) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) We considered the (Usable) Rules: dropLast(nil) -> nil isEmpty(nil) -> true dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) isEmpty(cons(z0, z1)) -> false And the Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(x0, nil) -> c11(IF(isEmpty(x0), dropLast(x0), last(x0), nil), DROPLAST(x0)) REV(x0, cons(z0, z1)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1)), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(nil, x1) -> c11(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, cons(z0, z1)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1)), APPEND(cons(z0, z1), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, nil) -> c12(IF(isEmpty(x0), dropLast(x0), last(x0), nil), LAST(x0)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(x0, nil) -> c10(IF(isEmpty(x0), dropLast(x0), last(x0), nil)) REV(x0, cons(z0, z1)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) The order we found is given by the following interpretation: Polynomial interpretation : POL(APPEND(x_1, x_2)) = 0 POL(DROPLAST(x_1)) = 0 POL(IF(x_1, x_2, x_3, x_4)) = x_1 + x_2 POL(LAST(x_1)) = 0 POL(REV(x_1, x_2)) = [1] + x_1 POL(append(x_1, x_2)) = [1] + x_1 POL(c10(x_1)) = x_1 POL(c11(x_1)) = x_1 POL(c11(x_1, x_2)) = x_1 + x_2 POL(c12(x_1)) = x_1 POL(c12(x_1, x_2)) = x_1 + x_2 POL(c12(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c14(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(cons(x_1, x_2)) = [1] POL(dropLast(x_1)) = [1] POL(false) = [1] POL(isEmpty(x_1)) = x_1 POL(last(x_1)) = 0 POL(nil) = 0 POL(true) = 0 ---------------------------------------- (86) Obligation: Complexity Dependency Tuples Problem Rules: isEmpty(nil) -> true isEmpty(cons(z0, z1)) -> false dropLast(nil) -> nil dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) append(nil, z0) -> z0 append(cons(z0, z1), z2) -> cons(z0, append(z1, z2)) last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(x0, nil) -> c11(IF(isEmpty(x0), dropLast(x0), last(x0), nil), DROPLAST(x0)) REV(x0, cons(z0, z1)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1)), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(nil, x1) -> c11(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, cons(z0, z1)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1)), APPEND(cons(z0, z1), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, nil) -> c12(IF(isEmpty(x0), dropLast(x0), last(x0), nil), LAST(x0)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(x0, nil) -> c10(IF(isEmpty(x0), dropLast(x0), last(x0), nil)) REV(x0, cons(z0, z1)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) S tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(x0, nil) -> c11(IF(isEmpty(x0), dropLast(x0), last(x0), nil), DROPLAST(x0)) REV(x0, cons(z0, z1)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1)), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(x0, cons(z0, z1)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1)), APPEND(cons(z0, z1), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, nil) -> c12(IF(isEmpty(x0), dropLast(x0), last(x0), nil), LAST(x0)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(x0, nil) -> c10(IF(isEmpty(x0), dropLast(x0), last(x0), nil)) REV(x0, cons(z0, z1)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) K tuples: REV(nil, x1) -> c11(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) Defined Rule Symbols: isEmpty_1, dropLast_1, append_2, last_1 Defined Pair Symbols: LAST_1, DROPLAST_1, APPEND_2, IF_4, REV_2 Compound Symbols: c3_1, c6_1, c8_1, c14_1, c11_2, c11_1, c12_3, c12_2, c12_1, c10_1 ---------------------------------------- (87) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace REV(x0, nil) -> c11(IF(isEmpty(x0), dropLast(x0), last(x0), nil), DROPLAST(x0)) by REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil), DROPLAST(cons(z0, nil))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil), DROPLAST(nil)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil), DROPLAST(cons(z0, nil))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(nil, nil) -> c11(IF(true, dropLast(nil), last(nil), nil), DROPLAST(nil)) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) ---------------------------------------- (88) Obligation: Complexity Dependency Tuples Problem Rules: isEmpty(nil) -> true isEmpty(cons(z0, z1)) -> false dropLast(nil) -> nil dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) append(nil, z0) -> z0 append(cons(z0, z1), z2) -> cons(z0, append(z1, z2)) last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(x0, cons(z0, z1)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1)), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(nil, x1) -> c11(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, cons(z0, z1)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1)), APPEND(cons(z0, z1), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, nil) -> c12(IF(isEmpty(x0), dropLast(x0), last(x0), nil), LAST(x0)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(x0, nil) -> c10(IF(isEmpty(x0), dropLast(x0), last(x0), nil)) REV(x0, cons(z0, z1)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil), DROPLAST(cons(z0, nil))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil), DROPLAST(nil)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil), DROPLAST(cons(z0, nil))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(nil, nil) -> c11(IF(true, dropLast(nil), last(nil), nil), DROPLAST(nil)) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) S tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(x0, cons(z0, z1)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1)), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(x0, cons(z0, z1)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1)), APPEND(cons(z0, z1), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, nil) -> c12(IF(isEmpty(x0), dropLast(x0), last(x0), nil), LAST(x0)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(x0, nil) -> c10(IF(isEmpty(x0), dropLast(x0), last(x0), nil)) REV(x0, cons(z0, z1)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil), DROPLAST(cons(z0, nil))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil), DROPLAST(nil)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil), DROPLAST(cons(z0, nil))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(nil, nil) -> c11(IF(true, dropLast(nil), last(nil), nil), DROPLAST(nil)) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) K tuples: REV(nil, x1) -> c11(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) Defined Rule Symbols: isEmpty_1, dropLast_1, append_2, last_1 Defined Pair Symbols: LAST_1, DROPLAST_1, APPEND_2, IF_4, REV_2 Compound Symbols: c3_1, c6_1, c8_1, c14_1, c11_2, c11_1, c12_3, c12_2, c12_1, c10_1 ---------------------------------------- (89) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: REV(nil, nil) -> c11(IF(true, dropLast(nil), last(nil), nil), DROPLAST(nil)) ---------------------------------------- (90) Obligation: Complexity Dependency Tuples Problem Rules: isEmpty(nil) -> true isEmpty(cons(z0, z1)) -> false dropLast(nil) -> nil dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) append(nil, z0) -> z0 append(cons(z0, z1), z2) -> cons(z0, append(z1, z2)) last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(x0, cons(z0, z1)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1)), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(nil, x1) -> c11(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, cons(z0, z1)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1)), APPEND(cons(z0, z1), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, nil) -> c12(IF(isEmpty(x0), dropLast(x0), last(x0), nil), LAST(x0)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(x0, nil) -> c10(IF(isEmpty(x0), dropLast(x0), last(x0), nil)) REV(x0, cons(z0, z1)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil), DROPLAST(cons(z0, nil))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil), DROPLAST(nil)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil), DROPLAST(cons(z0, nil))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) S tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(x0, cons(z0, z1)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1)), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(x0, cons(z0, z1)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1)), APPEND(cons(z0, z1), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, nil) -> c12(IF(isEmpty(x0), dropLast(x0), last(x0), nil), LAST(x0)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(x0, nil) -> c10(IF(isEmpty(x0), dropLast(x0), last(x0), nil)) REV(x0, cons(z0, z1)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil), DROPLAST(cons(z0, nil))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil), DROPLAST(nil)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil), DROPLAST(cons(z0, nil))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) K tuples: REV(nil, x1) -> c11(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) Defined Rule Symbols: isEmpty_1, dropLast_1, append_2, last_1 Defined Pair Symbols: LAST_1, DROPLAST_1, APPEND_2, IF_4, REV_2 Compound Symbols: c3_1, c6_1, c8_1, c14_1, c11_2, c11_1, c12_3, c12_2, c12_1, c10_1 ---------------------------------------- (91) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (92) Obligation: Complexity Dependency Tuples Problem Rules: isEmpty(nil) -> true isEmpty(cons(z0, z1)) -> false dropLast(nil) -> nil dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) append(nil, z0) -> z0 append(cons(z0, z1), z2) -> cons(z0, append(z1, z2)) last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(x0, cons(z0, z1)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1)), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(nil, x1) -> c11(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, cons(z0, z1)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1)), APPEND(cons(z0, z1), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, nil) -> c12(IF(isEmpty(x0), dropLast(x0), last(x0), nil), LAST(x0)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(x0, nil) -> c10(IF(isEmpty(x0), dropLast(x0), last(x0), nil)) REV(x0, cons(z0, z1)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) S tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(x0, cons(z0, z1)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1)), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(x0, cons(z0, z1)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1)), APPEND(cons(z0, z1), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, nil) -> c12(IF(isEmpty(x0), dropLast(x0), last(x0), nil), LAST(x0)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(x0, nil) -> c10(IF(isEmpty(x0), dropLast(x0), last(x0), nil)) REV(x0, cons(z0, z1)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) K tuples: REV(nil, x1) -> c11(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) Defined Rule Symbols: isEmpty_1, dropLast_1, append_2, last_1 Defined Pair Symbols: LAST_1, DROPLAST_1, APPEND_2, IF_4, REV_2 Compound Symbols: c3_1, c6_1, c8_1, c14_1, c11_2, c11_1, c12_3, c12_2, c12_1, c10_1 ---------------------------------------- (93) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) We considered the (Usable) Rules: dropLast(nil) -> nil isEmpty(nil) -> true dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) isEmpty(cons(z0, z1)) -> false And the Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(x0, cons(z0, z1)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1)), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(nil, x1) -> c11(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, cons(z0, z1)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1)), APPEND(cons(z0, z1), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, nil) -> c12(IF(isEmpty(x0), dropLast(x0), last(x0), nil), LAST(x0)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(x0, nil) -> c10(IF(isEmpty(x0), dropLast(x0), last(x0), nil)) REV(x0, cons(z0, z1)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) The order we found is given by the following interpretation: Polynomial interpretation : POL(APPEND(x_1, x_2)) = 0 POL(DROPLAST(x_1)) = 0 POL(IF(x_1, x_2, x_3, x_4)) = x_1 + x_2 POL(LAST(x_1)) = 0 POL(REV(x_1, x_2)) = x_1 POL(append(x_1, x_2)) = [1] + x_1 POL(c10(x_1)) = x_1 POL(c11(x_1)) = x_1 POL(c11(x_1, x_2)) = x_1 + x_2 POL(c12(x_1)) = x_1 POL(c12(x_1, x_2)) = x_1 + x_2 POL(c12(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c14(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(cons(x_1, x_2)) = [1] + x_2 POL(dropLast(x_1)) = x_1 POL(false) = 0 POL(isEmpty(x_1)) = 0 POL(last(x_1)) = 0 POL(nil) = [1] POL(true) = 0 ---------------------------------------- (94) Obligation: Complexity Dependency Tuples Problem Rules: isEmpty(nil) -> true isEmpty(cons(z0, z1)) -> false dropLast(nil) -> nil dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) append(nil, z0) -> z0 append(cons(z0, z1), z2) -> cons(z0, append(z1, z2)) last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(x0, cons(z0, z1)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1)), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(nil, x1) -> c11(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, cons(z0, z1)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1)), APPEND(cons(z0, z1), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, nil) -> c12(IF(isEmpty(x0), dropLast(x0), last(x0), nil), LAST(x0)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(x0, nil) -> c10(IF(isEmpty(x0), dropLast(x0), last(x0), nil)) REV(x0, cons(z0, z1)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) S tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(x0, cons(z0, z1)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1)), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(x0, cons(z0, z1)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1)), APPEND(cons(z0, z1), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, nil) -> c12(IF(isEmpty(x0), dropLast(x0), last(x0), nil), LAST(x0)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(x0, nil) -> c10(IF(isEmpty(x0), dropLast(x0), last(x0), nil)) REV(x0, cons(z0, z1)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) K tuples: REV(nil, x1) -> c11(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) Defined Rule Symbols: isEmpty_1, dropLast_1, append_2, last_1 Defined Pair Symbols: LAST_1, DROPLAST_1, APPEND_2, IF_4, REV_2 Compound Symbols: c3_1, c6_1, c8_1, c14_1, c11_2, c11_1, c12_3, c12_2, c12_1, c10_1 ---------------------------------------- (95) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) We considered the (Usable) Rules: dropLast(nil) -> nil isEmpty(nil) -> true dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) isEmpty(cons(z0, z1)) -> false And the Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(x0, cons(z0, z1)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1)), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(nil, x1) -> c11(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, cons(z0, z1)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1)), APPEND(cons(z0, z1), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, nil) -> c12(IF(isEmpty(x0), dropLast(x0), last(x0), nil), LAST(x0)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(x0, nil) -> c10(IF(isEmpty(x0), dropLast(x0), last(x0), nil)) REV(x0, cons(z0, z1)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) The order we found is given by the following interpretation: Polynomial interpretation : POL(APPEND(x_1, x_2)) = 0 POL(DROPLAST(x_1)) = 0 POL(IF(x_1, x_2, x_3, x_4)) = [2]x_1 + [2]x_2 POL(LAST(x_1)) = 0 POL(REV(x_1, x_2)) = [2] + [2]x_1 POL(append(x_1, x_2)) = 0 POL(c10(x_1)) = x_1 POL(c11(x_1)) = x_1 POL(c11(x_1, x_2)) = x_1 + x_2 POL(c12(x_1)) = x_1 POL(c12(x_1, x_2)) = x_1 + x_2 POL(c12(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c14(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(cons(x_1, x_2)) = [1] POL(dropLast(x_1)) = [1] POL(false) = [1] POL(isEmpty(x_1)) = x_1 POL(last(x_1)) = 0 POL(nil) = 0 POL(true) = 0 ---------------------------------------- (96) Obligation: Complexity Dependency Tuples Problem Rules: isEmpty(nil) -> true isEmpty(cons(z0, z1)) -> false dropLast(nil) -> nil dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) append(nil, z0) -> z0 append(cons(z0, z1), z2) -> cons(z0, append(z1, z2)) last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(x0, cons(z0, z1)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1)), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(nil, x1) -> c11(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, cons(z0, z1)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1)), APPEND(cons(z0, z1), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, nil) -> c12(IF(isEmpty(x0), dropLast(x0), last(x0), nil), LAST(x0)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(x0, nil) -> c10(IF(isEmpty(x0), dropLast(x0), last(x0), nil)) REV(x0, cons(z0, z1)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) S tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(x0, cons(z0, z1)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1)), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(x0, cons(z0, z1)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1)), APPEND(cons(z0, z1), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, nil) -> c12(IF(isEmpty(x0), dropLast(x0), last(x0), nil), LAST(x0)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(x0, nil) -> c10(IF(isEmpty(x0), dropLast(x0), last(x0), nil)) REV(x0, cons(z0, z1)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) K tuples: REV(nil, x1) -> c11(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) Defined Rule Symbols: isEmpty_1, dropLast_1, append_2, last_1 Defined Pair Symbols: LAST_1, DROPLAST_1, APPEND_2, IF_4, REV_2 Compound Symbols: c3_1, c6_1, c8_1, c14_1, c11_2, c11_1, c12_3, c12_2, c12_1, c10_1 ---------------------------------------- (97) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace REV(x0, cons(z0, z1)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1)), DROPLAST(x0)) by REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), DROPLAST(cons(z0, nil))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), DROPLAST(nil)) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), DROPLAST(cons(z0, nil))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(nil, cons(x1, x2)) -> c11(IF(true, dropLast(nil), cons(x1, append(x2, last(nil))), cons(x1, x2)), DROPLAST(nil)) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) ---------------------------------------- (98) Obligation: Complexity Dependency Tuples Problem Rules: isEmpty(nil) -> true isEmpty(cons(z0, z1)) -> false dropLast(nil) -> nil dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) append(nil, z0) -> z0 append(cons(z0, z1), z2) -> cons(z0, append(z1, z2)) last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(nil, x1) -> c11(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, cons(z0, z1)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1)), APPEND(cons(z0, z1), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, nil) -> c12(IF(isEmpty(x0), dropLast(x0), last(x0), nil), LAST(x0)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(x0, nil) -> c10(IF(isEmpty(x0), dropLast(x0), last(x0), nil)) REV(x0, cons(z0, z1)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), DROPLAST(cons(z0, nil))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), DROPLAST(nil)) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), DROPLAST(cons(z0, nil))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(nil, cons(x1, x2)) -> c11(IF(true, dropLast(nil), cons(x1, append(x2, last(nil))), cons(x1, x2)), DROPLAST(nil)) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) S tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(x0, cons(z0, z1)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1)), APPEND(cons(z0, z1), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, nil) -> c12(IF(isEmpty(x0), dropLast(x0), last(x0), nil), LAST(x0)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(x0, nil) -> c10(IF(isEmpty(x0), dropLast(x0), last(x0), nil)) REV(x0, cons(z0, z1)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), DROPLAST(cons(z0, nil))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), DROPLAST(nil)) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), DROPLAST(cons(z0, nil))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(nil, cons(x1, x2)) -> c11(IF(true, dropLast(nil), cons(x1, append(x2, last(nil))), cons(x1, x2)), DROPLAST(nil)) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) K tuples: REV(nil, x1) -> c11(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) Defined Rule Symbols: isEmpty_1, dropLast_1, append_2, last_1 Defined Pair Symbols: LAST_1, DROPLAST_1, APPEND_2, IF_4, REV_2 Compound Symbols: c3_1, c6_1, c8_1, c14_1, c11_2, c11_1, c12_3, c12_2, c12_1, c10_1 ---------------------------------------- (99) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: REV(nil, cons(x1, x2)) -> c11(IF(true, dropLast(nil), cons(x1, append(x2, last(nil))), cons(x1, x2)), DROPLAST(nil)) ---------------------------------------- (100) Obligation: Complexity Dependency Tuples Problem Rules: isEmpty(nil) -> true isEmpty(cons(z0, z1)) -> false dropLast(nil) -> nil dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) append(nil, z0) -> z0 append(cons(z0, z1), z2) -> cons(z0, append(z1, z2)) last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(nil, x1) -> c11(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, cons(z0, z1)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1)), APPEND(cons(z0, z1), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, nil) -> c12(IF(isEmpty(x0), dropLast(x0), last(x0), nil), LAST(x0)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(x0, nil) -> c10(IF(isEmpty(x0), dropLast(x0), last(x0), nil)) REV(x0, cons(z0, z1)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), DROPLAST(cons(z0, nil))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), DROPLAST(nil)) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), DROPLAST(cons(z0, nil))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) S tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(x0, cons(z0, z1)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1)), APPEND(cons(z0, z1), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, nil) -> c12(IF(isEmpty(x0), dropLast(x0), last(x0), nil), LAST(x0)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(x0, nil) -> c10(IF(isEmpty(x0), dropLast(x0), last(x0), nil)) REV(x0, cons(z0, z1)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), DROPLAST(cons(z0, nil))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), DROPLAST(nil)) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), DROPLAST(cons(z0, nil))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) K tuples: REV(nil, x1) -> c11(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) Defined Rule Symbols: isEmpty_1, dropLast_1, append_2, last_1 Defined Pair Symbols: LAST_1, DROPLAST_1, APPEND_2, IF_4, REV_2 Compound Symbols: c3_1, c6_1, c8_1, c14_1, c11_2, c11_1, c12_3, c12_2, c12_1, c10_1 ---------------------------------------- (101) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (102) Obligation: Complexity Dependency Tuples Problem Rules: isEmpty(nil) -> true isEmpty(cons(z0, z1)) -> false dropLast(nil) -> nil dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) append(nil, z0) -> z0 append(cons(z0, z1), z2) -> cons(z0, append(z1, z2)) last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(nil, x1) -> c11(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, cons(z0, z1)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1)), APPEND(cons(z0, z1), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, nil) -> c12(IF(isEmpty(x0), dropLast(x0), last(x0), nil), LAST(x0)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(x0, nil) -> c10(IF(isEmpty(x0), dropLast(x0), last(x0), nil)) REV(x0, cons(z0, z1)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) S tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(x0, cons(z0, z1)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1)), APPEND(cons(z0, z1), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, nil) -> c12(IF(isEmpty(x0), dropLast(x0), last(x0), nil), LAST(x0)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(x0, nil) -> c10(IF(isEmpty(x0), dropLast(x0), last(x0), nil)) REV(x0, cons(z0, z1)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) K tuples: REV(nil, x1) -> c11(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) Defined Rule Symbols: isEmpty_1, dropLast_1, append_2, last_1 Defined Pair Symbols: LAST_1, DROPLAST_1, APPEND_2, IF_4, REV_2 Compound Symbols: c3_1, c6_1, c8_1, c14_1, c11_2, c11_1, c12_3, c12_2, c12_1, c10_1 ---------------------------------------- (103) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) We considered the (Usable) Rules: dropLast(nil) -> nil dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) And the Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(nil, x1) -> c11(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, cons(z0, z1)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1)), APPEND(cons(z0, z1), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, nil) -> c12(IF(isEmpty(x0), dropLast(x0), last(x0), nil), LAST(x0)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(x0, nil) -> c10(IF(isEmpty(x0), dropLast(x0), last(x0), nil)) REV(x0, cons(z0, z1)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) The order we found is given by the following interpretation: Polynomial interpretation : POL(APPEND(x_1, x_2)) = 0 POL(DROPLAST(x_1)) = 0 POL(IF(x_1, x_2, x_3, x_4)) = [2]x_2 POL(LAST(x_1)) = 0 POL(REV(x_1, x_2)) = [2]x_1 POL(append(x_1, x_2)) = 0 POL(c10(x_1)) = x_1 POL(c11(x_1)) = x_1 POL(c11(x_1, x_2)) = x_1 + x_2 POL(c12(x_1)) = x_1 POL(c12(x_1, x_2)) = x_1 + x_2 POL(c12(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c14(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(cons(x_1, x_2)) = [1] + x_2 POL(dropLast(x_1)) = x_1 POL(false) = [3] POL(isEmpty(x_1)) = [3] + [2]x_1 POL(last(x_1)) = 0 POL(nil) = 0 POL(true) = [3] ---------------------------------------- (104) Obligation: Complexity Dependency Tuples Problem Rules: isEmpty(nil) -> true isEmpty(cons(z0, z1)) -> false dropLast(nil) -> nil dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) append(nil, z0) -> z0 append(cons(z0, z1), z2) -> cons(z0, append(z1, z2)) last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(nil, x1) -> c11(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, cons(z0, z1)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1)), APPEND(cons(z0, z1), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, nil) -> c12(IF(isEmpty(x0), dropLast(x0), last(x0), nil), LAST(x0)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(x0, nil) -> c10(IF(isEmpty(x0), dropLast(x0), last(x0), nil)) REV(x0, cons(z0, z1)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) S tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(x0, cons(z0, z1)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1)), APPEND(cons(z0, z1), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, nil) -> c12(IF(isEmpty(x0), dropLast(x0), last(x0), nil), LAST(x0)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(x0, nil) -> c10(IF(isEmpty(x0), dropLast(x0), last(x0), nil)) REV(x0, cons(z0, z1)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) K tuples: REV(nil, x1) -> c11(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) Defined Rule Symbols: isEmpty_1, dropLast_1, append_2, last_1 Defined Pair Symbols: LAST_1, DROPLAST_1, APPEND_2, IF_4, REV_2 Compound Symbols: c3_1, c6_1, c8_1, c14_1, c11_2, c11_1, c12_3, c12_2, c12_1, c10_1 ---------------------------------------- (105) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) We considered the (Usable) Rules: dropLast(nil) -> nil isEmpty(nil) -> true dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) isEmpty(cons(z0, z1)) -> false And the Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(nil, x1) -> c11(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, cons(z0, z1)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1)), APPEND(cons(z0, z1), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, nil) -> c12(IF(isEmpty(x0), dropLast(x0), last(x0), nil), LAST(x0)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(x0, nil) -> c10(IF(isEmpty(x0), dropLast(x0), last(x0), nil)) REV(x0, cons(z0, z1)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) The order we found is given by the following interpretation: Polynomial interpretation : POL(APPEND(x_1, x_2)) = 0 POL(DROPLAST(x_1)) = 0 POL(IF(x_1, x_2, x_3, x_4)) = x_1 + x_2 POL(LAST(x_1)) = 0 POL(REV(x_1, x_2)) = [1] + x_1 POL(append(x_1, x_2)) = [1] + x_1 POL(c10(x_1)) = x_1 POL(c11(x_1)) = x_1 POL(c11(x_1, x_2)) = x_1 + x_2 POL(c12(x_1)) = x_1 POL(c12(x_1, x_2)) = x_1 + x_2 POL(c12(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c14(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(cons(x_1, x_2)) = [1] POL(dropLast(x_1)) = [1] POL(false) = [1] POL(isEmpty(x_1)) = x_1 POL(last(x_1)) = 0 POL(nil) = 0 POL(true) = 0 ---------------------------------------- (106) Obligation: Complexity Dependency Tuples Problem Rules: isEmpty(nil) -> true isEmpty(cons(z0, z1)) -> false dropLast(nil) -> nil dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) append(nil, z0) -> z0 append(cons(z0, z1), z2) -> cons(z0, append(z1, z2)) last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(nil, x1) -> c11(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, cons(z0, z1)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1)), APPEND(cons(z0, z1), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, nil) -> c12(IF(isEmpty(x0), dropLast(x0), last(x0), nil), LAST(x0)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(x0, nil) -> c10(IF(isEmpty(x0), dropLast(x0), last(x0), nil)) REV(x0, cons(z0, z1)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) S tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(x0, cons(z0, z1)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1)), APPEND(cons(z0, z1), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, nil) -> c12(IF(isEmpty(x0), dropLast(x0), last(x0), nil), LAST(x0)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(x0, nil) -> c10(IF(isEmpty(x0), dropLast(x0), last(x0), nil)) REV(x0, cons(z0, z1)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) K tuples: REV(nil, x1) -> c11(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) Defined Rule Symbols: isEmpty_1, dropLast_1, append_2, last_1 Defined Pair Symbols: LAST_1, DROPLAST_1, APPEND_2, IF_4, REV_2 Compound Symbols: c3_1, c6_1, c8_1, c14_1, c11_2, c11_1, c12_3, c12_2, c12_1, c10_1 ---------------------------------------- (107) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), DROPLAST(cons(z0, cons(z1, z2)))) by REV(cons(x0, cons(x1, x2)), nil) -> c11(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), DROPLAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), cons(z0, z1)) -> c11(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), cons(z0, append(z1, last(cons(x1, x2)))), cons(z0, z1)), DROPLAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) ---------------------------------------- (108) Obligation: Complexity Dependency Tuples Problem Rules: isEmpty(nil) -> true isEmpty(cons(z0, z1)) -> false dropLast(nil) -> nil dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) append(nil, z0) -> z0 append(cons(z0, z1), z2) -> cons(z0, append(z1, z2)) last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(nil, x1) -> c11(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, cons(z0, z1)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1)), APPEND(cons(z0, z1), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, nil) -> c12(IF(isEmpty(x0), dropLast(x0), last(x0), nil), LAST(x0)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(x0, nil) -> c10(IF(isEmpty(x0), dropLast(x0), last(x0), nil)) REV(x0, cons(z0, z1)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) S tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(x0, cons(z0, z1)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1)), APPEND(cons(z0, z1), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, nil) -> c12(IF(isEmpty(x0), dropLast(x0), last(x0), nil), LAST(x0)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(x0, nil) -> c10(IF(isEmpty(x0), dropLast(x0), last(x0), nil)) REV(x0, cons(z0, z1)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) K tuples: REV(nil, x1) -> c11(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) Defined Rule Symbols: isEmpty_1, dropLast_1, append_2, last_1 Defined Pair Symbols: LAST_1, DROPLAST_1, APPEND_2, IF_4, REV_2 Compound Symbols: c3_1, c6_1, c8_1, c14_1, c11_2, c11_1, c12_3, c12_2, c12_1, c10_1 ---------------------------------------- (109) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace REV(cons(z0, cons(z1, z2)), x1) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), DROPLAST(cons(z0, cons(z1, z2)))) by REV(cons(x0, cons(x1, x2)), nil) -> c11(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), DROPLAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), cons(z0, z1)) -> c11(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), cons(z0, append(z1, last(cons(x0, cons(x1, x2))))), cons(z0, z1)), DROPLAST(cons(x0, cons(x1, x2)))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) ---------------------------------------- (110) Obligation: Complexity Dependency Tuples Problem Rules: isEmpty(nil) -> true isEmpty(cons(z0, z1)) -> false dropLast(nil) -> nil dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) append(nil, z0) -> z0 append(cons(z0, z1), z2) -> cons(z0, append(z1, z2)) last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(nil, x1) -> c11(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, cons(z0, z1)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1)), APPEND(cons(z0, z1), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, nil) -> c12(IF(isEmpty(x0), dropLast(x0), last(x0), nil), LAST(x0)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(x0, nil) -> c10(IF(isEmpty(x0), dropLast(x0), last(x0), nil)) REV(x0, cons(z0, z1)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) S tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(x0, cons(z0, z1)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1)), APPEND(cons(z0, z1), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, nil) -> c12(IF(isEmpty(x0), dropLast(x0), last(x0), nil), LAST(x0)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(x0, nil) -> c10(IF(isEmpty(x0), dropLast(x0), last(x0), nil)) REV(x0, cons(z0, z1)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) K tuples: REV(nil, x1) -> c11(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) Defined Rule Symbols: isEmpty_1, dropLast_1, append_2, last_1 Defined Pair Symbols: LAST_1, DROPLAST_1, APPEND_2, IF_4, REV_2 Compound Symbols: c3_1, c6_1, c8_1, c14_1, c11_2, c11_1, c12_3, c12_2, c12_1, c10_1 ---------------------------------------- (111) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) We considered the (Usable) Rules: dropLast(nil) -> nil isEmpty(nil) -> true dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) isEmpty(cons(z0, z1)) -> false And the Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(nil, x1) -> c11(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, cons(z0, z1)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1)), APPEND(cons(z0, z1), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, nil) -> c12(IF(isEmpty(x0), dropLast(x0), last(x0), nil), LAST(x0)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(x0, nil) -> c10(IF(isEmpty(x0), dropLast(x0), last(x0), nil)) REV(x0, cons(z0, z1)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) The order we found is given by the following interpretation: Polynomial interpretation : POL(APPEND(x_1, x_2)) = 0 POL(DROPLAST(x_1)) = 0 POL(IF(x_1, x_2, x_3, x_4)) = x_1 + x_2 POL(LAST(x_1)) = 0 POL(REV(x_1, x_2)) = x_1 POL(append(x_1, x_2)) = [1] + x_1 POL(c10(x_1)) = x_1 POL(c11(x_1)) = x_1 POL(c11(x_1, x_2)) = x_1 + x_2 POL(c12(x_1)) = x_1 POL(c12(x_1, x_2)) = x_1 + x_2 POL(c12(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c14(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(cons(x_1, x_2)) = [1] + x_2 POL(dropLast(x_1)) = x_1 POL(false) = 0 POL(isEmpty(x_1)) = 0 POL(last(x_1)) = 0 POL(nil) = [1] POL(true) = 0 ---------------------------------------- (112) Obligation: Complexity Dependency Tuples Problem Rules: isEmpty(nil) -> true isEmpty(cons(z0, z1)) -> false dropLast(nil) -> nil dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) append(nil, z0) -> z0 append(cons(z0, z1), z2) -> cons(z0, append(z1, z2)) last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(nil, x1) -> c11(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, cons(z0, z1)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1)), APPEND(cons(z0, z1), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, nil) -> c12(IF(isEmpty(x0), dropLast(x0), last(x0), nil), LAST(x0)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(x0, nil) -> c10(IF(isEmpty(x0), dropLast(x0), last(x0), nil)) REV(x0, cons(z0, z1)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) S tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(x0, cons(z0, z1)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1)), APPEND(cons(z0, z1), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, nil) -> c12(IF(isEmpty(x0), dropLast(x0), last(x0), nil), LAST(x0)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(x0, nil) -> c10(IF(isEmpty(x0), dropLast(x0), last(x0), nil)) REV(x0, cons(z0, z1)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) K tuples: REV(nil, x1) -> c11(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) Defined Rule Symbols: isEmpty_1, dropLast_1, append_2, last_1 Defined Pair Symbols: LAST_1, DROPLAST_1, APPEND_2, IF_4, REV_2 Compound Symbols: c3_1, c6_1, c8_1, c14_1, c11_2, c11_1, c12_3, c12_2, c12_1, c10_1 ---------------------------------------- (113) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) by REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z2, nil), cons(z0, z1)) -> c11(IF(isEmpty(cons(z2, nil)), dropLast(cons(z2, nil)), cons(z0, append(z1, z2)), cons(z0, z1))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) ---------------------------------------- (114) Obligation: Complexity Dependency Tuples Problem Rules: isEmpty(nil) -> true isEmpty(cons(z0, z1)) -> false dropLast(nil) -> nil dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) append(nil, z0) -> z0 append(cons(z0, z1), z2) -> cons(z0, append(z1, z2)) last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(nil, x1) -> c11(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, cons(z0, z1)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1)), APPEND(cons(z0, z1), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, nil) -> c12(IF(isEmpty(x0), dropLast(x0), last(x0), nil), LAST(x0)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(x0, nil) -> c10(IF(isEmpty(x0), dropLast(x0), last(x0), nil)) REV(x0, cons(z0, z1)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) S tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(x0, cons(z0, z1)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1)), APPEND(cons(z0, z1), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, nil) -> c12(IF(isEmpty(x0), dropLast(x0), last(x0), nil), LAST(x0)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(x0, nil) -> c10(IF(isEmpty(x0), dropLast(x0), last(x0), nil)) REV(x0, cons(z0, z1)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) K tuples: REV(nil, x1) -> c11(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) Defined Rule Symbols: isEmpty_1, dropLast_1, append_2, last_1 Defined Pair Symbols: LAST_1, DROPLAST_1, APPEND_2, IF_4, REV_2 Compound Symbols: c3_1, c6_1, c8_1, c14_1, c11_2, c11_1, c12_3, c12_2, c12_1, c10_1 ---------------------------------------- (115) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) We considered the (Usable) Rules: dropLast(nil) -> nil isEmpty(nil) -> true dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) isEmpty(cons(z0, z1)) -> false And the Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(nil, x1) -> c11(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, cons(z0, z1)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1)), APPEND(cons(z0, z1), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, nil) -> c12(IF(isEmpty(x0), dropLast(x0), last(x0), nil), LAST(x0)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(x0, nil) -> c10(IF(isEmpty(x0), dropLast(x0), last(x0), nil)) REV(x0, cons(z0, z1)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) The order we found is given by the following interpretation: Polynomial interpretation : POL(APPEND(x_1, x_2)) = 0 POL(DROPLAST(x_1)) = 0 POL(IF(x_1, x_2, x_3, x_4)) = x_1 + x_2 POL(LAST(x_1)) = 0 POL(REV(x_1, x_2)) = x_1 POL(append(x_1, x_2)) = [1] + x_1 POL(c10(x_1)) = x_1 POL(c11(x_1)) = x_1 POL(c11(x_1, x_2)) = x_1 + x_2 POL(c12(x_1)) = x_1 POL(c12(x_1, x_2)) = x_1 + x_2 POL(c12(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c14(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(cons(x_1, x_2)) = [1] POL(dropLast(x_1)) = x_1 POL(false) = 0 POL(isEmpty(x_1)) = 0 POL(last(x_1)) = 0 POL(nil) = 0 POL(true) = 0 ---------------------------------------- (116) Obligation: Complexity Dependency Tuples Problem Rules: isEmpty(nil) -> true isEmpty(cons(z0, z1)) -> false dropLast(nil) -> nil dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) append(nil, z0) -> z0 append(cons(z0, z1), z2) -> cons(z0, append(z1, z2)) last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(nil, x1) -> c11(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, cons(z0, z1)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1)), APPEND(cons(z0, z1), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, nil) -> c12(IF(isEmpty(x0), dropLast(x0), last(x0), nil), LAST(x0)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(x0, nil) -> c10(IF(isEmpty(x0), dropLast(x0), last(x0), nil)) REV(x0, cons(z0, z1)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) S tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(x0, cons(z0, z1)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1)), APPEND(cons(z0, z1), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, nil) -> c12(IF(isEmpty(x0), dropLast(x0), last(x0), nil), LAST(x0)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(x0, nil) -> c10(IF(isEmpty(x0), dropLast(x0), last(x0), nil)) REV(x0, cons(z0, z1)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) K tuples: REV(nil, x1) -> c11(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) Defined Rule Symbols: isEmpty_1, dropLast_1, append_2, last_1 Defined Pair Symbols: LAST_1, DROPLAST_1, APPEND_2, IF_4, REV_2 Compound Symbols: c3_1, c6_1, c8_1, c14_1, c11_2, c11_1, c12_3, c12_2, c12_1, c10_1 ---------------------------------------- (117) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace REV(nil, x1) -> c11(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) by REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(nil, cons(z0, z1)) -> c11(IF(isEmpty(nil), nil, cons(z0, append(z1, last(nil))), cons(z0, z1))) REV(nil, x0) -> c11(IF(true, nil, append(x0, last(nil)), x0)) ---------------------------------------- (118) Obligation: Complexity Dependency Tuples Problem Rules: isEmpty(nil) -> true isEmpty(cons(z0, z1)) -> false dropLast(nil) -> nil dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) append(nil, z0) -> z0 append(cons(z0, z1), z2) -> cons(z0, append(z1, z2)) last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, cons(z0, z1)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1)), APPEND(cons(z0, z1), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, nil) -> c12(IF(isEmpty(x0), dropLast(x0), last(x0), nil), LAST(x0)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(x0, nil) -> c10(IF(isEmpty(x0), dropLast(x0), last(x0), nil)) REV(x0, cons(z0, z1)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(nil, x0) -> c11(IF(true, nil, append(x0, last(nil)), x0)) S tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(x0, cons(z0, z1)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1)), APPEND(cons(z0, z1), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, nil) -> c12(IF(isEmpty(x0), dropLast(x0), last(x0), nil), LAST(x0)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(x0, nil) -> c10(IF(isEmpty(x0), dropLast(x0), last(x0), nil)) REV(x0, cons(z0, z1)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) K tuples: REV(nil, x1) -> c11(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) Defined Rule Symbols: isEmpty_1, dropLast_1, append_2, last_1 Defined Pair Symbols: LAST_1, DROPLAST_1, APPEND_2, IF_4, REV_2 Compound Symbols: c3_1, c6_1, c8_1, c14_1, c11_2, c11_1, c12_3, c12_2, c12_1, c10_1 ---------------------------------------- (119) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: REV(nil, x0) -> c11(IF(true, nil, append(x0, last(nil)), x0)) ---------------------------------------- (120) Obligation: Complexity Dependency Tuples Problem Rules: isEmpty(nil) -> true isEmpty(cons(z0, z1)) -> false dropLast(nil) -> nil dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) append(nil, z0) -> z0 append(cons(z0, z1), z2) -> cons(z0, append(z1, z2)) last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, cons(z0, z1)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1)), APPEND(cons(z0, z1), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, nil) -> c12(IF(isEmpty(x0), dropLast(x0), last(x0), nil), LAST(x0)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(x0, nil) -> c10(IF(isEmpty(x0), dropLast(x0), last(x0), nil)) REV(x0, cons(z0, z1)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) S tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(x0, cons(z0, z1)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1)), APPEND(cons(z0, z1), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, nil) -> c12(IF(isEmpty(x0), dropLast(x0), last(x0), nil), LAST(x0)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(x0, nil) -> c10(IF(isEmpty(x0), dropLast(x0), last(x0), nil)) REV(x0, cons(z0, z1)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) K tuples: REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) Defined Rule Symbols: isEmpty_1, dropLast_1, append_2, last_1 Defined Pair Symbols: LAST_1, DROPLAST_1, APPEND_2, IF_4, REV_2 Compound Symbols: c3_1, c6_1, c8_1, c14_1, c11_2, c11_1, c12_3, c12_2, c12_1, c10_1 ---------------------------------------- (121) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) by REV(cons(x0, nil), nil) -> c11(IF(isEmpty(cons(x0, nil)), nil, last(cons(x0, nil)), nil)) REV(cons(x0, nil), cons(z0, z1)) -> c11(IF(isEmpty(cons(x0, nil)), nil, cons(z0, append(z1, last(cons(x0, nil)))), cons(z0, z1))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) ---------------------------------------- (122) Obligation: Complexity Dependency Tuples Problem Rules: isEmpty(nil) -> true isEmpty(cons(z0, z1)) -> false dropLast(nil) -> nil dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) append(nil, z0) -> z0 append(cons(z0, z1), z2) -> cons(z0, append(z1, z2)) last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(x0, cons(z0, z1)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1)), APPEND(cons(z0, z1), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, nil) -> c12(IF(isEmpty(x0), dropLast(x0), last(x0), nil), LAST(x0)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(x0, nil) -> c10(IF(isEmpty(x0), dropLast(x0), last(x0), nil)) REV(x0, cons(z0, z1)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) S tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(x0, cons(z0, z1)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1)), APPEND(cons(z0, z1), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, nil) -> c12(IF(isEmpty(x0), dropLast(x0), last(x0), nil), LAST(x0)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(x0, nil) -> c10(IF(isEmpty(x0), dropLast(x0), last(x0), nil)) REV(x0, cons(z0, z1)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) K tuples: REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) Defined Rule Symbols: isEmpty_1, dropLast_1, append_2, last_1 Defined Pair Symbols: LAST_1, DROPLAST_1, APPEND_2, IF_4, REV_2 Compound Symbols: c3_1, c6_1, c8_1, c14_1, c11_2, c12_3, c12_2, c12_1, c10_1, c11_1 ---------------------------------------- (123) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace REV(x0, cons(z0, z1)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1)), APPEND(cons(z0, z1), last(x0)), LAST(x0)) by REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil))), LAST(cons(z0, nil))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil)), LAST(nil)) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil))), LAST(cons(z0, nil))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(nil, cons(x1, x2)) -> c12(IF(true, dropLast(nil), cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil)), LAST(nil)) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) ---------------------------------------- (124) Obligation: Complexity Dependency Tuples Problem Rules: isEmpty(nil) -> true isEmpty(cons(z0, z1)) -> false dropLast(nil) -> nil dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) append(nil, z0) -> z0 append(cons(z0, z1), z2) -> cons(z0, append(z1, z2)) last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, nil) -> c12(IF(isEmpty(x0), dropLast(x0), last(x0), nil), LAST(x0)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(x0, nil) -> c10(IF(isEmpty(x0), dropLast(x0), last(x0), nil)) REV(x0, cons(z0, z1)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil))), LAST(cons(z0, nil))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil)), LAST(nil)) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil))), LAST(cons(z0, nil))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(nil, cons(x1, x2)) -> c12(IF(true, dropLast(nil), cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil)), LAST(nil)) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) S tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, nil) -> c12(IF(isEmpty(x0), dropLast(x0), last(x0), nil), LAST(x0)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(x0, nil) -> c10(IF(isEmpty(x0), dropLast(x0), last(x0), nil)) REV(x0, cons(z0, z1)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil))), LAST(cons(z0, nil))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil)), LAST(nil)) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil))), LAST(cons(z0, nil))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(nil, cons(x1, x2)) -> c12(IF(true, dropLast(nil), cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil)), LAST(nil)) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) K tuples: REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) Defined Rule Symbols: isEmpty_1, dropLast_1, append_2, last_1 Defined Pair Symbols: LAST_1, DROPLAST_1, APPEND_2, IF_4, REV_2 Compound Symbols: c3_1, c6_1, c8_1, c14_1, c11_2, c12_3, c12_2, c12_1, c10_1, c11_1 ---------------------------------------- (125) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 5 trailing tuple parts ---------------------------------------- (126) Obligation: Complexity Dependency Tuples Problem Rules: isEmpty(nil) -> true isEmpty(cons(z0, z1)) -> false dropLast(nil) -> nil dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) append(nil, z0) -> z0 append(cons(z0, z1), z2) -> cons(z0, append(z1, z2)) last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, nil) -> c12(IF(isEmpty(x0), dropLast(x0), last(x0), nil), LAST(x0)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(x0, nil) -> c10(IF(isEmpty(x0), dropLast(x0), last(x0), nil)) REV(x0, cons(z0, z1)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) S tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, nil) -> c12(IF(isEmpty(x0), dropLast(x0), last(x0), nil), LAST(x0)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(x0, nil) -> c10(IF(isEmpty(x0), dropLast(x0), last(x0), nil)) REV(x0, cons(z0, z1)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) K tuples: REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) Defined Rule Symbols: isEmpty_1, dropLast_1, append_2, last_1 Defined Pair Symbols: LAST_1, DROPLAST_1, APPEND_2, IF_4, REV_2 Compound Symbols: c3_1, c6_1, c8_1, c14_1, c11_2, c12_3, c12_2, c12_1, c10_1, c11_1 ---------------------------------------- (127) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) We considered the (Usable) Rules: dropLast(nil) -> nil isEmpty(nil) -> true dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) isEmpty(cons(z0, z1)) -> false And the Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, nil) -> c12(IF(isEmpty(x0), dropLast(x0), last(x0), nil), LAST(x0)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(x0, nil) -> c10(IF(isEmpty(x0), dropLast(x0), last(x0), nil)) REV(x0, cons(z0, z1)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) The order we found is given by the following interpretation: Polynomial interpretation : POL(APPEND(x_1, x_2)) = 0 POL(DROPLAST(x_1)) = 0 POL(IF(x_1, x_2, x_3, x_4)) = x_1 + x_2 POL(LAST(x_1)) = 0 POL(REV(x_1, x_2)) = x_1 POL(append(x_1, x_2)) = [1] POL(c10(x_1)) = x_1 POL(c11(x_1)) = x_1 POL(c11(x_1, x_2)) = x_1 + x_2 POL(c12(x_1)) = x_1 POL(c12(x_1, x_2)) = x_1 + x_2 POL(c12(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c14(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(cons(x_1, x_2)) = x_2 POL(dropLast(x_1)) = x_1 POL(false) = 0 POL(isEmpty(x_1)) = 0 POL(last(x_1)) = 0 POL(nil) = [1] POL(true) = 0 ---------------------------------------- (128) Obligation: Complexity Dependency Tuples Problem Rules: isEmpty(nil) -> true isEmpty(cons(z0, z1)) -> false dropLast(nil) -> nil dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) append(nil, z0) -> z0 append(cons(z0, z1), z2) -> cons(z0, append(z1, z2)) last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, nil) -> c12(IF(isEmpty(x0), dropLast(x0), last(x0), nil), LAST(x0)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(x0, nil) -> c10(IF(isEmpty(x0), dropLast(x0), last(x0), nil)) REV(x0, cons(z0, z1)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) S tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, nil) -> c12(IF(isEmpty(x0), dropLast(x0), last(x0), nil), LAST(x0)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(x0, nil) -> c10(IF(isEmpty(x0), dropLast(x0), last(x0), nil)) REV(x0, cons(z0, z1)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) K tuples: REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) Defined Rule Symbols: isEmpty_1, dropLast_1, append_2, last_1 Defined Pair Symbols: LAST_1, DROPLAST_1, APPEND_2, IF_4, REV_2 Compound Symbols: c3_1, c6_1, c8_1, c14_1, c11_2, c12_3, c12_2, c12_1, c10_1, c11_1 ---------------------------------------- (129) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) We considered the (Usable) Rules: dropLast(nil) -> nil isEmpty(nil) -> true dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) isEmpty(cons(z0, z1)) -> false And the Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, nil) -> c12(IF(isEmpty(x0), dropLast(x0), last(x0), nil), LAST(x0)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(x0, nil) -> c10(IF(isEmpty(x0), dropLast(x0), last(x0), nil)) REV(x0, cons(z0, z1)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) The order we found is given by the following interpretation: Polynomial interpretation : POL(APPEND(x_1, x_2)) = 0 POL(DROPLAST(x_1)) = 0 POL(IF(x_1, x_2, x_3, x_4)) = x_1 + x_2 POL(LAST(x_1)) = 0 POL(REV(x_1, x_2)) = [1] + x_1 POL(append(x_1, x_2)) = [1] POL(c10(x_1)) = x_1 POL(c11(x_1)) = x_1 POL(c11(x_1, x_2)) = x_1 + x_2 POL(c12(x_1)) = x_1 POL(c12(x_1, x_2)) = x_1 + x_2 POL(c12(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c14(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(cons(x_1, x_2)) = [1] POL(dropLast(x_1)) = x_1 POL(false) = [1] POL(isEmpty(x_1)) = [1] POL(last(x_1)) = 0 POL(nil) = 0 POL(true) = [1] ---------------------------------------- (130) Obligation: Complexity Dependency Tuples Problem Rules: isEmpty(nil) -> true isEmpty(cons(z0, z1)) -> false dropLast(nil) -> nil dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) append(nil, z0) -> z0 append(cons(z0, z1), z2) -> cons(z0, append(z1, z2)) last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, nil) -> c12(IF(isEmpty(x0), dropLast(x0), last(x0), nil), LAST(x0)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(x0, nil) -> c10(IF(isEmpty(x0), dropLast(x0), last(x0), nil)) REV(x0, cons(z0, z1)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) S tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, nil) -> c12(IF(isEmpty(x0), dropLast(x0), last(x0), nil), LAST(x0)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(x0, nil) -> c10(IF(isEmpty(x0), dropLast(x0), last(x0), nil)) REV(x0, cons(z0, z1)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) K tuples: REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) Defined Rule Symbols: isEmpty_1, dropLast_1, append_2, last_1 Defined Pair Symbols: LAST_1, DROPLAST_1, APPEND_2, IF_4, REV_2 Compound Symbols: c3_1, c6_1, c8_1, c14_1, c11_2, c12_3, c12_2, c12_1, c10_1, c11_1 ---------------------------------------- (131) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) We considered the (Usable) Rules: dropLast(nil) -> nil isEmpty(nil) -> true dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) isEmpty(cons(z0, z1)) -> false And the Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, nil) -> c12(IF(isEmpty(x0), dropLast(x0), last(x0), nil), LAST(x0)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(x0, nil) -> c10(IF(isEmpty(x0), dropLast(x0), last(x0), nil)) REV(x0, cons(z0, z1)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) The order we found is given by the following interpretation: Polynomial interpretation : POL(APPEND(x_1, x_2)) = 0 POL(DROPLAST(x_1)) = 0 POL(IF(x_1, x_2, x_3, x_4)) = x_1 + x_2 POL(LAST(x_1)) = 0 POL(REV(x_1, x_2)) = [1] + x_1 POL(append(x_1, x_2)) = [1] POL(c10(x_1)) = x_1 POL(c11(x_1)) = x_1 POL(c11(x_1, x_2)) = x_1 + x_2 POL(c12(x_1)) = x_1 POL(c12(x_1, x_2)) = x_1 + x_2 POL(c12(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c14(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(cons(x_1, x_2)) = [1] POL(dropLast(x_1)) = [1] POL(false) = [1] POL(isEmpty(x_1)) = x_1 POL(last(x_1)) = 0 POL(nil) = 0 POL(true) = 0 ---------------------------------------- (132) Obligation: Complexity Dependency Tuples Problem Rules: isEmpty(nil) -> true isEmpty(cons(z0, z1)) -> false dropLast(nil) -> nil dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) append(nil, z0) -> z0 append(cons(z0, z1), z2) -> cons(z0, append(z1, z2)) last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, nil) -> c12(IF(isEmpty(x0), dropLast(x0), last(x0), nil), LAST(x0)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(x0, nil) -> c10(IF(isEmpty(x0), dropLast(x0), last(x0), nil)) REV(x0, cons(z0, z1)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) S tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, nil) -> c12(IF(isEmpty(x0), dropLast(x0), last(x0), nil), LAST(x0)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(x0, nil) -> c10(IF(isEmpty(x0), dropLast(x0), last(x0), nil)) REV(x0, cons(z0, z1)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) K tuples: REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) Defined Rule Symbols: isEmpty_1, dropLast_1, append_2, last_1 Defined Pair Symbols: LAST_1, DROPLAST_1, APPEND_2, IF_4, REV_2 Compound Symbols: c3_1, c6_1, c8_1, c14_1, c11_2, c12_3, c12_2, c12_1, c10_1, c11_1 ---------------------------------------- (133) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) by REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), APPEND(nil, last(cons(x0, cons(x1, x2)))), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), cons(z0, z1)) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), cons(z0, append(z1, last(cons(x1, x2)))), cons(z0, z1)), APPEND(cons(z0, z1), last(cons(x0, cons(x1, x2)))), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) ---------------------------------------- (134) Obligation: Complexity Dependency Tuples Problem Rules: isEmpty(nil) -> true isEmpty(cons(z0, z1)) -> false dropLast(nil) -> nil dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) append(nil, z0) -> z0 append(cons(z0, z1), z2) -> cons(z0, append(z1, z2)) last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, nil) -> c12(IF(isEmpty(x0), dropLast(x0), last(x0), nil), LAST(x0)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(x0, nil) -> c10(IF(isEmpty(x0), dropLast(x0), last(x0), nil)) REV(x0, cons(z0, z1)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), APPEND(nil, last(cons(x0, cons(x1, x2)))), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) S tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, nil) -> c12(IF(isEmpty(x0), dropLast(x0), last(x0), nil), LAST(x0)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(x0, nil) -> c10(IF(isEmpty(x0), dropLast(x0), last(x0), nil)) REV(x0, cons(z0, z1)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), APPEND(nil, last(cons(x0, cons(x1, x2)))), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) K tuples: REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) Defined Rule Symbols: isEmpty_1, dropLast_1, append_2, last_1 Defined Pair Symbols: LAST_1, DROPLAST_1, APPEND_2, IF_4, REV_2 Compound Symbols: c3_1, c6_1, c8_1, c14_1, c11_2, c12_3, c12_2, c12_1, c10_1, c11_1 ---------------------------------------- (135) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (136) Obligation: Complexity Dependency Tuples Problem Rules: isEmpty(nil) -> true isEmpty(cons(z0, z1)) -> false dropLast(nil) -> nil dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) append(nil, z0) -> z0 append(cons(z0, z1), z2) -> cons(z0, append(z1, z2)) last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, nil) -> c12(IF(isEmpty(x0), dropLast(x0), last(x0), nil), LAST(x0)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(x0, nil) -> c10(IF(isEmpty(x0), dropLast(x0), last(x0), nil)) REV(x0, cons(z0, z1)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) S tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, nil) -> c12(IF(isEmpty(x0), dropLast(x0), last(x0), nil), LAST(x0)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(x0, nil) -> c10(IF(isEmpty(x0), dropLast(x0), last(x0), nil)) REV(x0, cons(z0, z1)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) K tuples: REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) Defined Rule Symbols: isEmpty_1, dropLast_1, append_2, last_1 Defined Pair Symbols: LAST_1, DROPLAST_1, APPEND_2, IF_4, REV_2 Compound Symbols: c3_1, c6_1, c8_1, c14_1, c11_2, c12_3, c12_2, c12_1, c10_1, c11_1 ---------------------------------------- (137) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace REV(cons(z0, cons(z1, z2)), x1) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1), APPEND(x1, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) by REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), APPEND(nil, last(cons(x0, cons(x1, x2)))), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), cons(z0, z1)) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), cons(z0, append(z1, last(cons(x0, cons(x1, x2))))), cons(z0, z1)), APPEND(cons(z0, z1), last(cons(x0, cons(x1, x2)))), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) ---------------------------------------- (138) Obligation: Complexity Dependency Tuples Problem Rules: isEmpty(nil) -> true isEmpty(cons(z0, z1)) -> false dropLast(nil) -> nil dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) append(nil, z0) -> z0 append(cons(z0, z1), z2) -> cons(z0, append(z1, z2)) last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, nil) -> c12(IF(isEmpty(x0), dropLast(x0), last(x0), nil), LAST(x0)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(x0, nil) -> c10(IF(isEmpty(x0), dropLast(x0), last(x0), nil)) REV(x0, cons(z0, z1)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), APPEND(nil, last(cons(x0, cons(x1, x2)))), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) S tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, nil) -> c12(IF(isEmpty(x0), dropLast(x0), last(x0), nil), LAST(x0)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(x0, nil) -> c10(IF(isEmpty(x0), dropLast(x0), last(x0), nil)) REV(x0, cons(z0, z1)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), APPEND(nil, last(cons(x0, cons(x1, x2)))), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) K tuples: REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) Defined Rule Symbols: isEmpty_1, dropLast_1, append_2, last_1 Defined Pair Symbols: LAST_1, DROPLAST_1, APPEND_2, IF_4, REV_2 Compound Symbols: c3_1, c6_1, c8_1, c14_1, c11_2, c12_3, c12_2, c12_1, c10_1, c11_1 ---------------------------------------- (139) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (140) Obligation: Complexity Dependency Tuples Problem Rules: isEmpty(nil) -> true isEmpty(cons(z0, z1)) -> false dropLast(nil) -> nil dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) append(nil, z0) -> z0 append(cons(z0, z1), z2) -> cons(z0, append(z1, z2)) last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, nil) -> c12(IF(isEmpty(x0), dropLast(x0), last(x0), nil), LAST(x0)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(x0, nil) -> c10(IF(isEmpty(x0), dropLast(x0), last(x0), nil)) REV(x0, cons(z0, z1)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) S tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, nil) -> c12(IF(isEmpty(x0), dropLast(x0), last(x0), nil), LAST(x0)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(x0, nil) -> c10(IF(isEmpty(x0), dropLast(x0), last(x0), nil)) REV(x0, cons(z0, z1)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) K tuples: REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) Defined Rule Symbols: isEmpty_1, dropLast_1, append_2, last_1 Defined Pair Symbols: LAST_1, DROPLAST_1, APPEND_2, IF_4, REV_2 Compound Symbols: c3_1, c6_1, c8_1, c14_1, c11_2, c12_3, c12_2, c12_1, c10_1, c11_1 ---------------------------------------- (141) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) We considered the (Usable) Rules: dropLast(nil) -> nil isEmpty(nil) -> true dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) isEmpty(cons(z0, z1)) -> false And the Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, nil) -> c12(IF(isEmpty(x0), dropLast(x0), last(x0), nil), LAST(x0)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(x0, nil) -> c10(IF(isEmpty(x0), dropLast(x0), last(x0), nil)) REV(x0, cons(z0, z1)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) The order we found is given by the following interpretation: Polynomial interpretation : POL(APPEND(x_1, x_2)) = 0 POL(DROPLAST(x_1)) = 0 POL(IF(x_1, x_2, x_3, x_4)) = [1] + x_1 + x_2 POL(LAST(x_1)) = 0 POL(REV(x_1, x_2)) = [1] + x_1 POL(append(x_1, x_2)) = [1] POL(c10(x_1)) = x_1 POL(c11(x_1)) = x_1 POL(c11(x_1, x_2)) = x_1 + x_2 POL(c12(x_1)) = x_1 POL(c12(x_1, x_2)) = x_1 + x_2 POL(c12(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c14(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(cons(x_1, x_2)) = [1] + x_2 POL(dropLast(x_1)) = x_1 POL(false) = 0 POL(isEmpty(x_1)) = 0 POL(last(x_1)) = 0 POL(nil) = [1] POL(true) = 0 ---------------------------------------- (142) Obligation: Complexity Dependency Tuples Problem Rules: isEmpty(nil) -> true isEmpty(cons(z0, z1)) -> false dropLast(nil) -> nil dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) append(nil, z0) -> z0 append(cons(z0, z1), z2) -> cons(z0, append(z1, z2)) last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, nil) -> c12(IF(isEmpty(x0), dropLast(x0), last(x0), nil), LAST(x0)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(x0, nil) -> c10(IF(isEmpty(x0), dropLast(x0), last(x0), nil)) REV(x0, cons(z0, z1)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) S tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, nil) -> c12(IF(isEmpty(x0), dropLast(x0), last(x0), nil), LAST(x0)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(x0, nil) -> c10(IF(isEmpty(x0), dropLast(x0), last(x0), nil)) REV(x0, cons(z0, z1)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) K tuples: REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) Defined Rule Symbols: isEmpty_1, dropLast_1, append_2, last_1 Defined Pair Symbols: LAST_1, DROPLAST_1, APPEND_2, IF_4, REV_2 Compound Symbols: c3_1, c6_1, c8_1, c14_1, c11_2, c12_3, c12_2, c12_1, c10_1, c11_1 ---------------------------------------- (143) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace REV(x0, nil) -> c12(IF(isEmpty(x0), dropLast(x0), last(x0), nil), LAST(x0)) by REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil), LAST(cons(z0, nil))) REV(cons(z0, cons(z1, z2)), nil) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), LAST(cons(z0, cons(z1, z2)))) REV(nil, nil) -> c12(IF(isEmpty(nil), nil, last(nil), nil), LAST(nil)) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil), LAST(cons(z0, nil))) REV(cons(z0, cons(z1, z2)), nil) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), LAST(cons(z0, cons(z1, z2)))) REV(nil, nil) -> c12(IF(true, dropLast(nil), last(nil), nil), LAST(nil)) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) ---------------------------------------- (144) Obligation: Complexity Dependency Tuples Problem Rules: isEmpty(nil) -> true isEmpty(cons(z0, z1)) -> false dropLast(nil) -> nil dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) append(nil, z0) -> z0 append(cons(z0, z1), z2) -> cons(z0, append(z1, z2)) last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(x0, nil) -> c10(IF(isEmpty(x0), dropLast(x0), last(x0), nil)) REV(x0, cons(z0, z1)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil), LAST(cons(z0, nil))) REV(nil, nil) -> c12(IF(isEmpty(nil), nil, last(nil), nil), LAST(nil)) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil), LAST(cons(z0, nil))) REV(nil, nil) -> c12(IF(true, dropLast(nil), last(nil), nil), LAST(nil)) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) S tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(x0, nil) -> c10(IF(isEmpty(x0), dropLast(x0), last(x0), nil)) REV(x0, cons(z0, z1)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil), LAST(cons(z0, nil))) REV(nil, nil) -> c12(IF(isEmpty(nil), nil, last(nil), nil), LAST(nil)) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil), LAST(cons(z0, nil))) REV(nil, nil) -> c12(IF(true, dropLast(nil), last(nil), nil), LAST(nil)) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) K tuples: REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) Defined Rule Symbols: isEmpty_1, dropLast_1, append_2, last_1 Defined Pair Symbols: LAST_1, DROPLAST_1, APPEND_2, IF_4, REV_2 Compound Symbols: c3_1, c6_1, c8_1, c14_1, c11_2, c12_3, c12_2, c12_1, c10_1, c11_1 ---------------------------------------- (145) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: REV(nil, nil) -> c12(IF(true, dropLast(nil), last(nil), nil), LAST(nil)) ---------------------------------------- (146) Obligation: Complexity Dependency Tuples Problem Rules: isEmpty(nil) -> true isEmpty(cons(z0, z1)) -> false dropLast(nil) -> nil dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) append(nil, z0) -> z0 append(cons(z0, z1), z2) -> cons(z0, append(z1, z2)) last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(x0, nil) -> c10(IF(isEmpty(x0), dropLast(x0), last(x0), nil)) REV(x0, cons(z0, z1)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil), LAST(cons(z0, nil))) REV(nil, nil) -> c12(IF(isEmpty(nil), nil, last(nil), nil), LAST(nil)) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil), LAST(cons(z0, nil))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) S tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(x0, nil) -> c10(IF(isEmpty(x0), dropLast(x0), last(x0), nil)) REV(x0, cons(z0, z1)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil), LAST(cons(z0, nil))) REV(nil, nil) -> c12(IF(isEmpty(nil), nil, last(nil), nil), LAST(nil)) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil), LAST(cons(z0, nil))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) K tuples: REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) Defined Rule Symbols: isEmpty_1, dropLast_1, append_2, last_1 Defined Pair Symbols: LAST_1, DROPLAST_1, APPEND_2, IF_4, REV_2 Compound Symbols: c3_1, c6_1, c8_1, c14_1, c11_2, c12_3, c12_2, c12_1, c10_1, c11_1 ---------------------------------------- (147) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (148) Obligation: Complexity Dependency Tuples Problem Rules: isEmpty(nil) -> true isEmpty(cons(z0, z1)) -> false dropLast(nil) -> nil dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) append(nil, z0) -> z0 append(cons(z0, z1), z2) -> cons(z0, append(z1, z2)) last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(x0, nil) -> c10(IF(isEmpty(x0), dropLast(x0), last(x0), nil)) REV(x0, cons(z0, z1)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c12(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) S tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(x0, nil) -> c10(IF(isEmpty(x0), dropLast(x0), last(x0), nil)) REV(x0, cons(z0, z1)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c12(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) K tuples: REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) Defined Rule Symbols: isEmpty_1, dropLast_1, append_2, last_1 Defined Pair Symbols: LAST_1, DROPLAST_1, APPEND_2, IF_4, REV_2 Compound Symbols: c3_1, c6_1, c8_1, c14_1, c11_2, c12_3, c12_2, c12_1, c10_1, c11_1 ---------------------------------------- (149) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) We considered the (Usable) Rules: dropLast(nil) -> nil isEmpty(nil) -> true dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) isEmpty(cons(z0, z1)) -> false And the Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(x0, nil) -> c10(IF(isEmpty(x0), dropLast(x0), last(x0), nil)) REV(x0, cons(z0, z1)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c12(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) The order we found is given by the following interpretation: Polynomial interpretation : POL(APPEND(x_1, x_2)) = 0 POL(DROPLAST(x_1)) = 0 POL(IF(x_1, x_2, x_3, x_4)) = [1] + x_1 + x_2 POL(LAST(x_1)) = 0 POL(REV(x_1, x_2)) = [1] + x_1 POL(append(x_1, x_2)) = [1] POL(c10(x_1)) = x_1 POL(c11(x_1)) = x_1 POL(c11(x_1, x_2)) = x_1 + x_2 POL(c12(x_1)) = x_1 POL(c12(x_1, x_2)) = x_1 + x_2 POL(c12(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c14(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(cons(x_1, x_2)) = [1] POL(dropLast(x_1)) = x_1 POL(false) = 0 POL(isEmpty(x_1)) = 0 POL(last(x_1)) = 0 POL(nil) = 0 POL(true) = 0 ---------------------------------------- (150) Obligation: Complexity Dependency Tuples Problem Rules: isEmpty(nil) -> true isEmpty(cons(z0, z1)) -> false dropLast(nil) -> nil dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) append(nil, z0) -> z0 append(cons(z0, z1), z2) -> cons(z0, append(z1, z2)) last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(x0, nil) -> c10(IF(isEmpty(x0), dropLast(x0), last(x0), nil)) REV(x0, cons(z0, z1)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c12(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) S tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(x0, nil) -> c10(IF(isEmpty(x0), dropLast(x0), last(x0), nil)) REV(x0, cons(z0, z1)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c12(IF(isEmpty(nil), nil, last(nil), nil)) K tuples: REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) Defined Rule Symbols: isEmpty_1, dropLast_1, append_2, last_1 Defined Pair Symbols: LAST_1, DROPLAST_1, APPEND_2, IF_4, REV_2 Compound Symbols: c3_1, c6_1, c8_1, c14_1, c11_2, c12_3, c12_2, c12_1, c10_1, c11_1 ---------------------------------------- (151) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. REV(nil, nil) -> c12(IF(isEmpty(nil), nil, last(nil), nil)) We considered the (Usable) Rules: dropLast(nil) -> nil isEmpty(nil) -> true dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) isEmpty(cons(z0, z1)) -> false And the Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(x0, nil) -> c10(IF(isEmpty(x0), dropLast(x0), last(x0), nil)) REV(x0, cons(z0, z1)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c12(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) The order we found is given by the following interpretation: Polynomial interpretation : POL(APPEND(x_1, x_2)) = 0 POL(DROPLAST(x_1)) = 0 POL(IF(x_1, x_2, x_3, x_4)) = x_1 + x_2 POL(LAST(x_1)) = 0 POL(REV(x_1, x_2)) = [1] + x_1 POL(append(x_1, x_2)) = [1] POL(c10(x_1)) = x_1 POL(c11(x_1)) = x_1 POL(c11(x_1, x_2)) = x_1 + x_2 POL(c12(x_1)) = x_1 POL(c12(x_1, x_2)) = x_1 + x_2 POL(c12(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c14(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(cons(x_1, x_2)) = [1] POL(dropLast(x_1)) = [1] POL(false) = [1] POL(isEmpty(x_1)) = x_1 POL(last(x_1)) = 0 POL(nil) = 0 POL(true) = 0 ---------------------------------------- (152) Obligation: Complexity Dependency Tuples Problem Rules: isEmpty(nil) -> true isEmpty(cons(z0, z1)) -> false dropLast(nil) -> nil dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) append(nil, z0) -> z0 append(cons(z0, z1), z2) -> cons(z0, append(z1, z2)) last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(x0, nil) -> c10(IF(isEmpty(x0), dropLast(x0), last(x0), nil)) REV(x0, cons(z0, z1)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c12(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) S tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(x0, nil) -> c10(IF(isEmpty(x0), dropLast(x0), last(x0), nil)) REV(x0, cons(z0, z1)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) K tuples: REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(nil, nil) -> c12(IF(isEmpty(nil), nil, last(nil), nil)) Defined Rule Symbols: isEmpty_1, dropLast_1, append_2, last_1 Defined Pair Symbols: LAST_1, DROPLAST_1, APPEND_2, IF_4, REV_2 Compound Symbols: c3_1, c6_1, c8_1, c14_1, c11_2, c12_3, c12_2, c12_1, c10_1, c11_1 ---------------------------------------- (153) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) by REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil), APPEND(nil, last(cons(z0, nil)))) REV(cons(z2, nil), cons(z0, z1)) -> c12(IF(isEmpty(cons(z2, nil)), dropLast(cons(z2, nil)), cons(z0, append(z1, z2)), cons(z0, z1)), APPEND(cons(z0, z1), last(cons(z2, nil)))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) ---------------------------------------- (154) Obligation: Complexity Dependency Tuples Problem Rules: isEmpty(nil) -> true isEmpty(cons(z0, z1)) -> false dropLast(nil) -> nil dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) append(nil, z0) -> z0 append(cons(z0, z1), z2) -> cons(z0, append(z1, z2)) last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(nil, x1) -> c12(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(x0, nil) -> c10(IF(isEmpty(x0), dropLast(x0), last(x0), nil)) REV(x0, cons(z0, z1)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c12(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil), APPEND(nil, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) S tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, nil) -> c10(IF(isEmpty(x0), dropLast(x0), last(x0), nil)) REV(x0, cons(z0, z1)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil), APPEND(nil, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) K tuples: REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(nil, nil) -> c12(IF(isEmpty(nil), nil, last(nil), nil)) Defined Rule Symbols: isEmpty_1, dropLast_1, append_2, last_1 Defined Pair Symbols: LAST_1, DROPLAST_1, APPEND_2, IF_4, REV_2 Compound Symbols: c3_1, c6_1, c8_1, c14_1, c11_2, c12_3, c12_2, c12_1, c10_1, c11_1 ---------------------------------------- (155) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (156) Obligation: Complexity Dependency Tuples Problem Rules: isEmpty(nil) -> true isEmpty(cons(z0, z1)) -> false dropLast(nil) -> nil dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) append(nil, z0) -> z0 append(cons(z0, z1), z2) -> cons(z0, append(z1, z2)) last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(nil, x1) -> c12(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(x0, nil) -> c10(IF(isEmpty(x0), dropLast(x0), last(x0), nil)) REV(x0, cons(z0, z1)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c12(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) S tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, nil) -> c10(IF(isEmpty(x0), dropLast(x0), last(x0), nil)) REV(x0, cons(z0, z1)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) K tuples: REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(nil, nil) -> c12(IF(isEmpty(nil), nil, last(nil), nil)) Defined Rule Symbols: isEmpty_1, dropLast_1, append_2, last_1 Defined Pair Symbols: LAST_1, DROPLAST_1, APPEND_2, IF_4, REV_2 Compound Symbols: c3_1, c6_1, c8_1, c14_1, c11_2, c12_3, c12_2, c12_1, c10_1, c11_1 ---------------------------------------- (157) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) We considered the (Usable) Rules: dropLast(nil) -> nil isEmpty(nil) -> true dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) isEmpty(cons(z0, z1)) -> false And the Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(nil, x1) -> c12(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(x0, nil) -> c10(IF(isEmpty(x0), dropLast(x0), last(x0), nil)) REV(x0, cons(z0, z1)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c12(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) The order we found is given by the following interpretation: Polynomial interpretation : POL(APPEND(x_1, x_2)) = 0 POL(DROPLAST(x_1)) = 0 POL(IF(x_1, x_2, x_3, x_4)) = x_1 + x_2 POL(LAST(x_1)) = 0 POL(REV(x_1, x_2)) = [1] + x_1 POL(append(x_1, x_2)) = [1] POL(c10(x_1)) = x_1 POL(c11(x_1)) = x_1 POL(c11(x_1, x_2)) = x_1 + x_2 POL(c12(x_1)) = x_1 POL(c12(x_1, x_2)) = x_1 + x_2 POL(c12(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c14(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(cons(x_1, x_2)) = [1] POL(dropLast(x_1)) = [1] POL(false) = [1] POL(isEmpty(x_1)) = x_1 POL(last(x_1)) = 0 POL(nil) = 0 POL(true) = 0 ---------------------------------------- (158) Obligation: Complexity Dependency Tuples Problem Rules: isEmpty(nil) -> true isEmpty(cons(z0, z1)) -> false dropLast(nil) -> nil dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) append(nil, z0) -> z0 append(cons(z0, z1), z2) -> cons(z0, append(z1, z2)) last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(nil, x1) -> c12(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(x0, nil) -> c10(IF(isEmpty(x0), dropLast(x0), last(x0), nil)) REV(x0, cons(z0, z1)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c12(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) S tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, nil) -> c10(IF(isEmpty(x0), dropLast(x0), last(x0), nil)) REV(x0, cons(z0, z1)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, nil), x1) -> c12(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) K tuples: REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(nil, nil) -> c12(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) Defined Rule Symbols: isEmpty_1, dropLast_1, append_2, last_1 Defined Pair Symbols: LAST_1, DROPLAST_1, APPEND_2, IF_4, REV_2 Compound Symbols: c3_1, c6_1, c8_1, c14_1, c11_2, c12_3, c12_2, c12_1, c10_1, c11_1 ---------------------------------------- (159) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace REV(nil, x1) -> c12(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), APPEND(x1, last(nil))) by REV(nil, nil) -> c12(IF(isEmpty(nil), nil, last(nil), nil), APPEND(nil, last(nil))) REV(nil, cons(z0, z1)) -> c12(IF(isEmpty(nil), nil, cons(z0, append(z1, last(nil))), cons(z0, z1)), APPEND(cons(z0, z1), last(nil))) REV(nil, x0) -> c12(IF(true, nil, append(x0, last(nil)), x0), APPEND(x0, last(nil))) ---------------------------------------- (160) Obligation: Complexity Dependency Tuples Problem Rules: isEmpty(nil) -> true isEmpty(cons(z0, z1)) -> false dropLast(nil) -> nil dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) append(nil, z0) -> z0 append(cons(z0, z1), z2) -> cons(z0, append(z1, z2)) last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(x0, nil) -> c10(IF(isEmpty(x0), dropLast(x0), last(x0), nil)) REV(x0, cons(z0, z1)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c12(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, nil) -> c12(IF(isEmpty(nil), nil, last(nil), nil), APPEND(nil, last(nil))) REV(nil, x0) -> c12(IF(true, nil, append(x0, last(nil)), x0), APPEND(x0, last(nil))) S tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, nil) -> c10(IF(isEmpty(x0), dropLast(x0), last(x0), nil)) REV(x0, cons(z0, z1)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, nil), x1) -> c12(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) K tuples: REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(nil, nil) -> c12(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) Defined Rule Symbols: isEmpty_1, dropLast_1, append_2, last_1 Defined Pair Symbols: LAST_1, DROPLAST_1, APPEND_2, IF_4, REV_2 Compound Symbols: c3_1, c6_1, c8_1, c14_1, c11_2, c12_3, c12_2, c12_1, c10_1, c11_1 ---------------------------------------- (161) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (162) Obligation: Complexity Dependency Tuples Problem Rules: isEmpty(nil) -> true isEmpty(cons(z0, z1)) -> false dropLast(nil) -> nil dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) append(nil, z0) -> z0 append(cons(z0, z1), z2) -> cons(z0, append(z1, z2)) last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(x0, nil) -> c10(IF(isEmpty(x0), dropLast(x0), last(x0), nil)) REV(x0, cons(z0, z1)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c12(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) S tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, nil) -> c10(IF(isEmpty(x0), dropLast(x0), last(x0), nil)) REV(x0, cons(z0, z1)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, nil), x1) -> c12(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) K tuples: REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(nil, nil) -> c12(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) Defined Rule Symbols: isEmpty_1, dropLast_1, append_2, last_1 Defined Pair Symbols: LAST_1, DROPLAST_1, APPEND_2, IF_4, REV_2 Compound Symbols: c3_1, c6_1, c8_1, c14_1, c11_2, c12_3, c12_2, c12_1, c10_1, c11_1 ---------------------------------------- (163) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) by REV(cons(x0, nil), nil) -> c12(IF(isEmpty(cons(x0, nil)), nil, last(cons(x0, nil)), nil), APPEND(nil, last(cons(x0, nil)))) REV(cons(x0, nil), cons(z0, z1)) -> c12(IF(isEmpty(cons(x0, nil)), nil, cons(z0, append(z1, last(cons(x0, nil)))), cons(z0, z1)), APPEND(cons(z0, z1), last(cons(x0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(false, nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) ---------------------------------------- (164) Obligation: Complexity Dependency Tuples Problem Rules: isEmpty(nil) -> true isEmpty(cons(z0, z1)) -> false dropLast(nil) -> nil dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) append(nil, z0) -> z0 append(cons(z0, z1), z2) -> cons(z0, append(z1, z2)) last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(x0, nil) -> c10(IF(isEmpty(x0), dropLast(x0), last(x0), nil)) REV(x0, cons(z0, z1)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c12(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(x0, nil), nil) -> c12(IF(isEmpty(cons(x0, nil)), nil, last(cons(x0, nil)), nil), APPEND(nil, last(cons(x0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(false, nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) S tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, nil) -> c10(IF(isEmpty(x0), dropLast(x0), last(x0), nil)) REV(x0, cons(z0, z1)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, nil), x1) -> c12(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) K tuples: REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(nil, nil) -> c12(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) Defined Rule Symbols: isEmpty_1, dropLast_1, append_2, last_1 Defined Pair Symbols: LAST_1, DROPLAST_1, APPEND_2, IF_4, REV_2 Compound Symbols: c3_1, c6_1, c8_1, c14_1, c11_2, c12_3, c12_1, c10_1, c11_1, c12_2 ---------------------------------------- (165) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (166) Obligation: Complexity Dependency Tuples Problem Rules: isEmpty(nil) -> true isEmpty(cons(z0, z1)) -> false dropLast(nil) -> nil dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) append(nil, z0) -> z0 append(cons(z0, z1), z2) -> cons(z0, append(z1, z2)) last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(x0, nil) -> c10(IF(isEmpty(x0), dropLast(x0), last(x0), nil)) REV(x0, cons(z0, z1)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c12(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(false, nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) S tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, nil) -> c10(IF(isEmpty(x0), dropLast(x0), last(x0), nil)) REV(x0, cons(z0, z1)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, nil), x1) -> c12(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) K tuples: REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(nil, nil) -> c12(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) Defined Rule Symbols: isEmpty_1, dropLast_1, append_2, last_1 Defined Pair Symbols: LAST_1, DROPLAST_1, APPEND_2, IF_4, REV_2 Compound Symbols: c3_1, c6_1, c8_1, c14_1, c11_2, c12_3, c12_1, c10_1, c11_1, c12_2 ---------------------------------------- (167) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace REV(x0, nil) -> c10(IF(isEmpty(x0), dropLast(x0), last(x0), nil)) by REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil)) REV(nil, nil) -> c10(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil)) REV(nil, nil) -> c10(IF(true, dropLast(nil), last(nil), nil)) REV(cons(z0, z1), nil) -> c10(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil)) ---------------------------------------- (168) Obligation: Complexity Dependency Tuples Problem Rules: isEmpty(nil) -> true isEmpty(cons(z0, z1)) -> false dropLast(nil) -> nil dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) append(nil, z0) -> z0 append(cons(z0, z1), z2) -> cons(z0, append(z1, z2)) last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(x0, cons(z0, z1)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c12(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(false, nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil)) REV(nil, nil) -> c10(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil)) REV(nil, nil) -> c10(IF(true, dropLast(nil), last(nil), nil)) REV(cons(z0, z1), nil) -> c10(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil)) S tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, cons(z0, z1)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, nil), x1) -> c12(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil)) REV(nil, nil) -> c10(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil)) REV(nil, nil) -> c10(IF(true, dropLast(nil), last(nil), nil)) REV(cons(z0, z1), nil) -> c10(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil)) K tuples: REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, x1) -> c12(IF(isEmpty(nil), nil, append(x1, last(nil)), x1), APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(nil, nil) -> c12(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) Defined Rule Symbols: isEmpty_1, dropLast_1, append_2, last_1 Defined Pair Symbols: LAST_1, DROPLAST_1, APPEND_2, IF_4, REV_2 Compound Symbols: c3_1, c6_1, c8_1, c14_1, c11_2, c12_3, c12_1, c10_1, c11_1, c12_2 ---------------------------------------- (169) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: REV(nil, nil) -> c10(IF(true, dropLast(nil), last(nil), nil)) ---------------------------------------- (170) Obligation: Complexity Dependency Tuples Problem Rules: isEmpty(nil) -> true isEmpty(cons(z0, z1)) -> false dropLast(nil) -> nil dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) append(nil, z0) -> z0 append(cons(z0, z1), z2) -> cons(z0, append(z1, z2)) last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(x0, cons(z0, z1)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c12(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(false, nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil)) REV(nil, nil) -> c10(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil)) REV(cons(z0, z1), nil) -> c10(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil)) S tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, cons(z0, z1)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, nil), x1) -> c12(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil)) REV(nil, nil) -> c10(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil)) REV(cons(z0, z1), nil) -> c10(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil)) K tuples: REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(nil, nil) -> c12(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) Defined Rule Symbols: isEmpty_1, dropLast_1, append_2, last_1 Defined Pair Symbols: LAST_1, DROPLAST_1, APPEND_2, IF_4, REV_2 Compound Symbols: c3_1, c6_1, c8_1, c14_1, c11_2, c12_3, c12_1, c10_1, c11_1, c12_2 ---------------------------------------- (171) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) We considered the (Usable) Rules: dropLast(nil) -> nil isEmpty(nil) -> true dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) isEmpty(cons(z0, z1)) -> false And the Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(x0, cons(z0, z1)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c12(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(false, nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil)) REV(nil, nil) -> c10(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil)) REV(cons(z0, z1), nil) -> c10(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil)) The order we found is given by the following interpretation: Polynomial interpretation : POL(APPEND(x_1, x_2)) = 0 POL(DROPLAST(x_1)) = 0 POL(IF(x_1, x_2, x_3, x_4)) = x_1 + x_2 POL(LAST(x_1)) = 0 POL(REV(x_1, x_2)) = x_1 POL(append(x_1, x_2)) = [1] POL(c10(x_1)) = x_1 POL(c11(x_1)) = x_1 POL(c11(x_1, x_2)) = x_1 + x_2 POL(c12(x_1)) = x_1 POL(c12(x_1, x_2)) = x_1 + x_2 POL(c12(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c14(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(cons(x_1, x_2)) = [1] POL(dropLast(x_1)) = x_1 POL(false) = 0 POL(isEmpty(x_1)) = 0 POL(last(x_1)) = 0 POL(nil) = 0 POL(true) = 0 ---------------------------------------- (172) Obligation: Complexity Dependency Tuples Problem Rules: isEmpty(nil) -> true isEmpty(cons(z0, z1)) -> false dropLast(nil) -> nil dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) append(nil, z0) -> z0 append(cons(z0, z1), z2) -> cons(z0, append(z1, z2)) last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(x0, cons(z0, z1)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c12(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(false, nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil)) REV(nil, nil) -> c10(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil)) REV(cons(z0, z1), nil) -> c10(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil)) S tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, cons(z0, z1)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, nil), x1) -> c12(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil)) REV(nil, nil) -> c10(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil)) REV(cons(z0, z1), nil) -> c10(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil)) K tuples: REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(nil, nil) -> c12(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) Defined Rule Symbols: isEmpty_1, dropLast_1, append_2, last_1 Defined Pair Symbols: LAST_1, DROPLAST_1, APPEND_2, IF_4, REV_2 Compound Symbols: c3_1, c6_1, c8_1, c14_1, c11_2, c12_3, c12_1, c10_1, c11_1, c12_2 ---------------------------------------- (173) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. REV(nil, nil) -> c10(IF(isEmpty(nil), nil, last(nil), nil)) We considered the (Usable) Rules: dropLast(nil) -> nil isEmpty(nil) -> true dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) isEmpty(cons(z0, z1)) -> false And the Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(x0, cons(z0, z1)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c12(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(false, nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil)) REV(nil, nil) -> c10(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil)) REV(cons(z0, z1), nil) -> c10(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil)) The order we found is given by the following interpretation: Polynomial interpretation : POL(APPEND(x_1, x_2)) = 0 POL(DROPLAST(x_1)) = 0 POL(IF(x_1, x_2, x_3, x_4)) = x_1 + x_2 POL(LAST(x_1)) = 0 POL(REV(x_1, x_2)) = [1] + x_1 POL(append(x_1, x_2)) = [1] POL(c10(x_1)) = x_1 POL(c11(x_1)) = x_1 POL(c11(x_1, x_2)) = x_1 + x_2 POL(c12(x_1)) = x_1 POL(c12(x_1, x_2)) = x_1 + x_2 POL(c12(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c14(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(cons(x_1, x_2)) = [1] POL(dropLast(x_1)) = [1] POL(false) = [1] POL(isEmpty(x_1)) = x_1 POL(last(x_1)) = 0 POL(nil) = 0 POL(true) = 0 ---------------------------------------- (174) Obligation: Complexity Dependency Tuples Problem Rules: isEmpty(nil) -> true isEmpty(cons(z0, z1)) -> false dropLast(nil) -> nil dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) append(nil, z0) -> z0 append(cons(z0, z1), z2) -> cons(z0, append(z1, z2)) last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(x0, cons(z0, z1)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c12(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(false, nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil)) REV(nil, nil) -> c10(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil)) REV(cons(z0, z1), nil) -> c10(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil)) S tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, cons(z0, z1)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, nil), x1) -> c12(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil)) REV(cons(z0, z1), nil) -> c10(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil)) K tuples: REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(nil, nil) -> c12(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(nil, nil) -> c10(IF(isEmpty(nil), nil, last(nil), nil)) Defined Rule Symbols: isEmpty_1, dropLast_1, append_2, last_1 Defined Pair Symbols: LAST_1, DROPLAST_1, APPEND_2, IF_4, REV_2 Compound Symbols: c3_1, c6_1, c8_1, c14_1, c11_2, c12_3, c12_1, c10_1, c11_1, c12_2 ---------------------------------------- (175) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace REV(x0, cons(z0, z1)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(z0, append(z1, last(x0))), cons(z0, z1))) by REV(x0, cons(x1, nil)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil))) REV(x0, cons(x1, cons(z0, z1))) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1)))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c10(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c10(IF(true, dropLast(nil), cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, z1), cons(x1, x2)) -> c10(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2))) ---------------------------------------- (176) Obligation: Complexity Dependency Tuples Problem Rules: isEmpty(nil) -> true isEmpty(cons(z0, z1)) -> false dropLast(nil) -> nil dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) append(nil, z0) -> z0 append(cons(z0, z1), z2) -> cons(z0, append(z1, z2)) last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c12(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(false, nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil)) REV(nil, nil) -> c10(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil)) REV(cons(z0, z1), nil) -> c10(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil)) REV(x0, cons(x1, nil)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil))) REV(x0, cons(x1, cons(z0, z1))) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1)))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c10(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c10(IF(true, dropLast(nil), cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, z1), cons(x1, x2)) -> c10(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2))) S tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, nil), x1) -> c12(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil)) REV(cons(z0, z1), nil) -> c10(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil)) REV(x0, cons(x1, nil)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil))) REV(x0, cons(x1, cons(z0, z1))) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1)))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c10(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c10(IF(true, dropLast(nil), cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, z1), cons(x1, x2)) -> c10(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2))) K tuples: REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(nil, nil) -> c12(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(nil, nil) -> c10(IF(isEmpty(nil), nil, last(nil), nil)) Defined Rule Symbols: isEmpty_1, dropLast_1, append_2, last_1 Defined Pair Symbols: LAST_1, DROPLAST_1, APPEND_2, IF_4, REV_2 Compound Symbols: c3_1, c6_1, c8_1, c14_1, c11_2, c12_3, c12_1, c10_1, c11_1, c12_2 ---------------------------------------- (177) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: REV(nil, cons(x1, x2)) -> c10(IF(true, dropLast(nil), cons(x1, append(x2, last(nil))), cons(x1, x2))) ---------------------------------------- (178) Obligation: Complexity Dependency Tuples Problem Rules: isEmpty(nil) -> true isEmpty(cons(z0, z1)) -> false dropLast(nil) -> nil dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) append(nil, z0) -> z0 append(cons(z0, z1), z2) -> cons(z0, append(z1, z2)) last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c12(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(false, nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil)) REV(nil, nil) -> c10(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil)) REV(cons(z0, z1), nil) -> c10(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil)) REV(x0, cons(x1, nil)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil))) REV(x0, cons(x1, cons(z0, z1))) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1)))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c10(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2))) REV(cons(z0, z1), cons(x1, x2)) -> c10(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2))) S tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, nil), x1) -> c12(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil)) REV(cons(z0, z1), nil) -> c10(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil)) REV(x0, cons(x1, nil)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil))) REV(x0, cons(x1, cons(z0, z1))) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1)))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c10(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2))) REV(cons(z0, z1), cons(x1, x2)) -> c10(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2))) K tuples: REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(nil, nil) -> c12(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(nil, nil) -> c10(IF(isEmpty(nil), nil, last(nil), nil)) Defined Rule Symbols: isEmpty_1, dropLast_1, append_2, last_1 Defined Pair Symbols: LAST_1, DROPLAST_1, APPEND_2, IF_4, REV_2 Compound Symbols: c3_1, c6_1, c8_1, c14_1, c11_2, c12_3, c12_1, c10_1, c11_1, c12_2 ---------------------------------------- (179) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) We considered the (Usable) Rules: dropLast(nil) -> nil isEmpty(nil) -> true dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) isEmpty(cons(z0, z1)) -> false And the Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c12(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(false, nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil)) REV(nil, nil) -> c10(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil)) REV(cons(z0, z1), nil) -> c10(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil)) REV(x0, cons(x1, nil)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil))) REV(x0, cons(x1, cons(z0, z1))) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1)))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c10(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2))) REV(cons(z0, z1), cons(x1, x2)) -> c10(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2))) The order we found is given by the following interpretation: Polynomial interpretation : POL(APPEND(x_1, x_2)) = 0 POL(DROPLAST(x_1)) = 0 POL(IF(x_1, x_2, x_3, x_4)) = x_1 + x_2 POL(LAST(x_1)) = 0 POL(REV(x_1, x_2)) = x_1 POL(append(x_1, x_2)) = [1] POL(c10(x_1)) = x_1 POL(c11(x_1)) = x_1 POL(c11(x_1, x_2)) = x_1 + x_2 POL(c12(x_1)) = x_1 POL(c12(x_1, x_2)) = x_1 + x_2 POL(c12(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c14(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(cons(x_1, x_2)) = [1] POL(dropLast(x_1)) = x_1 POL(false) = 0 POL(isEmpty(x_1)) = 0 POL(last(x_1)) = 0 POL(nil) = 0 POL(true) = 0 ---------------------------------------- (180) Obligation: Complexity Dependency Tuples Problem Rules: isEmpty(nil) -> true isEmpty(cons(z0, z1)) -> false dropLast(nil) -> nil dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) append(nil, z0) -> z0 append(cons(z0, z1), z2) -> cons(z0, append(z1, z2)) last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c12(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(false, nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil)) REV(nil, nil) -> c10(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil)) REV(cons(z0, z1), nil) -> c10(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil)) REV(x0, cons(x1, nil)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil))) REV(x0, cons(x1, cons(z0, z1))) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1)))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c10(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2))) REV(cons(z0, z1), cons(x1, x2)) -> c10(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2))) S tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, nil), x1) -> c12(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil)) REV(cons(z0, z1), nil) -> c10(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil)) REV(x0, cons(x1, nil)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil))) REV(x0, cons(x1, cons(z0, z1))) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1)))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c10(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2))) REV(cons(z0, z1), cons(x1, x2)) -> c10(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2))) K tuples: REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(nil, nil) -> c12(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(nil, nil) -> c10(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) Defined Rule Symbols: isEmpty_1, dropLast_1, append_2, last_1 Defined Pair Symbols: LAST_1, DROPLAST_1, APPEND_2, IF_4, REV_2 Compound Symbols: c3_1, c6_1, c8_1, c14_1, c11_2, c12_3, c12_1, c10_1, c11_1, c12_2 ---------------------------------------- (181) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. REV(nil, cons(x1, x2)) -> c10(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) We considered the (Usable) Rules: dropLast(nil) -> nil isEmpty(nil) -> true dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) isEmpty(cons(z0, z1)) -> false And the Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c12(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(false, nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil)) REV(nil, nil) -> c10(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil)) REV(cons(z0, z1), nil) -> c10(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil)) REV(x0, cons(x1, nil)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil))) REV(x0, cons(x1, cons(z0, z1))) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1)))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c10(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2))) REV(cons(z0, z1), cons(x1, x2)) -> c10(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2))) The order we found is given by the following interpretation: Polynomial interpretation : POL(APPEND(x_1, x_2)) = 0 POL(DROPLAST(x_1)) = 0 POL(IF(x_1, x_2, x_3, x_4)) = x_1 + x_2 POL(LAST(x_1)) = 0 POL(REV(x_1, x_2)) = [1] + x_1 POL(append(x_1, x_2)) = [1] POL(c10(x_1)) = x_1 POL(c11(x_1)) = x_1 POL(c11(x_1, x_2)) = x_1 + x_2 POL(c12(x_1)) = x_1 POL(c12(x_1, x_2)) = x_1 + x_2 POL(c12(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c14(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(cons(x_1, x_2)) = [1] POL(dropLast(x_1)) = [1] POL(false) = [1] POL(isEmpty(x_1)) = x_1 POL(last(x_1)) = 0 POL(nil) = 0 POL(true) = 0 ---------------------------------------- (182) Obligation: Complexity Dependency Tuples Problem Rules: isEmpty(nil) -> true isEmpty(cons(z0, z1)) -> false dropLast(nil) -> nil dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) append(nil, z0) -> z0 append(cons(z0, z1), z2) -> cons(z0, append(z1, z2)) last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c12(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(false, nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil)) REV(nil, nil) -> c10(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil)) REV(cons(z0, z1), nil) -> c10(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil)) REV(x0, cons(x1, nil)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil))) REV(x0, cons(x1, cons(z0, z1))) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1)))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c10(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2))) REV(cons(z0, z1), cons(x1, x2)) -> c10(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2))) S tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, nil), x1) -> c12(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil)) REV(cons(z0, z1), nil) -> c10(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil)) REV(x0, cons(x1, nil)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil))) REV(x0, cons(x1, cons(z0, z1))) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1)))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2))) REV(cons(z0, z1), cons(x1, x2)) -> c10(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2))) K tuples: REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(nil, nil) -> c12(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(nil, nil) -> c10(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c10(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) Defined Rule Symbols: isEmpty_1, dropLast_1, append_2, last_1 Defined Pair Symbols: LAST_1, DROPLAST_1, APPEND_2, IF_4, REV_2 Compound Symbols: c3_1, c6_1, c8_1, c14_1, c11_2, c12_3, c12_1, c10_1, c11_1, c12_2 ---------------------------------------- (183) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), append(x1, z0), x1)) by REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z2, nil), cons(z0, z1)) -> c10(IF(isEmpty(cons(z2, nil)), dropLast(cons(z2, nil)), cons(z0, append(z1, z2)), cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c10(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) ---------------------------------------- (184) Obligation: Complexity Dependency Tuples Problem Rules: isEmpty(nil) -> true isEmpty(cons(z0, z1)) -> false dropLast(nil) -> nil dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) append(nil, z0) -> z0 append(cons(z0, z1), z2) -> cons(z0, append(z1, z2)) last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c12(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(false, nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil)) REV(nil, nil) -> c10(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil)) REV(cons(z0, z1), nil) -> c10(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil)) REV(x0, cons(x1, nil)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil))) REV(x0, cons(x1, cons(z0, z1))) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1)))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c10(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2))) REV(cons(z0, z1), cons(x1, x2)) -> c10(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c10(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) S tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, nil), x1) -> c12(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil)) REV(cons(z0, z1), nil) -> c10(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil)) REV(x0, cons(x1, nil)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil))) REV(x0, cons(x1, cons(z0, z1))) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1)))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2))) REV(cons(z0, z1), cons(x1, x2)) -> c10(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c10(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) K tuples: REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(nil, nil) -> c12(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(nil, nil) -> c10(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c10(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) Defined Rule Symbols: isEmpty_1, dropLast_1, append_2, last_1 Defined Pair Symbols: LAST_1, DROPLAST_1, APPEND_2, IF_4, REV_2 Compound Symbols: c3_1, c6_1, c8_1, c14_1, c11_2, c12_3, c12_1, c10_1, c11_1, c12_2 ---------------------------------------- (185) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) We considered the (Usable) Rules: dropLast(nil) -> nil dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) And the Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c12(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(false, nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil)) REV(nil, nil) -> c10(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil)) REV(cons(z0, z1), nil) -> c10(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil)) REV(x0, cons(x1, nil)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil))) REV(x0, cons(x1, cons(z0, z1))) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1)))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c10(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2))) REV(cons(z0, z1), cons(x1, x2)) -> c10(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c10(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) The order we found is given by the following interpretation: Polynomial interpretation : POL(APPEND(x_1, x_2)) = 0 POL(DROPLAST(x_1)) = 0 POL(IF(x_1, x_2, x_3, x_4)) = [2]x_2 POL(LAST(x_1)) = 0 POL(REV(x_1, x_2)) = [2]x_1 POL(append(x_1, x_2)) = [2]x_1 + [2]x_2 POL(c10(x_1)) = x_1 POL(c11(x_1)) = x_1 POL(c11(x_1, x_2)) = x_1 + x_2 POL(c12(x_1)) = x_1 POL(c12(x_1, x_2)) = x_1 + x_2 POL(c12(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c14(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(cons(x_1, x_2)) = [2] POL(dropLast(x_1)) = x_1 POL(false) = [3] POL(isEmpty(x_1)) = [3] + [2]x_1 POL(last(x_1)) = 0 POL(nil) = 0 POL(true) = [3] ---------------------------------------- (186) Obligation: Complexity Dependency Tuples Problem Rules: isEmpty(nil) -> true isEmpty(cons(z0, z1)) -> false dropLast(nil) -> nil dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) append(nil, z0) -> z0 append(cons(z0, z1), z2) -> cons(z0, append(z1, z2)) last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c12(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(false, nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil)) REV(nil, nil) -> c10(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil)) REV(cons(z0, z1), nil) -> c10(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil)) REV(x0, cons(x1, nil)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil))) REV(x0, cons(x1, cons(z0, z1))) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1)))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c10(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2))) REV(cons(z0, z1), cons(x1, x2)) -> c10(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c10(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) S tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, nil), x1) -> c12(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil)) REV(cons(z0, z1), nil) -> c10(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil)) REV(x0, cons(x1, nil)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil))) REV(x0, cons(x1, cons(z0, z1))) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1)))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2))) REV(cons(z0, z1), cons(x1, x2)) -> c10(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2))) REV(cons(z0, nil), x1) -> c10(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) K tuples: REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(nil, nil) -> c12(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(nil, nil) -> c10(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c10(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) Defined Rule Symbols: isEmpty_1, dropLast_1, append_2, last_1 Defined Pair Symbols: LAST_1, DROPLAST_1, APPEND_2, IF_4, REV_2 Compound Symbols: c3_1, c6_1, c8_1, c14_1, c11_2, c12_3, c12_1, c10_1, c11_1, c12_2 ---------------------------------------- (187) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), append(x1, last(cons(z1, z2))), x1)) by REV(cons(x0, cons(x1, x2)), nil) -> c10(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil)) REV(cons(x0, cons(x1, x2)), cons(z0, z1)) -> c10(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), cons(z0, append(z1, last(cons(x1, x2)))), cons(z0, z1))) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(z1, z2)), x3) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3)) ---------------------------------------- (188) Obligation: Complexity Dependency Tuples Problem Rules: isEmpty(nil) -> true isEmpty(cons(z0, z1)) -> false dropLast(nil) -> nil dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) append(nil, z0) -> z0 append(cons(z0, z1), z2) -> cons(z0, append(z1, z2)) last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c12(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(false, nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil)) REV(nil, nil) -> c10(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil)) REV(cons(z0, z1), nil) -> c10(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil)) REV(x0, cons(x1, nil)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil))) REV(x0, cons(x1, cons(z0, z1))) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1)))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c10(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2))) REV(cons(z0, z1), cons(x1, x2)) -> c10(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c10(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(z1, z2)), x3) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3)) S tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, nil), x1) -> c12(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil)) REV(cons(z0, z1), nil) -> c10(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil)) REV(x0, cons(x1, nil)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil))) REV(x0, cons(x1, cons(z0, z1))) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1)))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2))) REV(cons(z0, z1), cons(x1, x2)) -> c10(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2))) REV(cons(z0, nil), x1) -> c10(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(z1, z2)), x3) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3)) K tuples: REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(nil, nil) -> c12(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(nil, nil) -> c10(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c10(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) Defined Rule Symbols: isEmpty_1, dropLast_1, append_2, last_1 Defined Pair Symbols: LAST_1, DROPLAST_1, APPEND_2, IF_4, REV_2 Compound Symbols: c3_1, c6_1, c8_1, c14_1, c11_2, c12_3, c12_1, c10_1, c11_1, c12_2 ---------------------------------------- (189) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) by REV(nil, nil) -> c10(IF(isEmpty(nil), nil, last(nil), nil)) REV(nil, cons(z0, z1)) -> c10(IF(isEmpty(nil), nil, cons(z0, append(z1, last(nil))), cons(z0, z1))) REV(nil, x0) -> c10(IF(true, nil, append(x0, last(nil)), x0)) ---------------------------------------- (190) Obligation: Complexity Dependency Tuples Problem Rules: isEmpty(nil) -> true isEmpty(cons(z0, z1)) -> false dropLast(nil) -> nil dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) append(nil, z0) -> z0 append(cons(z0, z1), z2) -> cons(z0, append(z1, z2)) last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c12(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(false, nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil)) REV(nil, nil) -> c10(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil)) REV(cons(z0, z1), nil) -> c10(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil)) REV(x0, cons(x1, nil)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil))) REV(x0, cons(x1, cons(z0, z1))) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1)))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c10(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2))) REV(cons(z0, z1), cons(x1, x2)) -> c10(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c10(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(z1, z2)), x3) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3)) REV(nil, x0) -> c10(IF(true, nil, append(x0, last(nil)), x0)) S tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, nil), x1) -> c12(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil)) REV(cons(z0, z1), nil) -> c10(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil)) REV(x0, cons(x1, nil)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil))) REV(x0, cons(x1, cons(z0, z1))) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1)))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2))) REV(cons(z0, z1), cons(x1, x2)) -> c10(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2))) REV(cons(z0, nil), x1) -> c10(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(z1, z2)), x3) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3)) K tuples: REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(nil, x1) -> c10(IF(isEmpty(nil), nil, append(x1, last(nil)), x1)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(nil, nil) -> c12(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(nil, nil) -> c10(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c10(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) Defined Rule Symbols: isEmpty_1, dropLast_1, append_2, last_1 Defined Pair Symbols: LAST_1, DROPLAST_1, APPEND_2, IF_4, REV_2 Compound Symbols: c3_1, c6_1, c8_1, c14_1, c11_2, c12_3, c12_1, c10_1, c11_1, c12_2 ---------------------------------------- (191) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: REV(nil, x0) -> c10(IF(true, nil, append(x0, last(nil)), x0)) ---------------------------------------- (192) Obligation: Complexity Dependency Tuples Problem Rules: isEmpty(nil) -> true isEmpty(cons(z0, z1)) -> false dropLast(nil) -> nil dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) append(nil, z0) -> z0 append(cons(z0, z1), z2) -> cons(z0, append(z1, z2)) last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c12(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(false, nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil)) REV(nil, nil) -> c10(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil)) REV(cons(z0, z1), nil) -> c10(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil)) REV(x0, cons(x1, nil)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil))) REV(x0, cons(x1, cons(z0, z1))) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1)))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c10(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2))) REV(cons(z0, z1), cons(x1, x2)) -> c10(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c10(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(z1, z2)), x3) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3)) S tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, nil), x1) -> c12(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil)) REV(cons(z0, z1), nil) -> c10(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil)) REV(x0, cons(x1, nil)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil))) REV(x0, cons(x1, cons(z0, z1))) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1)))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2))) REV(cons(z0, z1), cons(x1, x2)) -> c10(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2))) REV(cons(z0, nil), x1) -> c10(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(z1, z2)), x3) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3)) K tuples: REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(nil, nil) -> c12(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(nil, nil) -> c10(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c10(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) Defined Rule Symbols: isEmpty_1, dropLast_1, append_2, last_1 Defined Pair Symbols: LAST_1, DROPLAST_1, APPEND_2, IF_4, REV_2 Compound Symbols: c3_1, c6_1, c8_1, c14_1, c11_2, c12_3, c12_1, c10_1, c11_1, c12_2 ---------------------------------------- (193) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) by REV(cons(x0, nil), nil) -> c10(IF(isEmpty(cons(x0, nil)), nil, last(cons(x0, nil)), nil)) REV(cons(x0, nil), cons(z0, z1)) -> c10(IF(isEmpty(cons(x0, nil)), nil, cons(z0, append(z1, last(cons(x0, nil)))), cons(z0, z1))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c10(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) ---------------------------------------- (194) Obligation: Complexity Dependency Tuples Problem Rules: isEmpty(nil) -> true isEmpty(cons(z0, z1)) -> false dropLast(nil) -> nil dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) append(nil, z0) -> z0 append(cons(z0, z1), z2) -> cons(z0, append(z1, z2)) last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c12(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(false, nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil)) REV(nil, nil) -> c10(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil)) REV(cons(z0, z1), nil) -> c10(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil)) REV(x0, cons(x1, nil)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil))) REV(x0, cons(x1, cons(z0, z1))) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1)))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c10(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2))) REV(cons(z0, z1), cons(x1, x2)) -> c10(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c10(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(z1, z2)), x3) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3)) REV(cons(z0, nil), x1) -> c10(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) S tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, nil), x1) -> c12(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil)) REV(cons(z0, z1), nil) -> c10(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil)) REV(x0, cons(x1, nil)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil))) REV(x0, cons(x1, cons(z0, z1))) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1)))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2))) REV(cons(z0, z1), cons(x1, x2)) -> c10(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2))) REV(cons(z0, nil), x1) -> c10(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(z1, z2)), x3) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3)) K tuples: REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(nil, nil) -> c12(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(nil, nil) -> c10(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c10(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) Defined Rule Symbols: isEmpty_1, dropLast_1, append_2, last_1 Defined Pair Symbols: LAST_1, DROPLAST_1, APPEND_2, IF_4, REV_2 Compound Symbols: c3_1, c6_1, c8_1, c14_1, c11_2, c12_3, c12_1, c10_1, c11_1, c12_2 ---------------------------------------- (195) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace REV(cons(z0, cons(z1, z2)), x1) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x1, last(cons(z0, cons(z1, z2)))), x1)) by REV(cons(x0, cons(x1, x2)), nil) -> c10(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil)) REV(cons(x0, cons(x1, x2)), cons(z0, z1)) -> c10(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), cons(z0, append(z1, last(cons(x0, cons(x1, x2))))), cons(z0, z1))) REV(cons(z0, cons(z1, z2)), x3) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3)) ---------------------------------------- (196) Obligation: Complexity Dependency Tuples Problem Rules: isEmpty(nil) -> true isEmpty(cons(z0, z1)) -> false dropLast(nil) -> nil dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) append(nil, z0) -> z0 append(cons(z0, z1), z2) -> cons(z0, append(z1, z2)) last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c12(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(false, nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil)) REV(nil, nil) -> c10(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil)) REV(cons(z0, z1), nil) -> c10(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil)) REV(x0, cons(x1, nil)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil))) REV(x0, cons(x1, cons(z0, z1))) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1)))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c10(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2))) REV(cons(z0, z1), cons(x1, x2)) -> c10(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c10(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(z1, z2)), x3) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3)) REV(cons(z0, nil), x1) -> c10(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3)) S tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, nil), x1) -> c12(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil)) REV(cons(z0, z1), nil) -> c10(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil)) REV(x0, cons(x1, nil)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil))) REV(x0, cons(x1, cons(z0, z1))) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1)))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2))) REV(cons(z0, z1), cons(x1, x2)) -> c10(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2))) REV(cons(z0, nil), x1) -> c10(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(z1, z2)), x3) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3)) K tuples: REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(nil, nil) -> c12(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(nil, nil) -> c10(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c10(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) Defined Rule Symbols: isEmpty_1, dropLast_1, append_2, last_1 Defined Pair Symbols: LAST_1, DROPLAST_1, APPEND_2, IF_4, REV_2 Compound Symbols: c3_1, c6_1, c8_1, c14_1, c11_2, c12_3, c12_1, c10_1, c11_1, c12_2 ---------------------------------------- (197) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3)) We considered the (Usable) Rules: dropLast(nil) -> nil isEmpty(nil) -> true dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) isEmpty(cons(z0, z1)) -> false And the Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c12(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(false, nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil)) REV(nil, nil) -> c10(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil)) REV(cons(z0, z1), nil) -> c10(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil)) REV(x0, cons(x1, nil)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil))) REV(x0, cons(x1, cons(z0, z1))) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1)))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c10(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2))) REV(cons(z0, z1), cons(x1, x2)) -> c10(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c10(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(z1, z2)), x3) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3)) REV(cons(z0, nil), x1) -> c10(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3)) The order we found is given by the following interpretation: Polynomial interpretation : POL(APPEND(x_1, x_2)) = 0 POL(DROPLAST(x_1)) = 0 POL(IF(x_1, x_2, x_3, x_4)) = x_1 + x_2 POL(LAST(x_1)) = 0 POL(REV(x_1, x_2)) = x_1 POL(append(x_1, x_2)) = [1] POL(c10(x_1)) = x_1 POL(c11(x_1)) = x_1 POL(c11(x_1, x_2)) = x_1 + x_2 POL(c12(x_1)) = x_1 POL(c12(x_1, x_2)) = x_1 + x_2 POL(c12(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c14(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(cons(x_1, x_2)) = [1] + x_2 POL(dropLast(x_1)) = x_1 POL(false) = 0 POL(isEmpty(x_1)) = 0 POL(last(x_1)) = 0 POL(nil) = 0 POL(true) = 0 ---------------------------------------- (198) Obligation: Complexity Dependency Tuples Problem Rules: isEmpty(nil) -> true isEmpty(cons(z0, z1)) -> false dropLast(nil) -> nil dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) append(nil, z0) -> z0 append(cons(z0, z1), z2) -> cons(z0, append(z1, z2)) last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c12(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(false, nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil)) REV(nil, nil) -> c10(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil)) REV(cons(z0, z1), nil) -> c10(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil)) REV(x0, cons(x1, nil)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil))) REV(x0, cons(x1, cons(z0, z1))) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1)))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c10(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2))) REV(cons(z0, z1), cons(x1, x2)) -> c10(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c10(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(z1, z2)), x3) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3)) REV(cons(z0, nil), x1) -> c10(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3)) S tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, nil), x1) -> c12(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil)) REV(cons(z0, z1), nil) -> c10(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil)) REV(x0, cons(x1, nil)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil))) REV(x0, cons(x1, cons(z0, z1))) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1)))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2))) REV(cons(z0, z1), cons(x1, x2)) -> c10(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2))) REV(cons(z0, nil), x1) -> c10(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(z1, z2)), x3) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3)) K tuples: REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(nil, nil) -> c12(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(nil, nil) -> c10(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c10(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3)) Defined Rule Symbols: isEmpty_1, dropLast_1, append_2, last_1 Defined Pair Symbols: LAST_1, DROPLAST_1, APPEND_2, IF_4, REV_2 Compound Symbols: c3_1, c6_1, c8_1, c14_1, c11_2, c12_3, c12_1, c10_1, c11_1, c12_2 ---------------------------------------- (199) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) by REV(nil, nil) -> c11(IF(true, nil, last(nil), nil)) ---------------------------------------- (200) Obligation: Complexity Dependency Tuples Problem Rules: isEmpty(nil) -> true isEmpty(cons(z0, z1)) -> false dropLast(nil) -> nil dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) append(nil, z0) -> z0 append(cons(z0, z1), z2) -> cons(z0, append(z1, z2)) last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c12(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(false, nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil)) REV(nil, nil) -> c10(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil)) REV(cons(z0, z1), nil) -> c10(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil)) REV(x0, cons(x1, nil)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil))) REV(x0, cons(x1, cons(z0, z1))) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1)))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c10(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2))) REV(cons(z0, z1), cons(x1, x2)) -> c10(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c10(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(z1, z2)), x3) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3)) REV(cons(z0, nil), x1) -> c10(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3)) REV(nil, nil) -> c11(IF(true, nil, last(nil), nil)) S tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, nil), x1) -> c12(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil)) REV(cons(z0, z1), nil) -> c10(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil)) REV(x0, cons(x1, nil)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil))) REV(x0, cons(x1, cons(z0, z1))) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1)))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2))) REV(cons(z0, z1), cons(x1, x2)) -> c10(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2))) REV(cons(z0, nil), x1) -> c10(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(z1, z2)), x3) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3)) K tuples: REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(nil, nil) -> c12(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(nil, nil) -> c10(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c10(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3)) Defined Rule Symbols: isEmpty_1, dropLast_1, append_2, last_1 Defined Pair Symbols: LAST_1, DROPLAST_1, APPEND_2, IF_4, REV_2 Compound Symbols: c3_1, c6_1, c8_1, c14_1, c11_2, c12_3, c12_1, c10_1, c11_1, c12_2 ---------------------------------------- (201) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: REV(nil, nil) -> c11(IF(true, nil, last(nil), nil)) ---------------------------------------- (202) Obligation: Complexity Dependency Tuples Problem Rules: isEmpty(nil) -> true isEmpty(cons(z0, z1)) -> false dropLast(nil) -> nil dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) append(nil, z0) -> z0 append(cons(z0, z1), z2) -> cons(z0, append(z1, z2)) last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), x1) -> c11(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c12(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(false, nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil)) REV(nil, nil) -> c10(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil)) REV(cons(z0, z1), nil) -> c10(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil)) REV(x0, cons(x1, nil)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil))) REV(x0, cons(x1, cons(z0, z1))) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1)))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c10(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2))) REV(cons(z0, z1), cons(x1, x2)) -> c10(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c10(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(z1, z2)), x3) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3)) REV(cons(z0, nil), x1) -> c10(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3)) S tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, nil), x1) -> c12(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil)) REV(cons(z0, z1), nil) -> c10(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil)) REV(x0, cons(x1, nil)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil))) REV(x0, cons(x1, cons(z0, z1))) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1)))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2))) REV(cons(z0, z1), cons(x1, x2)) -> c10(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2))) REV(cons(z0, nil), x1) -> c10(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(z1, z2)), x3) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3)) K tuples: REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(nil, nil) -> c12(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(nil, nil) -> c10(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c10(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3)) Defined Rule Symbols: isEmpty_1, dropLast_1, append_2, last_1 Defined Pair Symbols: LAST_1, DROPLAST_1, APPEND_2, IF_4, REV_2 Compound Symbols: c3_1, c6_1, c8_1, c14_1, c11_2, c12_3, c12_1, c10_1, c11_1, c12_2 ---------------------------------------- (203) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) by REV(nil, nil) -> c11(IF(true, nil, last(nil), nil)) ---------------------------------------- (204) Obligation: Complexity Dependency Tuples Problem Rules: isEmpty(nil) -> true isEmpty(cons(z0, z1)) -> false dropLast(nil) -> nil dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) append(nil, z0) -> z0 append(cons(z0, z1), z2) -> cons(z0, append(z1, z2)) last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c12(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(false, nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil)) REV(nil, nil) -> c10(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil)) REV(cons(z0, z1), nil) -> c10(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil)) REV(x0, cons(x1, nil)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil))) REV(x0, cons(x1, cons(z0, z1))) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1)))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c10(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2))) REV(cons(z0, z1), cons(x1, x2)) -> c10(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c10(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(z1, z2)), x3) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3)) REV(cons(z0, nil), x1) -> c10(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3)) REV(nil, nil) -> c11(IF(true, nil, last(nil), nil)) S tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, nil), x1) -> c12(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil)) REV(cons(z0, z1), nil) -> c10(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil)) REV(x0, cons(x1, nil)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil))) REV(x0, cons(x1, cons(z0, z1))) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1)))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2))) REV(cons(z0, z1), cons(x1, x2)) -> c10(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2))) REV(cons(z0, nil), x1) -> c10(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(z1, z2)), x3) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3)) K tuples: REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(nil, nil) -> c11(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(nil, nil) -> c12(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(nil, nil) -> c10(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c10(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3)) Defined Rule Symbols: isEmpty_1, dropLast_1, append_2, last_1 Defined Pair Symbols: LAST_1, DROPLAST_1, APPEND_2, IF_4, REV_2 Compound Symbols: c3_1, c6_1, c8_1, c14_1, c11_2, c12_3, c12_1, c10_1, c11_1, c12_2 ---------------------------------------- (205) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: REV(nil, nil) -> c11(IF(true, nil, last(nil), nil)) ---------------------------------------- (206) Obligation: Complexity Dependency Tuples Problem Rules: isEmpty(nil) -> true isEmpty(cons(z0, z1)) -> false dropLast(nil) -> nil dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) append(nil, z0) -> z0 append(cons(z0, z1), z2) -> cons(z0, append(z1, z2)) last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c12(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(false, nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil)) REV(nil, nil) -> c10(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil)) REV(cons(z0, z1), nil) -> c10(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil)) REV(x0, cons(x1, nil)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil))) REV(x0, cons(x1, cons(z0, z1))) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1)))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c10(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2))) REV(cons(z0, z1), cons(x1, x2)) -> c10(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c10(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(z1, z2)), x3) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3)) REV(cons(z0, nil), x1) -> c10(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3)) S tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, nil), x1) -> c12(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil)) REV(cons(z0, z1), nil) -> c10(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil)) REV(x0, cons(x1, nil)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil))) REV(x0, cons(x1, cons(z0, z1))) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1)))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2))) REV(cons(z0, z1), cons(x1, x2)) -> c10(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2))) REV(cons(z0, nil), x1) -> c10(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(z1, z2)), x3) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3)) K tuples: REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(nil, nil) -> c12(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(nil, nil) -> c10(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c10(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3)) Defined Rule Symbols: isEmpty_1, dropLast_1, append_2, last_1 Defined Pair Symbols: LAST_1, DROPLAST_1, APPEND_2, IF_4, REV_2 Compound Symbols: c3_1, c6_1, c8_1, c14_1, c11_2, c12_3, c12_1, c10_1, c11_1, c12_2 ---------------------------------------- (207) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace REV(nil, nil) -> c12(IF(isEmpty(nil), nil, last(nil), nil)) by REV(nil, nil) -> c12(IF(true, nil, last(nil), nil)) ---------------------------------------- (208) Obligation: Complexity Dependency Tuples Problem Rules: isEmpty(nil) -> true isEmpty(cons(z0, z1)) -> false dropLast(nil) -> nil dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) append(nil, z0) -> z0 append(cons(z0, z1), z2) -> cons(z0, append(z1, z2)) last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(nil, nil) -> c12(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(false, nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil)) REV(nil, nil) -> c10(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil)) REV(cons(z0, z1), nil) -> c10(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil)) REV(x0, cons(x1, nil)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil))) REV(x0, cons(x1, cons(z0, z1))) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1)))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c10(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2))) REV(cons(z0, z1), cons(x1, x2)) -> c10(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c10(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(z1, z2)), x3) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3)) REV(cons(z0, nil), x1) -> c10(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3)) REV(nil, nil) -> c12(IF(true, nil, last(nil), nil)) S tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, nil), x1) -> c12(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil)) REV(cons(z0, z1), nil) -> c10(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil)) REV(x0, cons(x1, nil)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil))) REV(x0, cons(x1, cons(z0, z1))) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1)))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2))) REV(cons(z0, z1), cons(x1, x2)) -> c10(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2))) REV(cons(z0, nil), x1) -> c10(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(z1, z2)), x3) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3)) K tuples: REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(nil, nil) -> c12(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(nil, nil) -> c10(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c10(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3)) Defined Rule Symbols: isEmpty_1, dropLast_1, append_2, last_1 Defined Pair Symbols: LAST_1, DROPLAST_1, APPEND_2, IF_4, REV_2 Compound Symbols: c3_1, c6_1, c8_1, c14_1, c11_2, c12_3, c12_1, c10_1, c11_1, c12_2 ---------------------------------------- (209) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: REV(nil, nil) -> c12(IF(true, nil, last(nil), nil)) ---------------------------------------- (210) Obligation: Complexity Dependency Tuples Problem Rules: isEmpty(nil) -> true isEmpty(cons(z0, z1)) -> false dropLast(nil) -> nil dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) append(nil, z0) -> z0 append(cons(z0, z1), z2) -> cons(z0, append(z1, z2)) last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(nil, nil) -> c12(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), x1) -> c12(IF(false, nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil)) REV(nil, nil) -> c10(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil)) REV(cons(z0, z1), nil) -> c10(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil)) REV(x0, cons(x1, nil)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil))) REV(x0, cons(x1, cons(z0, z1))) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1)))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c10(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2))) REV(cons(z0, z1), cons(x1, x2)) -> c10(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c10(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(z1, z2)), x3) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3)) REV(cons(z0, nil), x1) -> c10(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3)) S tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, nil), x1) -> c12(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil)) REV(cons(z0, z1), nil) -> c10(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil)) REV(x0, cons(x1, nil)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil))) REV(x0, cons(x1, cons(z0, z1))) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1)))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2))) REV(cons(z0, z1), cons(x1, x2)) -> c10(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2))) REV(cons(z0, nil), x1) -> c10(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(z1, z2)), x3) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3)) K tuples: REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(nil, nil) -> c12(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(nil, nil) -> c10(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c10(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3)) Defined Rule Symbols: isEmpty_1, dropLast_1, append_2, last_1 Defined Pair Symbols: LAST_1, DROPLAST_1, APPEND_2, IF_4, REV_2 Compound Symbols: c3_1, c6_1, c8_1, c14_1, c11_2, c12_3, c12_1, c10_1, c11_1, c12_2 ---------------------------------------- (211) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace REV(nil, nil) -> c12(IF(isEmpty(nil), nil, last(nil), nil)) by REV(nil, nil) -> c12(IF(true, nil, last(nil), nil)) ---------------------------------------- (212) Obligation: Complexity Dependency Tuples Problem Rules: isEmpty(nil) -> true isEmpty(cons(z0, z1)) -> false dropLast(nil) -> nil dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) append(nil, z0) -> z0 append(cons(z0, z1), z2) -> cons(z0, append(z1, z2)) last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(false, nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil)) REV(nil, nil) -> c10(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil)) REV(cons(z0, z1), nil) -> c10(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil)) REV(x0, cons(x1, nil)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil))) REV(x0, cons(x1, cons(z0, z1))) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1)))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c10(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2))) REV(cons(z0, z1), cons(x1, x2)) -> c10(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c10(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(z1, z2)), x3) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3)) REV(cons(z0, nil), x1) -> c10(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3)) REV(nil, nil) -> c12(IF(true, nil, last(nil), nil)) S tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, nil), x1) -> c12(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil)) REV(cons(z0, z1), nil) -> c10(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil)) REV(x0, cons(x1, nil)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil))) REV(x0, cons(x1, cons(z0, z1))) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1)))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2))) REV(cons(z0, z1), cons(x1, x2)) -> c10(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2))) REV(cons(z0, nil), x1) -> c10(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(z1, z2)), x3) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3)) K tuples: REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(nil, nil) -> c12(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(nil, nil) -> c10(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c10(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3)) Defined Rule Symbols: isEmpty_1, dropLast_1, append_2, last_1 Defined Pair Symbols: LAST_1, DROPLAST_1, APPEND_2, IF_4, REV_2 Compound Symbols: c3_1, c6_1, c8_1, c14_1, c11_2, c12_3, c12_1, c10_1, c11_1, c12_2 ---------------------------------------- (213) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: REV(nil, nil) -> c12(IF(true, nil, last(nil), nil)) ---------------------------------------- (214) Obligation: Complexity Dependency Tuples Problem Rules: isEmpty(nil) -> true isEmpty(cons(z0, z1)) -> false dropLast(nil) -> nil dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) append(nil, z0) -> z0 append(cons(z0, z1), z2) -> cons(z0, append(z1, z2)) last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(false, nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil)) REV(nil, nil) -> c10(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil)) REV(cons(z0, z1), nil) -> c10(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil)) REV(x0, cons(x1, nil)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil))) REV(x0, cons(x1, cons(z0, z1))) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1)))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c10(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2))) REV(cons(z0, z1), cons(x1, x2)) -> c10(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c10(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(z1, z2)), x3) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3)) REV(cons(z0, nil), x1) -> c10(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3)) S tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, nil), x1) -> c12(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil)) REV(cons(z0, z1), nil) -> c10(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil)) REV(x0, cons(x1, nil)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil))) REV(x0, cons(x1, cons(z0, z1))) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1)))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2))) REV(cons(z0, z1), cons(x1, x2)) -> c10(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2))) REV(cons(z0, nil), x1) -> c10(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(z1, z2)), x3) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3)) K tuples: REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(nil, nil) -> c10(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c10(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3)) Defined Rule Symbols: isEmpty_1, dropLast_1, append_2, last_1 Defined Pair Symbols: LAST_1, DROPLAST_1, APPEND_2, IF_4, REV_2 Compound Symbols: c3_1, c6_1, c8_1, c14_1, c11_2, c12_3, c12_1, c10_1, c11_1, c12_2 ---------------------------------------- (215) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace REV(nil, nil) -> c10(IF(isEmpty(nil), nil, last(nil), nil)) by REV(nil, nil) -> c10(IF(true, nil, last(nil), nil)) ---------------------------------------- (216) Obligation: Complexity Dependency Tuples Problem Rules: isEmpty(nil) -> true isEmpty(cons(z0, z1)) -> false dropLast(nil) -> nil dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) append(nil, z0) -> z0 append(cons(z0, z1), z2) -> cons(z0, append(z1, z2)) last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(false, nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil)) REV(nil, nil) -> c10(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil)) REV(cons(z0, z1), nil) -> c10(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil)) REV(x0, cons(x1, nil)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil))) REV(x0, cons(x1, cons(z0, z1))) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1)))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c10(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2))) REV(cons(z0, z1), cons(x1, x2)) -> c10(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c10(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(z1, z2)), x3) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3)) REV(cons(z0, nil), x1) -> c10(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3)) REV(nil, nil) -> c10(IF(true, nil, last(nil), nil)) S tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, nil), x1) -> c12(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil)) REV(cons(z0, z1), nil) -> c10(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil)) REV(x0, cons(x1, nil)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil))) REV(x0, cons(x1, cons(z0, z1))) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1)))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2))) REV(cons(z0, z1), cons(x1, x2)) -> c10(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2))) REV(cons(z0, nil), x1) -> c10(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(z1, z2)), x3) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3)) K tuples: REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(nil, nil) -> c10(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c10(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3)) Defined Rule Symbols: isEmpty_1, dropLast_1, append_2, last_1 Defined Pair Symbols: LAST_1, DROPLAST_1, APPEND_2, IF_4, REV_2 Compound Symbols: c3_1, c6_1, c8_1, c14_1, c11_2, c12_3, c12_1, c10_1, c11_1, c12_2 ---------------------------------------- (217) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: REV(nil, nil) -> c10(IF(true, nil, last(nil), nil)) ---------------------------------------- (218) Obligation: Complexity Dependency Tuples Problem Rules: isEmpty(nil) -> true isEmpty(cons(z0, z1)) -> false dropLast(nil) -> nil dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) append(nil, z0) -> z0 append(cons(z0, z1), z2) -> cons(z0, append(z1, z2)) last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(false, nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil)) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil)) REV(cons(z0, z1), nil) -> c10(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil)) REV(x0, cons(x1, nil)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil))) REV(x0, cons(x1, cons(z0, z1))) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1)))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c10(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2))) REV(cons(z0, z1), cons(x1, x2)) -> c10(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c10(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(z1, z2)), x3) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3)) REV(nil, nil) -> c10(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), x1) -> c10(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3)) S tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, nil), x1) -> c12(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil)) REV(cons(z0, z1), nil) -> c10(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil)) REV(x0, cons(x1, nil)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil))) REV(x0, cons(x1, cons(z0, z1))) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1)))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2))) REV(cons(z0, z1), cons(x1, x2)) -> c10(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2))) REV(cons(z0, nil), x1) -> c10(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(z1, z2)), x3) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3)) K tuples: REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(nil, nil) -> c10(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c10(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3)) Defined Rule Symbols: isEmpty_1, dropLast_1, append_2, last_1 Defined Pair Symbols: LAST_1, DROPLAST_1, APPEND_2, IF_4, REV_2 Compound Symbols: c3_1, c6_1, c8_1, c14_1, c11_2, c12_3, c12_1, c10_1, c11_1, c12_2 ---------------------------------------- (219) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace REV(nil, nil) -> c10(IF(isEmpty(nil), nil, last(nil), nil)) by REV(nil, nil) -> c10(IF(true, nil, last(nil), nil)) ---------------------------------------- (220) Obligation: Complexity Dependency Tuples Problem Rules: isEmpty(nil) -> true isEmpty(cons(z0, z1)) -> false dropLast(nil) -> nil dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) append(nil, z0) -> z0 append(cons(z0, z1), z2) -> cons(z0, append(z1, z2)) last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(false, nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil)) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil)) REV(cons(z0, z1), nil) -> c10(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil)) REV(x0, cons(x1, nil)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil))) REV(x0, cons(x1, cons(z0, z1))) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1)))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c10(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2))) REV(cons(z0, z1), cons(x1, x2)) -> c10(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c10(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(z1, z2)), x3) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3)) REV(cons(z0, nil), x1) -> c10(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3)) REV(nil, nil) -> c10(IF(true, nil, last(nil), nil)) S tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, nil), x1) -> c12(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil)) REV(cons(z0, z1), nil) -> c10(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil)) REV(x0, cons(x1, nil)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil))) REV(x0, cons(x1, cons(z0, z1))) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1)))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2))) REV(cons(z0, z1), cons(x1, x2)) -> c10(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2))) REV(cons(z0, nil), x1) -> c10(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(z1, z2)), x3) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3)) K tuples: REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(nil, nil) -> c10(IF(isEmpty(nil), nil, last(nil), nil)) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c10(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3)) Defined Rule Symbols: isEmpty_1, dropLast_1, append_2, last_1 Defined Pair Symbols: LAST_1, DROPLAST_1, APPEND_2, IF_4, REV_2 Compound Symbols: c3_1, c6_1, c8_1, c14_1, c11_2, c12_3, c12_1, c10_1, c11_1, c12_2 ---------------------------------------- (221) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: REV(nil, nil) -> c10(IF(true, nil, last(nil), nil)) ---------------------------------------- (222) Obligation: Complexity Dependency Tuples Problem Rules: isEmpty(nil) -> true isEmpty(cons(z0, z1)) -> false dropLast(nil) -> nil dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) append(nil, z0) -> z0 append(cons(z0, z1), z2) -> cons(z0, append(z1, z2)) last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(false, nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil)) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil)) REV(cons(z0, z1), nil) -> c10(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil)) REV(x0, cons(x1, nil)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil))) REV(x0, cons(x1, cons(z0, z1))) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1)))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c10(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2))) REV(cons(z0, z1), cons(x1, x2)) -> c10(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c10(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(z1, z2)), x3) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3)) REV(cons(z0, nil), x1) -> c10(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3)) S tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, nil), x1) -> c12(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil)) REV(cons(z0, z1), nil) -> c10(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil)) REV(x0, cons(x1, nil)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil))) REV(x0, cons(x1, cons(z0, z1))) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1)))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2))) REV(cons(z0, z1), cons(x1, x2)) -> c10(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2))) REV(cons(z0, nil), x1) -> c10(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(z1, z2)), x3) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3)) K tuples: REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c10(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3)) Defined Rule Symbols: isEmpty_1, dropLast_1, append_2, last_1 Defined Pair Symbols: LAST_1, DROPLAST_1, APPEND_2, IF_4, REV_2 Compound Symbols: c3_1, c6_1, c8_1, c14_1, c11_2, c12_3, c12_1, c10_1, c11_1, c12_2 ---------------------------------------- (223) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) by REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) ---------------------------------------- (224) Obligation: Complexity Dependency Tuples Problem Rules: isEmpty(nil) -> true isEmpty(cons(z0, z1)) -> false dropLast(nil) -> nil dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) append(nil, z0) -> z0 append(cons(z0, z1), z2) -> cons(z0, append(z1, z2)) last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(false, nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil)) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil)) REV(cons(z0, z1), nil) -> c10(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil)) REV(x0, cons(x1, nil)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil))) REV(x0, cons(x1, cons(z0, z1))) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1)))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c10(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2))) REV(cons(z0, z1), cons(x1, x2)) -> c10(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c10(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(z1, z2)), x3) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3)) REV(cons(z0, nil), x1) -> c10(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) S tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, nil), x1) -> c12(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil)) REV(cons(z0, z1), nil) -> c10(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil)) REV(x0, cons(x1, nil)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil))) REV(x0, cons(x1, cons(z0, z1))) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1)))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2))) REV(cons(z0, z1), cons(x1, x2)) -> c10(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2))) REV(cons(z0, nil), x1) -> c10(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(z1, z2)), x3) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) K tuples: REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c10(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3)) Defined Rule Symbols: isEmpty_1, dropLast_1, append_2, last_1 Defined Pair Symbols: LAST_1, DROPLAST_1, APPEND_2, IF_4, REV_2 Compound Symbols: c3_1, c6_1, c8_1, c14_1, c11_2, c12_3, c12_1, c10_1, c11_1, c12_2 ---------------------------------------- (225) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) by REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) ---------------------------------------- (226) Obligation: Complexity Dependency Tuples Problem Rules: isEmpty(nil) -> true isEmpty(cons(z0, z1)) -> false dropLast(nil) -> nil dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) append(nil, z0) -> z0 append(cons(z0, z1), z2) -> cons(z0, append(z1, z2)) last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(false, nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil)) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil)) REV(cons(z0, z1), nil) -> c10(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil)) REV(x0, cons(x1, nil)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil))) REV(x0, cons(x1, cons(z0, z1))) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1)))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c10(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2))) REV(cons(z0, z1), cons(x1, x2)) -> c10(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c10(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(z1, z2)), x3) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3)) REV(cons(z0, nil), x1) -> c10(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) S tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, nil), x1) -> c12(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil)) REV(cons(z0, z1), nil) -> c10(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil)) REV(x0, cons(x1, nil)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil))) REV(x0, cons(x1, cons(z0, z1))) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1)))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2))) REV(cons(z0, z1), cons(x1, x2)) -> c10(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2))) REV(cons(z0, nil), x1) -> c10(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(z1, z2)), x3) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) K tuples: REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c10(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3)) Defined Rule Symbols: isEmpty_1, dropLast_1, append_2, last_1 Defined Pair Symbols: LAST_1, DROPLAST_1, APPEND_2, IF_4, REV_2 Compound Symbols: c3_1, c6_1, c8_1, c14_1, c11_2, c12_3, c12_1, c10_1, c11_1, c12_2 ---------------------------------------- (227) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) by REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, z0, nil)) ---------------------------------------- (228) Obligation: Complexity Dependency Tuples Problem Rules: isEmpty(nil) -> true isEmpty(cons(z0, z1)) -> false dropLast(nil) -> nil dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) append(nil, z0) -> z0 append(cons(z0, z1), z2) -> cons(z0, append(z1, z2)) last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(false, nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil)) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil)) REV(cons(z0, z1), nil) -> c10(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil)) REV(x0, cons(x1, nil)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil))) REV(x0, cons(x1, cons(z0, z1))) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1)))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c10(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2))) REV(cons(z0, z1), cons(x1, x2)) -> c10(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c10(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(z1, z2)), x3) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3)) REV(cons(z0, nil), x1) -> c10(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, z0, nil)) S tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, nil), x1) -> c12(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil)) REV(cons(z0, z1), nil) -> c10(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil)) REV(x0, cons(x1, nil)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil))) REV(x0, cons(x1, cons(z0, z1))) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1)))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2))) REV(cons(z0, z1), cons(x1, x2)) -> c10(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2))) REV(cons(z0, nil), x1) -> c10(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(z1, z2)), x3) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, z0, nil)) K tuples: REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c10(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3)) Defined Rule Symbols: isEmpty_1, dropLast_1, append_2, last_1 Defined Pair Symbols: LAST_1, DROPLAST_1, APPEND_2, IF_4, REV_2 Compound Symbols: c3_1, c6_1, c8_1, c14_1, c11_2, c12_3, c12_1, c10_1, c11_1, c12_2 ---------------------------------------- (229) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, z0, nil)) We considered the (Usable) Rules: dropLast(nil) -> nil isEmpty(nil) -> true dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) isEmpty(cons(z0, z1)) -> false And the Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(false, nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil)) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil)) REV(cons(z0, z1), nil) -> c10(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil)) REV(x0, cons(x1, nil)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil))) REV(x0, cons(x1, cons(z0, z1))) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1)))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c10(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2))) REV(cons(z0, z1), cons(x1, x2)) -> c10(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c10(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(z1, z2)), x3) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3)) REV(cons(z0, nil), x1) -> c10(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, z0, nil)) The order we found is given by the following interpretation: Polynomial interpretation : POL(APPEND(x_1, x_2)) = 0 POL(DROPLAST(x_1)) = 0 POL(IF(x_1, x_2, x_3, x_4)) = x_1 + x_2 POL(LAST(x_1)) = 0 POL(REV(x_1, x_2)) = x_1 POL(append(x_1, x_2)) = [1] POL(c10(x_1)) = x_1 POL(c11(x_1)) = x_1 POL(c11(x_1, x_2)) = x_1 + x_2 POL(c12(x_1)) = x_1 POL(c12(x_1, x_2)) = x_1 + x_2 POL(c12(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c14(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(cons(x_1, x_2)) = [1] POL(dropLast(x_1)) = x_1 POL(false) = 0 POL(isEmpty(x_1)) = 0 POL(last(x_1)) = 0 POL(nil) = 0 POL(true) = 0 ---------------------------------------- (230) Obligation: Complexity Dependency Tuples Problem Rules: isEmpty(nil) -> true isEmpty(cons(z0, z1)) -> false dropLast(nil) -> nil dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) append(nil, z0) -> z0 append(cons(z0, z1), z2) -> cons(z0, append(z1, z2)) last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(false, nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil)) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil)) REV(cons(z0, z1), nil) -> c10(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil)) REV(x0, cons(x1, nil)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil))) REV(x0, cons(x1, cons(z0, z1))) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1)))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c10(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2))) REV(cons(z0, z1), cons(x1, x2)) -> c10(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c10(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(z1, z2)), x3) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3)) REV(cons(z0, nil), x1) -> c10(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, z0, nil)) S tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, nil), x1) -> c12(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil)) REV(cons(z0, z1), nil) -> c10(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil)) REV(x0, cons(x1, nil)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil))) REV(x0, cons(x1, cons(z0, z1))) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1)))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2))) REV(cons(z0, z1), cons(x1, x2)) -> c10(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2))) REV(cons(z0, nil), x1) -> c10(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(z1, z2)), x3) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) K tuples: REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c10(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, z0, nil)) Defined Rule Symbols: isEmpty_1, dropLast_1, append_2, last_1 Defined Pair Symbols: LAST_1, DROPLAST_1, APPEND_2, IF_4, REV_2 Compound Symbols: c3_1, c6_1, c8_1, c14_1, c11_2, c12_3, c12_1, c10_1, c11_1, c12_2 ---------------------------------------- (231) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) by REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, z0, nil)) ---------------------------------------- (232) Obligation: Complexity Dependency Tuples Problem Rules: isEmpty(nil) -> true isEmpty(cons(z0, z1)) -> false dropLast(nil) -> nil dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) append(nil, z0) -> z0 append(cons(z0, z1), z2) -> cons(z0, append(z1, z2)) last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(false, nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil)) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil)) REV(cons(z0, z1), nil) -> c10(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil)) REV(x0, cons(x1, nil)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil))) REV(x0, cons(x1, cons(z0, z1))) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1)))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c10(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2))) REV(cons(z0, z1), cons(x1, x2)) -> c10(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c10(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(z1, z2)), x3) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3)) REV(cons(z0, nil), x1) -> c10(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, z0, nil)) S tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, nil), x1) -> c12(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil)) REV(cons(z0, z1), nil) -> c10(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil)) REV(x0, cons(x1, nil)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil))) REV(x0, cons(x1, cons(z0, z1))) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1)))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2))) REV(cons(z0, z1), cons(x1, x2)) -> c10(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2))) REV(cons(z0, nil), x1) -> c10(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(z1, z2)), x3) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) K tuples: REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c10(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, z0, nil)) Defined Rule Symbols: isEmpty_1, dropLast_1, append_2, last_1 Defined Pair Symbols: LAST_1, DROPLAST_1, APPEND_2, IF_4, REV_2 Compound Symbols: c3_1, c6_1, c8_1, c14_1, c11_2, c12_3, c12_1, c10_1, c11_1, c12_2 ---------------------------------------- (233) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) by REV(cons(z0, cons(z1, z2)), cons(z3, z4)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(z3, append(z4, last(cons(z1, z2)))), cons(z3, z4)), DROPLAST(cons(z0, cons(z1, z2)))) ---------------------------------------- (234) Obligation: Complexity Dependency Tuples Problem Rules: isEmpty(nil) -> true isEmpty(cons(z0, z1)) -> false dropLast(nil) -> nil dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) append(nil, z0) -> z0 append(cons(z0, z1), z2) -> cons(z0, append(z1, z2)) last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(false, nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil)) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil)) REV(cons(z0, z1), nil) -> c10(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil)) REV(x0, cons(x1, nil)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil))) REV(x0, cons(x1, cons(z0, z1))) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1)))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c10(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2))) REV(cons(z0, z1), cons(x1, x2)) -> c10(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c10(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(z1, z2)), x3) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3)) REV(cons(z0, nil), x1) -> c10(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, z0, nil)) REV(cons(z0, cons(z1, z2)), cons(z3, z4)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(z3, append(z4, last(cons(z1, z2)))), cons(z3, z4)), DROPLAST(cons(z0, cons(z1, z2)))) S tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, nil), x1) -> c12(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil)) REV(cons(z0, z1), nil) -> c10(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil)) REV(x0, cons(x1, nil)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil))) REV(x0, cons(x1, cons(z0, z1))) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1)))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2))) REV(cons(z0, z1), cons(x1, x2)) -> c10(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2))) REV(cons(z0, nil), x1) -> c10(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(z1, z2)), x3) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(z3, z4)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(z3, append(z4, last(cons(z1, z2)))), cons(z3, z4)), DROPLAST(cons(z0, cons(z1, z2)))) K tuples: REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c10(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, z0, nil)) Defined Rule Symbols: isEmpty_1, dropLast_1, append_2, last_1 Defined Pair Symbols: LAST_1, DROPLAST_1, APPEND_2, IF_4, REV_2 Compound Symbols: c3_1, c6_1, c8_1, c14_1, c11_2, c12_3, c12_1, c10_1, c11_1, c12_2 ---------------------------------------- (235) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) by REV(cons(z0, cons(z1, z2)), cons(z3, z4)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(z3, append(z4, last(cons(z1, z2)))), cons(z3, z4)), DROPLAST(cons(z0, cons(z1, z2)))) ---------------------------------------- (236) Obligation: Complexity Dependency Tuples Problem Rules: isEmpty(nil) -> true isEmpty(cons(z0, z1)) -> false dropLast(nil) -> nil dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) append(nil, z0) -> z0 append(cons(z0, z1), z2) -> cons(z0, append(z1, z2)) last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(false, nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil)) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil)) REV(cons(z0, z1), nil) -> c10(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil)) REV(x0, cons(x1, nil)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil))) REV(x0, cons(x1, cons(z0, z1))) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1)))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c10(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2))) REV(cons(z0, z1), cons(x1, x2)) -> c10(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c10(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(z1, z2)), x3) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3)) REV(cons(z0, nil), x1) -> c10(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, z0, nil)) REV(cons(z0, cons(z1, z2)), cons(z3, z4)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(z3, append(z4, last(cons(z1, z2)))), cons(z3, z4)), DROPLAST(cons(z0, cons(z1, z2)))) S tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, nil), x1) -> c12(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil)) REV(cons(z0, z1), nil) -> c10(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil)) REV(x0, cons(x1, nil)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil))) REV(x0, cons(x1, cons(z0, z1))) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1)))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2))) REV(cons(z0, z1), cons(x1, x2)) -> c10(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2))) REV(cons(z0, nil), x1) -> c10(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(z1, z2)), x3) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(z3, z4)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(z3, append(z4, last(cons(z1, z2)))), cons(z3, z4)), DROPLAST(cons(z0, cons(z1, z2)))) K tuples: REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c10(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, z0, nil)) Defined Rule Symbols: isEmpty_1, dropLast_1, append_2, last_1 Defined Pair Symbols: LAST_1, DROPLAST_1, APPEND_2, IF_4, REV_2 Compound Symbols: c3_1, c6_1, c8_1, c14_1, c11_2, c12_3, c12_1, c10_1, c11_1, c12_2 ---------------------------------------- (237) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) by REV(cons(z0, nil), cons(z1, z2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(z1, append(z2, z0)), cons(z1, z2))) ---------------------------------------- (238) Obligation: Complexity Dependency Tuples Problem Rules: isEmpty(nil) -> true isEmpty(cons(z0, z1)) -> false dropLast(nil) -> nil dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) append(nil, z0) -> z0 append(cons(z0, z1), z2) -> cons(z0, append(z1, z2)) last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(false, nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil)) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil)) REV(cons(z0, z1), nil) -> c10(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil)) REV(x0, cons(x1, nil)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil))) REV(x0, cons(x1, cons(z0, z1))) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1)))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c10(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2))) REV(cons(z0, z1), cons(x1, x2)) -> c10(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c10(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(z1, z2)), x3) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3)) REV(cons(z0, nil), x1) -> c10(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, z0, nil)) REV(cons(z0, cons(z1, z2)), cons(z3, z4)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(z3, append(z4, last(cons(z1, z2)))), cons(z3, z4)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, nil), cons(z1, z2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(z1, append(z2, z0)), cons(z1, z2))) S tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, nil), x1) -> c12(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil)) REV(cons(z0, z1), nil) -> c10(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil)) REV(x0, cons(x1, nil)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil))) REV(x0, cons(x1, cons(z0, z1))) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1)))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2))) REV(cons(z0, z1), cons(x1, x2)) -> c10(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2))) REV(cons(z0, nil), x1) -> c10(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(z1, z2)), x3) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(z3, z4)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(z3, append(z4, last(cons(z1, z2)))), cons(z3, z4)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, nil), cons(z1, z2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(z1, append(z2, z0)), cons(z1, z2))) K tuples: REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c10(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, z0, nil)) Defined Rule Symbols: isEmpty_1, dropLast_1, append_2, last_1 Defined Pair Symbols: LAST_1, DROPLAST_1, APPEND_2, IF_4, REV_2 Compound Symbols: c3_1, c6_1, c8_1, c14_1, c11_2, c12_3, c12_1, c10_1, c11_1, c12_2 ---------------------------------------- (239) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. REV(cons(z0, nil), cons(z1, z2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(z1, append(z2, z0)), cons(z1, z2))) We considered the (Usable) Rules: dropLast(nil) -> nil isEmpty(nil) -> true dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) isEmpty(cons(z0, z1)) -> false And the Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(false, nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil)) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil)) REV(cons(z0, z1), nil) -> c10(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil)) REV(x0, cons(x1, nil)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil))) REV(x0, cons(x1, cons(z0, z1))) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1)))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c10(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2))) REV(cons(z0, z1), cons(x1, x2)) -> c10(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c10(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(z1, z2)), x3) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3)) REV(cons(z0, nil), x1) -> c10(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, z0, nil)) REV(cons(z0, cons(z1, z2)), cons(z3, z4)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(z3, append(z4, last(cons(z1, z2)))), cons(z3, z4)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, nil), cons(z1, z2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(z1, append(z2, z0)), cons(z1, z2))) The order we found is given by the following interpretation: Polynomial interpretation : POL(APPEND(x_1, x_2)) = 0 POL(DROPLAST(x_1)) = 0 POL(IF(x_1, x_2, x_3, x_4)) = x_1 + x_2 POL(LAST(x_1)) = 0 POL(REV(x_1, x_2)) = x_1 POL(append(x_1, x_2)) = [1] POL(c10(x_1)) = x_1 POL(c11(x_1)) = x_1 POL(c11(x_1, x_2)) = x_1 + x_2 POL(c12(x_1)) = x_1 POL(c12(x_1, x_2)) = x_1 + x_2 POL(c12(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c14(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(cons(x_1, x_2)) = [1] POL(dropLast(x_1)) = x_1 POL(false) = 0 POL(isEmpty(x_1)) = 0 POL(last(x_1)) = 0 POL(nil) = 0 POL(true) = 0 ---------------------------------------- (240) Obligation: Complexity Dependency Tuples Problem Rules: isEmpty(nil) -> true isEmpty(cons(z0, z1)) -> false dropLast(nil) -> nil dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) append(nil, z0) -> z0 append(cons(z0, z1), z2) -> cons(z0, append(z1, z2)) last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(false, nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil)) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil)) REV(cons(z0, z1), nil) -> c10(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil)) REV(x0, cons(x1, nil)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil))) REV(x0, cons(x1, cons(z0, z1))) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1)))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c10(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2))) REV(cons(z0, z1), cons(x1, x2)) -> c10(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c10(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(z1, z2)), x3) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3)) REV(cons(z0, nil), x1) -> c10(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, z0, nil)) REV(cons(z0, cons(z1, z2)), cons(z3, z4)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(z3, append(z4, last(cons(z1, z2)))), cons(z3, z4)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, nil), cons(z1, z2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(z1, append(z2, z0)), cons(z1, z2))) S tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, nil), x1) -> c12(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil)) REV(cons(z0, z1), nil) -> c10(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil)) REV(x0, cons(x1, nil)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil))) REV(x0, cons(x1, cons(z0, z1))) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1)))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2))) REV(cons(z0, z1), cons(x1, x2)) -> c10(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2))) REV(cons(z0, nil), x1) -> c10(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(z1, z2)), x3) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(z3, z4)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(z3, append(z4, last(cons(z1, z2)))), cons(z3, z4)), DROPLAST(cons(z0, cons(z1, z2)))) K tuples: REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c10(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, z0, nil)) REV(cons(z0, nil), cons(z1, z2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(z1, append(z2, z0)), cons(z1, z2))) Defined Rule Symbols: isEmpty_1, dropLast_1, append_2, last_1 Defined Pair Symbols: LAST_1, DROPLAST_1, APPEND_2, IF_4, REV_2 Compound Symbols: c3_1, c6_1, c8_1, c14_1, c11_2, c12_3, c12_1, c10_1, c11_1, c12_2 ---------------------------------------- (241) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) by REV(nil, cons(z0, z1)) -> c11(IF(true, nil, cons(z0, append(z1, last(nil))), cons(z0, z1))) ---------------------------------------- (242) Obligation: Complexity Dependency Tuples Problem Rules: isEmpty(nil) -> true isEmpty(cons(z0, z1)) -> false dropLast(nil) -> nil dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) append(nil, z0) -> z0 append(cons(z0, z1), z2) -> cons(z0, append(z1, z2)) last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(false, nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil)) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil)) REV(cons(z0, z1), nil) -> c10(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil)) REV(x0, cons(x1, nil)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil))) REV(x0, cons(x1, cons(z0, z1))) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1)))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c10(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2))) REV(cons(z0, z1), cons(x1, x2)) -> c10(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c10(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(z1, z2)), x3) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3)) REV(cons(z0, nil), x1) -> c10(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, z0, nil)) REV(cons(z0, cons(z1, z2)), cons(z3, z4)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(z3, append(z4, last(cons(z1, z2)))), cons(z3, z4)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, nil), cons(z1, z2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(z1, append(z2, z0)), cons(z1, z2))) REV(nil, cons(z0, z1)) -> c11(IF(true, nil, cons(z0, append(z1, last(nil))), cons(z0, z1))) S tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, nil), x1) -> c12(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil)) REV(cons(z0, z1), nil) -> c10(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil)) REV(x0, cons(x1, nil)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil))) REV(x0, cons(x1, cons(z0, z1))) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1)))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2))) REV(cons(z0, z1), cons(x1, x2)) -> c10(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2))) REV(cons(z0, nil), x1) -> c10(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(z1, z2)), x3) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(z3, z4)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(z3, append(z4, last(cons(z1, z2)))), cons(z3, z4)), DROPLAST(cons(z0, cons(z1, z2)))) K tuples: REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c10(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, z0, nil)) REV(cons(z0, nil), cons(z1, z2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(z1, append(z2, z0)), cons(z1, z2))) Defined Rule Symbols: isEmpty_1, dropLast_1, append_2, last_1 Defined Pair Symbols: LAST_1, DROPLAST_1, APPEND_2, IF_4, REV_2 Compound Symbols: c3_1, c6_1, c8_1, c14_1, c11_2, c12_3, c12_1, c10_1, c11_1, c12_2 ---------------------------------------- (243) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: REV(nil, cons(z0, z1)) -> c11(IF(true, nil, cons(z0, append(z1, last(nil))), cons(z0, z1))) ---------------------------------------- (244) Obligation: Complexity Dependency Tuples Problem Rules: isEmpty(nil) -> true isEmpty(cons(z0, z1)) -> false dropLast(nil) -> nil dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) append(nil, z0) -> z0 append(cons(z0, z1), z2) -> cons(z0, append(z1, z2)) last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(x0, cons(x1, x2)), nil) -> c11(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), DROPLAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), cons(z0, z1)) -> c11(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), cons(z0, append(z1, last(cons(x1, x2)))), cons(z0, z1)), DROPLAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c11(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), DROPLAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), cons(z0, z1)) -> c11(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), cons(z0, append(z1, last(cons(x0, cons(x1, x2))))), cons(z0, z1)), DROPLAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z2, nil), cons(z0, z1)) -> c11(IF(isEmpty(cons(z2, nil)), dropLast(cons(z2, nil)), cons(z0, append(z1, z2)), cons(z0, z1))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(nil, cons(z0, z1)) -> c11(IF(isEmpty(nil), nil, cons(z0, append(z1, last(nil))), cons(z0, z1))) REV(cons(x0, nil), nil) -> c11(IF(isEmpty(cons(x0, nil)), nil, last(cons(x0, nil)), nil)) REV(cons(z0, nil), x1) -> c11(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(false, nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil)) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil)) REV(cons(z0, z1), nil) -> c10(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil)) REV(x0, cons(x1, nil)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil))) REV(x0, cons(x1, cons(z0, z1))) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1)))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c10(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2))) REV(cons(z0, z1), cons(x1, x2)) -> c10(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c10(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(z1, z2)), x3) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3)) REV(cons(z0, nil), x1) -> c10(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, z0, nil)) REV(cons(z0, cons(z1, z2)), cons(z3, z4)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(z3, append(z4, last(cons(z1, z2)))), cons(z3, z4)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, nil), cons(z1, z2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(z1, append(z2, z0)), cons(z1, z2))) S tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, nil), x1) -> c12(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil)) REV(cons(z0, z1), nil) -> c10(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil)) REV(x0, cons(x1, nil)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil))) REV(x0, cons(x1, cons(z0, z1))) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1)))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2))) REV(cons(z0, z1), cons(x1, x2)) -> c10(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2))) REV(cons(z0, nil), x1) -> c10(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(z1, z2)), x3) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(z3, z4)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(z3, append(z4, last(cons(z1, z2)))), cons(z3, z4)), DROPLAST(cons(z0, cons(z1, z2)))) K tuples: REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c10(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, z0, nil)) REV(cons(z0, nil), cons(z1, z2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(z1, append(z2, z0)), cons(z1, z2))) Defined Rule Symbols: isEmpty_1, dropLast_1, append_2, last_1 Defined Pair Symbols: LAST_1, DROPLAST_1, APPEND_2, IF_4, REV_2 Compound Symbols: c3_1, c6_1, c8_1, c14_1, c11_2, c12_3, c12_1, c10_1, c11_1, c12_2 ---------------------------------------- (245) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) by REV(cons(z0, nil), cons(z1, z2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(z1, append(z2, z0)), cons(z1, z2))) ---------------------------------------- (246) Obligation: Complexity Dependency Tuples Problem Rules: isEmpty(nil) -> true isEmpty(cons(z0, z1)) -> false dropLast(nil) -> nil dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) append(nil, z0) -> z0 append(cons(z0, z1), z2) -> cons(z0, append(z1, z2)) last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(x0, cons(x1, x2)), nil) -> c11(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), DROPLAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), cons(z0, z1)) -> c11(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), cons(z0, append(z1, last(cons(x1, x2)))), cons(z0, z1)), DROPLAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c11(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), DROPLAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), cons(z0, z1)) -> c11(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), cons(z0, append(z1, last(cons(x0, cons(x1, x2))))), cons(z0, z1)), DROPLAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z2, nil), cons(z0, z1)) -> c11(IF(isEmpty(cons(z2, nil)), dropLast(cons(z2, nil)), cons(z0, append(z1, z2)), cons(z0, z1))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(nil, cons(z0, z1)) -> c11(IF(isEmpty(nil), nil, cons(z0, append(z1, last(nil))), cons(z0, z1))) REV(cons(x0, nil), nil) -> c11(IF(isEmpty(cons(x0, nil)), nil, last(cons(x0, nil)), nil)) REV(cons(z0, nil), x1) -> c11(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(false, nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil)) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil)) REV(cons(z0, z1), nil) -> c10(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil)) REV(x0, cons(x1, nil)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil))) REV(x0, cons(x1, cons(z0, z1))) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1)))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c10(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2))) REV(cons(z0, z1), cons(x1, x2)) -> c10(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c10(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(z1, z2)), x3) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3)) REV(cons(z0, nil), x1) -> c10(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, z0, nil)) REV(cons(z0, cons(z1, z2)), cons(z3, z4)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(z3, append(z4, last(cons(z1, z2)))), cons(z3, z4)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, nil), cons(z1, z2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(z1, append(z2, z0)), cons(z1, z2))) S tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, nil), x1) -> c12(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil)) REV(cons(z0, z1), nil) -> c10(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil)) REV(x0, cons(x1, nil)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil))) REV(x0, cons(x1, cons(z0, z1))) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1)))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2))) REV(cons(z0, z1), cons(x1, x2)) -> c10(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2))) REV(cons(z0, nil), x1) -> c10(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(z1, z2)), x3) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(z3, z4)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(z3, append(z4, last(cons(z1, z2)))), cons(z3, z4)), DROPLAST(cons(z0, cons(z1, z2)))) K tuples: REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c10(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, z0, nil)) REV(cons(z0, nil), cons(z1, z2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(z1, append(z2, z0)), cons(z1, z2))) Defined Rule Symbols: isEmpty_1, dropLast_1, append_2, last_1 Defined Pair Symbols: LAST_1, DROPLAST_1, APPEND_2, IF_4, REV_2 Compound Symbols: c3_1, c6_1, c8_1, c14_1, c11_2, c12_3, c12_1, c10_1, c11_1, c12_2 ---------------------------------------- (247) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace REV(cons(x0, cons(x1, x2)), nil) -> c11(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), DROPLAST(cons(x0, cons(x1, x2)))) by REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) ---------------------------------------- (248) Obligation: Complexity Dependency Tuples Problem Rules: isEmpty(nil) -> true isEmpty(cons(z0, z1)) -> false dropLast(nil) -> nil dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) append(nil, z0) -> z0 append(cons(z0, z1), z2) -> cons(z0, append(z1, z2)) last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(x0, cons(x1, x2)), cons(z0, z1)) -> c11(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), cons(z0, append(z1, last(cons(x1, x2)))), cons(z0, z1)), DROPLAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c11(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), DROPLAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), cons(z0, z1)) -> c11(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), cons(z0, append(z1, last(cons(x0, cons(x1, x2))))), cons(z0, z1)), DROPLAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z2, nil), cons(z0, z1)) -> c11(IF(isEmpty(cons(z2, nil)), dropLast(cons(z2, nil)), cons(z0, append(z1, z2)), cons(z0, z1))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(nil, cons(z0, z1)) -> c11(IF(isEmpty(nil), nil, cons(z0, append(z1, last(nil))), cons(z0, z1))) REV(cons(x0, nil), nil) -> c11(IF(isEmpty(cons(x0, nil)), nil, last(cons(x0, nil)), nil)) REV(cons(z0, nil), x1) -> c11(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(false, nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil)) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil)) REV(cons(z0, z1), nil) -> c10(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil)) REV(x0, cons(x1, nil)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil))) REV(x0, cons(x1, cons(z0, z1))) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1)))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c10(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2))) REV(cons(z0, z1), cons(x1, x2)) -> c10(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c10(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(z1, z2)), x3) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3)) REV(cons(z0, nil), x1) -> c10(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, z0, nil)) REV(cons(z0, cons(z1, z2)), cons(z3, z4)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(z3, append(z4, last(cons(z1, z2)))), cons(z3, z4)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, nil), cons(z1, z2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(z1, append(z2, z0)), cons(z1, z2))) S tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, nil), x1) -> c12(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil)) REV(cons(z0, z1), nil) -> c10(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil)) REV(x0, cons(x1, nil)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil))) REV(x0, cons(x1, cons(z0, z1))) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1)))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2))) REV(cons(z0, z1), cons(x1, x2)) -> c10(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2))) REV(cons(z0, nil), x1) -> c10(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(z1, z2)), x3) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(z3, z4)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(z3, append(z4, last(cons(z1, z2)))), cons(z3, z4)), DROPLAST(cons(z0, cons(z1, z2)))) K tuples: REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c10(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, z0, nil)) REV(cons(z0, nil), cons(z1, z2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(z1, append(z2, z0)), cons(z1, z2))) Defined Rule Symbols: isEmpty_1, dropLast_1, append_2, last_1 Defined Pair Symbols: LAST_1, DROPLAST_1, APPEND_2, IF_4, REV_2 Compound Symbols: c3_1, c6_1, c8_1, c14_1, c11_2, c12_3, c12_1, c10_1, c11_1, c12_2 ---------------------------------------- (249) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace REV(cons(x0, cons(x1, x2)), cons(z0, z1)) -> c11(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), cons(z0, append(z1, last(cons(x1, x2)))), cons(z0, z1)), DROPLAST(cons(x0, cons(x1, x2)))) by REV(cons(z0, cons(z1, z2)), cons(z3, z4)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(z3, append(z4, last(cons(z1, z2)))), cons(z3, z4)), DROPLAST(cons(z0, cons(z1, z2)))) ---------------------------------------- (250) Obligation: Complexity Dependency Tuples Problem Rules: isEmpty(nil) -> true isEmpty(cons(z0, z1)) -> false dropLast(nil) -> nil dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) append(nil, z0) -> z0 append(cons(z0, z1), z2) -> cons(z0, append(z1, z2)) last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c11(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), DROPLAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), cons(z0, z1)) -> c11(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), cons(z0, append(z1, last(cons(x0, cons(x1, x2))))), cons(z0, z1)), DROPLAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z2, nil), cons(z0, z1)) -> c11(IF(isEmpty(cons(z2, nil)), dropLast(cons(z2, nil)), cons(z0, append(z1, z2)), cons(z0, z1))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(nil, cons(z0, z1)) -> c11(IF(isEmpty(nil), nil, cons(z0, append(z1, last(nil))), cons(z0, z1))) REV(cons(x0, nil), nil) -> c11(IF(isEmpty(cons(x0, nil)), nil, last(cons(x0, nil)), nil)) REV(cons(z0, nil), x1) -> c11(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(false, nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil)) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil)) REV(cons(z0, z1), nil) -> c10(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil)) REV(x0, cons(x1, nil)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil))) REV(x0, cons(x1, cons(z0, z1))) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1)))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c10(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2))) REV(cons(z0, z1), cons(x1, x2)) -> c10(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c10(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(z1, z2)), x3) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3)) REV(cons(z0, nil), x1) -> c10(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, z0, nil)) REV(cons(z0, cons(z1, z2)), cons(z3, z4)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(z3, append(z4, last(cons(z1, z2)))), cons(z3, z4)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, nil), cons(z1, z2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(z1, append(z2, z0)), cons(z1, z2))) S tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, nil), x1) -> c12(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil)) REV(cons(z0, z1), nil) -> c10(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil)) REV(x0, cons(x1, nil)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil))) REV(x0, cons(x1, cons(z0, z1))) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1)))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2))) REV(cons(z0, z1), cons(x1, x2)) -> c10(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2))) REV(cons(z0, nil), x1) -> c10(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(z1, z2)), x3) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(z3, z4)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(z3, append(z4, last(cons(z1, z2)))), cons(z3, z4)), DROPLAST(cons(z0, cons(z1, z2)))) K tuples: REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c10(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, z0, nil)) REV(cons(z0, nil), cons(z1, z2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(z1, append(z2, z0)), cons(z1, z2))) Defined Rule Symbols: isEmpty_1, dropLast_1, append_2, last_1 Defined Pair Symbols: LAST_1, DROPLAST_1, APPEND_2, IF_4, REV_2 Compound Symbols: c3_1, c6_1, c8_1, c14_1, c11_2, c12_3, c12_1, c10_1, c11_1, c12_2 ---------------------------------------- (251) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), DROPLAST(cons(x0, cons(z0, nil)))) by REV(cons(z0, cons(z1, nil)), z2) -> c11(IF(isEmpty(cons(z0, cons(z1, nil))), cons(z0, dropLast(cons(z1, nil))), append(z2, z1), z2), DROPLAST(cons(z0, cons(z1, nil)))) ---------------------------------------- (252) Obligation: Complexity Dependency Tuples Problem Rules: isEmpty(nil) -> true isEmpty(cons(z0, z1)) -> false dropLast(nil) -> nil dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) append(nil, z0) -> z0 append(cons(z0, z1), z2) -> cons(z0, append(z1, z2)) last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c11(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), DROPLAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), cons(z0, z1)) -> c11(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), cons(z0, append(z1, last(cons(x0, cons(x1, x2))))), cons(z0, z1)), DROPLAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z2, nil), cons(z0, z1)) -> c11(IF(isEmpty(cons(z2, nil)), dropLast(cons(z2, nil)), cons(z0, append(z1, z2)), cons(z0, z1))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(nil, cons(z0, z1)) -> c11(IF(isEmpty(nil), nil, cons(z0, append(z1, last(nil))), cons(z0, z1))) REV(cons(x0, nil), nil) -> c11(IF(isEmpty(cons(x0, nil)), nil, last(cons(x0, nil)), nil)) REV(cons(z0, nil), x1) -> c11(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(false, nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil)) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil)) REV(cons(z0, z1), nil) -> c10(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil)) REV(x0, cons(x1, nil)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil))) REV(x0, cons(x1, cons(z0, z1))) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1)))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c10(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2))) REV(cons(z0, z1), cons(x1, x2)) -> c10(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c10(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(z1, z2)), x3) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3)) REV(cons(z0, nil), x1) -> c10(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, z0, nil)) REV(cons(z0, cons(z1, z2)), cons(z3, z4)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(z3, append(z4, last(cons(z1, z2)))), cons(z3, z4)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, nil), cons(z1, z2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(z1, append(z2, z0)), cons(z1, z2))) REV(cons(z0, cons(z1, nil)), z2) -> c11(IF(isEmpty(cons(z0, cons(z1, nil))), cons(z0, dropLast(cons(z1, nil))), append(z2, z1), z2), DROPLAST(cons(z0, cons(z1, nil)))) S tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, nil), x1) -> c12(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil)) REV(cons(z0, z1), nil) -> c10(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil)) REV(x0, cons(x1, nil)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil))) REV(x0, cons(x1, cons(z0, z1))) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1)))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2))) REV(cons(z0, z1), cons(x1, x2)) -> c10(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2))) REV(cons(z0, nil), x1) -> c10(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(z1, z2)), x3) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(z3, z4)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(z3, append(z4, last(cons(z1, z2)))), cons(z3, z4)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, nil)), z2) -> c11(IF(isEmpty(cons(z0, cons(z1, nil))), cons(z0, dropLast(cons(z1, nil))), append(z2, z1), z2), DROPLAST(cons(z0, cons(z1, nil)))) K tuples: REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c10(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, z0, nil)) REV(cons(z0, nil), cons(z1, z2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(z1, append(z2, z0)), cons(z1, z2))) Defined Rule Symbols: isEmpty_1, dropLast_1, append_2, last_1 Defined Pair Symbols: LAST_1, DROPLAST_1, APPEND_2, IF_4, REV_2 Compound Symbols: c3_1, c6_1, c8_1, c14_1, c11_2, c12_3, c12_1, c10_1, c11_1, c12_2 ---------------------------------------- (253) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) by REV(cons(z0, cons(z1, cons(z2, z3))), z4) -> c11(IF(isEmpty(cons(z0, cons(z1, cons(z2, z3)))), cons(z0, dropLast(cons(z1, cons(z2, z3)))), append(z4, last(cons(z2, z3))), z4), DROPLAST(cons(z0, cons(z1, cons(z2, z3))))) ---------------------------------------- (254) Obligation: Complexity Dependency Tuples Problem Rules: isEmpty(nil) -> true isEmpty(cons(z0, z1)) -> false dropLast(nil) -> nil dropLast(cons(z0, nil)) -> nil dropLast(cons(z0, cons(z1, z2))) -> cons(z0, dropLast(cons(z1, z2))) append(nil, z0) -> z0 append(cons(z0, z1), z2) -> cons(z0, append(z1, z2)) last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c11(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), DROPLAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), cons(z0, z1)) -> c11(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), cons(z0, append(z1, last(cons(x0, cons(x1, x2))))), cons(z0, z1)), DROPLAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z2, nil), cons(z0, z1)) -> c11(IF(isEmpty(cons(z2, nil)), dropLast(cons(z2, nil)), cons(z0, append(z1, z2)), cons(z0, z1))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(nil, cons(z0, z1)) -> c11(IF(isEmpty(nil), nil, cons(z0, append(z1, last(nil))), cons(z0, z1))) REV(cons(x0, nil), nil) -> c11(IF(isEmpty(cons(x0, nil)), nil, last(cons(x0, nil)), nil)) REV(cons(z0, nil), x1) -> c11(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), x1) -> c12(IF(false, nil, append(x1, last(cons(z0, nil))), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil)) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil)) REV(cons(z0, z1), nil) -> c10(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil)) REV(x0, cons(x1, nil)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil))) REV(x0, cons(x1, cons(z0, z1))) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1)))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c10(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2))) REV(cons(z0, z1), cons(x1, x2)) -> c10(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(z0, nil), x1) -> c10(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(z1, z2)), x3) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3)) REV(cons(z0, nil), x1) -> c10(IF(false, nil, append(x1, last(cons(z0, nil))), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, z0, nil)) REV(cons(z0, cons(z1, z2)), cons(z3, z4)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(z3, append(z4, last(cons(z1, z2)))), cons(z3, z4)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, nil), cons(z1, z2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(z1, append(z2, z0)), cons(z1, z2))) REV(cons(z0, cons(z1, nil)), z2) -> c11(IF(isEmpty(cons(z0, cons(z1, nil))), cons(z0, dropLast(cons(z1, nil))), append(z2, z1), z2), DROPLAST(cons(z0, cons(z1, nil)))) REV(cons(z0, cons(z1, cons(z2, z3))), z4) -> c11(IF(isEmpty(cons(z0, cons(z1, cons(z2, z3)))), cons(z0, dropLast(cons(z1, cons(z2, z3)))), append(z4, last(cons(z2, z3))), z4), DROPLAST(cons(z0, cons(z1, cons(z2, z3))))) S tuples: LAST(cons(z0, cons(z1, z2))) -> c3(LAST(cons(z1, z2))) DROPLAST(cons(z0, cons(z1, z2))) -> c6(DROPLAST(cons(z1, z2))) APPEND(cons(z0, z1), z2) -> c8(APPEND(z1, z2)) IF(false, z0, z1, z2) -> c14(REV(z0, z1)) REV(cons(z0, z1), x1) -> c11(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), DROPLAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c12(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1), APPEND(x1, last(cons(z0, z1))), LAST(cons(z0, z1))) REV(cons(z0, z1), x1) -> c10(IF(false, dropLast(cons(z0, z1)), append(x1, last(cons(z0, z1))), x1)) REV(cons(z0, z1), nil) -> c11(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), DROPLAST(cons(z0, z1))) REV(x0, cons(x1, nil)) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), DROPLAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c11(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), DROPLAST(x0)) REV(cons(z0, z1), cons(x1, x2)) -> c11(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), DROPLAST(cons(z0, z1))) REV(cons(z0, cons(z1, z2)), x3) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), DROPLAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c11(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), DROPLAST(cons(z0, cons(x1, x2)))) REV(cons(z0, nil), x1) -> c11(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(x0, cons(x1, nil)) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil)), APPEND(cons(x1, nil), last(x0)), LAST(x0)) REV(x0, cons(x1, cons(z0, z1))) -> c12(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1))), APPEND(cons(x1, cons(z0, z1)), last(x0)), LAST(x0)) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, z1), cons(x1, x2)) -> c12(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, z1))), LAST(cons(z0, z1))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(z1, z2)), x3) -> c12(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3), APPEND(x3, last(cons(z0, cons(z1, z2)))), LAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), dropLast(cons(x0, cons(x1, x2))), last(cons(x1, x2)), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3), APPEND(x3, last(cons(x0, cons(z0, cons(z1, z2))))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REV(cons(z0, cons(x1, x2)), x3) -> c12(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3), APPEND(x3, last(cons(z0, cons(x1, x2)))), LAST(cons(z0, cons(x1, x2)))) REV(cons(x0, cons(x1, x2)), nil) -> c12(IF(isEmpty(cons(x0, cons(x1, x2))), cons(x0, dropLast(cons(x1, x2))), last(cons(x0, cons(x1, x2))), nil), LAST(cons(x0, cons(x1, x2)))) REV(cons(z0, z1), nil) -> c12(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil), LAST(cons(z0, z1))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, nil), x1) -> c12(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), z0, nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), last(cons(z1, z2)), nil)) REV(cons(z0, cons(z1, z2)), nil) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z0, cons(z1, z2))), nil)) REV(cons(z0, z1), nil) -> c10(IF(false, dropLast(cons(z0, z1)), last(cons(z0, z1)), nil)) REV(x0, cons(x1, nil)) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, last(x0)), cons(x1, nil))) REV(x0, cons(x1, cons(z0, z1))) -> c10(IF(isEmpty(x0), dropLast(x0), cons(x1, cons(z0, append(z1, last(x0)))), cons(x1, cons(z0, z1)))) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), dropLast(cons(z0, nil)), cons(x1, append(x2, z0)), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), dropLast(cons(z0, cons(z1, z2))), cons(x1, append(x2, last(cons(z1, z2)))), cons(x1, x2))) REV(cons(z0, cons(z1, z2)), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(x1, append(x2, last(cons(z0, cons(z1, z2))))), cons(x1, x2))) REV(cons(z0, z1), cons(x1, x2)) -> c10(IF(false, dropLast(cons(z0, z1)), cons(x1, append(x2, last(cons(z0, z1)))), cons(x1, x2))) REV(cons(z0, nil), x1) -> c10(IF(false, dropLast(cons(z0, nil)), append(x1, z0), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), dropLast(cons(x0, cons(z0, nil))), append(x3, z0), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), dropLast(cons(x0, cons(z0, cons(z1, z2)))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(z1, z2)), x3) -> c10(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), append(x3, last(cons(z1, z2))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, dropLast(cons(z0, cons(x1, x2))), append(x3, last(cons(x1, x2))), x3)) REV(cons(x0, cons(z0, cons(z1, z2))), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, dropLast(cons(z1, z2)))), append(x3, last(cons(x0, cons(z0, cons(z1, z2))))), x3)) REV(cons(z0, cons(x1, x2)), x3) -> c10(IF(false, cons(z0, dropLast(cons(x1, x2))), append(x3, last(cons(z0, cons(x1, x2)))), x3)) REV(cons(z0, cons(z1, z2)), nil) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), last(cons(z1, z2)), nil), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, z2)), cons(z3, z4)) -> c11(IF(isEmpty(cons(z0, cons(z1, z2))), cons(z0, dropLast(cons(z1, z2))), cons(z3, append(z4, last(cons(z1, z2)))), cons(z3, z4)), DROPLAST(cons(z0, cons(z1, z2)))) REV(cons(z0, cons(z1, nil)), z2) -> c11(IF(isEmpty(cons(z0, cons(z1, nil))), cons(z0, dropLast(cons(z1, nil))), append(z2, z1), z2), DROPLAST(cons(z0, cons(z1, nil)))) REV(cons(z0, cons(z1, cons(z2, z3))), z4) -> c11(IF(isEmpty(cons(z0, cons(z1, cons(z2, z3)))), cons(z0, dropLast(cons(z1, cons(z2, z3)))), append(z4, last(cons(z2, z3))), z4), DROPLAST(cons(z0, cons(z1, cons(z2, z3))))) K tuples: REV(nil, x1) -> c12(APPEND(x1, last(nil))) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), cons(x1, x2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c11(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(x0, cons(z0, nil)), x3) -> c11(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), DROPLAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), x1) -> c11(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(nil, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(nil))) REV(x0, cons(x1, x2)) -> c12(APPEND(cons(x1, x2), last(x0))) REV(cons(z0, nil), cons(x1, x2)) -> c12(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2)), APPEND(cons(x1, x2), last(cons(z0, nil)))) REV(nil, cons(x1, x2)) -> c12(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2)), APPEND(cons(x1, x2), last(nil))) REV(cons(x0, cons(z0, nil)), x3) -> c12(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3), APPEND(x3, last(cons(x0, cons(z0, nil)))), LAST(cons(x0, cons(z0, nil)))) REV(cons(z0, nil), nil) -> c12(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), x1) -> c12(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1), APPEND(x1, last(cons(z0, nil)))) REV(cons(z0, nil), nil) -> c10(IF(isEmpty(cons(z0, nil)), nil, last(cons(z0, nil)), nil)) REV(cons(z0, nil), cons(x1, x2)) -> c10(IF(isEmpty(cons(z0, nil)), nil, cons(x1, append(x2, last(cons(z0, nil)))), cons(x1, x2))) REV(nil, cons(x1, x2)) -> c10(IF(isEmpty(nil), nil, cons(x1, append(x2, last(nil))), cons(x1, x2))) REV(cons(z0, nil), x1) -> c10(IF(isEmpty(cons(z0, nil)), nil, append(x1, z0), x1)) REV(cons(x0, cons(z0, nil)), x3) -> c10(IF(isEmpty(cons(x0, cons(z0, nil))), cons(x0, nil), append(x3, last(cons(x0, cons(z0, nil)))), x3)) REV(cons(z0, nil), nil) -> c11(IF(isEmpty(cons(z0, nil)), nil, z0, nil)) REV(cons(z0, nil), cons(z1, z2)) -> c11(IF(isEmpty(cons(z0, nil)), nil, cons(z1, append(z2, z0)), cons(z1, z2))) Defined Rule Symbols: isEmpty_1, dropLast_1, append_2, last_1 Defined Pair Symbols: LAST_1, DROPLAST_1, APPEND_2, IF_4, REV_2 Compound Symbols: c3_1, c6_1, c8_1, c14_1, c11_2, c12_3, c12_1, c10_1, c11_1, c12_2