KILLED proof of input_sNn5jQ81Et.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).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) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (12) CpxRNTS (13) CompletionProof [UPPER BOUND(ID), 0 ms] (14) CpxTypedWeightedCompleteTrs (15) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (16) CpxTypedWeightedCompleteTrs (17) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (18) CpxRNTS (19) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (20) CpxRNTS (21) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms] (22) CpxRNTS (23) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (24) CpxRNTS (25) IntTrsBoundProof [UPPER BOUND(ID), 258 ms] (26) CpxRNTS (27) IntTrsBoundProof [UPPER BOUND(ID), 42 ms] (28) CpxRNTS (29) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (30) CpxRNTS (31) IntTrsBoundProof [UPPER BOUND(ID), 1515 ms] (32) CpxRNTS (33) IntTrsBoundProof [UPPER BOUND(ID), 148 ms] (34) CpxRNTS (35) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (36) CpxRNTS (37) IntTrsBoundProof [UPPER BOUND(ID), 255 ms] (38) CpxRNTS (39) IntTrsBoundProof [UPPER BOUND(ID), 116 ms] (40) CpxRNTS (41) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (42) CpxRNTS (43) IntTrsBoundProof [UPPER BOUND(ID), 3391 ms] (44) CpxRNTS (45) IntTrsBoundProof [UPPER BOUND(ID), 831 ms] (46) CpxRNTS (47) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (48) CdtProblem (49) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (50) CdtProblem (51) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (52) CdtProblem (53) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (54) CdtProblem (55) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 167 ms] (56) CdtProblem (57) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (58) CdtProblem (59) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (60) CdtProblem (61) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 62 ms] (62) CdtProblem (63) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 125 ms] (64) CdtProblem (65) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (66) CdtProblem (67) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (68) CdtProblem (69) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 3 ms] (70) CdtProblem (71) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (72) CdtProblem (73) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (74) CdtProblem (75) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (76) CdtProblem (77) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (78) CdtProblem (79) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (80) CdtProblem (81) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (82) CdtProblem (83) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 3 ms] (84) CdtProblem (85) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (86) CdtProblem (87) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (88) CdtProblem (89) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (90) CdtProblem (91) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 44 ms] (92) CdtProblem (93) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (94) CdtProblem (95) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (96) CdtProblem (97) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (98) CdtProblem (99) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (100) CdtProblem (101) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (102) CdtProblem (103) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (104) CdtProblem (105) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (106) CdtProblem (107) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (108) CdtProblem (109) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (110) CdtProblem (111) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (112) CdtProblem (113) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (114) CdtProblem (115) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (116) CdtProblem (117) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (118) CdtProblem (119) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (120) CdtProblem (121) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (122) CdtProblem (123) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (124) CdtProblem (125) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (126) CdtProblem (127) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (128) CdtProblem (129) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (130) CdtProblem (131) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 55 ms] (132) CdtProblem (133) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (134) CdtProblem (135) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (136) CdtProblem (137) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (138) CdtProblem (139) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (140) CdtProblem (141) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (142) CdtProblem (143) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (144) CdtProblem (145) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (146) CdtProblem (147) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (148) CdtProblem (149) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (150) CdtProblem (151) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (152) CdtProblem (153) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (154) CdtProblem (155) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (156) CdtProblem (157) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (158) CdtProblem (159) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (160) CdtProblem (161) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (162) CdtProblem (163) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (164) CdtProblem (165) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (166) CdtProblem (167) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (168) CdtProblem (169) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (170) CdtProblem (171) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (172) CdtProblem (173) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (174) CdtProblem (175) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (176) CdtProblem (177) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (178) CdtProblem (179) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (180) CdtProblem (181) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (182) CdtProblem (183) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (184) CdtProblem (185) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (186) CdtProblem (187) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (188) CdtProblem (189) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (190) CdtProblem (191) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (192) CdtProblem (193) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (194) CdtProblem (195) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (196) CdtProblem (197) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (198) CdtProblem (199) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (200) CdtProblem (201) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (202) CdtProblem (203) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (204) CdtProblem (205) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (206) CdtProblem (207) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (208) CdtProblem (209) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 36 ms] (210) CdtProblem (211) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (212) CdtProblem (213) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (214) CdtProblem (215) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (216) 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: bsort(nil) -> nil bsort(.(x, y)) -> last(.(bubble(.(x, y)), bsort(butlast(bubble(.(x, y)))))) bubble(nil) -> nil bubble(.(x, nil)) -> .(x, nil) bubble(.(x, .(y, z))) -> if(<=(x, y), .(y, bubble(.(x, z))), .(x, bubble(.(y, z)))) last(nil) -> 0 last(.(x, nil)) -> x last(.(x, .(y, z))) -> last(.(y, z)) butlast(nil) -> nil butlast(.(x, nil)) -> nil butlast(.(x, .(y, z))) -> .(x, butlast(.(y, z))) S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (1) RenamingProof (BOTH BOUNDS(ID, ID)) Renamed function symbols to avoid clashes with predefined symbol. ---------------------------------------- (2) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: bsort(nil) -> nil bsort(.(x, y)) -> last(.(bubble(.(x, y)), bsort(butlast(bubble(.(x, y)))))) bubble(nil) -> nil bubble(.(x, nil)) -> .(x, nil) bubble(.(x, .(y, z))) -> if(<=(x, y), .(y, bubble(.(x, z))), .(x, bubble(.(y, z)))) last(nil) -> 0' last(.(x, nil)) -> x last(.(x, .(y, z))) -> last(.(y, z)) butlast(nil) -> nil butlast(.(x, nil)) -> nil butlast(.(x, .(y, z))) -> .(x, butlast(.(y, z))) S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (3) RelTrsToTrsProof (UPPER BOUND(ID)) transformed relative TRS to TRS ---------------------------------------- (4) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: bsort(nil) -> nil bsort(.(x, y)) -> last(.(bubble(.(x, y)), bsort(butlast(bubble(.(x, y)))))) bubble(nil) -> nil bubble(.(x, nil)) -> .(x, nil) bubble(.(x, .(y, z))) -> if(<=(x, y), .(y, bubble(.(x, z))), .(x, bubble(.(y, z)))) last(nil) -> 0 last(.(x, nil)) -> x last(.(x, .(y, z))) -> last(.(y, z)) butlast(nil) -> nil butlast(.(x, nil)) -> nil butlast(.(x, .(y, z))) -> .(x, butlast(.(y, z))) S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (5) RelTrsToWeightedTrsProof (UPPER BOUND(ID)) Transformed relative TRS to weighted TRS ---------------------------------------- (6) Obligation: The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: bsort(nil) -> nil [1] bsort(.(x, y)) -> last(.(bubble(.(x, y)), bsort(butlast(bubble(.(x, y)))))) [1] bubble(nil) -> nil [1] bubble(.(x, nil)) -> .(x, nil) [1] bubble(.(x, .(y, z))) -> if(<=(x, y), .(y, bubble(.(x, z))), .(x, bubble(.(y, z)))) [1] last(nil) -> 0 [1] last(.(x, nil)) -> x [1] last(.(x, .(y, z))) -> last(.(y, z)) [1] butlast(nil) -> nil [1] butlast(.(x, nil)) -> nil [1] butlast(.(x, .(y, z))) -> .(x, butlast(.(y, z))) [1] Rewrite Strategy: INNERMOST ---------------------------------------- (7) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (8) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: bsort(nil) -> nil [1] bsort(.(x, y)) -> last(.(bubble(.(x, y)), bsort(butlast(bubble(.(x, y)))))) [1] bubble(nil) -> nil [1] bubble(.(x, nil)) -> .(x, nil) [1] bubble(.(x, .(y, z))) -> if(<=(x, y), .(y, bubble(.(x, z))), .(x, bubble(.(y, z)))) [1] last(nil) -> 0 [1] last(.(x, nil)) -> x [1] last(.(x, .(y, z))) -> last(.(y, z)) [1] butlast(nil) -> nil [1] butlast(.(x, nil)) -> nil [1] butlast(.(x, .(y, z))) -> .(x, butlast(.(y, z))) [1] The TRS has the following type information: bsort :: nil:.:if:0 -> nil:.:if:0 nil :: nil:.:if:0 . :: nil:.:if:0 -> nil:.:if:0 -> nil:.:if:0 last :: nil:.:if:0 -> nil:.:if:0 bubble :: nil:.:if:0 -> nil:.:if:0 butlast :: nil:.:if:0 -> nil:.:if:0 if :: <= -> nil:.:if:0 -> nil:.:if:0 -> nil:.:if:0 <= :: nil:.:if:0 -> nil:.:if:0 -> <= 0 :: nil:.:if:0 Rewrite Strategy: INNERMOST ---------------------------------------- (9) 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: bsort(v0) -> null_bsort [0] bubble(v0) -> null_bubble [0] last(v0) -> null_last [0] butlast(v0) -> null_butlast [0] And the following fresh constants: null_bsort, null_bubble, null_last, null_butlast, const ---------------------------------------- (10) 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: bsort(nil) -> nil [1] bsort(.(x, y)) -> last(.(bubble(.(x, y)), bsort(butlast(bubble(.(x, y)))))) [1] bubble(nil) -> nil [1] bubble(.(x, nil)) -> .(x, nil) [1] bubble(.(x, .(y, z))) -> if(<=(x, y), .(y, bubble(.(x, z))), .(x, bubble(.(y, z)))) [1] last(nil) -> 0 [1] last(.(x, nil)) -> x [1] last(.(x, .(y, z))) -> last(.(y, z)) [1] butlast(nil) -> nil [1] butlast(.(x, nil)) -> nil [1] butlast(.(x, .(y, z))) -> .(x, butlast(.(y, z))) [1] bsort(v0) -> null_bsort [0] bubble(v0) -> null_bubble [0] last(v0) -> null_last [0] butlast(v0) -> null_butlast [0] The TRS has the following type information: bsort :: nil:.:if:0:null_bsort:null_bubble:null_last:null_butlast -> nil:.:if:0:null_bsort:null_bubble:null_last:null_butlast nil :: nil:.:if:0:null_bsort:null_bubble:null_last:null_butlast . :: nil:.:if:0:null_bsort:null_bubble:null_last:null_butlast -> nil:.:if:0:null_bsort:null_bubble:null_last:null_butlast -> nil:.:if:0:null_bsort:null_bubble:null_last:null_butlast last :: nil:.:if:0:null_bsort:null_bubble:null_last:null_butlast -> nil:.:if:0:null_bsort:null_bubble:null_last:null_butlast bubble :: nil:.:if:0:null_bsort:null_bubble:null_last:null_butlast -> nil:.:if:0:null_bsort:null_bubble:null_last:null_butlast butlast :: nil:.:if:0:null_bsort:null_bubble:null_last:null_butlast -> nil:.:if:0:null_bsort:null_bubble:null_last:null_butlast if :: <= -> nil:.:if:0:null_bsort:null_bubble:null_last:null_butlast -> nil:.:if:0:null_bsort:null_bubble:null_last:null_butlast -> nil:.:if:0:null_bsort:null_bubble:null_last:null_butlast <= :: nil:.:if:0:null_bsort:null_bubble:null_last:null_butlast -> nil:.:if:0:null_bsort:null_bubble:null_last:null_butlast -> <= 0 :: nil:.:if:0:null_bsort:null_bubble:null_last:null_butlast null_bsort :: nil:.:if:0:null_bsort:null_bubble:null_last:null_butlast null_bubble :: nil:.:if:0:null_bsort:null_bubble:null_last:null_butlast null_last :: nil:.:if:0:null_bsort:null_bubble:null_last:null_butlast null_butlast :: nil:.:if:0:null_bsort:null_bubble:null_last:null_butlast const :: <= Rewrite Strategy: INNERMOST ---------------------------------------- (11) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: nil => 1 0 => 0 null_bsort => 0 null_bubble => 0 null_last => 0 null_butlast => 0 const => 0 ---------------------------------------- (12) Obligation: Complexity RNTS consisting of the following rules: bsort(z') -{ 1 }-> last(1 + bubble(1 + x + y) + bsort(butlast(bubble(1 + x + y)))) :|: z' = 1 + x + y, x >= 0, y >= 0 bsort(z') -{ 1 }-> 1 :|: z' = 1 bsort(z') -{ 0 }-> 0 :|: v0 >= 0, z' = v0 bubble(z') -{ 1 }-> 1 :|: z' = 1 bubble(z') -{ 0 }-> 0 :|: v0 >= 0, z' = v0 bubble(z') -{ 1 }-> 1 + x + 1 :|: x >= 0, z' = 1 + x + 1 bubble(z') -{ 1 }-> 1 + (1 + x + y) + (1 + y + bubble(1 + x + z)) + (1 + x + bubble(1 + y + z)) :|: z' = 1 + x + (1 + y + z), z >= 0, x >= 0, y >= 0 butlast(z') -{ 1 }-> 1 :|: z' = 1 butlast(z') -{ 1 }-> 1 :|: x >= 0, z' = 1 + x + 1 butlast(z') -{ 0 }-> 0 :|: v0 >= 0, z' = v0 butlast(z') -{ 1 }-> 1 + x + butlast(1 + y + z) :|: z' = 1 + x + (1 + y + z), z >= 0, x >= 0, y >= 0 last(z') -{ 1 }-> x :|: x >= 0, z' = 1 + x + 1 last(z') -{ 1 }-> last(1 + y + z) :|: z' = 1 + x + (1 + y + z), z >= 0, x >= 0, y >= 0 last(z') -{ 1 }-> 0 :|: z' = 1 last(z') -{ 0 }-> 0 :|: v0 >= 0, z' = v0 Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (13) CompletionProof (UPPER BOUND(ID)) The transformation into a RNTS is sound, since: (a) The obligation is a constructor system where every type has a constant constructor, (b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols: none (c) The following functions are completely defined: bubble_1 bsort_1 butlast_1 last_1 Due to the following rules being added: bubble(v0) -> null_bubble [0] bsort(v0) -> null_bsort [0] butlast(v0) -> null_butlast [0] last(v0) -> null_last [0] And the following fresh constants: null_bubble, null_bsort, null_butlast, null_last, const ---------------------------------------- (14) 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: bsort(nil) -> nil [1] bsort(.(x, y)) -> last(.(bubble(.(x, y)), bsort(butlast(bubble(.(x, y)))))) [1] bubble(nil) -> nil [1] bubble(.(x, nil)) -> .(x, nil) [1] bubble(.(x, .(y, z))) -> if(<=(x, y), .(y, bubble(.(x, z))), .(x, bubble(.(y, z)))) [1] last(nil) -> 0 [1] last(.(x, nil)) -> x [1] last(.(x, .(y, z))) -> last(.(y, z)) [1] butlast(nil) -> nil [1] butlast(.(x, nil)) -> nil [1] butlast(.(x, .(y, z))) -> .(x, butlast(.(y, z))) [1] bubble(v0) -> null_bubble [0] bsort(v0) -> null_bsort [0] butlast(v0) -> null_butlast [0] last(v0) -> null_last [0] The TRS has the following type information: bsort :: nil:.:if:0:null_bubble:null_bsort:null_butlast:null_last -> nil:.:if:0:null_bubble:null_bsort:null_butlast:null_last nil :: nil:.:if:0:null_bubble:null_bsort:null_butlast:null_last . :: nil:.:if:0:null_bubble:null_bsort:null_butlast:null_last -> nil:.:if:0:null_bubble:null_bsort:null_butlast:null_last -> nil:.:if:0:null_bubble:null_bsort:null_butlast:null_last last :: nil:.:if:0:null_bubble:null_bsort:null_butlast:null_last -> nil:.:if:0:null_bubble:null_bsort:null_butlast:null_last bubble :: nil:.:if:0:null_bubble:null_bsort:null_butlast:null_last -> nil:.:if:0:null_bubble:null_bsort:null_butlast:null_last butlast :: nil:.:if:0:null_bubble:null_bsort:null_butlast:null_last -> nil:.:if:0:null_bubble:null_bsort:null_butlast:null_last if :: <= -> nil:.:if:0:null_bubble:null_bsort:null_butlast:null_last -> nil:.:if:0:null_bubble:null_bsort:null_butlast:null_last -> nil:.:if:0:null_bubble:null_bsort:null_butlast:null_last <= :: nil:.:if:0:null_bubble:null_bsort:null_butlast:null_last -> nil:.:if:0:null_bubble:null_bsort:null_butlast:null_last -> <= 0 :: nil:.:if:0:null_bubble:null_bsort:null_butlast:null_last null_bubble :: nil:.:if:0:null_bubble:null_bsort:null_butlast:null_last null_bsort :: nil:.:if:0:null_bubble:null_bsort:null_butlast:null_last null_butlast :: nil:.:if:0:null_bubble:null_bsort:null_butlast:null_last null_last :: nil:.:if:0:null_bubble:null_bsort:null_butlast:null_last const :: <= Rewrite Strategy: INNERMOST ---------------------------------------- (15) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (16) 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: bsort(nil) -> nil [1] bsort(.(x, nil)) -> last(.(.(x, nil), bsort(butlast(.(x, nil))))) [3] bsort(.(x, nil)) -> last(.(.(x, nil), bsort(butlast(null_bubble)))) [2] bsort(.(x, .(y', z'))) -> last(.(if(<=(x, y'), .(y', bubble(.(x, z'))), .(x, bubble(.(y', z')))), bsort(butlast(if(<=(x, y'), .(y', bubble(.(x, z'))), .(x, bubble(.(y', z')))))))) [3] bsort(.(x, .(y', z'))) -> last(.(if(<=(x, y'), .(y', bubble(.(x, z'))), .(x, bubble(.(y', z')))), bsort(butlast(null_bubble)))) [2] bsort(.(x, nil)) -> last(.(null_bubble, bsort(butlast(.(x, nil))))) [2] bsort(.(x, .(y'', z''))) -> last(.(null_bubble, bsort(butlast(if(<=(x, y''), .(y'', bubble(.(x, z''))), .(x, bubble(.(y'', z'')))))))) [2] bsort(.(x, y)) -> last(.(null_bubble, bsort(butlast(null_bubble)))) [1] bubble(nil) -> nil [1] bubble(.(x, nil)) -> .(x, nil) [1] bubble(.(x, .(y, z))) -> if(<=(x, y), .(y, bubble(.(x, z))), .(x, bubble(.(y, z)))) [1] last(nil) -> 0 [1] last(.(x, nil)) -> x [1] last(.(x, .(y, z))) -> last(.(y, z)) [1] butlast(nil) -> nil [1] butlast(.(x, nil)) -> nil [1] butlast(.(x, .(y, z))) -> .(x, butlast(.(y, z))) [1] bubble(v0) -> null_bubble [0] bsort(v0) -> null_bsort [0] butlast(v0) -> null_butlast [0] last(v0) -> null_last [0] The TRS has the following type information: bsort :: nil:.:if:0:null_bubble:null_bsort:null_butlast:null_last -> nil:.:if:0:null_bubble:null_bsort:null_butlast:null_last nil :: nil:.:if:0:null_bubble:null_bsort:null_butlast:null_last . :: nil:.:if:0:null_bubble:null_bsort:null_butlast:null_last -> nil:.:if:0:null_bubble:null_bsort:null_butlast:null_last -> nil:.:if:0:null_bubble:null_bsort:null_butlast:null_last last :: nil:.:if:0:null_bubble:null_bsort:null_butlast:null_last -> nil:.:if:0:null_bubble:null_bsort:null_butlast:null_last bubble :: nil:.:if:0:null_bubble:null_bsort:null_butlast:null_last -> nil:.:if:0:null_bubble:null_bsort:null_butlast:null_last butlast :: nil:.:if:0:null_bubble:null_bsort:null_butlast:null_last -> nil:.:if:0:null_bubble:null_bsort:null_butlast:null_last if :: <= -> nil:.:if:0:null_bubble:null_bsort:null_butlast:null_last -> nil:.:if:0:null_bubble:null_bsort:null_butlast:null_last -> nil:.:if:0:null_bubble:null_bsort:null_butlast:null_last <= :: nil:.:if:0:null_bubble:null_bsort:null_butlast:null_last -> nil:.:if:0:null_bubble:null_bsort:null_butlast:null_last -> <= 0 :: nil:.:if:0:null_bubble:null_bsort:null_butlast:null_last null_bubble :: nil:.:if:0:null_bubble:null_bsort:null_butlast:null_last null_bsort :: nil:.:if:0:null_bubble:null_bsort:null_butlast:null_last null_butlast :: nil:.:if:0:null_bubble:null_bsort:null_butlast:null_last null_last :: nil:.:if:0:null_bubble:null_bsort:null_butlast:null_last const :: <= Rewrite Strategy: INNERMOST ---------------------------------------- (17) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: nil => 1 0 => 0 null_bubble => 0 null_bsort => 0 null_butlast => 0 null_last => 0 const => 0 ---------------------------------------- (18) Obligation: Complexity RNTS consisting of the following rules: bsort(z1) -{ 1 }-> last(1 + 0 + bsort(butlast(0))) :|: x >= 0, y >= 0, z1 = 1 + x + y bsort(z1) -{ 2 }-> last(1 + 0 + bsort(butlast(1 + x + 1))) :|: x >= 0, z1 = 1 + x + 1 bsort(z1) -{ 2 }-> last(1 + 0 + bsort(butlast(1 + (1 + x + y'') + (1 + y'' + bubble(1 + x + z'')) + (1 + x + bubble(1 + y'' + z''))))) :|: z'' >= 0, x >= 0, y'' >= 0, z1 = 1 + x + (1 + y'' + z'') bsort(z1) -{ 2 }-> last(1 + (1 + x + 1) + bsort(butlast(0))) :|: x >= 0, z1 = 1 + x + 1 bsort(z1) -{ 3 }-> last(1 + (1 + x + 1) + bsort(butlast(1 + x + 1))) :|: x >= 0, z1 = 1 + x + 1 bsort(z1) -{ 2 }-> last(1 + (1 + (1 + x + y') + (1 + y' + bubble(1 + x + z')) + (1 + x + bubble(1 + y' + z'))) + bsort(butlast(0))) :|: x >= 0, z' >= 0, y' >= 0, z1 = 1 + x + (1 + y' + z') bsort(z1) -{ 3 }-> last(1 + (1 + (1 + x + y') + (1 + y' + bubble(1 + x + z')) + (1 + x + bubble(1 + y' + z'))) + bsort(butlast(1 + (1 + x + y') + (1 + y' + bubble(1 + x + z')) + (1 + x + bubble(1 + y' + z'))))) :|: x >= 0, z' >= 0, y' >= 0, z1 = 1 + x + (1 + y' + z') bsort(z1) -{ 1 }-> 1 :|: z1 = 1 bsort(z1) -{ 0 }-> 0 :|: z1 = v0, v0 >= 0 bubble(z1) -{ 1 }-> 1 :|: z1 = 1 bubble(z1) -{ 0 }-> 0 :|: z1 = v0, v0 >= 0 bubble(z1) -{ 1 }-> 1 + x + 1 :|: x >= 0, z1 = 1 + x + 1 bubble(z1) -{ 1 }-> 1 + (1 + x + y) + (1 + y + bubble(1 + x + z)) + (1 + x + bubble(1 + y + z)) :|: z >= 0, z1 = 1 + x + (1 + y + z), x >= 0, y >= 0 butlast(z1) -{ 1 }-> 1 :|: z1 = 1 butlast(z1) -{ 1 }-> 1 :|: x >= 0, z1 = 1 + x + 1 butlast(z1) -{ 0 }-> 0 :|: z1 = v0, v0 >= 0 butlast(z1) -{ 1 }-> 1 + x + butlast(1 + y + z) :|: z >= 0, z1 = 1 + x + (1 + y + z), x >= 0, y >= 0 last(z1) -{ 1 }-> x :|: x >= 0, z1 = 1 + x + 1 last(z1) -{ 1 }-> last(1 + y + z) :|: z >= 0, z1 = 1 + x + (1 + y + z), x >= 0, y >= 0 last(z1) -{ 1 }-> 0 :|: z1 = 1 last(z1) -{ 0 }-> 0 :|: z1 = v0, v0 >= 0 ---------------------------------------- (19) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: bsort(z1) -{ 1 }-> last(1 + 0 + bsort(butlast(0))) :|: x >= 0, y >= 0, z1 = 1 + x + y bsort(z1) -{ 2 }-> last(1 + 0 + bsort(butlast(1 + (z1 - 2) + 1))) :|: z1 - 2 >= 0 bsort(z1) -{ 2 }-> last(1 + 0 + bsort(butlast(1 + (1 + x + y'') + (1 + y'' + bubble(1 + x + z'')) + (1 + x + bubble(1 + y'' + z''))))) :|: z'' >= 0, x >= 0, y'' >= 0, z1 = 1 + x + (1 + y'' + z'') bsort(z1) -{ 2 }-> last(1 + (1 + (z1 - 2) + 1) + bsort(butlast(0))) :|: z1 - 2 >= 0 bsort(z1) -{ 3 }-> last(1 + (1 + (z1 - 2) + 1) + bsort(butlast(1 + (z1 - 2) + 1))) :|: z1 - 2 >= 0 bsort(z1) -{ 2 }-> last(1 + (1 + (1 + x + y') + (1 + y' + bubble(1 + x + z')) + (1 + x + bubble(1 + y' + z'))) + bsort(butlast(0))) :|: x >= 0, z' >= 0, y' >= 0, z1 = 1 + x + (1 + y' + z') bsort(z1) -{ 3 }-> last(1 + (1 + (1 + x + y') + (1 + y' + bubble(1 + x + z')) + (1 + x + bubble(1 + y' + z'))) + bsort(butlast(1 + (1 + x + y') + (1 + y' + bubble(1 + x + z')) + (1 + x + bubble(1 + y' + z'))))) :|: x >= 0, z' >= 0, y' >= 0, z1 = 1 + x + (1 + y' + z') bsort(z1) -{ 1 }-> 1 :|: z1 = 1 bsort(z1) -{ 0 }-> 0 :|: z1 >= 0 bubble(z1) -{ 1 }-> 1 :|: z1 = 1 bubble(z1) -{ 0 }-> 0 :|: z1 >= 0 bubble(z1) -{ 1 }-> 1 + (z1 - 2) + 1 :|: z1 - 2 >= 0 bubble(z1) -{ 1 }-> 1 + (1 + x + y) + (1 + y + bubble(1 + x + z)) + (1 + x + bubble(1 + y + z)) :|: z >= 0, z1 = 1 + x + (1 + y + z), x >= 0, y >= 0 butlast(z1) -{ 1 }-> 1 :|: z1 = 1 butlast(z1) -{ 1 }-> 1 :|: z1 - 2 >= 0 butlast(z1) -{ 0 }-> 0 :|: z1 >= 0 butlast(z1) -{ 1 }-> 1 + x + butlast(1 + y + z) :|: z >= 0, z1 = 1 + x + (1 + y + z), x >= 0, y >= 0 last(z1) -{ 1 }-> last(1 + y + z) :|: z >= 0, z1 = 1 + x + (1 + y + z), x >= 0, y >= 0 last(z1) -{ 1 }-> 0 :|: z1 = 1 last(z1) -{ 0 }-> 0 :|: z1 >= 0 last(z1) -{ 1 }-> z1 - 2 :|: z1 - 2 >= 0 ---------------------------------------- (21) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { last } { bubble } { butlast } { bsort } ---------------------------------------- (22) Obligation: Complexity RNTS consisting of the following rules: bsort(z1) -{ 1 }-> last(1 + 0 + bsort(butlast(0))) :|: x >= 0, y >= 0, z1 = 1 + x + y bsort(z1) -{ 2 }-> last(1 + 0 + bsort(butlast(1 + (z1 - 2) + 1))) :|: z1 - 2 >= 0 bsort(z1) -{ 2 }-> last(1 + 0 + bsort(butlast(1 + (1 + x + y'') + (1 + y'' + bubble(1 + x + z'')) + (1 + x + bubble(1 + y'' + z''))))) :|: z'' >= 0, x >= 0, y'' >= 0, z1 = 1 + x + (1 + y'' + z'') bsort(z1) -{ 2 }-> last(1 + (1 + (z1 - 2) + 1) + bsort(butlast(0))) :|: z1 - 2 >= 0 bsort(z1) -{ 3 }-> last(1 + (1 + (z1 - 2) + 1) + bsort(butlast(1 + (z1 - 2) + 1))) :|: z1 - 2 >= 0 bsort(z1) -{ 2 }-> last(1 + (1 + (1 + x + y') + (1 + y' + bubble(1 + x + z')) + (1 + x + bubble(1 + y' + z'))) + bsort(butlast(0))) :|: x >= 0, z' >= 0, y' >= 0, z1 = 1 + x + (1 + y' + z') bsort(z1) -{ 3 }-> last(1 + (1 + (1 + x + y') + (1 + y' + bubble(1 + x + z')) + (1 + x + bubble(1 + y' + z'))) + bsort(butlast(1 + (1 + x + y') + (1 + y' + bubble(1 + x + z')) + (1 + x + bubble(1 + y' + z'))))) :|: x >= 0, z' >= 0, y' >= 0, z1 = 1 + x + (1 + y' + z') bsort(z1) -{ 1 }-> 1 :|: z1 = 1 bsort(z1) -{ 0 }-> 0 :|: z1 >= 0 bubble(z1) -{ 1 }-> 1 :|: z1 = 1 bubble(z1) -{ 0 }-> 0 :|: z1 >= 0 bubble(z1) -{ 1 }-> 1 + (z1 - 2) + 1 :|: z1 - 2 >= 0 bubble(z1) -{ 1 }-> 1 + (1 + x + y) + (1 + y + bubble(1 + x + z)) + (1 + x + bubble(1 + y + z)) :|: z >= 0, z1 = 1 + x + (1 + y + z), x >= 0, y >= 0 butlast(z1) -{ 1 }-> 1 :|: z1 = 1 butlast(z1) -{ 1 }-> 1 :|: z1 - 2 >= 0 butlast(z1) -{ 0 }-> 0 :|: z1 >= 0 butlast(z1) -{ 1 }-> 1 + x + butlast(1 + y + z) :|: z >= 0, z1 = 1 + x + (1 + y + z), x >= 0, y >= 0 last(z1) -{ 1 }-> last(1 + y + z) :|: z >= 0, z1 = 1 + x + (1 + y + z), x >= 0, y >= 0 last(z1) -{ 1 }-> 0 :|: z1 = 1 last(z1) -{ 0 }-> 0 :|: z1 >= 0 last(z1) -{ 1 }-> z1 - 2 :|: z1 - 2 >= 0 Function symbols to be analyzed: {last}, {bubble}, {butlast}, {bsort} ---------------------------------------- (23) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (24) Obligation: Complexity RNTS consisting of the following rules: bsort(z1) -{ 1 }-> last(1 + 0 + bsort(butlast(0))) :|: x >= 0, y >= 0, z1 = 1 + x + y bsort(z1) -{ 2 }-> last(1 + 0 + bsort(butlast(1 + (z1 - 2) + 1))) :|: z1 - 2 >= 0 bsort(z1) -{ 2 }-> last(1 + 0 + bsort(butlast(1 + (1 + x + y'') + (1 + y'' + bubble(1 + x + z'')) + (1 + x + bubble(1 + y'' + z''))))) :|: z'' >= 0, x >= 0, y'' >= 0, z1 = 1 + x + (1 + y'' + z'') bsort(z1) -{ 2 }-> last(1 + (1 + (z1 - 2) + 1) + bsort(butlast(0))) :|: z1 - 2 >= 0 bsort(z1) -{ 3 }-> last(1 + (1 + (z1 - 2) + 1) + bsort(butlast(1 + (z1 - 2) + 1))) :|: z1 - 2 >= 0 bsort(z1) -{ 2 }-> last(1 + (1 + (1 + x + y') + (1 + y' + bubble(1 + x + z')) + (1 + x + bubble(1 + y' + z'))) + bsort(butlast(0))) :|: x >= 0, z' >= 0, y' >= 0, z1 = 1 + x + (1 + y' + z') bsort(z1) -{ 3 }-> last(1 + (1 + (1 + x + y') + (1 + y' + bubble(1 + x + z')) + (1 + x + bubble(1 + y' + z'))) + bsort(butlast(1 + (1 + x + y') + (1 + y' + bubble(1 + x + z')) + (1 + x + bubble(1 + y' + z'))))) :|: x >= 0, z' >= 0, y' >= 0, z1 = 1 + x + (1 + y' + z') bsort(z1) -{ 1 }-> 1 :|: z1 = 1 bsort(z1) -{ 0 }-> 0 :|: z1 >= 0 bubble(z1) -{ 1 }-> 1 :|: z1 = 1 bubble(z1) -{ 0 }-> 0 :|: z1 >= 0 bubble(z1) -{ 1 }-> 1 + (z1 - 2) + 1 :|: z1 - 2 >= 0 bubble(z1) -{ 1 }-> 1 + (1 + x + y) + (1 + y + bubble(1 + x + z)) + (1 + x + bubble(1 + y + z)) :|: z >= 0, z1 = 1 + x + (1 + y + z), x >= 0, y >= 0 butlast(z1) -{ 1 }-> 1 :|: z1 = 1 butlast(z1) -{ 1 }-> 1 :|: z1 - 2 >= 0 butlast(z1) -{ 0 }-> 0 :|: z1 >= 0 butlast(z1) -{ 1 }-> 1 + x + butlast(1 + y + z) :|: z >= 0, z1 = 1 + x + (1 + y + z), x >= 0, y >= 0 last(z1) -{ 1 }-> last(1 + y + z) :|: z >= 0, z1 = 1 + x + (1 + y + z), x >= 0, y >= 0 last(z1) -{ 1 }-> 0 :|: z1 = 1 last(z1) -{ 0 }-> 0 :|: z1 >= 0 last(z1) -{ 1 }-> z1 - 2 :|: z1 - 2 >= 0 Function symbols to be analyzed: {last}, {bubble}, {butlast}, {bsort} ---------------------------------------- (25) 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: z1 ---------------------------------------- (26) Obligation: Complexity RNTS consisting of the following rules: bsort(z1) -{ 1 }-> last(1 + 0 + bsort(butlast(0))) :|: x >= 0, y >= 0, z1 = 1 + x + y bsort(z1) -{ 2 }-> last(1 + 0 + bsort(butlast(1 + (z1 - 2) + 1))) :|: z1 - 2 >= 0 bsort(z1) -{ 2 }-> last(1 + 0 + bsort(butlast(1 + (1 + x + y'') + (1 + y'' + bubble(1 + x + z'')) + (1 + x + bubble(1 + y'' + z''))))) :|: z'' >= 0, x >= 0, y'' >= 0, z1 = 1 + x + (1 + y'' + z'') bsort(z1) -{ 2 }-> last(1 + (1 + (z1 - 2) + 1) + bsort(butlast(0))) :|: z1 - 2 >= 0 bsort(z1) -{ 3 }-> last(1 + (1 + (z1 - 2) + 1) + bsort(butlast(1 + (z1 - 2) + 1))) :|: z1 - 2 >= 0 bsort(z1) -{ 2 }-> last(1 + (1 + (1 + x + y') + (1 + y' + bubble(1 + x + z')) + (1 + x + bubble(1 + y' + z'))) + bsort(butlast(0))) :|: x >= 0, z' >= 0, y' >= 0, z1 = 1 + x + (1 + y' + z') bsort(z1) -{ 3 }-> last(1 + (1 + (1 + x + y') + (1 + y' + bubble(1 + x + z')) + (1 + x + bubble(1 + y' + z'))) + bsort(butlast(1 + (1 + x + y') + (1 + y' + bubble(1 + x + z')) + (1 + x + bubble(1 + y' + z'))))) :|: x >= 0, z' >= 0, y' >= 0, z1 = 1 + x + (1 + y' + z') bsort(z1) -{ 1 }-> 1 :|: z1 = 1 bsort(z1) -{ 0 }-> 0 :|: z1 >= 0 bubble(z1) -{ 1 }-> 1 :|: z1 = 1 bubble(z1) -{ 0 }-> 0 :|: z1 >= 0 bubble(z1) -{ 1 }-> 1 + (z1 - 2) + 1 :|: z1 - 2 >= 0 bubble(z1) -{ 1 }-> 1 + (1 + x + y) + (1 + y + bubble(1 + x + z)) + (1 + x + bubble(1 + y + z)) :|: z >= 0, z1 = 1 + x + (1 + y + z), x >= 0, y >= 0 butlast(z1) -{ 1 }-> 1 :|: z1 = 1 butlast(z1) -{ 1 }-> 1 :|: z1 - 2 >= 0 butlast(z1) -{ 0 }-> 0 :|: z1 >= 0 butlast(z1) -{ 1 }-> 1 + x + butlast(1 + y + z) :|: z >= 0, z1 = 1 + x + (1 + y + z), x >= 0, y >= 0 last(z1) -{ 1 }-> last(1 + y + z) :|: z >= 0, z1 = 1 + x + (1 + y + z), x >= 0, y >= 0 last(z1) -{ 1 }-> 0 :|: z1 = 1 last(z1) -{ 0 }-> 0 :|: z1 >= 0 last(z1) -{ 1 }-> z1 - 2 :|: z1 - 2 >= 0 Function symbols to be analyzed: {last}, {bubble}, {butlast}, {bsort} Previous analysis results are: last: runtime: ?, size: O(n^1) [z1] ---------------------------------------- (27) 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: 2 + z1 ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: bsort(z1) -{ 1 }-> last(1 + 0 + bsort(butlast(0))) :|: x >= 0, y >= 0, z1 = 1 + x + y bsort(z1) -{ 2 }-> last(1 + 0 + bsort(butlast(1 + (z1 - 2) + 1))) :|: z1 - 2 >= 0 bsort(z1) -{ 2 }-> last(1 + 0 + bsort(butlast(1 + (1 + x + y'') + (1 + y'' + bubble(1 + x + z'')) + (1 + x + bubble(1 + y'' + z''))))) :|: z'' >= 0, x >= 0, y'' >= 0, z1 = 1 + x + (1 + y'' + z'') bsort(z1) -{ 2 }-> last(1 + (1 + (z1 - 2) + 1) + bsort(butlast(0))) :|: z1 - 2 >= 0 bsort(z1) -{ 3 }-> last(1 + (1 + (z1 - 2) + 1) + bsort(butlast(1 + (z1 - 2) + 1))) :|: z1 - 2 >= 0 bsort(z1) -{ 2 }-> last(1 + (1 + (1 + x + y') + (1 + y' + bubble(1 + x + z')) + (1 + x + bubble(1 + y' + z'))) + bsort(butlast(0))) :|: x >= 0, z' >= 0, y' >= 0, z1 = 1 + x + (1 + y' + z') bsort(z1) -{ 3 }-> last(1 + (1 + (1 + x + y') + (1 + y' + bubble(1 + x + z')) + (1 + x + bubble(1 + y' + z'))) + bsort(butlast(1 + (1 + x + y') + (1 + y' + bubble(1 + x + z')) + (1 + x + bubble(1 + y' + z'))))) :|: x >= 0, z' >= 0, y' >= 0, z1 = 1 + x + (1 + y' + z') bsort(z1) -{ 1 }-> 1 :|: z1 = 1 bsort(z1) -{ 0 }-> 0 :|: z1 >= 0 bubble(z1) -{ 1 }-> 1 :|: z1 = 1 bubble(z1) -{ 0 }-> 0 :|: z1 >= 0 bubble(z1) -{ 1 }-> 1 + (z1 - 2) + 1 :|: z1 - 2 >= 0 bubble(z1) -{ 1 }-> 1 + (1 + x + y) + (1 + y + bubble(1 + x + z)) + (1 + x + bubble(1 + y + z)) :|: z >= 0, z1 = 1 + x + (1 + y + z), x >= 0, y >= 0 butlast(z1) -{ 1 }-> 1 :|: z1 = 1 butlast(z1) -{ 1 }-> 1 :|: z1 - 2 >= 0 butlast(z1) -{ 0 }-> 0 :|: z1 >= 0 butlast(z1) -{ 1 }-> 1 + x + butlast(1 + y + z) :|: z >= 0, z1 = 1 + x + (1 + y + z), x >= 0, y >= 0 last(z1) -{ 1 }-> last(1 + y + z) :|: z >= 0, z1 = 1 + x + (1 + y + z), x >= 0, y >= 0 last(z1) -{ 1 }-> 0 :|: z1 = 1 last(z1) -{ 0 }-> 0 :|: z1 >= 0 last(z1) -{ 1 }-> z1 - 2 :|: z1 - 2 >= 0 Function symbols to be analyzed: {bubble}, {butlast}, {bsort} Previous analysis results are: last: runtime: O(n^1) [2 + z1], size: O(n^1) [z1] ---------------------------------------- (29) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (30) Obligation: Complexity RNTS consisting of the following rules: bsort(z1) -{ 1 }-> last(1 + 0 + bsort(butlast(0))) :|: x >= 0, y >= 0, z1 = 1 + x + y bsort(z1) -{ 2 }-> last(1 + 0 + bsort(butlast(1 + (z1 - 2) + 1))) :|: z1 - 2 >= 0 bsort(z1) -{ 2 }-> last(1 + 0 + bsort(butlast(1 + (1 + x + y'') + (1 + y'' + bubble(1 + x + z'')) + (1 + x + bubble(1 + y'' + z''))))) :|: z'' >= 0, x >= 0, y'' >= 0, z1 = 1 + x + (1 + y'' + z'') bsort(z1) -{ 2 }-> last(1 + (1 + (z1 - 2) + 1) + bsort(butlast(0))) :|: z1 - 2 >= 0 bsort(z1) -{ 3 }-> last(1 + (1 + (z1 - 2) + 1) + bsort(butlast(1 + (z1 - 2) + 1))) :|: z1 - 2 >= 0 bsort(z1) -{ 2 }-> last(1 + (1 + (1 + x + y') + (1 + y' + bubble(1 + x + z')) + (1 + x + bubble(1 + y' + z'))) + bsort(butlast(0))) :|: x >= 0, z' >= 0, y' >= 0, z1 = 1 + x + (1 + y' + z') bsort(z1) -{ 3 }-> last(1 + (1 + (1 + x + y') + (1 + y' + bubble(1 + x + z')) + (1 + x + bubble(1 + y' + z'))) + bsort(butlast(1 + (1 + x + y') + (1 + y' + bubble(1 + x + z')) + (1 + x + bubble(1 + y' + z'))))) :|: x >= 0, z' >= 0, y' >= 0, z1 = 1 + x + (1 + y' + z') bsort(z1) -{ 1 }-> 1 :|: z1 = 1 bsort(z1) -{ 0 }-> 0 :|: z1 >= 0 bubble(z1) -{ 1 }-> 1 :|: z1 = 1 bubble(z1) -{ 0 }-> 0 :|: z1 >= 0 bubble(z1) -{ 1 }-> 1 + (z1 - 2) + 1 :|: z1 - 2 >= 0 bubble(z1) -{ 1 }-> 1 + (1 + x + y) + (1 + y + bubble(1 + x + z)) + (1 + x + bubble(1 + y + z)) :|: z >= 0, z1 = 1 + x + (1 + y + z), x >= 0, y >= 0 butlast(z1) -{ 1 }-> 1 :|: z1 = 1 butlast(z1) -{ 1 }-> 1 :|: z1 - 2 >= 0 butlast(z1) -{ 0 }-> 0 :|: z1 >= 0 butlast(z1) -{ 1 }-> 1 + x + butlast(1 + y + z) :|: z >= 0, z1 = 1 + x + (1 + y + z), x >= 0, y >= 0 last(z1) -{ 4 + y + z }-> s :|: s >= 0, s <= 1 + y + z, z >= 0, z1 = 1 + x + (1 + y + z), x >= 0, y >= 0 last(z1) -{ 1 }-> 0 :|: z1 = 1 last(z1) -{ 0 }-> 0 :|: z1 >= 0 last(z1) -{ 1 }-> z1 - 2 :|: z1 - 2 >= 0 Function symbols to be analyzed: {bubble}, {butlast}, {bsort} Previous analysis results are: last: runtime: O(n^1) [2 + z1], size: O(n^1) [z1] ---------------------------------------- (31) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: bubble after applying outer abstraction to obtain an ITS, resulting in: EXP with polynomial bound: ? ---------------------------------------- (32) Obligation: Complexity RNTS consisting of the following rules: bsort(z1) -{ 1 }-> last(1 + 0 + bsort(butlast(0))) :|: x >= 0, y >= 0, z1 = 1 + x + y bsort(z1) -{ 2 }-> last(1 + 0 + bsort(butlast(1 + (z1 - 2) + 1))) :|: z1 - 2 >= 0 bsort(z1) -{ 2 }-> last(1 + 0 + bsort(butlast(1 + (1 + x + y'') + (1 + y'' + bubble(1 + x + z'')) + (1 + x + bubble(1 + y'' + z''))))) :|: z'' >= 0, x >= 0, y'' >= 0, z1 = 1 + x + (1 + y'' + z'') bsort(z1) -{ 2 }-> last(1 + (1 + (z1 - 2) + 1) + bsort(butlast(0))) :|: z1 - 2 >= 0 bsort(z1) -{ 3 }-> last(1 + (1 + (z1 - 2) + 1) + bsort(butlast(1 + (z1 - 2) + 1))) :|: z1 - 2 >= 0 bsort(z1) -{ 2 }-> last(1 + (1 + (1 + x + y') + (1 + y' + bubble(1 + x + z')) + (1 + x + bubble(1 + y' + z'))) + bsort(butlast(0))) :|: x >= 0, z' >= 0, y' >= 0, z1 = 1 + x + (1 + y' + z') bsort(z1) -{ 3 }-> last(1 + (1 + (1 + x + y') + (1 + y' + bubble(1 + x + z')) + (1 + x + bubble(1 + y' + z'))) + bsort(butlast(1 + (1 + x + y') + (1 + y' + bubble(1 + x + z')) + (1 + x + bubble(1 + y' + z'))))) :|: x >= 0, z' >= 0, y' >= 0, z1 = 1 + x + (1 + y' + z') bsort(z1) -{ 1 }-> 1 :|: z1 = 1 bsort(z1) -{ 0 }-> 0 :|: z1 >= 0 bubble(z1) -{ 1 }-> 1 :|: z1 = 1 bubble(z1) -{ 0 }-> 0 :|: z1 >= 0 bubble(z1) -{ 1 }-> 1 + (z1 - 2) + 1 :|: z1 - 2 >= 0 bubble(z1) -{ 1 }-> 1 + (1 + x + y) + (1 + y + bubble(1 + x + z)) + (1 + x + bubble(1 + y + z)) :|: z >= 0, z1 = 1 + x + (1 + y + z), x >= 0, y >= 0 butlast(z1) -{ 1 }-> 1 :|: z1 = 1 butlast(z1) -{ 1 }-> 1 :|: z1 - 2 >= 0 butlast(z1) -{ 0 }-> 0 :|: z1 >= 0 butlast(z1) -{ 1 }-> 1 + x + butlast(1 + y + z) :|: z >= 0, z1 = 1 + x + (1 + y + z), x >= 0, y >= 0 last(z1) -{ 4 + y + z }-> s :|: s >= 0, s <= 1 + y + z, z >= 0, z1 = 1 + x + (1 + y + z), x >= 0, y >= 0 last(z1) -{ 1 }-> 0 :|: z1 = 1 last(z1) -{ 0 }-> 0 :|: z1 >= 0 last(z1) -{ 1 }-> z1 - 2 :|: z1 - 2 >= 0 Function symbols to be analyzed: {bubble}, {butlast}, {bsort} Previous analysis results are: last: runtime: O(n^1) [2 + z1], size: O(n^1) [z1] bubble: runtime: ?, size: EXP ---------------------------------------- (33) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: bubble after applying outer abstraction to obtain an ITS, resulting in: EXP with polynomial bound: ? ---------------------------------------- (34) Obligation: Complexity RNTS consisting of the following rules: bsort(z1) -{ 1 }-> last(1 + 0 + bsort(butlast(0))) :|: x >= 0, y >= 0, z1 = 1 + x + y bsort(z1) -{ 2 }-> last(1 + 0 + bsort(butlast(1 + (z1 - 2) + 1))) :|: z1 - 2 >= 0 bsort(z1) -{ 2 }-> last(1 + 0 + bsort(butlast(1 + (1 + x + y'') + (1 + y'' + bubble(1 + x + z'')) + (1 + x + bubble(1 + y'' + z''))))) :|: z'' >= 0, x >= 0, y'' >= 0, z1 = 1 + x + (1 + y'' + z'') bsort(z1) -{ 2 }-> last(1 + (1 + (z1 - 2) + 1) + bsort(butlast(0))) :|: z1 - 2 >= 0 bsort(z1) -{ 3 }-> last(1 + (1 + (z1 - 2) + 1) + bsort(butlast(1 + (z1 - 2) + 1))) :|: z1 - 2 >= 0 bsort(z1) -{ 2 }-> last(1 + (1 + (1 + x + y') + (1 + y' + bubble(1 + x + z')) + (1 + x + bubble(1 + y' + z'))) + bsort(butlast(0))) :|: x >= 0, z' >= 0, y' >= 0, z1 = 1 + x + (1 + y' + z') bsort(z1) -{ 3 }-> last(1 + (1 + (1 + x + y') + (1 + y' + bubble(1 + x + z')) + (1 + x + bubble(1 + y' + z'))) + bsort(butlast(1 + (1 + x + y') + (1 + y' + bubble(1 + x + z')) + (1 + x + bubble(1 + y' + z'))))) :|: x >= 0, z' >= 0, y' >= 0, z1 = 1 + x + (1 + y' + z') bsort(z1) -{ 1 }-> 1 :|: z1 = 1 bsort(z1) -{ 0 }-> 0 :|: z1 >= 0 bubble(z1) -{ 1 }-> 1 :|: z1 = 1 bubble(z1) -{ 0 }-> 0 :|: z1 >= 0 bubble(z1) -{ 1 }-> 1 + (z1 - 2) + 1 :|: z1 - 2 >= 0 bubble(z1) -{ 1 }-> 1 + (1 + x + y) + (1 + y + bubble(1 + x + z)) + (1 + x + bubble(1 + y + z)) :|: z >= 0, z1 = 1 + x + (1 + y + z), x >= 0, y >= 0 butlast(z1) -{ 1 }-> 1 :|: z1 = 1 butlast(z1) -{ 1 }-> 1 :|: z1 - 2 >= 0 butlast(z1) -{ 0 }-> 0 :|: z1 >= 0 butlast(z1) -{ 1 }-> 1 + x + butlast(1 + y + z) :|: z >= 0, z1 = 1 + x + (1 + y + z), x >= 0, y >= 0 last(z1) -{ 4 + y + z }-> s :|: s >= 0, s <= 1 + y + z, z >= 0, z1 = 1 + x + (1 + y + z), x >= 0, y >= 0 last(z1) -{ 1 }-> 0 :|: z1 = 1 last(z1) -{ 0 }-> 0 :|: z1 >= 0 last(z1) -{ 1 }-> z1 - 2 :|: z1 - 2 >= 0 Function symbols to be analyzed: {butlast}, {bsort} Previous analysis results are: last: runtime: O(n^1) [2 + z1], size: O(n^1) [z1] bubble: runtime: EXP, size: EXP ---------------------------------------- (35) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (36) Obligation: Complexity RNTS consisting of the following rules: bsort(z1) -{ 1 }-> last(1 + 0 + bsort(butlast(0))) :|: x >= 0, y >= 0, z1 = 1 + x + y bsort(z1) -{ 2 }-> last(1 + 0 + bsort(butlast(1 + (z1 - 2) + 1))) :|: z1 - 2 >= 0 bsort(z1) -{ 2 + inf10 + inf12 }-> last(1 + 0 + bsort(butlast(1 + (1 + x + y'') + (1 + y'' + s5) + (1 + x + s6)))) :|: s5 >= 0, s5 <= inf11, s6 >= 0, s6 <= inf13, z'' >= 0, x >= 0, y'' >= 0, z1 = 1 + x + (1 + y'' + z'') bsort(z1) -{ 2 }-> last(1 + (1 + (z1 - 2) + 1) + bsort(butlast(0))) :|: z1 - 2 >= 0 bsort(z1) -{ 3 }-> last(1 + (1 + (z1 - 2) + 1) + bsort(butlast(1 + (z1 - 2) + 1))) :|: z1 - 2 >= 0 bsort(z1) -{ 3 + inf + inf'' + inf2 + inf4 }-> last(1 + (1 + (1 + x + y') + (1 + y' + s') + (1 + x + s'')) + bsort(butlast(1 + (1 + x + y') + (1 + y' + s1) + (1 + x + s2)))) :|: s' >= 0, s' <= inf', s'' >= 0, s'' <= inf1, s1 >= 0, s1 <= inf3, s2 >= 0, s2 <= inf5, x >= 0, z' >= 0, y' >= 0, z1 = 1 + x + (1 + y' + z') bsort(z1) -{ 2 + inf6 + inf8 }-> last(1 + (1 + (1 + x + y') + (1 + y' + s3) + (1 + x + s4)) + bsort(butlast(0))) :|: s3 >= 0, s3 <= inf7, s4 >= 0, s4 <= inf9, x >= 0, z' >= 0, y' >= 0, z1 = 1 + x + (1 + y' + z') bsort(z1) -{ 1 }-> 1 :|: z1 = 1 bsort(z1) -{ 0 }-> 0 :|: z1 >= 0 bubble(z1) -{ 1 }-> 1 :|: z1 = 1 bubble(z1) -{ 0 }-> 0 :|: z1 >= 0 bubble(z1) -{ 1 }-> 1 + (z1 - 2) + 1 :|: z1 - 2 >= 0 bubble(z1) -{ 1 + inf14 + inf16 }-> 1 + (1 + x + y) + (1 + y + s7) + (1 + x + s8) :|: s7 >= 0, s7 <= inf15, s8 >= 0, s8 <= inf17, z >= 0, z1 = 1 + x + (1 + y + z), x >= 0, y >= 0 butlast(z1) -{ 1 }-> 1 :|: z1 = 1 butlast(z1) -{ 1 }-> 1 :|: z1 - 2 >= 0 butlast(z1) -{ 0 }-> 0 :|: z1 >= 0 butlast(z1) -{ 1 }-> 1 + x + butlast(1 + y + z) :|: z >= 0, z1 = 1 + x + (1 + y + z), x >= 0, y >= 0 last(z1) -{ 4 + y + z }-> s :|: s >= 0, s <= 1 + y + z, z >= 0, z1 = 1 + x + (1 + y + z), x >= 0, y >= 0 last(z1) -{ 1 }-> 0 :|: z1 = 1 last(z1) -{ 0 }-> 0 :|: z1 >= 0 last(z1) -{ 1 }-> z1 - 2 :|: z1 - 2 >= 0 Function symbols to be analyzed: {butlast}, {bsort} Previous analysis results are: last: runtime: O(n^1) [2 + z1], size: O(n^1) [z1] bubble: runtime: EXP, size: EXP ---------------------------------------- (37) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: butlast after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z1 ---------------------------------------- (38) Obligation: Complexity RNTS consisting of the following rules: bsort(z1) -{ 1 }-> last(1 + 0 + bsort(butlast(0))) :|: x >= 0, y >= 0, z1 = 1 + x + y bsort(z1) -{ 2 }-> last(1 + 0 + bsort(butlast(1 + (z1 - 2) + 1))) :|: z1 - 2 >= 0 bsort(z1) -{ 2 + inf10 + inf12 }-> last(1 + 0 + bsort(butlast(1 + (1 + x + y'') + (1 + y'' + s5) + (1 + x + s6)))) :|: s5 >= 0, s5 <= inf11, s6 >= 0, s6 <= inf13, z'' >= 0, x >= 0, y'' >= 0, z1 = 1 + x + (1 + y'' + z'') bsort(z1) -{ 2 }-> last(1 + (1 + (z1 - 2) + 1) + bsort(butlast(0))) :|: z1 - 2 >= 0 bsort(z1) -{ 3 }-> last(1 + (1 + (z1 - 2) + 1) + bsort(butlast(1 + (z1 - 2) + 1))) :|: z1 - 2 >= 0 bsort(z1) -{ 3 + inf + inf'' + inf2 + inf4 }-> last(1 + (1 + (1 + x + y') + (1 + y' + s') + (1 + x + s'')) + bsort(butlast(1 + (1 + x + y') + (1 + y' + s1) + (1 + x + s2)))) :|: s' >= 0, s' <= inf', s'' >= 0, s'' <= inf1, s1 >= 0, s1 <= inf3, s2 >= 0, s2 <= inf5, x >= 0, z' >= 0, y' >= 0, z1 = 1 + x + (1 + y' + z') bsort(z1) -{ 2 + inf6 + inf8 }-> last(1 + (1 + (1 + x + y') + (1 + y' + s3) + (1 + x + s4)) + bsort(butlast(0))) :|: s3 >= 0, s3 <= inf7, s4 >= 0, s4 <= inf9, x >= 0, z' >= 0, y' >= 0, z1 = 1 + x + (1 + y' + z') bsort(z1) -{ 1 }-> 1 :|: z1 = 1 bsort(z1) -{ 0 }-> 0 :|: z1 >= 0 bubble(z1) -{ 1 }-> 1 :|: z1 = 1 bubble(z1) -{ 0 }-> 0 :|: z1 >= 0 bubble(z1) -{ 1 }-> 1 + (z1 - 2) + 1 :|: z1 - 2 >= 0 bubble(z1) -{ 1 + inf14 + inf16 }-> 1 + (1 + x + y) + (1 + y + s7) + (1 + x + s8) :|: s7 >= 0, s7 <= inf15, s8 >= 0, s8 <= inf17, z >= 0, z1 = 1 + x + (1 + y + z), x >= 0, y >= 0 butlast(z1) -{ 1 }-> 1 :|: z1 = 1 butlast(z1) -{ 1 }-> 1 :|: z1 - 2 >= 0 butlast(z1) -{ 0 }-> 0 :|: z1 >= 0 butlast(z1) -{ 1 }-> 1 + x + butlast(1 + y + z) :|: z >= 0, z1 = 1 + x + (1 + y + z), x >= 0, y >= 0 last(z1) -{ 4 + y + z }-> s :|: s >= 0, s <= 1 + y + z, z >= 0, z1 = 1 + x + (1 + y + z), x >= 0, y >= 0 last(z1) -{ 1 }-> 0 :|: z1 = 1 last(z1) -{ 0 }-> 0 :|: z1 >= 0 last(z1) -{ 1 }-> z1 - 2 :|: z1 - 2 >= 0 Function symbols to be analyzed: {butlast}, {bsort} Previous analysis results are: last: runtime: O(n^1) [2 + z1], size: O(n^1) [z1] bubble: runtime: EXP, size: EXP butlast: runtime: ?, size: O(n^1) [z1] ---------------------------------------- (39) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: butlast after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2 + z1 ---------------------------------------- (40) Obligation: Complexity RNTS consisting of the following rules: bsort(z1) -{ 1 }-> last(1 + 0 + bsort(butlast(0))) :|: x >= 0, y >= 0, z1 = 1 + x + y bsort(z1) -{ 2 }-> last(1 + 0 + bsort(butlast(1 + (z1 - 2) + 1))) :|: z1 - 2 >= 0 bsort(z1) -{ 2 + inf10 + inf12 }-> last(1 + 0 + bsort(butlast(1 + (1 + x + y'') + (1 + y'' + s5) + (1 + x + s6)))) :|: s5 >= 0, s5 <= inf11, s6 >= 0, s6 <= inf13, z'' >= 0, x >= 0, y'' >= 0, z1 = 1 + x + (1 + y'' + z'') bsort(z1) -{ 2 }-> last(1 + (1 + (z1 - 2) + 1) + bsort(butlast(0))) :|: z1 - 2 >= 0 bsort(z1) -{ 3 }-> last(1 + (1 + (z1 - 2) + 1) + bsort(butlast(1 + (z1 - 2) + 1))) :|: z1 - 2 >= 0 bsort(z1) -{ 3 + inf + inf'' + inf2 + inf4 }-> last(1 + (1 + (1 + x + y') + (1 + y' + s') + (1 + x + s'')) + bsort(butlast(1 + (1 + x + y') + (1 + y' + s1) + (1 + x + s2)))) :|: s' >= 0, s' <= inf', s'' >= 0, s'' <= inf1, s1 >= 0, s1 <= inf3, s2 >= 0, s2 <= inf5, x >= 0, z' >= 0, y' >= 0, z1 = 1 + x + (1 + y' + z') bsort(z1) -{ 2 + inf6 + inf8 }-> last(1 + (1 + (1 + x + y') + (1 + y' + s3) + (1 + x + s4)) + bsort(butlast(0))) :|: s3 >= 0, s3 <= inf7, s4 >= 0, s4 <= inf9, x >= 0, z' >= 0, y' >= 0, z1 = 1 + x + (1 + y' + z') bsort(z1) -{ 1 }-> 1 :|: z1 = 1 bsort(z1) -{ 0 }-> 0 :|: z1 >= 0 bubble(z1) -{ 1 }-> 1 :|: z1 = 1 bubble(z1) -{ 0 }-> 0 :|: z1 >= 0 bubble(z1) -{ 1 }-> 1 + (z1 - 2) + 1 :|: z1 - 2 >= 0 bubble(z1) -{ 1 + inf14 + inf16 }-> 1 + (1 + x + y) + (1 + y + s7) + (1 + x + s8) :|: s7 >= 0, s7 <= inf15, s8 >= 0, s8 <= inf17, z >= 0, z1 = 1 + x + (1 + y + z), x >= 0, y >= 0 butlast(z1) -{ 1 }-> 1 :|: z1 = 1 butlast(z1) -{ 1 }-> 1 :|: z1 - 2 >= 0 butlast(z1) -{ 0 }-> 0 :|: z1 >= 0 butlast(z1) -{ 1 }-> 1 + x + butlast(1 + y + z) :|: z >= 0, z1 = 1 + x + (1 + y + z), x >= 0, y >= 0 last(z1) -{ 4 + y + z }-> s :|: s >= 0, s <= 1 + y + z, z >= 0, z1 = 1 + x + (1 + y + z), x >= 0, y >= 0 last(z1) -{ 1 }-> 0 :|: z1 = 1 last(z1) -{ 0 }-> 0 :|: z1 >= 0 last(z1) -{ 1 }-> z1 - 2 :|: z1 - 2 >= 0 Function symbols to be analyzed: {bsort} Previous analysis results are: last: runtime: O(n^1) [2 + z1], size: O(n^1) [z1] bubble: runtime: EXP, size: EXP butlast: runtime: O(n^1) [2 + z1], size: O(n^1) [z1] ---------------------------------------- (41) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (42) Obligation: Complexity RNTS consisting of the following rules: bsort(z1) -{ 4 + z1 }-> last(1 + 0 + bsort(s13)) :|: s13 >= 0, s13 <= 1 + (z1 - 2) + 1, z1 - 2 >= 0 bsort(z1) -{ 8 + inf10 + inf12 + s5 + s6 + 2*x + 2*y'' }-> last(1 + 0 + bsort(s14)) :|: s14 >= 0, s14 <= 1 + (1 + x + y'') + (1 + y'' + s5) + (1 + x + s6), s5 >= 0, s5 <= inf11, s6 >= 0, s6 <= inf13, z'' >= 0, x >= 0, y'' >= 0, z1 = 1 + x + (1 + y'' + z'') bsort(z1) -{ 3 }-> last(1 + 0 + bsort(s15)) :|: s15 >= 0, s15 <= 0, x >= 0, y >= 0, z1 = 1 + x + y bsort(z1) -{ 4 }-> last(1 + (1 + (z1 - 2) + 1) + bsort(s10)) :|: s10 >= 0, s10 <= 0, z1 - 2 >= 0 bsort(z1) -{ 5 + z1 }-> last(1 + (1 + (z1 - 2) + 1) + bsort(s9)) :|: s9 >= 0, s9 <= 1 + (z1 - 2) + 1, z1 - 2 >= 0 bsort(z1) -{ 9 + inf + inf'' + inf2 + inf4 + s1 + s2 + 2*x + 2*y' }-> last(1 + (1 + (1 + x + y') + (1 + y' + s') + (1 + x + s'')) + bsort(s11)) :|: s11 >= 0, s11 <= 1 + (1 + x + y') + (1 + y' + s1) + (1 + x + s2), s' >= 0, s' <= inf', s'' >= 0, s'' <= inf1, s1 >= 0, s1 <= inf3, s2 >= 0, s2 <= inf5, x >= 0, z' >= 0, y' >= 0, z1 = 1 + x + (1 + y' + z') bsort(z1) -{ 4 + inf6 + inf8 }-> last(1 + (1 + (1 + x + y') + (1 + y' + s3) + (1 + x + s4)) + bsort(s12)) :|: s12 >= 0, s12 <= 0, s3 >= 0, s3 <= inf7, s4 >= 0, s4 <= inf9, x >= 0, z' >= 0, y' >= 0, z1 = 1 + x + (1 + y' + z') bsort(z1) -{ 1 }-> 1 :|: z1 = 1 bsort(z1) -{ 0 }-> 0 :|: z1 >= 0 bubble(z1) -{ 1 }-> 1 :|: z1 = 1 bubble(z1) -{ 0 }-> 0 :|: z1 >= 0 bubble(z1) -{ 1 }-> 1 + (z1 - 2) + 1 :|: z1 - 2 >= 0 bubble(z1) -{ 1 + inf14 + inf16 }-> 1 + (1 + x + y) + (1 + y + s7) + (1 + x + s8) :|: s7 >= 0, s7 <= inf15, s8 >= 0, s8 <= inf17, z >= 0, z1 = 1 + x + (1 + y + z), x >= 0, y >= 0 butlast(z1) -{ 1 }-> 1 :|: z1 = 1 butlast(z1) -{ 1 }-> 1 :|: z1 - 2 >= 0 butlast(z1) -{ 0 }-> 0 :|: z1 >= 0 butlast(z1) -{ 4 + y + z }-> 1 + x + s16 :|: s16 >= 0, s16 <= 1 + y + z, z >= 0, z1 = 1 + x + (1 + y + z), x >= 0, y >= 0 last(z1) -{ 4 + y + z }-> s :|: s >= 0, s <= 1 + y + z, z >= 0, z1 = 1 + x + (1 + y + z), x >= 0, y >= 0 last(z1) -{ 1 }-> 0 :|: z1 = 1 last(z1) -{ 0 }-> 0 :|: z1 >= 0 last(z1) -{ 1 }-> z1 - 2 :|: z1 - 2 >= 0 Function symbols to be analyzed: {bsort} Previous analysis results are: last: runtime: O(n^1) [2 + z1], size: O(n^1) [z1] bubble: runtime: EXP, size: EXP butlast: runtime: O(n^1) [2 + z1], size: O(n^1) [z1] ---------------------------------------- (43) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: bsort after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (44) Obligation: Complexity RNTS consisting of the following rules: bsort(z1) -{ 4 + z1 }-> last(1 + 0 + bsort(s13)) :|: s13 >= 0, s13 <= 1 + (z1 - 2) + 1, z1 - 2 >= 0 bsort(z1) -{ 8 + inf10 + inf12 + s5 + s6 + 2*x + 2*y'' }-> last(1 + 0 + bsort(s14)) :|: s14 >= 0, s14 <= 1 + (1 + x + y'') + (1 + y'' + s5) + (1 + x + s6), s5 >= 0, s5 <= inf11, s6 >= 0, s6 <= inf13, z'' >= 0, x >= 0, y'' >= 0, z1 = 1 + x + (1 + y'' + z'') bsort(z1) -{ 3 }-> last(1 + 0 + bsort(s15)) :|: s15 >= 0, s15 <= 0, x >= 0, y >= 0, z1 = 1 + x + y bsort(z1) -{ 4 }-> last(1 + (1 + (z1 - 2) + 1) + bsort(s10)) :|: s10 >= 0, s10 <= 0, z1 - 2 >= 0 bsort(z1) -{ 5 + z1 }-> last(1 + (1 + (z1 - 2) + 1) + bsort(s9)) :|: s9 >= 0, s9 <= 1 + (z1 - 2) + 1, z1 - 2 >= 0 bsort(z1) -{ 9 + inf + inf'' + inf2 + inf4 + s1 + s2 + 2*x + 2*y' }-> last(1 + (1 + (1 + x + y') + (1 + y' + s') + (1 + x + s'')) + bsort(s11)) :|: s11 >= 0, s11 <= 1 + (1 + x + y') + (1 + y' + s1) + (1 + x + s2), s' >= 0, s' <= inf', s'' >= 0, s'' <= inf1, s1 >= 0, s1 <= inf3, s2 >= 0, s2 <= inf5, x >= 0, z' >= 0, y' >= 0, z1 = 1 + x + (1 + y' + z') bsort(z1) -{ 4 + inf6 + inf8 }-> last(1 + (1 + (1 + x + y') + (1 + y' + s3) + (1 + x + s4)) + bsort(s12)) :|: s12 >= 0, s12 <= 0, s3 >= 0, s3 <= inf7, s4 >= 0, s4 <= inf9, x >= 0, z' >= 0, y' >= 0, z1 = 1 + x + (1 + y' + z') bsort(z1) -{ 1 }-> 1 :|: z1 = 1 bsort(z1) -{ 0 }-> 0 :|: z1 >= 0 bubble(z1) -{ 1 }-> 1 :|: z1 = 1 bubble(z1) -{ 0 }-> 0 :|: z1 >= 0 bubble(z1) -{ 1 }-> 1 + (z1 - 2) + 1 :|: z1 - 2 >= 0 bubble(z1) -{ 1 + inf14 + inf16 }-> 1 + (1 + x + y) + (1 + y + s7) + (1 + x + s8) :|: s7 >= 0, s7 <= inf15, s8 >= 0, s8 <= inf17, z >= 0, z1 = 1 + x + (1 + y + z), x >= 0, y >= 0 butlast(z1) -{ 1 }-> 1 :|: z1 = 1 butlast(z1) -{ 1 }-> 1 :|: z1 - 2 >= 0 butlast(z1) -{ 0 }-> 0 :|: z1 >= 0 butlast(z1) -{ 4 + y + z }-> 1 + x + s16 :|: s16 >= 0, s16 <= 1 + y + z, z >= 0, z1 = 1 + x + (1 + y + z), x >= 0, y >= 0 last(z1) -{ 4 + y + z }-> s :|: s >= 0, s <= 1 + y + z, z >= 0, z1 = 1 + x + (1 + y + z), x >= 0, y >= 0 last(z1) -{ 1 }-> 0 :|: z1 = 1 last(z1) -{ 0 }-> 0 :|: z1 >= 0 last(z1) -{ 1 }-> z1 - 2 :|: z1 - 2 >= 0 Function symbols to be analyzed: {bsort} Previous analysis results are: last: runtime: O(n^1) [2 + z1], size: O(n^1) [z1] bubble: runtime: EXP, size: EXP butlast: runtime: O(n^1) [2 + z1], size: O(n^1) [z1] bsort: runtime: ?, size: INF ---------------------------------------- (45) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: bsort after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (46) Obligation: Complexity RNTS consisting of the following rules: bsort(z1) -{ 4 + z1 }-> last(1 + 0 + bsort(s13)) :|: s13 >= 0, s13 <= 1 + (z1 - 2) + 1, z1 - 2 >= 0 bsort(z1) -{ 8 + inf10 + inf12 + s5 + s6 + 2*x + 2*y'' }-> last(1 + 0 + bsort(s14)) :|: s14 >= 0, s14 <= 1 + (1 + x + y'') + (1 + y'' + s5) + (1 + x + s6), s5 >= 0, s5 <= inf11, s6 >= 0, s6 <= inf13, z'' >= 0, x >= 0, y'' >= 0, z1 = 1 + x + (1 + y'' + z'') bsort(z1) -{ 3 }-> last(1 + 0 + bsort(s15)) :|: s15 >= 0, s15 <= 0, x >= 0, y >= 0, z1 = 1 + x + y bsort(z1) -{ 4 }-> last(1 + (1 + (z1 - 2) + 1) + bsort(s10)) :|: s10 >= 0, s10 <= 0, z1 - 2 >= 0 bsort(z1) -{ 5 + z1 }-> last(1 + (1 + (z1 - 2) + 1) + bsort(s9)) :|: s9 >= 0, s9 <= 1 + (z1 - 2) + 1, z1 - 2 >= 0 bsort(z1) -{ 9 + inf + inf'' + inf2 + inf4 + s1 + s2 + 2*x + 2*y' }-> last(1 + (1 + (1 + x + y') + (1 + y' + s') + (1 + x + s'')) + bsort(s11)) :|: s11 >= 0, s11 <= 1 + (1 + x + y') + (1 + y' + s1) + (1 + x + s2), s' >= 0, s' <= inf', s'' >= 0, s'' <= inf1, s1 >= 0, s1 <= inf3, s2 >= 0, s2 <= inf5, x >= 0, z' >= 0, y' >= 0, z1 = 1 + x + (1 + y' + z') bsort(z1) -{ 4 + inf6 + inf8 }-> last(1 + (1 + (1 + x + y') + (1 + y' + s3) + (1 + x + s4)) + bsort(s12)) :|: s12 >= 0, s12 <= 0, s3 >= 0, s3 <= inf7, s4 >= 0, s4 <= inf9, x >= 0, z' >= 0, y' >= 0, z1 = 1 + x + (1 + y' + z') bsort(z1) -{ 1 }-> 1 :|: z1 = 1 bsort(z1) -{ 0 }-> 0 :|: z1 >= 0 bubble(z1) -{ 1 }-> 1 :|: z1 = 1 bubble(z1) -{ 0 }-> 0 :|: z1 >= 0 bubble(z1) -{ 1 }-> 1 + (z1 - 2) + 1 :|: z1 - 2 >= 0 bubble(z1) -{ 1 + inf14 + inf16 }-> 1 + (1 + x + y) + (1 + y + s7) + (1 + x + s8) :|: s7 >= 0, s7 <= inf15, s8 >= 0, s8 <= inf17, z >= 0, z1 = 1 + x + (1 + y + z), x >= 0, y >= 0 butlast(z1) -{ 1 }-> 1 :|: z1 = 1 butlast(z1) -{ 1 }-> 1 :|: z1 - 2 >= 0 butlast(z1) -{ 0 }-> 0 :|: z1 >= 0 butlast(z1) -{ 4 + y + z }-> 1 + x + s16 :|: s16 >= 0, s16 <= 1 + y + z, z >= 0, z1 = 1 + x + (1 + y + z), x >= 0, y >= 0 last(z1) -{ 4 + y + z }-> s :|: s >= 0, s <= 1 + y + z, z >= 0, z1 = 1 + x + (1 + y + z), x >= 0, y >= 0 last(z1) -{ 1 }-> 0 :|: z1 = 1 last(z1) -{ 0 }-> 0 :|: z1 >= 0 last(z1) -{ 1 }-> z1 - 2 :|: z1 - 2 >= 0 Function symbols to be analyzed: {bsort} Previous analysis results are: last: runtime: O(n^1) [2 + z1], size: O(n^1) [z1] bubble: runtime: EXP, size: EXP butlast: runtime: O(n^1) [2 + z1], size: O(n^1) [z1] bsort: runtime: INF, size: INF ---------------------------------------- (47) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (48) Obligation: Complexity Dependency Tuples Problem Rules: bsort(nil) -> nil bsort(.(z0, z1)) -> last(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1)))))) bubble(nil) -> nil bubble(.(z0, nil)) -> .(z0, nil) bubble(.(z0, .(z1, z2))) -> if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))) last(nil) -> 0 last(.(z0, nil)) -> z0 last(.(z0, .(z1, z2))) -> last(.(z1, z2)) butlast(nil) -> nil butlast(.(z0, nil)) -> nil butlast(.(z0, .(z1, z2))) -> .(z0, butlast(.(z1, z2))) Tuples: BSORT(nil) -> c BSORT(.(z0, z1)) -> c1(LAST(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1)))))), BUBBLE(.(z0, z1))) BSORT(.(z0, z1)) -> c2(LAST(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1)))))), BSORT(butlast(bubble(.(z0, z1)))), BUTLAST(bubble(.(z0, z1))), BUBBLE(.(z0, z1))) BUBBLE(nil) -> c3 BUBBLE(.(z0, nil)) -> c4 BUBBLE(.(z0, .(z1, z2))) -> c5(BUBBLE(.(z0, z2))) BUBBLE(.(z0, .(z1, z2))) -> c6(BUBBLE(.(z1, z2))) LAST(nil) -> c7 LAST(.(z0, nil)) -> c8 LAST(.(z0, .(z1, z2))) -> c9(LAST(.(z1, z2))) BUTLAST(nil) -> c10 BUTLAST(.(z0, nil)) -> c11 BUTLAST(.(z0, .(z1, z2))) -> c12(BUTLAST(.(z1, z2))) S tuples: BSORT(nil) -> c BSORT(.(z0, z1)) -> c1(LAST(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1)))))), BUBBLE(.(z0, z1))) BSORT(.(z0, z1)) -> c2(LAST(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1)))))), BSORT(butlast(bubble(.(z0, z1)))), BUTLAST(bubble(.(z0, z1))), BUBBLE(.(z0, z1))) BUBBLE(nil) -> c3 BUBBLE(.(z0, nil)) -> c4 BUBBLE(.(z0, .(z1, z2))) -> c5(BUBBLE(.(z0, z2))) BUBBLE(.(z0, .(z1, z2))) -> c6(BUBBLE(.(z1, z2))) LAST(nil) -> c7 LAST(.(z0, nil)) -> c8 LAST(.(z0, .(z1, z2))) -> c9(LAST(.(z1, z2))) BUTLAST(nil) -> c10 BUTLAST(.(z0, nil)) -> c11 BUTLAST(.(z0, .(z1, z2))) -> c12(BUTLAST(.(z1, z2))) K tuples:none Defined Rule Symbols: bsort_1, bubble_1, last_1, butlast_1 Defined Pair Symbols: BSORT_1, BUBBLE_1, LAST_1, BUTLAST_1 Compound Symbols: c, c1_2, c2_4, c3, c4, c5_1, c6_1, c7, c8, c9_1, c10, c11, c12_1 ---------------------------------------- (49) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 7 trailing nodes: BUTLAST(nil) -> c10 BUTLAST(.(z0, nil)) -> c11 LAST(nil) -> c7 LAST(.(z0, nil)) -> c8 BUBBLE(.(z0, nil)) -> c4 BSORT(nil) -> c BUBBLE(nil) -> c3 ---------------------------------------- (50) Obligation: Complexity Dependency Tuples Problem Rules: bsort(nil) -> nil bsort(.(z0, z1)) -> last(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1)))))) bubble(nil) -> nil bubble(.(z0, nil)) -> .(z0, nil) bubble(.(z0, .(z1, z2))) -> if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))) last(nil) -> 0 last(.(z0, nil)) -> z0 last(.(z0, .(z1, z2))) -> last(.(z1, z2)) butlast(nil) -> nil butlast(.(z0, nil)) -> nil butlast(.(z0, .(z1, z2))) -> .(z0, butlast(.(z1, z2))) Tuples: BSORT(.(z0, z1)) -> c1(LAST(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1)))))), BUBBLE(.(z0, z1))) BSORT(.(z0, z1)) -> c2(LAST(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1)))))), BSORT(butlast(bubble(.(z0, z1)))), BUTLAST(bubble(.(z0, z1))), BUBBLE(.(z0, z1))) BUBBLE(.(z0, .(z1, z2))) -> c5(BUBBLE(.(z0, z2))) BUBBLE(.(z0, .(z1, z2))) -> c6(BUBBLE(.(z1, z2))) LAST(.(z0, .(z1, z2))) -> c9(LAST(.(z1, z2))) BUTLAST(.(z0, .(z1, z2))) -> c12(BUTLAST(.(z1, z2))) S tuples: BSORT(.(z0, z1)) -> c1(LAST(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1)))))), BUBBLE(.(z0, z1))) BSORT(.(z0, z1)) -> c2(LAST(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1)))))), BSORT(butlast(bubble(.(z0, z1)))), BUTLAST(bubble(.(z0, z1))), BUBBLE(.(z0, z1))) BUBBLE(.(z0, .(z1, z2))) -> c5(BUBBLE(.(z0, z2))) BUBBLE(.(z0, .(z1, z2))) -> c6(BUBBLE(.(z1, z2))) LAST(.(z0, .(z1, z2))) -> c9(LAST(.(z1, z2))) BUTLAST(.(z0, .(z1, z2))) -> c12(BUTLAST(.(z1, z2))) K tuples:none Defined Rule Symbols: bsort_1, bubble_1, last_1, butlast_1 Defined Pair Symbols: BSORT_1, BUBBLE_1, LAST_1, BUTLAST_1 Compound Symbols: c1_2, c2_4, c5_1, c6_1, c9_1, c12_1 ---------------------------------------- (51) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (52) Obligation: Complexity Dependency Tuples Problem Rules: bsort(nil) -> nil bsort(.(z0, z1)) -> last(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1)))))) bubble(nil) -> nil bubble(.(z0, nil)) -> .(z0, nil) bubble(.(z0, .(z1, z2))) -> if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))) last(nil) -> 0 last(.(z0, nil)) -> z0 last(.(z0, .(z1, z2))) -> last(.(z1, z2)) butlast(nil) -> nil butlast(.(z0, nil)) -> nil butlast(.(z0, .(z1, z2))) -> .(z0, butlast(.(z1, z2))) Tuples: BSORT(.(z0, z1)) -> c2(LAST(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1)))))), BSORT(butlast(bubble(.(z0, z1)))), BUTLAST(bubble(.(z0, z1))), BUBBLE(.(z0, z1))) BUBBLE(.(z0, .(z1, z2))) -> c5(BUBBLE(.(z0, z2))) BUBBLE(.(z0, .(z1, z2))) -> c6(BUBBLE(.(z1, z2))) LAST(.(z0, .(z1, z2))) -> c9(LAST(.(z1, z2))) BUTLAST(.(z0, .(z1, z2))) -> c12(BUTLAST(.(z1, z2))) BSORT(.(z0, z1)) -> c(LAST(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1))))))) BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) S tuples: BSORT(.(z0, z1)) -> c2(LAST(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1)))))), BSORT(butlast(bubble(.(z0, z1)))), BUTLAST(bubble(.(z0, z1))), BUBBLE(.(z0, z1))) BUBBLE(.(z0, .(z1, z2))) -> c5(BUBBLE(.(z0, z2))) BUBBLE(.(z0, .(z1, z2))) -> c6(BUBBLE(.(z1, z2))) LAST(.(z0, .(z1, z2))) -> c9(LAST(.(z1, z2))) BUTLAST(.(z0, .(z1, z2))) -> c12(BUTLAST(.(z1, z2))) BSORT(.(z0, z1)) -> c(LAST(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1))))))) BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) K tuples:none Defined Rule Symbols: bsort_1, bubble_1, last_1, butlast_1 Defined Pair Symbols: BSORT_1, BUBBLE_1, LAST_1, BUTLAST_1 Compound Symbols: c2_4, c5_1, c6_1, c9_1, c12_1, c_1 ---------------------------------------- (53) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: bubble(nil) -> nil last(nil) -> 0 ---------------------------------------- (54) Obligation: Complexity Dependency Tuples Problem Rules: bubble(.(z0, nil)) -> .(z0, nil) bubble(.(z0, .(z1, z2))) -> if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))) bsort(nil) -> nil bsort(.(z0, z1)) -> last(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1)))))) butlast(nil) -> nil butlast(.(z0, nil)) -> nil butlast(.(z0, .(z1, z2))) -> .(z0, butlast(.(z1, z2))) last(.(z0, nil)) -> z0 last(.(z0, .(z1, z2))) -> last(.(z1, z2)) Tuples: BSORT(.(z0, z1)) -> c2(LAST(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1)))))), BSORT(butlast(bubble(.(z0, z1)))), BUTLAST(bubble(.(z0, z1))), BUBBLE(.(z0, z1))) BUBBLE(.(z0, .(z1, z2))) -> c5(BUBBLE(.(z0, z2))) BUBBLE(.(z0, .(z1, z2))) -> c6(BUBBLE(.(z1, z2))) LAST(.(z0, .(z1, z2))) -> c9(LAST(.(z1, z2))) BUTLAST(.(z0, .(z1, z2))) -> c12(BUTLAST(.(z1, z2))) BSORT(.(z0, z1)) -> c(LAST(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1))))))) BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) S tuples: BSORT(.(z0, z1)) -> c2(LAST(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1)))))), BSORT(butlast(bubble(.(z0, z1)))), BUTLAST(bubble(.(z0, z1))), BUBBLE(.(z0, z1))) BUBBLE(.(z0, .(z1, z2))) -> c5(BUBBLE(.(z0, z2))) BUBBLE(.(z0, .(z1, z2))) -> c6(BUBBLE(.(z1, z2))) LAST(.(z0, .(z1, z2))) -> c9(LAST(.(z1, z2))) BUTLAST(.(z0, .(z1, z2))) -> c12(BUTLAST(.(z1, z2))) BSORT(.(z0, z1)) -> c(LAST(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1))))))) BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) K tuples:none Defined Rule Symbols: bubble_1, bsort_1, butlast_1, last_1 Defined Pair Symbols: BSORT_1, BUBBLE_1, LAST_1, BUTLAST_1 Compound Symbols: c2_4, c5_1, c6_1, c9_1, c12_1, c_1 ---------------------------------------- (55) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. BSORT(.(z0, z1)) -> c(LAST(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1))))))) BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) We considered the (Usable) Rules:none And the Tuples: BSORT(.(z0, z1)) -> c2(LAST(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1)))))), BSORT(butlast(bubble(.(z0, z1)))), BUTLAST(bubble(.(z0, z1))), BUBBLE(.(z0, z1))) BUBBLE(.(z0, .(z1, z2))) -> c5(BUBBLE(.(z0, z2))) BUBBLE(.(z0, .(z1, z2))) -> c6(BUBBLE(.(z1, z2))) LAST(.(z0, .(z1, z2))) -> c9(LAST(.(z1, z2))) BUTLAST(.(z0, .(z1, z2))) -> c12(BUTLAST(.(z1, z2))) BSORT(.(z0, z1)) -> c(LAST(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1))))))) BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) The order we found is given by the following interpretation: Polynomial interpretation : POL(.(x_1, x_2)) = 0 POL(<=(x_1, x_2)) = [1] + x_1 + x_2 POL(BSORT(x_1)) = [1] POL(BUBBLE(x_1)) = x_1 POL(BUTLAST(x_1)) = 0 POL(LAST(x_1)) = 0 POL(bsort(x_1)) = [1] POL(bubble(x_1)) = 0 POL(butlast(x_1)) = 0 POL(c(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c2(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4 POL(c5(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c9(x_1)) = x_1 POL(if(x_1, x_2, x_3)) = [1] + x_1 POL(last(x_1)) = [1] POL(nil) = [1] ---------------------------------------- (56) Obligation: Complexity Dependency Tuples Problem Rules: bubble(.(z0, nil)) -> .(z0, nil) bubble(.(z0, .(z1, z2))) -> if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))) bsort(nil) -> nil bsort(.(z0, z1)) -> last(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1)))))) butlast(nil) -> nil butlast(.(z0, nil)) -> nil butlast(.(z0, .(z1, z2))) -> .(z0, butlast(.(z1, z2))) last(.(z0, nil)) -> z0 last(.(z0, .(z1, z2))) -> last(.(z1, z2)) Tuples: BSORT(.(z0, z1)) -> c2(LAST(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1)))))), BSORT(butlast(bubble(.(z0, z1)))), BUTLAST(bubble(.(z0, z1))), BUBBLE(.(z0, z1))) BUBBLE(.(z0, .(z1, z2))) -> c5(BUBBLE(.(z0, z2))) BUBBLE(.(z0, .(z1, z2))) -> c6(BUBBLE(.(z1, z2))) LAST(.(z0, .(z1, z2))) -> c9(LAST(.(z1, z2))) BUTLAST(.(z0, .(z1, z2))) -> c12(BUTLAST(.(z1, z2))) BSORT(.(z0, z1)) -> c(LAST(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1))))))) BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) S tuples: BSORT(.(z0, z1)) -> c2(LAST(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1)))))), BSORT(butlast(bubble(.(z0, z1)))), BUTLAST(bubble(.(z0, z1))), BUBBLE(.(z0, z1))) BUBBLE(.(z0, .(z1, z2))) -> c5(BUBBLE(.(z0, z2))) BUBBLE(.(z0, .(z1, z2))) -> c6(BUBBLE(.(z1, z2))) LAST(.(z0, .(z1, z2))) -> c9(LAST(.(z1, z2))) BUTLAST(.(z0, .(z1, z2))) -> c12(BUTLAST(.(z1, z2))) K tuples: BSORT(.(z0, z1)) -> c(LAST(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1))))))) BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) Defined Rule Symbols: bubble_1, bsort_1, butlast_1, last_1 Defined Pair Symbols: BSORT_1, BUBBLE_1, LAST_1, BUTLAST_1 Compound Symbols: c2_4, c5_1, c6_1, c9_1, c12_1, c_1 ---------------------------------------- (57) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace BSORT(.(z0, z1)) -> c2(LAST(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1)))))), BSORT(butlast(bubble(.(z0, z1)))), BUTLAST(bubble(.(z0, z1))), BUBBLE(.(z0, z1))) by BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil))), BUBBLE(.(z0, nil))) BSORT(.(z0, .(z1, z2))) -> c2(LAST(.(bubble(.(z0, .(z1, z2))), bsort(butlast(if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))))))), BSORT(butlast(bubble(.(z0, .(z1, z2))))), BUTLAST(bubble(.(z0, .(z1, z2)))), BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(bubble(.(z0, nil)))))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil))), BUBBLE(.(z0, nil))) BSORT(.(z0, .(z1, z2))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))), bsort(butlast(bubble(.(z0, .(z1, z2))))))), BSORT(butlast(bubble(.(z0, .(z1, z2))))), BUTLAST(bubble(.(z0, .(z1, z2)))), BUBBLE(.(z0, .(z1, z2)))) ---------------------------------------- (58) Obligation: Complexity Dependency Tuples Problem Rules: bubble(.(z0, nil)) -> .(z0, nil) bubble(.(z0, .(z1, z2))) -> if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))) bsort(nil) -> nil bsort(.(z0, z1)) -> last(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1)))))) butlast(nil) -> nil butlast(.(z0, nil)) -> nil butlast(.(z0, .(z1, z2))) -> .(z0, butlast(.(z1, z2))) last(.(z0, nil)) -> z0 last(.(z0, .(z1, z2))) -> last(.(z1, z2)) Tuples: BUBBLE(.(z0, .(z1, z2))) -> c5(BUBBLE(.(z0, z2))) BUBBLE(.(z0, .(z1, z2))) -> c6(BUBBLE(.(z1, z2))) LAST(.(z0, .(z1, z2))) -> c9(LAST(.(z1, z2))) BUTLAST(.(z0, .(z1, z2))) -> c12(BUTLAST(.(z1, z2))) BSORT(.(z0, z1)) -> c(LAST(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1))))))) BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil))), BUBBLE(.(z0, nil))) BSORT(.(z0, .(z1, z2))) -> c2(LAST(.(bubble(.(z0, .(z1, z2))), bsort(butlast(if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))))))), BSORT(butlast(bubble(.(z0, .(z1, z2))))), BUTLAST(bubble(.(z0, .(z1, z2)))), BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(bubble(.(z0, nil)))))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil))), BUBBLE(.(z0, nil))) BSORT(.(z0, .(z1, z2))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))), bsort(butlast(bubble(.(z0, .(z1, z2))))))), BSORT(butlast(bubble(.(z0, .(z1, z2))))), BUTLAST(bubble(.(z0, .(z1, z2)))), BUBBLE(.(z0, .(z1, z2)))) S tuples: BUBBLE(.(z0, .(z1, z2))) -> c5(BUBBLE(.(z0, z2))) BUBBLE(.(z0, .(z1, z2))) -> c6(BUBBLE(.(z1, z2))) LAST(.(z0, .(z1, z2))) -> c9(LAST(.(z1, z2))) BUTLAST(.(z0, .(z1, z2))) -> c12(BUTLAST(.(z1, z2))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil))), BUBBLE(.(z0, nil))) BSORT(.(z0, .(z1, z2))) -> c2(LAST(.(bubble(.(z0, .(z1, z2))), bsort(butlast(if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))))))), BSORT(butlast(bubble(.(z0, .(z1, z2))))), BUTLAST(bubble(.(z0, .(z1, z2)))), BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(bubble(.(z0, nil)))))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil))), BUBBLE(.(z0, nil))) BSORT(.(z0, .(z1, z2))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))), bsort(butlast(bubble(.(z0, .(z1, z2))))))), BSORT(butlast(bubble(.(z0, .(z1, z2))))), BUTLAST(bubble(.(z0, .(z1, z2)))), BUBBLE(.(z0, .(z1, z2)))) K tuples: BSORT(.(z0, z1)) -> c(LAST(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1))))))) BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) Defined Rule Symbols: bubble_1, bsort_1, butlast_1, last_1 Defined Pair Symbols: BUBBLE_1, LAST_1, BUTLAST_1, BSORT_1 Compound Symbols: c5_1, c6_1, c9_1, c12_1, c_1, c2_4 ---------------------------------------- (59) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (60) Obligation: Complexity Dependency Tuples Problem Rules: bubble(.(z0, nil)) -> .(z0, nil) bubble(.(z0, .(z1, z2))) -> if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))) bsort(nil) -> nil bsort(.(z0, z1)) -> last(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1)))))) butlast(nil) -> nil butlast(.(z0, nil)) -> nil butlast(.(z0, .(z1, z2))) -> .(z0, butlast(.(z1, z2))) last(.(z0, nil)) -> z0 last(.(z0, .(z1, z2))) -> last(.(z1, z2)) Tuples: BUBBLE(.(z0, .(z1, z2))) -> c5(BUBBLE(.(z0, z2))) BUBBLE(.(z0, .(z1, z2))) -> c6(BUBBLE(.(z1, z2))) LAST(.(z0, .(z1, z2))) -> c9(LAST(.(z1, z2))) BUTLAST(.(z0, .(z1, z2))) -> c12(BUTLAST(.(z1, z2))) BSORT(.(z0, z1)) -> c(LAST(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1))))))) BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BSORT(.(z0, .(z1, z2))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))), bsort(butlast(bubble(.(z0, .(z1, z2))))))), BSORT(butlast(bubble(.(z0, .(z1, z2))))), BUTLAST(bubble(.(z0, .(z1, z2)))), BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) BSORT(.(z0, .(z1, z2))) -> c2(BSORT(butlast(bubble(.(z0, .(z1, z2))))), BUTLAST(bubble(.(z0, .(z1, z2)))), BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(bubble(.(z0, nil)))))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) S tuples: BUBBLE(.(z0, .(z1, z2))) -> c5(BUBBLE(.(z0, z2))) BUBBLE(.(z0, .(z1, z2))) -> c6(BUBBLE(.(z1, z2))) LAST(.(z0, .(z1, z2))) -> c9(LAST(.(z1, z2))) BUTLAST(.(z0, .(z1, z2))) -> c12(BUTLAST(.(z1, z2))) BSORT(.(z0, .(z1, z2))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))), bsort(butlast(bubble(.(z0, .(z1, z2))))))), BSORT(butlast(bubble(.(z0, .(z1, z2))))), BUTLAST(bubble(.(z0, .(z1, z2)))), BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) BSORT(.(z0, .(z1, z2))) -> c2(BSORT(butlast(bubble(.(z0, .(z1, z2))))), BUTLAST(bubble(.(z0, .(z1, z2)))), BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(bubble(.(z0, nil)))))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) K tuples: BSORT(.(z0, z1)) -> c(LAST(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1))))))) BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) Defined Rule Symbols: bubble_1, bsort_1, butlast_1, last_1 Defined Pair Symbols: BUBBLE_1, LAST_1, BUTLAST_1, BSORT_1 Compound Symbols: c5_1, c6_1, c9_1, c12_1, c_1, c2_4, c2_3 ---------------------------------------- (61) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. BSORT(.(z0, .(z1, z2))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))), bsort(butlast(bubble(.(z0, .(z1, z2))))))), BSORT(butlast(bubble(.(z0, .(z1, z2))))), BUTLAST(bubble(.(z0, .(z1, z2)))), BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, .(z1, z2))) -> c2(BSORT(butlast(bubble(.(z0, .(z1, z2))))), BUTLAST(bubble(.(z0, .(z1, z2)))), BUBBLE(.(z0, .(z1, z2)))) We considered the (Usable) Rules: butlast(nil) -> nil butlast(.(z0, nil)) -> nil bubble(.(z0, nil)) -> .(z0, nil) bubble(.(z0, .(z1, z2))) -> if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))) butlast(.(z0, .(z1, z2))) -> .(z0, butlast(.(z1, z2))) And the Tuples: BUBBLE(.(z0, .(z1, z2))) -> c5(BUBBLE(.(z0, z2))) BUBBLE(.(z0, .(z1, z2))) -> c6(BUBBLE(.(z1, z2))) LAST(.(z0, .(z1, z2))) -> c9(LAST(.(z1, z2))) BUTLAST(.(z0, .(z1, z2))) -> c12(BUTLAST(.(z1, z2))) BSORT(.(z0, z1)) -> c(LAST(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1))))))) BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BSORT(.(z0, .(z1, z2))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))), bsort(butlast(bubble(.(z0, .(z1, z2))))))), BSORT(butlast(bubble(.(z0, .(z1, z2))))), BUTLAST(bubble(.(z0, .(z1, z2)))), BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) BSORT(.(z0, .(z1, z2))) -> c2(BSORT(butlast(bubble(.(z0, .(z1, z2))))), BUTLAST(bubble(.(z0, .(z1, z2)))), BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(bubble(.(z0, nil)))))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) The order we found is given by the following interpretation: Polynomial interpretation : POL(.(x_1, x_2)) = [1] + x_2 POL(<=(x_1, x_2)) = [1] + x_1 + x_2 POL(BSORT(x_1)) = [1] + x_1 POL(BUBBLE(x_1)) = 0 POL(BUTLAST(x_1)) = 0 POL(LAST(x_1)) = 0 POL(bsort(x_1)) = [1] POL(bubble(x_1)) = [1] POL(butlast(x_1)) = x_1 POL(c(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c2(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c2(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4 POL(c5(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c9(x_1)) = x_1 POL(if(x_1, x_2, x_3)) = 0 POL(last(x_1)) = [1] POL(nil) = 0 ---------------------------------------- (62) Obligation: Complexity Dependency Tuples Problem Rules: bubble(.(z0, nil)) -> .(z0, nil) bubble(.(z0, .(z1, z2))) -> if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))) bsort(nil) -> nil bsort(.(z0, z1)) -> last(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1)))))) butlast(nil) -> nil butlast(.(z0, nil)) -> nil butlast(.(z0, .(z1, z2))) -> .(z0, butlast(.(z1, z2))) last(.(z0, nil)) -> z0 last(.(z0, .(z1, z2))) -> last(.(z1, z2)) Tuples: BUBBLE(.(z0, .(z1, z2))) -> c5(BUBBLE(.(z0, z2))) BUBBLE(.(z0, .(z1, z2))) -> c6(BUBBLE(.(z1, z2))) LAST(.(z0, .(z1, z2))) -> c9(LAST(.(z1, z2))) BUTLAST(.(z0, .(z1, z2))) -> c12(BUTLAST(.(z1, z2))) BSORT(.(z0, z1)) -> c(LAST(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1))))))) BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BSORT(.(z0, .(z1, z2))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))), bsort(butlast(bubble(.(z0, .(z1, z2))))))), BSORT(butlast(bubble(.(z0, .(z1, z2))))), BUTLAST(bubble(.(z0, .(z1, z2)))), BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) BSORT(.(z0, .(z1, z2))) -> c2(BSORT(butlast(bubble(.(z0, .(z1, z2))))), BUTLAST(bubble(.(z0, .(z1, z2)))), BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(bubble(.(z0, nil)))))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) S tuples: BUBBLE(.(z0, .(z1, z2))) -> c5(BUBBLE(.(z0, z2))) BUBBLE(.(z0, .(z1, z2))) -> c6(BUBBLE(.(z1, z2))) LAST(.(z0, .(z1, z2))) -> c9(LAST(.(z1, z2))) BUTLAST(.(z0, .(z1, z2))) -> c12(BUTLAST(.(z1, z2))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(bubble(.(z0, nil)))))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) K tuples: BSORT(.(z0, z1)) -> c(LAST(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1))))))) BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BSORT(.(z0, .(z1, z2))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))), bsort(butlast(bubble(.(z0, .(z1, z2))))))), BSORT(butlast(bubble(.(z0, .(z1, z2))))), BUTLAST(bubble(.(z0, .(z1, z2)))), BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, .(z1, z2))) -> c2(BSORT(butlast(bubble(.(z0, .(z1, z2))))), BUTLAST(bubble(.(z0, .(z1, z2)))), BUBBLE(.(z0, .(z1, z2)))) Defined Rule Symbols: bubble_1, bsort_1, butlast_1, last_1 Defined Pair Symbols: BUBBLE_1, LAST_1, BUTLAST_1, BSORT_1 Compound Symbols: c5_1, c6_1, c9_1, c12_1, c_1, c2_4, c2_3 ---------------------------------------- (63) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. BUBBLE(.(z0, .(z1, z2))) -> c5(BUBBLE(.(z0, z2))) BUBBLE(.(z0, .(z1, z2))) -> c6(BUBBLE(.(z1, z2))) We considered the (Usable) Rules: butlast(nil) -> nil butlast(.(z0, nil)) -> nil bubble(.(z0, nil)) -> .(z0, nil) bubble(.(z0, .(z1, z2))) -> if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))) butlast(.(z0, .(z1, z2))) -> .(z0, butlast(.(z1, z2))) And the Tuples: BUBBLE(.(z0, .(z1, z2))) -> c5(BUBBLE(.(z0, z2))) BUBBLE(.(z0, .(z1, z2))) -> c6(BUBBLE(.(z1, z2))) LAST(.(z0, .(z1, z2))) -> c9(LAST(.(z1, z2))) BUTLAST(.(z0, .(z1, z2))) -> c12(BUTLAST(.(z1, z2))) BSORT(.(z0, z1)) -> c(LAST(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1))))))) BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BSORT(.(z0, .(z1, z2))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))), bsort(butlast(bubble(.(z0, .(z1, z2))))))), BSORT(butlast(bubble(.(z0, .(z1, z2))))), BUTLAST(bubble(.(z0, .(z1, z2)))), BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) BSORT(.(z0, .(z1, z2))) -> c2(BSORT(butlast(bubble(.(z0, .(z1, z2))))), BUTLAST(bubble(.(z0, .(z1, z2)))), BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(bubble(.(z0, nil)))))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) The order we found is given by the following interpretation: Polynomial interpretation : POL(.(x_1, x_2)) = [2] + x_2 POL(<=(x_1, x_2)) = [3] + x_1 + x_2 POL(BSORT(x_1)) = [2]x_1 POL(BUBBLE(x_1)) = x_1 POL(BUTLAST(x_1)) = 0 POL(LAST(x_1)) = 0 POL(bsort(x_1)) = 0 POL(bubble(x_1)) = [2] POL(butlast(x_1)) = x_1 POL(c(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c2(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c2(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4 POL(c5(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c9(x_1)) = x_1 POL(if(x_1, x_2, x_3)) = 0 POL(last(x_1)) = [3] POL(nil) = 0 ---------------------------------------- (64) Obligation: Complexity Dependency Tuples Problem Rules: bubble(.(z0, nil)) -> .(z0, nil) bubble(.(z0, .(z1, z2))) -> if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))) bsort(nil) -> nil bsort(.(z0, z1)) -> last(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1)))))) butlast(nil) -> nil butlast(.(z0, nil)) -> nil butlast(.(z0, .(z1, z2))) -> .(z0, butlast(.(z1, z2))) last(.(z0, nil)) -> z0 last(.(z0, .(z1, z2))) -> last(.(z1, z2)) Tuples: BUBBLE(.(z0, .(z1, z2))) -> c5(BUBBLE(.(z0, z2))) BUBBLE(.(z0, .(z1, z2))) -> c6(BUBBLE(.(z1, z2))) LAST(.(z0, .(z1, z2))) -> c9(LAST(.(z1, z2))) BUTLAST(.(z0, .(z1, z2))) -> c12(BUTLAST(.(z1, z2))) BSORT(.(z0, z1)) -> c(LAST(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1))))))) BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BSORT(.(z0, .(z1, z2))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))), bsort(butlast(bubble(.(z0, .(z1, z2))))))), BSORT(butlast(bubble(.(z0, .(z1, z2))))), BUTLAST(bubble(.(z0, .(z1, z2)))), BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) BSORT(.(z0, .(z1, z2))) -> c2(BSORT(butlast(bubble(.(z0, .(z1, z2))))), BUTLAST(bubble(.(z0, .(z1, z2)))), BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(bubble(.(z0, nil)))))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) S tuples: LAST(.(z0, .(z1, z2))) -> c9(LAST(.(z1, z2))) BUTLAST(.(z0, .(z1, z2))) -> c12(BUTLAST(.(z1, z2))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(bubble(.(z0, nil)))))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) K tuples: BSORT(.(z0, z1)) -> c(LAST(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1))))))) BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BSORT(.(z0, .(z1, z2))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))), bsort(butlast(bubble(.(z0, .(z1, z2))))))), BSORT(butlast(bubble(.(z0, .(z1, z2))))), BUTLAST(bubble(.(z0, .(z1, z2)))), BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, .(z1, z2))) -> c2(BSORT(butlast(bubble(.(z0, .(z1, z2))))), BUTLAST(bubble(.(z0, .(z1, z2)))), BUBBLE(.(z0, .(z1, z2)))) BUBBLE(.(z0, .(z1, z2))) -> c5(BUBBLE(.(z0, z2))) BUBBLE(.(z0, .(z1, z2))) -> c6(BUBBLE(.(z1, z2))) Defined Rule Symbols: bubble_1, bsort_1, butlast_1, last_1 Defined Pair Symbols: BUBBLE_1, LAST_1, BUTLAST_1, BSORT_1 Compound Symbols: c5_1, c6_1, c9_1, c12_1, c_1, c2_4, c2_3 ---------------------------------------- (65) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace BSORT(.(z0, z1)) -> c(LAST(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1))))))) by BSORT(.(z0, nil)) -> c(LAST(.(bubble(.(z0, nil)), bsort(butlast(.(z0, nil)))))) BSORT(.(z0, .(z1, z2))) -> c(LAST(.(bubble(.(z0, .(z1, z2))), bsort(butlast(if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2))))))))) BSORT(.(z0, nil)) -> c(LAST(.(.(z0, nil), bsort(butlast(bubble(.(z0, nil))))))) BSORT(.(z0, .(z1, z2))) -> c(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))), bsort(butlast(bubble(.(z0, .(z1, z2)))))))) ---------------------------------------- (66) Obligation: Complexity Dependency Tuples Problem Rules: bubble(.(z0, nil)) -> .(z0, nil) bubble(.(z0, .(z1, z2))) -> if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))) bsort(nil) -> nil bsort(.(z0, z1)) -> last(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1)))))) butlast(nil) -> nil butlast(.(z0, nil)) -> nil butlast(.(z0, .(z1, z2))) -> .(z0, butlast(.(z1, z2))) last(.(z0, nil)) -> z0 last(.(z0, .(z1, z2))) -> last(.(z1, z2)) Tuples: BUBBLE(.(z0, .(z1, z2))) -> c5(BUBBLE(.(z0, z2))) BUBBLE(.(z0, .(z1, z2))) -> c6(BUBBLE(.(z1, z2))) LAST(.(z0, .(z1, z2))) -> c9(LAST(.(z1, z2))) BUTLAST(.(z0, .(z1, z2))) -> c12(BUTLAST(.(z1, z2))) BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BSORT(.(z0, .(z1, z2))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))), bsort(butlast(bubble(.(z0, .(z1, z2))))))), BSORT(butlast(bubble(.(z0, .(z1, z2))))), BUTLAST(bubble(.(z0, .(z1, z2)))), BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) BSORT(.(z0, .(z1, z2))) -> c2(BSORT(butlast(bubble(.(z0, .(z1, z2))))), BUTLAST(bubble(.(z0, .(z1, z2)))), BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(bubble(.(z0, nil)))))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) BSORT(.(z0, nil)) -> c(LAST(.(bubble(.(z0, nil)), bsort(butlast(.(z0, nil)))))) BSORT(.(z0, .(z1, z2))) -> c(LAST(.(bubble(.(z0, .(z1, z2))), bsort(butlast(if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2))))))))) BSORT(.(z0, nil)) -> c(LAST(.(.(z0, nil), bsort(butlast(bubble(.(z0, nil))))))) BSORT(.(z0, .(z1, z2))) -> c(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))), bsort(butlast(bubble(.(z0, .(z1, z2)))))))) S tuples: LAST(.(z0, .(z1, z2))) -> c9(LAST(.(z1, z2))) BUTLAST(.(z0, .(z1, z2))) -> c12(BUTLAST(.(z1, z2))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(bubble(.(z0, nil)))))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) K tuples: BSORT(.(z0, z1)) -> c(LAST(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1))))))) BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BSORT(.(z0, .(z1, z2))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))), bsort(butlast(bubble(.(z0, .(z1, z2))))))), BSORT(butlast(bubble(.(z0, .(z1, z2))))), BUTLAST(bubble(.(z0, .(z1, z2)))), BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, .(z1, z2))) -> c2(BSORT(butlast(bubble(.(z0, .(z1, z2))))), BUTLAST(bubble(.(z0, .(z1, z2)))), BUBBLE(.(z0, .(z1, z2)))) BUBBLE(.(z0, .(z1, z2))) -> c5(BUBBLE(.(z0, z2))) BUBBLE(.(z0, .(z1, z2))) -> c6(BUBBLE(.(z1, z2))) Defined Rule Symbols: bubble_1, bsort_1, butlast_1, last_1 Defined Pair Symbols: BUBBLE_1, LAST_1, BUTLAST_1, BSORT_1 Compound Symbols: c5_1, c6_1, c9_1, c12_1, c_1, c2_4, c2_3 ---------------------------------------- (67) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: BSORT(.(z0, .(z1, z2))) -> c(LAST(.(bubble(.(z0, .(z1, z2))), bsort(butlast(if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2))))))))) ---------------------------------------- (68) Obligation: Complexity Dependency Tuples Problem Rules: bubble(.(z0, nil)) -> .(z0, nil) bubble(.(z0, .(z1, z2))) -> if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))) bsort(nil) -> nil bsort(.(z0, z1)) -> last(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1)))))) butlast(nil) -> nil butlast(.(z0, nil)) -> nil butlast(.(z0, .(z1, z2))) -> .(z0, butlast(.(z1, z2))) last(.(z0, nil)) -> z0 last(.(z0, .(z1, z2))) -> last(.(z1, z2)) Tuples: BUBBLE(.(z0, .(z1, z2))) -> c5(BUBBLE(.(z0, z2))) BUBBLE(.(z0, .(z1, z2))) -> c6(BUBBLE(.(z1, z2))) LAST(.(z0, .(z1, z2))) -> c9(LAST(.(z1, z2))) BUTLAST(.(z0, .(z1, z2))) -> c12(BUTLAST(.(z1, z2))) BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BSORT(.(z0, .(z1, z2))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))), bsort(butlast(bubble(.(z0, .(z1, z2))))))), BSORT(butlast(bubble(.(z0, .(z1, z2))))), BUTLAST(bubble(.(z0, .(z1, z2)))), BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) BSORT(.(z0, .(z1, z2))) -> c2(BSORT(butlast(bubble(.(z0, .(z1, z2))))), BUTLAST(bubble(.(z0, .(z1, z2)))), BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(bubble(.(z0, nil)))))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) BSORT(.(z0, nil)) -> c(LAST(.(bubble(.(z0, nil)), bsort(butlast(.(z0, nil)))))) BSORT(.(z0, nil)) -> c(LAST(.(.(z0, nil), bsort(butlast(bubble(.(z0, nil))))))) BSORT(.(z0, .(z1, z2))) -> c(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))), bsort(butlast(bubble(.(z0, .(z1, z2)))))))) S tuples: LAST(.(z0, .(z1, z2))) -> c9(LAST(.(z1, z2))) BUTLAST(.(z0, .(z1, z2))) -> c12(BUTLAST(.(z1, z2))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(bubble(.(z0, nil)))))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) K tuples: BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BSORT(.(z0, .(z1, z2))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))), bsort(butlast(bubble(.(z0, .(z1, z2))))))), BSORT(butlast(bubble(.(z0, .(z1, z2))))), BUTLAST(bubble(.(z0, .(z1, z2)))), BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, .(z1, z2))) -> c2(BSORT(butlast(bubble(.(z0, .(z1, z2))))), BUTLAST(bubble(.(z0, .(z1, z2)))), BUBBLE(.(z0, .(z1, z2)))) BUBBLE(.(z0, .(z1, z2))) -> c5(BUBBLE(.(z0, z2))) BUBBLE(.(z0, .(z1, z2))) -> c6(BUBBLE(.(z1, z2))) Defined Rule Symbols: bubble_1, bsort_1, butlast_1, last_1 Defined Pair Symbols: BUBBLE_1, LAST_1, BUTLAST_1, BSORT_1 Compound Symbols: c5_1, c6_1, c9_1, c12_1, c_1, c2_4, c2_3 ---------------------------------------- (69) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace BSORT(.(z0, .(z1, z2))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))), bsort(butlast(bubble(.(z0, .(z1, z2))))))), BSORT(butlast(bubble(.(z0, .(z1, z2))))), BUTLAST(bubble(.(z0, .(z1, z2)))), BUBBLE(.(z0, .(z1, z2)))) by BSORT(.(z0, .(z1, z2))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))), bsort(butlast(if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))))))), BSORT(butlast(bubble(.(z0, .(z1, z2))))), BUTLAST(bubble(.(z0, .(z1, z2)))), BUBBLE(.(z0, .(z1, z2)))) BSORT(.(x0, .(z0, nil))) -> c2(LAST(.(if(<=(x0, z0), .(z0, bubble(.(x0, nil))), .(x0, .(z0, nil))), bsort(butlast(bubble(.(x0, .(z0, nil))))))), BSORT(butlast(bubble(.(x0, .(z0, nil))))), BUTLAST(bubble(.(x0, .(z0, nil)))), BUBBLE(.(x0, .(z0, nil)))) BSORT(.(x0, .(z0, .(z1, z2)))) -> c2(LAST(.(if(<=(x0, z0), .(z0, bubble(.(x0, .(z1, z2)))), .(x0, if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))))), bsort(butlast(bubble(.(x0, .(z0, .(z1, z2)))))))), BSORT(butlast(bubble(.(x0, .(z0, .(z1, z2)))))), BUTLAST(bubble(.(x0, .(z0, .(z1, z2))))), BUBBLE(.(x0, .(z0, .(z1, z2))))) BSORT(.(z0, .(x1, nil))) -> c2(LAST(.(if(<=(z0, x1), .(x1, .(z0, nil)), .(z0, bubble(.(x1, nil)))), bsort(butlast(bubble(.(z0, .(x1, nil))))))), BSORT(butlast(bubble(.(z0, .(x1, nil))))), BUTLAST(bubble(.(z0, .(x1, nil)))), BUBBLE(.(z0, .(x1, nil)))) BSORT(.(z0, .(x1, .(z1, z2)))) -> c2(LAST(.(if(<=(z0, x1), .(x1, if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2))))), .(z0, bubble(.(x1, .(z1, z2))))), bsort(butlast(bubble(.(z0, .(x1, .(z1, z2)))))))), BSORT(butlast(bubble(.(z0, .(x1, .(z1, z2)))))), BUTLAST(bubble(.(z0, .(x1, .(z1, z2))))), BUBBLE(.(z0, .(x1, .(z1, z2))))) ---------------------------------------- (70) Obligation: Complexity Dependency Tuples Problem Rules: bubble(.(z0, nil)) -> .(z0, nil) bubble(.(z0, .(z1, z2))) -> if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))) bsort(nil) -> nil bsort(.(z0, z1)) -> last(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1)))))) butlast(nil) -> nil butlast(.(z0, nil)) -> nil butlast(.(z0, .(z1, z2))) -> .(z0, butlast(.(z1, z2))) last(.(z0, nil)) -> z0 last(.(z0, .(z1, z2))) -> last(.(z1, z2)) Tuples: BUBBLE(.(z0, .(z1, z2))) -> c5(BUBBLE(.(z0, z2))) BUBBLE(.(z0, .(z1, z2))) -> c6(BUBBLE(.(z1, z2))) LAST(.(z0, .(z1, z2))) -> c9(LAST(.(z1, z2))) BUTLAST(.(z0, .(z1, z2))) -> c12(BUTLAST(.(z1, z2))) BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) BSORT(.(z0, .(z1, z2))) -> c2(BSORT(butlast(bubble(.(z0, .(z1, z2))))), BUTLAST(bubble(.(z0, .(z1, z2)))), BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(bubble(.(z0, nil)))))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) BSORT(.(z0, nil)) -> c(LAST(.(bubble(.(z0, nil)), bsort(butlast(.(z0, nil)))))) BSORT(.(z0, nil)) -> c(LAST(.(.(z0, nil), bsort(butlast(bubble(.(z0, nil))))))) BSORT(.(z0, .(z1, z2))) -> c(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))), bsort(butlast(bubble(.(z0, .(z1, z2)))))))) BSORT(.(z0, .(z1, z2))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))), bsort(butlast(if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))))))), BSORT(butlast(bubble(.(z0, .(z1, z2))))), BUTLAST(bubble(.(z0, .(z1, z2)))), BUBBLE(.(z0, .(z1, z2)))) BSORT(.(x0, .(z0, nil))) -> c2(LAST(.(if(<=(x0, z0), .(z0, bubble(.(x0, nil))), .(x0, .(z0, nil))), bsort(butlast(bubble(.(x0, .(z0, nil))))))), BSORT(butlast(bubble(.(x0, .(z0, nil))))), BUTLAST(bubble(.(x0, .(z0, nil)))), BUBBLE(.(x0, .(z0, nil)))) BSORT(.(x0, .(z0, .(z1, z2)))) -> c2(LAST(.(if(<=(x0, z0), .(z0, bubble(.(x0, .(z1, z2)))), .(x0, if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))))), bsort(butlast(bubble(.(x0, .(z0, .(z1, z2)))))))), BSORT(butlast(bubble(.(x0, .(z0, .(z1, z2)))))), BUTLAST(bubble(.(x0, .(z0, .(z1, z2))))), BUBBLE(.(x0, .(z0, .(z1, z2))))) BSORT(.(z0, .(x1, nil))) -> c2(LAST(.(if(<=(z0, x1), .(x1, .(z0, nil)), .(z0, bubble(.(x1, nil)))), bsort(butlast(bubble(.(z0, .(x1, nil))))))), BSORT(butlast(bubble(.(z0, .(x1, nil))))), BUTLAST(bubble(.(z0, .(x1, nil)))), BUBBLE(.(z0, .(x1, nil)))) BSORT(.(z0, .(x1, .(z1, z2)))) -> c2(LAST(.(if(<=(z0, x1), .(x1, if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2))))), .(z0, bubble(.(x1, .(z1, z2))))), bsort(butlast(bubble(.(z0, .(x1, .(z1, z2)))))))), BSORT(butlast(bubble(.(z0, .(x1, .(z1, z2)))))), BUTLAST(bubble(.(z0, .(x1, .(z1, z2))))), BUBBLE(.(z0, .(x1, .(z1, z2))))) S tuples: LAST(.(z0, .(z1, z2))) -> c9(LAST(.(z1, z2))) BUTLAST(.(z0, .(z1, z2))) -> c12(BUTLAST(.(z1, z2))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(bubble(.(z0, nil)))))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) K tuples: BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BSORT(.(z0, .(z1, z2))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))), bsort(butlast(bubble(.(z0, .(z1, z2))))))), BSORT(butlast(bubble(.(z0, .(z1, z2))))), BUTLAST(bubble(.(z0, .(z1, z2)))), BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, .(z1, z2))) -> c2(BSORT(butlast(bubble(.(z0, .(z1, z2))))), BUTLAST(bubble(.(z0, .(z1, z2)))), BUBBLE(.(z0, .(z1, z2)))) BUBBLE(.(z0, .(z1, z2))) -> c5(BUBBLE(.(z0, z2))) BUBBLE(.(z0, .(z1, z2))) -> c6(BUBBLE(.(z1, z2))) Defined Rule Symbols: bubble_1, bsort_1, butlast_1, last_1 Defined Pair Symbols: BUBBLE_1, LAST_1, BUTLAST_1, BSORT_1 Compound Symbols: c5_1, c6_1, c9_1, c12_1, c_1, c2_3, c2_4 ---------------------------------------- (71) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (72) Obligation: Complexity Dependency Tuples Problem Rules: bubble(.(z0, nil)) -> .(z0, nil) bubble(.(z0, .(z1, z2))) -> if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))) bsort(nil) -> nil bsort(.(z0, z1)) -> last(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1)))))) butlast(nil) -> nil butlast(.(z0, nil)) -> nil butlast(.(z0, .(z1, z2))) -> .(z0, butlast(.(z1, z2))) last(.(z0, nil)) -> z0 last(.(z0, .(z1, z2))) -> last(.(z1, z2)) Tuples: BUBBLE(.(z0, .(z1, z2))) -> c5(BUBBLE(.(z0, z2))) BUBBLE(.(z0, .(z1, z2))) -> c6(BUBBLE(.(z1, z2))) LAST(.(z0, .(z1, z2))) -> c9(LAST(.(z1, z2))) BUTLAST(.(z0, .(z1, z2))) -> c12(BUTLAST(.(z1, z2))) BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) BSORT(.(z0, .(z1, z2))) -> c2(BSORT(butlast(bubble(.(z0, .(z1, z2))))), BUTLAST(bubble(.(z0, .(z1, z2)))), BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(bubble(.(z0, nil)))))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) BSORT(.(z0, nil)) -> c(LAST(.(bubble(.(z0, nil)), bsort(butlast(.(z0, nil)))))) BSORT(.(z0, nil)) -> c(LAST(.(.(z0, nil), bsort(butlast(bubble(.(z0, nil))))))) BSORT(.(z0, .(z1, z2))) -> c(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))), bsort(butlast(bubble(.(z0, .(z1, z2)))))))) BSORT(.(x0, .(z0, nil))) -> c2(LAST(.(if(<=(x0, z0), .(z0, bubble(.(x0, nil))), .(x0, .(z0, nil))), bsort(butlast(bubble(.(x0, .(z0, nil))))))), BSORT(butlast(bubble(.(x0, .(z0, nil))))), BUTLAST(bubble(.(x0, .(z0, nil)))), BUBBLE(.(x0, .(z0, nil)))) BSORT(.(x0, .(z0, .(z1, z2)))) -> c2(LAST(.(if(<=(x0, z0), .(z0, bubble(.(x0, .(z1, z2)))), .(x0, if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))))), bsort(butlast(bubble(.(x0, .(z0, .(z1, z2)))))))), BSORT(butlast(bubble(.(x0, .(z0, .(z1, z2)))))), BUTLAST(bubble(.(x0, .(z0, .(z1, z2))))), BUBBLE(.(x0, .(z0, .(z1, z2))))) BSORT(.(z0, .(x1, nil))) -> c2(LAST(.(if(<=(z0, x1), .(x1, .(z0, nil)), .(z0, bubble(.(x1, nil)))), bsort(butlast(bubble(.(z0, .(x1, nil))))))), BSORT(butlast(bubble(.(z0, .(x1, nil))))), BUTLAST(bubble(.(z0, .(x1, nil)))), BUBBLE(.(z0, .(x1, nil)))) BSORT(.(z0, .(x1, .(z1, z2)))) -> c2(LAST(.(if(<=(z0, x1), .(x1, if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2))))), .(z0, bubble(.(x1, .(z1, z2))))), bsort(butlast(bubble(.(z0, .(x1, .(z1, z2)))))))), BSORT(butlast(bubble(.(z0, .(x1, .(z1, z2)))))), BUTLAST(bubble(.(z0, .(x1, .(z1, z2))))), BUBBLE(.(z0, .(x1, .(z1, z2))))) S tuples: LAST(.(z0, .(z1, z2))) -> c9(LAST(.(z1, z2))) BUTLAST(.(z0, .(z1, z2))) -> c12(BUTLAST(.(z1, z2))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(bubble(.(z0, nil)))))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) K tuples: BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BSORT(.(z0, .(z1, z2))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))), bsort(butlast(bubble(.(z0, .(z1, z2))))))), BSORT(butlast(bubble(.(z0, .(z1, z2))))), BUTLAST(bubble(.(z0, .(z1, z2)))), BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, .(z1, z2))) -> c2(BSORT(butlast(bubble(.(z0, .(z1, z2))))), BUTLAST(bubble(.(z0, .(z1, z2)))), BUBBLE(.(z0, .(z1, z2)))) BUBBLE(.(z0, .(z1, z2))) -> c5(BUBBLE(.(z0, z2))) BUBBLE(.(z0, .(z1, z2))) -> c6(BUBBLE(.(z1, z2))) Defined Rule Symbols: bubble_1, bsort_1, butlast_1, last_1 Defined Pair Symbols: BUBBLE_1, LAST_1, BUTLAST_1, BSORT_1 Compound Symbols: c5_1, c6_1, c9_1, c12_1, c_1, c2_3, c2_4 ---------------------------------------- (73) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) by BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) ---------------------------------------- (74) Obligation: Complexity Dependency Tuples Problem Rules: bubble(.(z0, nil)) -> .(z0, nil) bubble(.(z0, .(z1, z2))) -> if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))) bsort(nil) -> nil bsort(.(z0, z1)) -> last(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1)))))) butlast(nil) -> nil butlast(.(z0, nil)) -> nil butlast(.(z0, .(z1, z2))) -> .(z0, butlast(.(z1, z2))) last(.(z0, nil)) -> z0 last(.(z0, .(z1, z2))) -> last(.(z1, z2)) Tuples: BUBBLE(.(z0, .(z1, z2))) -> c5(BUBBLE(.(z0, z2))) BUBBLE(.(z0, .(z1, z2))) -> c6(BUBBLE(.(z1, z2))) LAST(.(z0, .(z1, z2))) -> c9(LAST(.(z1, z2))) BUTLAST(.(z0, .(z1, z2))) -> c12(BUTLAST(.(z1, z2))) BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BSORT(.(z0, .(z1, z2))) -> c2(BSORT(butlast(bubble(.(z0, .(z1, z2))))), BUTLAST(bubble(.(z0, .(z1, z2)))), BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(bubble(.(z0, nil)))))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) BSORT(.(z0, nil)) -> c(LAST(.(bubble(.(z0, nil)), bsort(butlast(.(z0, nil)))))) BSORT(.(z0, nil)) -> c(LAST(.(.(z0, nil), bsort(butlast(bubble(.(z0, nil))))))) BSORT(.(z0, .(z1, z2))) -> c(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))), bsort(butlast(bubble(.(z0, .(z1, z2)))))))) BSORT(.(x0, .(z0, nil))) -> c2(LAST(.(if(<=(x0, z0), .(z0, bubble(.(x0, nil))), .(x0, .(z0, nil))), bsort(butlast(bubble(.(x0, .(z0, nil))))))), BSORT(butlast(bubble(.(x0, .(z0, nil))))), BUTLAST(bubble(.(x0, .(z0, nil)))), BUBBLE(.(x0, .(z0, nil)))) BSORT(.(x0, .(z0, .(z1, z2)))) -> c2(LAST(.(if(<=(x0, z0), .(z0, bubble(.(x0, .(z1, z2)))), .(x0, if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))))), bsort(butlast(bubble(.(x0, .(z0, .(z1, z2)))))))), BSORT(butlast(bubble(.(x0, .(z0, .(z1, z2)))))), BUTLAST(bubble(.(x0, .(z0, .(z1, z2))))), BUBBLE(.(x0, .(z0, .(z1, z2))))) BSORT(.(z0, .(x1, nil))) -> c2(LAST(.(if(<=(z0, x1), .(x1, .(z0, nil)), .(z0, bubble(.(x1, nil)))), bsort(butlast(bubble(.(z0, .(x1, nil))))))), BSORT(butlast(bubble(.(z0, .(x1, nil))))), BUTLAST(bubble(.(z0, .(x1, nil)))), BUBBLE(.(z0, .(x1, nil)))) BSORT(.(z0, .(x1, .(z1, z2)))) -> c2(LAST(.(if(<=(z0, x1), .(x1, if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2))))), .(z0, bubble(.(x1, .(z1, z2))))), bsort(butlast(bubble(.(z0, .(x1, .(z1, z2)))))))), BSORT(butlast(bubble(.(z0, .(x1, .(z1, z2)))))), BUTLAST(bubble(.(z0, .(x1, .(z1, z2))))), BUBBLE(.(z0, .(x1, .(z1, z2))))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) S tuples: LAST(.(z0, .(z1, z2))) -> c9(LAST(.(z1, z2))) BUTLAST(.(z0, .(z1, z2))) -> c12(BUTLAST(.(z1, z2))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(bubble(.(z0, nil)))))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) K tuples: BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BSORT(.(z0, .(z1, z2))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))), bsort(butlast(bubble(.(z0, .(z1, z2))))))), BSORT(butlast(bubble(.(z0, .(z1, z2))))), BUTLAST(bubble(.(z0, .(z1, z2)))), BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, .(z1, z2))) -> c2(BSORT(butlast(bubble(.(z0, .(z1, z2))))), BUTLAST(bubble(.(z0, .(z1, z2)))), BUBBLE(.(z0, .(z1, z2)))) BUBBLE(.(z0, .(z1, z2))) -> c5(BUBBLE(.(z0, z2))) BUBBLE(.(z0, .(z1, z2))) -> c6(BUBBLE(.(z1, z2))) Defined Rule Symbols: bubble_1, bsort_1, butlast_1, last_1 Defined Pair Symbols: BUBBLE_1, LAST_1, BUTLAST_1, BSORT_1 Compound Symbols: c5_1, c6_1, c9_1, c12_1, c_1, c2_3, c2_4 ---------------------------------------- (75) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace BSORT(.(z0, .(z1, z2))) -> c2(BSORT(butlast(bubble(.(z0, .(z1, z2))))), BUTLAST(bubble(.(z0, .(z1, z2)))), BUBBLE(.(z0, .(z1, z2)))) by BSORT(.(z0, .(z1, z2))) -> c2(BSORT(butlast(if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))))), BUTLAST(bubble(.(z0, .(z1, z2)))), BUBBLE(.(z0, .(z1, z2)))) ---------------------------------------- (76) Obligation: Complexity Dependency Tuples Problem Rules: bubble(.(z0, nil)) -> .(z0, nil) bubble(.(z0, .(z1, z2))) -> if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))) bsort(nil) -> nil bsort(.(z0, z1)) -> last(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1)))))) butlast(nil) -> nil butlast(.(z0, nil)) -> nil butlast(.(z0, .(z1, z2))) -> .(z0, butlast(.(z1, z2))) last(.(z0, nil)) -> z0 last(.(z0, .(z1, z2))) -> last(.(z1, z2)) Tuples: BUBBLE(.(z0, .(z1, z2))) -> c5(BUBBLE(.(z0, z2))) BUBBLE(.(z0, .(z1, z2))) -> c6(BUBBLE(.(z1, z2))) LAST(.(z0, .(z1, z2))) -> c9(LAST(.(z1, z2))) BUTLAST(.(z0, .(z1, z2))) -> c12(BUTLAST(.(z1, z2))) BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BSORT(.(z0, .(z1, z2))) -> c2(BSORT(butlast(bubble(.(z0, .(z1, z2))))), BUTLAST(bubble(.(z0, .(z1, z2)))), BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(bubble(.(z0, nil)))))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) BSORT(.(z0, nil)) -> c(LAST(.(bubble(.(z0, nil)), bsort(butlast(.(z0, nil)))))) BSORT(.(z0, nil)) -> c(LAST(.(.(z0, nil), bsort(butlast(bubble(.(z0, nil))))))) BSORT(.(z0, .(z1, z2))) -> c(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))), bsort(butlast(bubble(.(z0, .(z1, z2)))))))) BSORT(.(x0, .(z0, nil))) -> c2(LAST(.(if(<=(x0, z0), .(z0, bubble(.(x0, nil))), .(x0, .(z0, nil))), bsort(butlast(bubble(.(x0, .(z0, nil))))))), BSORT(butlast(bubble(.(x0, .(z0, nil))))), BUTLAST(bubble(.(x0, .(z0, nil)))), BUBBLE(.(x0, .(z0, nil)))) BSORT(.(x0, .(z0, .(z1, z2)))) -> c2(LAST(.(if(<=(x0, z0), .(z0, bubble(.(x0, .(z1, z2)))), .(x0, if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))))), bsort(butlast(bubble(.(x0, .(z0, .(z1, z2)))))))), BSORT(butlast(bubble(.(x0, .(z0, .(z1, z2)))))), BUTLAST(bubble(.(x0, .(z0, .(z1, z2))))), BUBBLE(.(x0, .(z0, .(z1, z2))))) BSORT(.(z0, .(x1, nil))) -> c2(LAST(.(if(<=(z0, x1), .(x1, .(z0, nil)), .(z0, bubble(.(x1, nil)))), bsort(butlast(bubble(.(z0, .(x1, nil))))))), BSORT(butlast(bubble(.(z0, .(x1, nil))))), BUTLAST(bubble(.(z0, .(x1, nil)))), BUBBLE(.(z0, .(x1, nil)))) BSORT(.(z0, .(x1, .(z1, z2)))) -> c2(LAST(.(if(<=(z0, x1), .(x1, if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2))))), .(z0, bubble(.(x1, .(z1, z2))))), bsort(butlast(bubble(.(z0, .(x1, .(z1, z2)))))))), BSORT(butlast(bubble(.(z0, .(x1, .(z1, z2)))))), BUTLAST(bubble(.(z0, .(x1, .(z1, z2))))), BUBBLE(.(z0, .(x1, .(z1, z2))))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) BSORT(.(z0, .(z1, z2))) -> c2(BSORT(butlast(if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))))), BUTLAST(bubble(.(z0, .(z1, z2)))), BUBBLE(.(z0, .(z1, z2)))) S tuples: LAST(.(z0, .(z1, z2))) -> c9(LAST(.(z1, z2))) BUTLAST(.(z0, .(z1, z2))) -> c12(BUTLAST(.(z1, z2))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(bubble(.(z0, nil)))))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) K tuples: BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BSORT(.(z0, .(z1, z2))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))), bsort(butlast(bubble(.(z0, .(z1, z2))))))), BSORT(butlast(bubble(.(z0, .(z1, z2))))), BUTLAST(bubble(.(z0, .(z1, z2)))), BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, .(z1, z2))) -> c2(BSORT(butlast(bubble(.(z0, .(z1, z2))))), BUTLAST(bubble(.(z0, .(z1, z2)))), BUBBLE(.(z0, .(z1, z2)))) BUBBLE(.(z0, .(z1, z2))) -> c5(BUBBLE(.(z0, z2))) BUBBLE(.(z0, .(z1, z2))) -> c6(BUBBLE(.(z1, z2))) Defined Rule Symbols: bubble_1, bsort_1, butlast_1, last_1 Defined Pair Symbols: BUBBLE_1, LAST_1, BUTLAST_1, BSORT_1 Compound Symbols: c5_1, c6_1, c9_1, c12_1, c_1, c2_3, c2_4 ---------------------------------------- (77) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (78) Obligation: Complexity Dependency Tuples Problem Rules: bubble(.(z0, nil)) -> .(z0, nil) bubble(.(z0, .(z1, z2))) -> if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))) bsort(nil) -> nil bsort(.(z0, z1)) -> last(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1)))))) butlast(nil) -> nil butlast(.(z0, nil)) -> nil butlast(.(z0, .(z1, z2))) -> .(z0, butlast(.(z1, z2))) last(.(z0, nil)) -> z0 last(.(z0, .(z1, z2))) -> last(.(z1, z2)) Tuples: BUBBLE(.(z0, .(z1, z2))) -> c5(BUBBLE(.(z0, z2))) BUBBLE(.(z0, .(z1, z2))) -> c6(BUBBLE(.(z1, z2))) LAST(.(z0, .(z1, z2))) -> c9(LAST(.(z1, z2))) BUTLAST(.(z0, .(z1, z2))) -> c12(BUTLAST(.(z1, z2))) BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BSORT(.(z0, .(z1, z2))) -> c2(BSORT(butlast(bubble(.(z0, .(z1, z2))))), BUTLAST(bubble(.(z0, .(z1, z2)))), BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(bubble(.(z0, nil)))))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) BSORT(.(z0, nil)) -> c(LAST(.(bubble(.(z0, nil)), bsort(butlast(.(z0, nil)))))) BSORT(.(z0, nil)) -> c(LAST(.(.(z0, nil), bsort(butlast(bubble(.(z0, nil))))))) BSORT(.(z0, .(z1, z2))) -> c(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))), bsort(butlast(bubble(.(z0, .(z1, z2)))))))) BSORT(.(x0, .(z0, nil))) -> c2(LAST(.(if(<=(x0, z0), .(z0, bubble(.(x0, nil))), .(x0, .(z0, nil))), bsort(butlast(bubble(.(x0, .(z0, nil))))))), BSORT(butlast(bubble(.(x0, .(z0, nil))))), BUTLAST(bubble(.(x0, .(z0, nil)))), BUBBLE(.(x0, .(z0, nil)))) BSORT(.(x0, .(z0, .(z1, z2)))) -> c2(LAST(.(if(<=(x0, z0), .(z0, bubble(.(x0, .(z1, z2)))), .(x0, if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))))), bsort(butlast(bubble(.(x0, .(z0, .(z1, z2)))))))), BSORT(butlast(bubble(.(x0, .(z0, .(z1, z2)))))), BUTLAST(bubble(.(x0, .(z0, .(z1, z2))))), BUBBLE(.(x0, .(z0, .(z1, z2))))) BSORT(.(z0, .(x1, nil))) -> c2(LAST(.(if(<=(z0, x1), .(x1, .(z0, nil)), .(z0, bubble(.(x1, nil)))), bsort(butlast(bubble(.(z0, .(x1, nil))))))), BSORT(butlast(bubble(.(z0, .(x1, nil))))), BUTLAST(bubble(.(z0, .(x1, nil)))), BUBBLE(.(z0, .(x1, nil)))) BSORT(.(z0, .(x1, .(z1, z2)))) -> c2(LAST(.(if(<=(z0, x1), .(x1, if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2))))), .(z0, bubble(.(x1, .(z1, z2))))), bsort(butlast(bubble(.(z0, .(x1, .(z1, z2)))))))), BSORT(butlast(bubble(.(z0, .(x1, .(z1, z2)))))), BUTLAST(bubble(.(z0, .(x1, .(z1, z2))))), BUBBLE(.(z0, .(x1, .(z1, z2))))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) BSORT(.(z0, .(z1, z2))) -> c2(BUTLAST(bubble(.(z0, .(z1, z2)))), BUBBLE(.(z0, .(z1, z2)))) S tuples: LAST(.(z0, .(z1, z2))) -> c9(LAST(.(z1, z2))) BUTLAST(.(z0, .(z1, z2))) -> c12(BUTLAST(.(z1, z2))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(bubble(.(z0, nil)))))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) K tuples: BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BSORT(.(z0, .(z1, z2))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))), bsort(butlast(bubble(.(z0, .(z1, z2))))))), BSORT(butlast(bubble(.(z0, .(z1, z2))))), BUTLAST(bubble(.(z0, .(z1, z2)))), BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, .(z1, z2))) -> c2(BSORT(butlast(bubble(.(z0, .(z1, z2))))), BUTLAST(bubble(.(z0, .(z1, z2)))), BUBBLE(.(z0, .(z1, z2)))) BUBBLE(.(z0, .(z1, z2))) -> c5(BUBBLE(.(z0, z2))) BUBBLE(.(z0, .(z1, z2))) -> c6(BUBBLE(.(z1, z2))) Defined Rule Symbols: bubble_1, bsort_1, butlast_1, last_1 Defined Pair Symbols: BUBBLE_1, LAST_1, BUTLAST_1, BSORT_1 Compound Symbols: c5_1, c6_1, c9_1, c12_1, c_1, c2_3, c2_4, c2_2 ---------------------------------------- (79) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (80) Obligation: Complexity Dependency Tuples Problem Rules: bubble(.(z0, nil)) -> .(z0, nil) bubble(.(z0, .(z1, z2))) -> if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))) bsort(nil) -> nil bsort(.(z0, z1)) -> last(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1)))))) butlast(nil) -> nil butlast(.(z0, nil)) -> nil butlast(.(z0, .(z1, z2))) -> .(z0, butlast(.(z1, z2))) last(.(z0, nil)) -> z0 last(.(z0, .(z1, z2))) -> last(.(z1, z2)) Tuples: BUBBLE(.(z0, .(z1, z2))) -> c5(BUBBLE(.(z0, z2))) BUBBLE(.(z0, .(z1, z2))) -> c6(BUBBLE(.(z1, z2))) LAST(.(z0, .(z1, z2))) -> c9(LAST(.(z1, z2))) BUTLAST(.(z0, .(z1, z2))) -> c12(BUTLAST(.(z1, z2))) BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BSORT(.(z0, .(z1, z2))) -> c2(BSORT(butlast(bubble(.(z0, .(z1, z2))))), BUTLAST(bubble(.(z0, .(z1, z2)))), BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(bubble(.(z0, nil)))))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) BSORT(.(z0, nil)) -> c(LAST(.(bubble(.(z0, nil)), bsort(butlast(.(z0, nil)))))) BSORT(.(z0, nil)) -> c(LAST(.(.(z0, nil), bsort(butlast(bubble(.(z0, nil))))))) BSORT(.(z0, .(z1, z2))) -> c(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))), bsort(butlast(bubble(.(z0, .(z1, z2)))))))) BSORT(.(x0, .(z0, nil))) -> c2(LAST(.(if(<=(x0, z0), .(z0, bubble(.(x0, nil))), .(x0, .(z0, nil))), bsort(butlast(bubble(.(x0, .(z0, nil))))))), BSORT(butlast(bubble(.(x0, .(z0, nil))))), BUTLAST(bubble(.(x0, .(z0, nil)))), BUBBLE(.(x0, .(z0, nil)))) BSORT(.(x0, .(z0, .(z1, z2)))) -> c2(LAST(.(if(<=(x0, z0), .(z0, bubble(.(x0, .(z1, z2)))), .(x0, if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))))), bsort(butlast(bubble(.(x0, .(z0, .(z1, z2)))))))), BSORT(butlast(bubble(.(x0, .(z0, .(z1, z2)))))), BUTLAST(bubble(.(x0, .(z0, .(z1, z2))))), BUBBLE(.(x0, .(z0, .(z1, z2))))) BSORT(.(z0, .(x1, nil))) -> c2(LAST(.(if(<=(z0, x1), .(x1, .(z0, nil)), .(z0, bubble(.(x1, nil)))), bsort(butlast(bubble(.(z0, .(x1, nil))))))), BSORT(butlast(bubble(.(z0, .(x1, nil))))), BUTLAST(bubble(.(z0, .(x1, nil)))), BUBBLE(.(z0, .(x1, nil)))) BSORT(.(z0, .(x1, .(z1, z2)))) -> c2(LAST(.(if(<=(z0, x1), .(x1, if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2))))), .(z0, bubble(.(x1, .(z1, z2))))), bsort(butlast(bubble(.(z0, .(x1, .(z1, z2)))))))), BSORT(butlast(bubble(.(z0, .(x1, .(z1, z2)))))), BUTLAST(bubble(.(z0, .(x1, .(z1, z2))))), BUBBLE(.(z0, .(x1, .(z1, z2))))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) BSORT(.(z0, .(z1, z2))) -> c1(BUTLAST(bubble(.(z0, .(z1, z2))))) BSORT(.(z0, .(z1, z2))) -> c1(BUBBLE(.(z0, .(z1, z2)))) S tuples: LAST(.(z0, .(z1, z2))) -> c9(LAST(.(z1, z2))) BUTLAST(.(z0, .(z1, z2))) -> c12(BUTLAST(.(z1, z2))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(bubble(.(z0, nil)))))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) K tuples: BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BSORT(.(z0, .(z1, z2))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))), bsort(butlast(bubble(.(z0, .(z1, z2))))))), BSORT(butlast(bubble(.(z0, .(z1, z2))))), BUTLAST(bubble(.(z0, .(z1, z2)))), BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, .(z1, z2))) -> c2(BSORT(butlast(bubble(.(z0, .(z1, z2))))), BUTLAST(bubble(.(z0, .(z1, z2)))), BUBBLE(.(z0, .(z1, z2)))) BUBBLE(.(z0, .(z1, z2))) -> c5(BUBBLE(.(z0, z2))) BUBBLE(.(z0, .(z1, z2))) -> c6(BUBBLE(.(z1, z2))) Defined Rule Symbols: bubble_1, bsort_1, butlast_1, last_1 Defined Pair Symbols: BUBBLE_1, LAST_1, BUTLAST_1, BSORT_1 Compound Symbols: c5_1, c6_1, c9_1, c12_1, c_1, c2_3, c2_4, c1_1 ---------------------------------------- (81) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(bubble(.(z0, nil)))))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) by BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) ---------------------------------------- (82) Obligation: Complexity Dependency Tuples Problem Rules: bubble(.(z0, nil)) -> .(z0, nil) bubble(.(z0, .(z1, z2))) -> if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))) bsort(nil) -> nil bsort(.(z0, z1)) -> last(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1)))))) butlast(nil) -> nil butlast(.(z0, nil)) -> nil butlast(.(z0, .(z1, z2))) -> .(z0, butlast(.(z1, z2))) last(.(z0, nil)) -> z0 last(.(z0, .(z1, z2))) -> last(.(z1, z2)) Tuples: BUBBLE(.(z0, .(z1, z2))) -> c5(BUBBLE(.(z0, z2))) BUBBLE(.(z0, .(z1, z2))) -> c6(BUBBLE(.(z1, z2))) LAST(.(z0, .(z1, z2))) -> c9(LAST(.(z1, z2))) BUTLAST(.(z0, .(z1, z2))) -> c12(BUTLAST(.(z1, z2))) BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BSORT(.(z0, .(z1, z2))) -> c2(BSORT(butlast(bubble(.(z0, .(z1, z2))))), BUTLAST(bubble(.(z0, .(z1, z2)))), BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, nil)) -> c(LAST(.(bubble(.(z0, nil)), bsort(butlast(.(z0, nil)))))) BSORT(.(z0, nil)) -> c(LAST(.(.(z0, nil), bsort(butlast(bubble(.(z0, nil))))))) BSORT(.(z0, .(z1, z2))) -> c(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))), bsort(butlast(bubble(.(z0, .(z1, z2)))))))) BSORT(.(x0, .(z0, nil))) -> c2(LAST(.(if(<=(x0, z0), .(z0, bubble(.(x0, nil))), .(x0, .(z0, nil))), bsort(butlast(bubble(.(x0, .(z0, nil))))))), BSORT(butlast(bubble(.(x0, .(z0, nil))))), BUTLAST(bubble(.(x0, .(z0, nil)))), BUBBLE(.(x0, .(z0, nil)))) BSORT(.(x0, .(z0, .(z1, z2)))) -> c2(LAST(.(if(<=(x0, z0), .(z0, bubble(.(x0, .(z1, z2)))), .(x0, if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))))), bsort(butlast(bubble(.(x0, .(z0, .(z1, z2)))))))), BSORT(butlast(bubble(.(x0, .(z0, .(z1, z2)))))), BUTLAST(bubble(.(x0, .(z0, .(z1, z2))))), BUBBLE(.(x0, .(z0, .(z1, z2))))) BSORT(.(z0, .(x1, nil))) -> c2(LAST(.(if(<=(z0, x1), .(x1, .(z0, nil)), .(z0, bubble(.(x1, nil)))), bsort(butlast(bubble(.(z0, .(x1, nil))))))), BSORT(butlast(bubble(.(z0, .(x1, nil))))), BUTLAST(bubble(.(z0, .(x1, nil)))), BUBBLE(.(z0, .(x1, nil)))) BSORT(.(z0, .(x1, .(z1, z2)))) -> c2(LAST(.(if(<=(z0, x1), .(x1, if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2))))), .(z0, bubble(.(x1, .(z1, z2))))), bsort(butlast(bubble(.(z0, .(x1, .(z1, z2)))))))), BSORT(butlast(bubble(.(z0, .(x1, .(z1, z2)))))), BUTLAST(bubble(.(z0, .(x1, .(z1, z2))))), BUBBLE(.(z0, .(x1, .(z1, z2))))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) BSORT(.(z0, .(z1, z2))) -> c1(BUTLAST(bubble(.(z0, .(z1, z2))))) BSORT(.(z0, .(z1, z2))) -> c1(BUBBLE(.(z0, .(z1, z2)))) S tuples: LAST(.(z0, .(z1, z2))) -> c9(LAST(.(z1, z2))) BUTLAST(.(z0, .(z1, z2))) -> c12(BUTLAST(.(z1, z2))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) K tuples: BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BSORT(.(z0, .(z1, z2))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))), bsort(butlast(bubble(.(z0, .(z1, z2))))))), BSORT(butlast(bubble(.(z0, .(z1, z2))))), BUTLAST(bubble(.(z0, .(z1, z2)))), BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, .(z1, z2))) -> c2(BSORT(butlast(bubble(.(z0, .(z1, z2))))), BUTLAST(bubble(.(z0, .(z1, z2)))), BUBBLE(.(z0, .(z1, z2)))) BUBBLE(.(z0, .(z1, z2))) -> c5(BUBBLE(.(z0, z2))) BUBBLE(.(z0, .(z1, z2))) -> c6(BUBBLE(.(z1, z2))) Defined Rule Symbols: bubble_1, bsort_1, butlast_1, last_1 Defined Pair Symbols: BUBBLE_1, LAST_1, BUTLAST_1, BSORT_1 Compound Symbols: c5_1, c6_1, c9_1, c12_1, c_1, c2_3, c2_4, c1_1 ---------------------------------------- (83) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace BSORT(.(z0, nil)) -> c(LAST(.(bubble(.(z0, nil)), bsort(butlast(.(z0, nil)))))) by BSORT(.(z0, nil)) -> c(LAST(.(bubble(.(z0, nil)), bsort(nil)))) BSORT(.(z0, nil)) -> c(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil)))))) ---------------------------------------- (84) Obligation: Complexity Dependency Tuples Problem Rules: bubble(.(z0, nil)) -> .(z0, nil) bubble(.(z0, .(z1, z2))) -> if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))) bsort(nil) -> nil bsort(.(z0, z1)) -> last(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1)))))) butlast(nil) -> nil butlast(.(z0, nil)) -> nil butlast(.(z0, .(z1, z2))) -> .(z0, butlast(.(z1, z2))) last(.(z0, nil)) -> z0 last(.(z0, .(z1, z2))) -> last(.(z1, z2)) Tuples: BUBBLE(.(z0, .(z1, z2))) -> c5(BUBBLE(.(z0, z2))) BUBBLE(.(z0, .(z1, z2))) -> c6(BUBBLE(.(z1, z2))) LAST(.(z0, .(z1, z2))) -> c9(LAST(.(z1, z2))) BUTLAST(.(z0, .(z1, z2))) -> c12(BUTLAST(.(z1, z2))) BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BSORT(.(z0, .(z1, z2))) -> c2(BSORT(butlast(bubble(.(z0, .(z1, z2))))), BUTLAST(bubble(.(z0, .(z1, z2)))), BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, nil)) -> c(LAST(.(.(z0, nil), bsort(butlast(bubble(.(z0, nil))))))) BSORT(.(z0, .(z1, z2))) -> c(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))), bsort(butlast(bubble(.(z0, .(z1, z2)))))))) BSORT(.(x0, .(z0, nil))) -> c2(LAST(.(if(<=(x0, z0), .(z0, bubble(.(x0, nil))), .(x0, .(z0, nil))), bsort(butlast(bubble(.(x0, .(z0, nil))))))), BSORT(butlast(bubble(.(x0, .(z0, nil))))), BUTLAST(bubble(.(x0, .(z0, nil)))), BUBBLE(.(x0, .(z0, nil)))) BSORT(.(x0, .(z0, .(z1, z2)))) -> c2(LAST(.(if(<=(x0, z0), .(z0, bubble(.(x0, .(z1, z2)))), .(x0, if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))))), bsort(butlast(bubble(.(x0, .(z0, .(z1, z2)))))))), BSORT(butlast(bubble(.(x0, .(z0, .(z1, z2)))))), BUTLAST(bubble(.(x0, .(z0, .(z1, z2))))), BUBBLE(.(x0, .(z0, .(z1, z2))))) BSORT(.(z0, .(x1, nil))) -> c2(LAST(.(if(<=(z0, x1), .(x1, .(z0, nil)), .(z0, bubble(.(x1, nil)))), bsort(butlast(bubble(.(z0, .(x1, nil))))))), BSORT(butlast(bubble(.(z0, .(x1, nil))))), BUTLAST(bubble(.(z0, .(x1, nil)))), BUBBLE(.(z0, .(x1, nil)))) BSORT(.(z0, .(x1, .(z1, z2)))) -> c2(LAST(.(if(<=(z0, x1), .(x1, if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2))))), .(z0, bubble(.(x1, .(z1, z2))))), bsort(butlast(bubble(.(z0, .(x1, .(z1, z2)))))))), BSORT(butlast(bubble(.(z0, .(x1, .(z1, z2)))))), BUTLAST(bubble(.(z0, .(x1, .(z1, z2))))), BUBBLE(.(z0, .(x1, .(z1, z2))))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) BSORT(.(z0, .(z1, z2))) -> c1(BUTLAST(bubble(.(z0, .(z1, z2))))) BSORT(.(z0, .(z1, z2))) -> c1(BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, nil)) -> c(LAST(.(bubble(.(z0, nil)), bsort(nil)))) BSORT(.(z0, nil)) -> c(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil)))))) S tuples: LAST(.(z0, .(z1, z2))) -> c9(LAST(.(z1, z2))) BUTLAST(.(z0, .(z1, z2))) -> c12(BUTLAST(.(z1, z2))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) K tuples: BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BSORT(.(z0, .(z1, z2))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))), bsort(butlast(bubble(.(z0, .(z1, z2))))))), BSORT(butlast(bubble(.(z0, .(z1, z2))))), BUTLAST(bubble(.(z0, .(z1, z2)))), BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, .(z1, z2))) -> c2(BSORT(butlast(bubble(.(z0, .(z1, z2))))), BUTLAST(bubble(.(z0, .(z1, z2)))), BUBBLE(.(z0, .(z1, z2)))) BUBBLE(.(z0, .(z1, z2))) -> c5(BUBBLE(.(z0, z2))) BUBBLE(.(z0, .(z1, z2))) -> c6(BUBBLE(.(z1, z2))) Defined Rule Symbols: bubble_1, bsort_1, butlast_1, last_1 Defined Pair Symbols: BUBBLE_1, LAST_1, BUTLAST_1, BSORT_1 Compound Symbols: c5_1, c6_1, c9_1, c12_1, c_1, c2_3, c2_4, c1_1 ---------------------------------------- (85) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace BSORT(.(z0, nil)) -> c(LAST(.(.(z0, nil), bsort(butlast(bubble(.(z0, nil))))))) by BSORT(.(z0, nil)) -> c(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil)))))) ---------------------------------------- (86) Obligation: Complexity Dependency Tuples Problem Rules: bubble(.(z0, nil)) -> .(z0, nil) bubble(.(z0, .(z1, z2))) -> if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))) bsort(nil) -> nil bsort(.(z0, z1)) -> last(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1)))))) butlast(nil) -> nil butlast(.(z0, nil)) -> nil butlast(.(z0, .(z1, z2))) -> .(z0, butlast(.(z1, z2))) last(.(z0, nil)) -> z0 last(.(z0, .(z1, z2))) -> last(.(z1, z2)) Tuples: BUBBLE(.(z0, .(z1, z2))) -> c5(BUBBLE(.(z0, z2))) BUBBLE(.(z0, .(z1, z2))) -> c6(BUBBLE(.(z1, z2))) LAST(.(z0, .(z1, z2))) -> c9(LAST(.(z1, z2))) BUTLAST(.(z0, .(z1, z2))) -> c12(BUTLAST(.(z1, z2))) BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BSORT(.(z0, .(z1, z2))) -> c2(BSORT(butlast(bubble(.(z0, .(z1, z2))))), BUTLAST(bubble(.(z0, .(z1, z2)))), BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, .(z1, z2))) -> c(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))), bsort(butlast(bubble(.(z0, .(z1, z2)))))))) BSORT(.(x0, .(z0, nil))) -> c2(LAST(.(if(<=(x0, z0), .(z0, bubble(.(x0, nil))), .(x0, .(z0, nil))), bsort(butlast(bubble(.(x0, .(z0, nil))))))), BSORT(butlast(bubble(.(x0, .(z0, nil))))), BUTLAST(bubble(.(x0, .(z0, nil)))), BUBBLE(.(x0, .(z0, nil)))) BSORT(.(x0, .(z0, .(z1, z2)))) -> c2(LAST(.(if(<=(x0, z0), .(z0, bubble(.(x0, .(z1, z2)))), .(x0, if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))))), bsort(butlast(bubble(.(x0, .(z0, .(z1, z2)))))))), BSORT(butlast(bubble(.(x0, .(z0, .(z1, z2)))))), BUTLAST(bubble(.(x0, .(z0, .(z1, z2))))), BUBBLE(.(x0, .(z0, .(z1, z2))))) BSORT(.(z0, .(x1, nil))) -> c2(LAST(.(if(<=(z0, x1), .(x1, .(z0, nil)), .(z0, bubble(.(x1, nil)))), bsort(butlast(bubble(.(z0, .(x1, nil))))))), BSORT(butlast(bubble(.(z0, .(x1, nil))))), BUTLAST(bubble(.(z0, .(x1, nil)))), BUBBLE(.(z0, .(x1, nil)))) BSORT(.(z0, .(x1, .(z1, z2)))) -> c2(LAST(.(if(<=(z0, x1), .(x1, if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2))))), .(z0, bubble(.(x1, .(z1, z2))))), bsort(butlast(bubble(.(z0, .(x1, .(z1, z2)))))))), BSORT(butlast(bubble(.(z0, .(x1, .(z1, z2)))))), BUTLAST(bubble(.(z0, .(x1, .(z1, z2))))), BUBBLE(.(z0, .(x1, .(z1, z2))))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) BSORT(.(z0, .(z1, z2))) -> c1(BUTLAST(bubble(.(z0, .(z1, z2))))) BSORT(.(z0, .(z1, z2))) -> c1(BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, nil)) -> c(LAST(.(bubble(.(z0, nil)), bsort(nil)))) BSORT(.(z0, nil)) -> c(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil)))))) S tuples: LAST(.(z0, .(z1, z2))) -> c9(LAST(.(z1, z2))) BUTLAST(.(z0, .(z1, z2))) -> c12(BUTLAST(.(z1, z2))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) K tuples: BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BSORT(.(z0, .(z1, z2))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))), bsort(butlast(bubble(.(z0, .(z1, z2))))))), BSORT(butlast(bubble(.(z0, .(z1, z2))))), BUTLAST(bubble(.(z0, .(z1, z2)))), BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, .(z1, z2))) -> c2(BSORT(butlast(bubble(.(z0, .(z1, z2))))), BUTLAST(bubble(.(z0, .(z1, z2)))), BUBBLE(.(z0, .(z1, z2)))) BUBBLE(.(z0, .(z1, z2))) -> c5(BUBBLE(.(z0, z2))) BUBBLE(.(z0, .(z1, z2))) -> c6(BUBBLE(.(z1, z2))) Defined Rule Symbols: bubble_1, bsort_1, butlast_1, last_1 Defined Pair Symbols: BUBBLE_1, LAST_1, BUTLAST_1, BSORT_1 Compound Symbols: c5_1, c6_1, c9_1, c12_1, c_1, c2_3, c2_4, c1_1 ---------------------------------------- (87) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace BSORT(.(z0, .(z1, z2))) -> c(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))), bsort(butlast(bubble(.(z0, .(z1, z2)))))))) by BSORT(.(z0, .(z1, z2))) -> c(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))), bsort(butlast(if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2))))))))) BSORT(.(x0, .(z0, nil))) -> c(LAST(.(if(<=(x0, z0), .(z0, bubble(.(x0, nil))), .(x0, .(z0, nil))), bsort(butlast(bubble(.(x0, .(z0, nil)))))))) BSORT(.(x0, .(z0, .(z1, z2)))) -> c(LAST(.(if(<=(x0, z0), .(z0, bubble(.(x0, .(z1, z2)))), .(x0, if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))))), bsort(butlast(bubble(.(x0, .(z0, .(z1, z2))))))))) BSORT(.(z0, .(x1, nil))) -> c(LAST(.(if(<=(z0, x1), .(x1, .(z0, nil)), .(z0, bubble(.(x1, nil)))), bsort(butlast(bubble(.(z0, .(x1, nil)))))))) BSORT(.(z0, .(x1, .(z1, z2)))) -> c(LAST(.(if(<=(z0, x1), .(x1, if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2))))), .(z0, bubble(.(x1, .(z1, z2))))), bsort(butlast(bubble(.(z0, .(x1, .(z1, z2))))))))) ---------------------------------------- (88) Obligation: Complexity Dependency Tuples Problem Rules: bubble(.(z0, nil)) -> .(z0, nil) bubble(.(z0, .(z1, z2))) -> if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))) bsort(nil) -> nil bsort(.(z0, z1)) -> last(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1)))))) butlast(nil) -> nil butlast(.(z0, nil)) -> nil butlast(.(z0, .(z1, z2))) -> .(z0, butlast(.(z1, z2))) last(.(z0, nil)) -> z0 last(.(z0, .(z1, z2))) -> last(.(z1, z2)) Tuples: BUBBLE(.(z0, .(z1, z2))) -> c5(BUBBLE(.(z0, z2))) BUBBLE(.(z0, .(z1, z2))) -> c6(BUBBLE(.(z1, z2))) LAST(.(z0, .(z1, z2))) -> c9(LAST(.(z1, z2))) BUTLAST(.(z0, .(z1, z2))) -> c12(BUTLAST(.(z1, z2))) BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BSORT(.(z0, .(z1, z2))) -> c2(BSORT(butlast(bubble(.(z0, .(z1, z2))))), BUTLAST(bubble(.(z0, .(z1, z2)))), BUBBLE(.(z0, .(z1, z2)))) BSORT(.(x0, .(z0, nil))) -> c2(LAST(.(if(<=(x0, z0), .(z0, bubble(.(x0, nil))), .(x0, .(z0, nil))), bsort(butlast(bubble(.(x0, .(z0, nil))))))), BSORT(butlast(bubble(.(x0, .(z0, nil))))), BUTLAST(bubble(.(x0, .(z0, nil)))), BUBBLE(.(x0, .(z0, nil)))) BSORT(.(x0, .(z0, .(z1, z2)))) -> c2(LAST(.(if(<=(x0, z0), .(z0, bubble(.(x0, .(z1, z2)))), .(x0, if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))))), bsort(butlast(bubble(.(x0, .(z0, .(z1, z2)))))))), BSORT(butlast(bubble(.(x0, .(z0, .(z1, z2)))))), BUTLAST(bubble(.(x0, .(z0, .(z1, z2))))), BUBBLE(.(x0, .(z0, .(z1, z2))))) BSORT(.(z0, .(x1, nil))) -> c2(LAST(.(if(<=(z0, x1), .(x1, .(z0, nil)), .(z0, bubble(.(x1, nil)))), bsort(butlast(bubble(.(z0, .(x1, nil))))))), BSORT(butlast(bubble(.(z0, .(x1, nil))))), BUTLAST(bubble(.(z0, .(x1, nil)))), BUBBLE(.(z0, .(x1, nil)))) BSORT(.(z0, .(x1, .(z1, z2)))) -> c2(LAST(.(if(<=(z0, x1), .(x1, if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2))))), .(z0, bubble(.(x1, .(z1, z2))))), bsort(butlast(bubble(.(z0, .(x1, .(z1, z2)))))))), BSORT(butlast(bubble(.(z0, .(x1, .(z1, z2)))))), BUTLAST(bubble(.(z0, .(x1, .(z1, z2))))), BUBBLE(.(z0, .(x1, .(z1, z2))))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) BSORT(.(z0, .(z1, z2))) -> c1(BUTLAST(bubble(.(z0, .(z1, z2))))) BSORT(.(z0, .(z1, z2))) -> c1(BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, nil)) -> c(LAST(.(bubble(.(z0, nil)), bsort(nil)))) BSORT(.(z0, nil)) -> c(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil)))))) BSORT(.(z0, .(z1, z2))) -> c(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))), bsort(butlast(if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2))))))))) BSORT(.(x0, .(z0, nil))) -> c(LAST(.(if(<=(x0, z0), .(z0, bubble(.(x0, nil))), .(x0, .(z0, nil))), bsort(butlast(bubble(.(x0, .(z0, nil)))))))) BSORT(.(x0, .(z0, .(z1, z2)))) -> c(LAST(.(if(<=(x0, z0), .(z0, bubble(.(x0, .(z1, z2)))), .(x0, if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))))), bsort(butlast(bubble(.(x0, .(z0, .(z1, z2))))))))) BSORT(.(z0, .(x1, nil))) -> c(LAST(.(if(<=(z0, x1), .(x1, .(z0, nil)), .(z0, bubble(.(x1, nil)))), bsort(butlast(bubble(.(z0, .(x1, nil)))))))) BSORT(.(z0, .(x1, .(z1, z2)))) -> c(LAST(.(if(<=(z0, x1), .(x1, if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2))))), .(z0, bubble(.(x1, .(z1, z2))))), bsort(butlast(bubble(.(z0, .(x1, .(z1, z2))))))))) S tuples: LAST(.(z0, .(z1, z2))) -> c9(LAST(.(z1, z2))) BUTLAST(.(z0, .(z1, z2))) -> c12(BUTLAST(.(z1, z2))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) K tuples: BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BSORT(.(z0, .(z1, z2))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))), bsort(butlast(bubble(.(z0, .(z1, z2))))))), BSORT(butlast(bubble(.(z0, .(z1, z2))))), BUTLAST(bubble(.(z0, .(z1, z2)))), BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, .(z1, z2))) -> c2(BSORT(butlast(bubble(.(z0, .(z1, z2))))), BUTLAST(bubble(.(z0, .(z1, z2)))), BUBBLE(.(z0, .(z1, z2)))) BUBBLE(.(z0, .(z1, z2))) -> c5(BUBBLE(.(z0, z2))) BUBBLE(.(z0, .(z1, z2))) -> c6(BUBBLE(.(z1, z2))) Defined Rule Symbols: bubble_1, bsort_1, butlast_1, last_1 Defined Pair Symbols: BUBBLE_1, LAST_1, BUTLAST_1, BSORT_1 Compound Symbols: c5_1, c6_1, c9_1, c12_1, c_1, c2_3, c2_4, c1_1 ---------------------------------------- (89) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: BSORT(.(z0, .(z1, z2))) -> c(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))), bsort(butlast(if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2))))))))) ---------------------------------------- (90) Obligation: Complexity Dependency Tuples Problem Rules: bubble(.(z0, nil)) -> .(z0, nil) bubble(.(z0, .(z1, z2))) -> if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))) bsort(nil) -> nil bsort(.(z0, z1)) -> last(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1)))))) butlast(nil) -> nil butlast(.(z0, nil)) -> nil butlast(.(z0, .(z1, z2))) -> .(z0, butlast(.(z1, z2))) last(.(z0, nil)) -> z0 last(.(z0, .(z1, z2))) -> last(.(z1, z2)) Tuples: BUBBLE(.(z0, .(z1, z2))) -> c5(BUBBLE(.(z0, z2))) BUBBLE(.(z0, .(z1, z2))) -> c6(BUBBLE(.(z1, z2))) LAST(.(z0, .(z1, z2))) -> c9(LAST(.(z1, z2))) BUTLAST(.(z0, .(z1, z2))) -> c12(BUTLAST(.(z1, z2))) BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BSORT(.(x0, .(z0, nil))) -> c2(LAST(.(if(<=(x0, z0), .(z0, bubble(.(x0, nil))), .(x0, .(z0, nil))), bsort(butlast(bubble(.(x0, .(z0, nil))))))), BSORT(butlast(bubble(.(x0, .(z0, nil))))), BUTLAST(bubble(.(x0, .(z0, nil)))), BUBBLE(.(x0, .(z0, nil)))) BSORT(.(x0, .(z0, .(z1, z2)))) -> c2(LAST(.(if(<=(x0, z0), .(z0, bubble(.(x0, .(z1, z2)))), .(x0, if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))))), bsort(butlast(bubble(.(x0, .(z0, .(z1, z2)))))))), BSORT(butlast(bubble(.(x0, .(z0, .(z1, z2)))))), BUTLAST(bubble(.(x0, .(z0, .(z1, z2))))), BUBBLE(.(x0, .(z0, .(z1, z2))))) BSORT(.(z0, .(x1, nil))) -> c2(LAST(.(if(<=(z0, x1), .(x1, .(z0, nil)), .(z0, bubble(.(x1, nil)))), bsort(butlast(bubble(.(z0, .(x1, nil))))))), BSORT(butlast(bubble(.(z0, .(x1, nil))))), BUTLAST(bubble(.(z0, .(x1, nil)))), BUBBLE(.(z0, .(x1, nil)))) BSORT(.(z0, .(x1, .(z1, z2)))) -> c2(LAST(.(if(<=(z0, x1), .(x1, if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2))))), .(z0, bubble(.(x1, .(z1, z2))))), bsort(butlast(bubble(.(z0, .(x1, .(z1, z2)))))))), BSORT(butlast(bubble(.(z0, .(x1, .(z1, z2)))))), BUTLAST(bubble(.(z0, .(x1, .(z1, z2))))), BUBBLE(.(z0, .(x1, .(z1, z2))))) BSORT(.(z0, .(z1, z2))) -> c2(BSORT(butlast(bubble(.(z0, .(z1, z2))))), BUTLAST(bubble(.(z0, .(z1, z2)))), BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) BSORT(.(z0, .(z1, z2))) -> c1(BUTLAST(bubble(.(z0, .(z1, z2))))) BSORT(.(z0, .(z1, z2))) -> c1(BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, nil)) -> c(LAST(.(bubble(.(z0, nil)), bsort(nil)))) BSORT(.(z0, nil)) -> c(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil)))))) BSORT(.(x0, .(z0, nil))) -> c(LAST(.(if(<=(x0, z0), .(z0, bubble(.(x0, nil))), .(x0, .(z0, nil))), bsort(butlast(bubble(.(x0, .(z0, nil)))))))) BSORT(.(x0, .(z0, .(z1, z2)))) -> c(LAST(.(if(<=(x0, z0), .(z0, bubble(.(x0, .(z1, z2)))), .(x0, if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))))), bsort(butlast(bubble(.(x0, .(z0, .(z1, z2))))))))) BSORT(.(z0, .(x1, nil))) -> c(LAST(.(if(<=(z0, x1), .(x1, .(z0, nil)), .(z0, bubble(.(x1, nil)))), bsort(butlast(bubble(.(z0, .(x1, nil)))))))) BSORT(.(z0, .(x1, .(z1, z2)))) -> c(LAST(.(if(<=(z0, x1), .(x1, if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2))))), .(z0, bubble(.(x1, .(z1, z2))))), bsort(butlast(bubble(.(z0, .(x1, .(z1, z2))))))))) S tuples: LAST(.(z0, .(z1, z2))) -> c9(LAST(.(z1, z2))) BUTLAST(.(z0, .(z1, z2))) -> c12(BUTLAST(.(z1, z2))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) K tuples: BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BSORT(.(z0, .(z1, z2))) -> c2(BSORT(butlast(bubble(.(z0, .(z1, z2))))), BUTLAST(bubble(.(z0, .(z1, z2)))), BUBBLE(.(z0, .(z1, z2)))) BUBBLE(.(z0, .(z1, z2))) -> c5(BUBBLE(.(z0, z2))) BUBBLE(.(z0, .(z1, z2))) -> c6(BUBBLE(.(z1, z2))) Defined Rule Symbols: bubble_1, bsort_1, butlast_1, last_1 Defined Pair Symbols: BUBBLE_1, LAST_1, BUTLAST_1, BSORT_1 Compound Symbols: c5_1, c6_1, c9_1, c12_1, c_1, c2_4, c2_3, c1_1 ---------------------------------------- (91) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace BSORT(.(z0, .(z1, z2))) -> c2(BSORT(butlast(bubble(.(z0, .(z1, z2))))), BUTLAST(bubble(.(z0, .(z1, z2)))), BUBBLE(.(z0, .(z1, z2)))) by BSORT(.(z0, .(z1, z2))) -> c2(BSORT(butlast(if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))))), BUTLAST(bubble(.(z0, .(z1, z2)))), BUBBLE(.(z0, .(z1, z2)))) ---------------------------------------- (92) Obligation: Complexity Dependency Tuples Problem Rules: bubble(.(z0, nil)) -> .(z0, nil) bubble(.(z0, .(z1, z2))) -> if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))) bsort(nil) -> nil bsort(.(z0, z1)) -> last(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1)))))) butlast(nil) -> nil butlast(.(z0, nil)) -> nil butlast(.(z0, .(z1, z2))) -> .(z0, butlast(.(z1, z2))) last(.(z0, nil)) -> z0 last(.(z0, .(z1, z2))) -> last(.(z1, z2)) Tuples: BUBBLE(.(z0, .(z1, z2))) -> c5(BUBBLE(.(z0, z2))) BUBBLE(.(z0, .(z1, z2))) -> c6(BUBBLE(.(z1, z2))) LAST(.(z0, .(z1, z2))) -> c9(LAST(.(z1, z2))) BUTLAST(.(z0, .(z1, z2))) -> c12(BUTLAST(.(z1, z2))) BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BSORT(.(x0, .(z0, nil))) -> c2(LAST(.(if(<=(x0, z0), .(z0, bubble(.(x0, nil))), .(x0, .(z0, nil))), bsort(butlast(bubble(.(x0, .(z0, nil))))))), BSORT(butlast(bubble(.(x0, .(z0, nil))))), BUTLAST(bubble(.(x0, .(z0, nil)))), BUBBLE(.(x0, .(z0, nil)))) BSORT(.(x0, .(z0, .(z1, z2)))) -> c2(LAST(.(if(<=(x0, z0), .(z0, bubble(.(x0, .(z1, z2)))), .(x0, if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))))), bsort(butlast(bubble(.(x0, .(z0, .(z1, z2)))))))), BSORT(butlast(bubble(.(x0, .(z0, .(z1, z2)))))), BUTLAST(bubble(.(x0, .(z0, .(z1, z2))))), BUBBLE(.(x0, .(z0, .(z1, z2))))) BSORT(.(z0, .(x1, nil))) -> c2(LAST(.(if(<=(z0, x1), .(x1, .(z0, nil)), .(z0, bubble(.(x1, nil)))), bsort(butlast(bubble(.(z0, .(x1, nil))))))), BSORT(butlast(bubble(.(z0, .(x1, nil))))), BUTLAST(bubble(.(z0, .(x1, nil)))), BUBBLE(.(z0, .(x1, nil)))) BSORT(.(z0, .(x1, .(z1, z2)))) -> c2(LAST(.(if(<=(z0, x1), .(x1, if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2))))), .(z0, bubble(.(x1, .(z1, z2))))), bsort(butlast(bubble(.(z0, .(x1, .(z1, z2)))))))), BSORT(butlast(bubble(.(z0, .(x1, .(z1, z2)))))), BUTLAST(bubble(.(z0, .(x1, .(z1, z2))))), BUBBLE(.(z0, .(x1, .(z1, z2))))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) BSORT(.(z0, .(z1, z2))) -> c1(BUTLAST(bubble(.(z0, .(z1, z2))))) BSORT(.(z0, .(z1, z2))) -> c1(BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, nil)) -> c(LAST(.(bubble(.(z0, nil)), bsort(nil)))) BSORT(.(z0, nil)) -> c(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil)))))) BSORT(.(x0, .(z0, nil))) -> c(LAST(.(if(<=(x0, z0), .(z0, bubble(.(x0, nil))), .(x0, .(z0, nil))), bsort(butlast(bubble(.(x0, .(z0, nil)))))))) BSORT(.(x0, .(z0, .(z1, z2)))) -> c(LAST(.(if(<=(x0, z0), .(z0, bubble(.(x0, .(z1, z2)))), .(x0, if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))))), bsort(butlast(bubble(.(x0, .(z0, .(z1, z2))))))))) BSORT(.(z0, .(x1, nil))) -> c(LAST(.(if(<=(z0, x1), .(x1, .(z0, nil)), .(z0, bubble(.(x1, nil)))), bsort(butlast(bubble(.(z0, .(x1, nil)))))))) BSORT(.(z0, .(x1, .(z1, z2)))) -> c(LAST(.(if(<=(z0, x1), .(x1, if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2))))), .(z0, bubble(.(x1, .(z1, z2))))), bsort(butlast(bubble(.(z0, .(x1, .(z1, z2))))))))) BSORT(.(z0, .(z1, z2))) -> c2(BSORT(butlast(if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))))), BUTLAST(bubble(.(z0, .(z1, z2)))), BUBBLE(.(z0, .(z1, z2)))) S tuples: LAST(.(z0, .(z1, z2))) -> c9(LAST(.(z1, z2))) BUTLAST(.(z0, .(z1, z2))) -> c12(BUTLAST(.(z1, z2))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) K tuples: BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BSORT(.(z0, .(z1, z2))) -> c2(BSORT(butlast(bubble(.(z0, .(z1, z2))))), BUTLAST(bubble(.(z0, .(z1, z2)))), BUBBLE(.(z0, .(z1, z2)))) BUBBLE(.(z0, .(z1, z2))) -> c5(BUBBLE(.(z0, z2))) BUBBLE(.(z0, .(z1, z2))) -> c6(BUBBLE(.(z1, z2))) Defined Rule Symbols: bubble_1, bsort_1, butlast_1, last_1 Defined Pair Symbols: BUBBLE_1, LAST_1, BUTLAST_1, BSORT_1 Compound Symbols: c5_1, c6_1, c9_1, c12_1, c_1, c2_4, c2_3, c1_1 ---------------------------------------- (93) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (94) Obligation: Complexity Dependency Tuples Problem Rules: bubble(.(z0, nil)) -> .(z0, nil) bubble(.(z0, .(z1, z2))) -> if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))) bsort(nil) -> nil bsort(.(z0, z1)) -> last(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1)))))) butlast(nil) -> nil butlast(.(z0, nil)) -> nil butlast(.(z0, .(z1, z2))) -> .(z0, butlast(.(z1, z2))) last(.(z0, nil)) -> z0 last(.(z0, .(z1, z2))) -> last(.(z1, z2)) Tuples: BUBBLE(.(z0, .(z1, z2))) -> c5(BUBBLE(.(z0, z2))) BUBBLE(.(z0, .(z1, z2))) -> c6(BUBBLE(.(z1, z2))) LAST(.(z0, .(z1, z2))) -> c9(LAST(.(z1, z2))) BUTLAST(.(z0, .(z1, z2))) -> c12(BUTLAST(.(z1, z2))) BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BSORT(.(x0, .(z0, nil))) -> c2(LAST(.(if(<=(x0, z0), .(z0, bubble(.(x0, nil))), .(x0, .(z0, nil))), bsort(butlast(bubble(.(x0, .(z0, nil))))))), BSORT(butlast(bubble(.(x0, .(z0, nil))))), BUTLAST(bubble(.(x0, .(z0, nil)))), BUBBLE(.(x0, .(z0, nil)))) BSORT(.(x0, .(z0, .(z1, z2)))) -> c2(LAST(.(if(<=(x0, z0), .(z0, bubble(.(x0, .(z1, z2)))), .(x0, if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))))), bsort(butlast(bubble(.(x0, .(z0, .(z1, z2)))))))), BSORT(butlast(bubble(.(x0, .(z0, .(z1, z2)))))), BUTLAST(bubble(.(x0, .(z0, .(z1, z2))))), BUBBLE(.(x0, .(z0, .(z1, z2))))) BSORT(.(z0, .(x1, nil))) -> c2(LAST(.(if(<=(z0, x1), .(x1, .(z0, nil)), .(z0, bubble(.(x1, nil)))), bsort(butlast(bubble(.(z0, .(x1, nil))))))), BSORT(butlast(bubble(.(z0, .(x1, nil))))), BUTLAST(bubble(.(z0, .(x1, nil)))), BUBBLE(.(z0, .(x1, nil)))) BSORT(.(z0, .(x1, .(z1, z2)))) -> c2(LAST(.(if(<=(z0, x1), .(x1, if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2))))), .(z0, bubble(.(x1, .(z1, z2))))), bsort(butlast(bubble(.(z0, .(x1, .(z1, z2)))))))), BSORT(butlast(bubble(.(z0, .(x1, .(z1, z2)))))), BUTLAST(bubble(.(z0, .(x1, .(z1, z2))))), BUBBLE(.(z0, .(x1, .(z1, z2))))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) BSORT(.(z0, .(z1, z2))) -> c1(BUTLAST(bubble(.(z0, .(z1, z2))))) BSORT(.(z0, .(z1, z2))) -> c1(BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, nil)) -> c(LAST(.(bubble(.(z0, nil)), bsort(nil)))) BSORT(.(z0, nil)) -> c(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil)))))) BSORT(.(x0, .(z0, nil))) -> c(LAST(.(if(<=(x0, z0), .(z0, bubble(.(x0, nil))), .(x0, .(z0, nil))), bsort(butlast(bubble(.(x0, .(z0, nil)))))))) BSORT(.(x0, .(z0, .(z1, z2)))) -> c(LAST(.(if(<=(x0, z0), .(z0, bubble(.(x0, .(z1, z2)))), .(x0, if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))))), bsort(butlast(bubble(.(x0, .(z0, .(z1, z2))))))))) BSORT(.(z0, .(x1, nil))) -> c(LAST(.(if(<=(z0, x1), .(x1, .(z0, nil)), .(z0, bubble(.(x1, nil)))), bsort(butlast(bubble(.(z0, .(x1, nil)))))))) BSORT(.(z0, .(x1, .(z1, z2)))) -> c(LAST(.(if(<=(z0, x1), .(x1, if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2))))), .(z0, bubble(.(x1, .(z1, z2))))), bsort(butlast(bubble(.(z0, .(x1, .(z1, z2))))))))) BSORT(.(z0, .(z1, z2))) -> c2(BUTLAST(bubble(.(z0, .(z1, z2)))), BUBBLE(.(z0, .(z1, z2)))) S tuples: LAST(.(z0, .(z1, z2))) -> c9(LAST(.(z1, z2))) BUTLAST(.(z0, .(z1, z2))) -> c12(BUTLAST(.(z1, z2))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) K tuples: BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BSORT(.(z0, .(z1, z2))) -> c2(BSORT(butlast(bubble(.(z0, .(z1, z2))))), BUTLAST(bubble(.(z0, .(z1, z2)))), BUBBLE(.(z0, .(z1, z2)))) BUBBLE(.(z0, .(z1, z2))) -> c5(BUBBLE(.(z0, z2))) BUBBLE(.(z0, .(z1, z2))) -> c6(BUBBLE(.(z1, z2))) Defined Rule Symbols: bubble_1, bsort_1, butlast_1, last_1 Defined Pair Symbols: BUBBLE_1, LAST_1, BUTLAST_1, BSORT_1 Compound Symbols: c5_1, c6_1, c9_1, c12_1, c_1, c2_4, c2_3, c1_1, c2_2 ---------------------------------------- (95) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (96) Obligation: Complexity Dependency Tuples Problem Rules: bubble(.(z0, nil)) -> .(z0, nil) bubble(.(z0, .(z1, z2))) -> if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))) bsort(nil) -> nil bsort(.(z0, z1)) -> last(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1)))))) butlast(nil) -> nil butlast(.(z0, nil)) -> nil butlast(.(z0, .(z1, z2))) -> .(z0, butlast(.(z1, z2))) last(.(z0, nil)) -> z0 last(.(z0, .(z1, z2))) -> last(.(z1, z2)) Tuples: BUBBLE(.(z0, .(z1, z2))) -> c5(BUBBLE(.(z0, z2))) BUBBLE(.(z0, .(z1, z2))) -> c6(BUBBLE(.(z1, z2))) LAST(.(z0, .(z1, z2))) -> c9(LAST(.(z1, z2))) BUTLAST(.(z0, .(z1, z2))) -> c12(BUTLAST(.(z1, z2))) BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BSORT(.(x0, .(z0, nil))) -> c2(LAST(.(if(<=(x0, z0), .(z0, bubble(.(x0, nil))), .(x0, .(z0, nil))), bsort(butlast(bubble(.(x0, .(z0, nil))))))), BSORT(butlast(bubble(.(x0, .(z0, nil))))), BUTLAST(bubble(.(x0, .(z0, nil)))), BUBBLE(.(x0, .(z0, nil)))) BSORT(.(x0, .(z0, .(z1, z2)))) -> c2(LAST(.(if(<=(x0, z0), .(z0, bubble(.(x0, .(z1, z2)))), .(x0, if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))))), bsort(butlast(bubble(.(x0, .(z0, .(z1, z2)))))))), BSORT(butlast(bubble(.(x0, .(z0, .(z1, z2)))))), BUTLAST(bubble(.(x0, .(z0, .(z1, z2))))), BUBBLE(.(x0, .(z0, .(z1, z2))))) BSORT(.(z0, .(x1, nil))) -> c2(LAST(.(if(<=(z0, x1), .(x1, .(z0, nil)), .(z0, bubble(.(x1, nil)))), bsort(butlast(bubble(.(z0, .(x1, nil))))))), BSORT(butlast(bubble(.(z0, .(x1, nil))))), BUTLAST(bubble(.(z0, .(x1, nil)))), BUBBLE(.(z0, .(x1, nil)))) BSORT(.(z0, .(x1, .(z1, z2)))) -> c2(LAST(.(if(<=(z0, x1), .(x1, if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2))))), .(z0, bubble(.(x1, .(z1, z2))))), bsort(butlast(bubble(.(z0, .(x1, .(z1, z2)))))))), BSORT(butlast(bubble(.(z0, .(x1, .(z1, z2)))))), BUTLAST(bubble(.(z0, .(x1, .(z1, z2))))), BUBBLE(.(z0, .(x1, .(z1, z2))))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) BSORT(.(z0, .(z1, z2))) -> c1(BUTLAST(bubble(.(z0, .(z1, z2))))) BSORT(.(z0, .(z1, z2))) -> c1(BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, nil)) -> c(LAST(.(bubble(.(z0, nil)), bsort(nil)))) BSORT(.(z0, nil)) -> c(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil)))))) BSORT(.(x0, .(z0, nil))) -> c(LAST(.(if(<=(x0, z0), .(z0, bubble(.(x0, nil))), .(x0, .(z0, nil))), bsort(butlast(bubble(.(x0, .(z0, nil)))))))) BSORT(.(x0, .(z0, .(z1, z2)))) -> c(LAST(.(if(<=(x0, z0), .(z0, bubble(.(x0, .(z1, z2)))), .(x0, if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))))), bsort(butlast(bubble(.(x0, .(z0, .(z1, z2))))))))) BSORT(.(z0, .(x1, nil))) -> c(LAST(.(if(<=(z0, x1), .(x1, .(z0, nil)), .(z0, bubble(.(x1, nil)))), bsort(butlast(bubble(.(z0, .(x1, nil)))))))) BSORT(.(z0, .(x1, .(z1, z2)))) -> c(LAST(.(if(<=(z0, x1), .(x1, if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2))))), .(z0, bubble(.(x1, .(z1, z2))))), bsort(butlast(bubble(.(z0, .(x1, .(z1, z2))))))))) BSORT(.(z0, .(z1, z2))) -> c3(BUTLAST(bubble(.(z0, .(z1, z2))))) BSORT(.(z0, .(z1, z2))) -> c3(BUBBLE(.(z0, .(z1, z2)))) S tuples: LAST(.(z0, .(z1, z2))) -> c9(LAST(.(z1, z2))) BUTLAST(.(z0, .(z1, z2))) -> c12(BUTLAST(.(z1, z2))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) K tuples: BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BSORT(.(z0, .(z1, z2))) -> c2(BSORT(butlast(bubble(.(z0, .(z1, z2))))), BUTLAST(bubble(.(z0, .(z1, z2)))), BUBBLE(.(z0, .(z1, z2)))) BUBBLE(.(z0, .(z1, z2))) -> c5(BUBBLE(.(z0, z2))) BUBBLE(.(z0, .(z1, z2))) -> c6(BUBBLE(.(z1, z2))) Defined Rule Symbols: bubble_1, bsort_1, butlast_1, last_1 Defined Pair Symbols: BUBBLE_1, LAST_1, BUTLAST_1, BSORT_1 Compound Symbols: c5_1, c6_1, c9_1, c12_1, c_1, c2_4, c2_3, c1_1, c3_1 ---------------------------------------- (97) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace BSORT(.(x0, .(z0, nil))) -> c2(LAST(.(if(<=(x0, z0), .(z0, bubble(.(x0, nil))), .(x0, .(z0, nil))), bsort(butlast(bubble(.(x0, .(z0, nil))))))), BSORT(butlast(bubble(.(x0, .(z0, nil))))), BUTLAST(bubble(.(x0, .(z0, nil)))), BUBBLE(.(x0, .(z0, nil)))) by BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, nil))), .(z0, .(z1, nil))), bsort(butlast(bubble(.(z0, .(z1, nil))))))), BSORT(butlast(bubble(.(z0, .(z1, nil))))), BUTLAST(if(<=(z0, z1), .(z1, bubble(.(z0, nil))), .(z0, bubble(.(z1, nil))))), BUBBLE(.(z0, .(z1, nil)))) ---------------------------------------- (98) Obligation: Complexity Dependency Tuples Problem Rules: bubble(.(z0, nil)) -> .(z0, nil) bubble(.(z0, .(z1, z2))) -> if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))) bsort(nil) -> nil bsort(.(z0, z1)) -> last(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1)))))) butlast(nil) -> nil butlast(.(z0, nil)) -> nil butlast(.(z0, .(z1, z2))) -> .(z0, butlast(.(z1, z2))) last(.(z0, nil)) -> z0 last(.(z0, .(z1, z2))) -> last(.(z1, z2)) Tuples: BUBBLE(.(z0, .(z1, z2))) -> c5(BUBBLE(.(z0, z2))) BUBBLE(.(z0, .(z1, z2))) -> c6(BUBBLE(.(z1, z2))) LAST(.(z0, .(z1, z2))) -> c9(LAST(.(z1, z2))) BUTLAST(.(z0, .(z1, z2))) -> c12(BUTLAST(.(z1, z2))) BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BSORT(.(x0, .(z0, .(z1, z2)))) -> c2(LAST(.(if(<=(x0, z0), .(z0, bubble(.(x0, .(z1, z2)))), .(x0, if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))))), bsort(butlast(bubble(.(x0, .(z0, .(z1, z2)))))))), BSORT(butlast(bubble(.(x0, .(z0, .(z1, z2)))))), BUTLAST(bubble(.(x0, .(z0, .(z1, z2))))), BUBBLE(.(x0, .(z0, .(z1, z2))))) BSORT(.(z0, .(x1, nil))) -> c2(LAST(.(if(<=(z0, x1), .(x1, .(z0, nil)), .(z0, bubble(.(x1, nil)))), bsort(butlast(bubble(.(z0, .(x1, nil))))))), BSORT(butlast(bubble(.(z0, .(x1, nil))))), BUTLAST(bubble(.(z0, .(x1, nil)))), BUBBLE(.(z0, .(x1, nil)))) BSORT(.(z0, .(x1, .(z1, z2)))) -> c2(LAST(.(if(<=(z0, x1), .(x1, if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2))))), .(z0, bubble(.(x1, .(z1, z2))))), bsort(butlast(bubble(.(z0, .(x1, .(z1, z2)))))))), BSORT(butlast(bubble(.(z0, .(x1, .(z1, z2)))))), BUTLAST(bubble(.(z0, .(x1, .(z1, z2))))), BUBBLE(.(z0, .(x1, .(z1, z2))))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) BSORT(.(z0, .(z1, z2))) -> c1(BUTLAST(bubble(.(z0, .(z1, z2))))) BSORT(.(z0, .(z1, z2))) -> c1(BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, nil)) -> c(LAST(.(bubble(.(z0, nil)), bsort(nil)))) BSORT(.(z0, nil)) -> c(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil)))))) BSORT(.(x0, .(z0, nil))) -> c(LAST(.(if(<=(x0, z0), .(z0, bubble(.(x0, nil))), .(x0, .(z0, nil))), bsort(butlast(bubble(.(x0, .(z0, nil)))))))) BSORT(.(x0, .(z0, .(z1, z2)))) -> c(LAST(.(if(<=(x0, z0), .(z0, bubble(.(x0, .(z1, z2)))), .(x0, if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))))), bsort(butlast(bubble(.(x0, .(z0, .(z1, z2))))))))) BSORT(.(z0, .(x1, nil))) -> c(LAST(.(if(<=(z0, x1), .(x1, .(z0, nil)), .(z0, bubble(.(x1, nil)))), bsort(butlast(bubble(.(z0, .(x1, nil)))))))) BSORT(.(z0, .(x1, .(z1, z2)))) -> c(LAST(.(if(<=(z0, x1), .(x1, if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2))))), .(z0, bubble(.(x1, .(z1, z2))))), bsort(butlast(bubble(.(z0, .(x1, .(z1, z2))))))))) BSORT(.(z0, .(z1, z2))) -> c3(BUTLAST(bubble(.(z0, .(z1, z2))))) BSORT(.(z0, .(z1, z2))) -> c3(BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, nil))), .(z0, .(z1, nil))), bsort(butlast(bubble(.(z0, .(z1, nil))))))), BSORT(butlast(bubble(.(z0, .(z1, nil))))), BUTLAST(if(<=(z0, z1), .(z1, bubble(.(z0, nil))), .(z0, bubble(.(z1, nil))))), BUBBLE(.(z0, .(z1, nil)))) S tuples: LAST(.(z0, .(z1, z2))) -> c9(LAST(.(z1, z2))) BUTLAST(.(z0, .(z1, z2))) -> c12(BUTLAST(.(z1, z2))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) K tuples: BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BSORT(.(z0, .(z1, z2))) -> c2(BSORT(butlast(bubble(.(z0, .(z1, z2))))), BUTLAST(bubble(.(z0, .(z1, z2)))), BUBBLE(.(z0, .(z1, z2)))) BUBBLE(.(z0, .(z1, z2))) -> c5(BUBBLE(.(z0, z2))) BUBBLE(.(z0, .(z1, z2))) -> c6(BUBBLE(.(z1, z2))) Defined Rule Symbols: bubble_1, bsort_1, butlast_1, last_1 Defined Pair Symbols: BUBBLE_1, LAST_1, BUTLAST_1, BSORT_1 Compound Symbols: c5_1, c6_1, c9_1, c12_1, c_1, c2_4, c2_3, c1_1, c3_1 ---------------------------------------- (99) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (100) Obligation: Complexity Dependency Tuples Problem Rules: bubble(.(z0, nil)) -> .(z0, nil) bubble(.(z0, .(z1, z2))) -> if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))) bsort(nil) -> nil bsort(.(z0, z1)) -> last(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1)))))) butlast(nil) -> nil butlast(.(z0, nil)) -> nil butlast(.(z0, .(z1, z2))) -> .(z0, butlast(.(z1, z2))) last(.(z0, nil)) -> z0 last(.(z0, .(z1, z2))) -> last(.(z1, z2)) Tuples: BUBBLE(.(z0, .(z1, z2))) -> c5(BUBBLE(.(z0, z2))) BUBBLE(.(z0, .(z1, z2))) -> c6(BUBBLE(.(z1, z2))) LAST(.(z0, .(z1, z2))) -> c9(LAST(.(z1, z2))) BUTLAST(.(z0, .(z1, z2))) -> c12(BUTLAST(.(z1, z2))) BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BSORT(.(x0, .(z0, .(z1, z2)))) -> c2(LAST(.(if(<=(x0, z0), .(z0, bubble(.(x0, .(z1, z2)))), .(x0, if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))))), bsort(butlast(bubble(.(x0, .(z0, .(z1, z2)))))))), BSORT(butlast(bubble(.(x0, .(z0, .(z1, z2)))))), BUTLAST(bubble(.(x0, .(z0, .(z1, z2))))), BUBBLE(.(x0, .(z0, .(z1, z2))))) BSORT(.(z0, .(x1, nil))) -> c2(LAST(.(if(<=(z0, x1), .(x1, .(z0, nil)), .(z0, bubble(.(x1, nil)))), bsort(butlast(bubble(.(z0, .(x1, nil))))))), BSORT(butlast(bubble(.(z0, .(x1, nil))))), BUTLAST(bubble(.(z0, .(x1, nil)))), BUBBLE(.(z0, .(x1, nil)))) BSORT(.(z0, .(x1, .(z1, z2)))) -> c2(LAST(.(if(<=(z0, x1), .(x1, if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2))))), .(z0, bubble(.(x1, .(z1, z2))))), bsort(butlast(bubble(.(z0, .(x1, .(z1, z2)))))))), BSORT(butlast(bubble(.(z0, .(x1, .(z1, z2)))))), BUTLAST(bubble(.(z0, .(x1, .(z1, z2))))), BUBBLE(.(z0, .(x1, .(z1, z2))))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) BSORT(.(z0, .(z1, z2))) -> c1(BUTLAST(bubble(.(z0, .(z1, z2))))) BSORT(.(z0, .(z1, z2))) -> c1(BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, nil)) -> c(LAST(.(bubble(.(z0, nil)), bsort(nil)))) BSORT(.(z0, nil)) -> c(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil)))))) BSORT(.(x0, .(z0, nil))) -> c(LAST(.(if(<=(x0, z0), .(z0, bubble(.(x0, nil))), .(x0, .(z0, nil))), bsort(butlast(bubble(.(x0, .(z0, nil)))))))) BSORT(.(x0, .(z0, .(z1, z2)))) -> c(LAST(.(if(<=(x0, z0), .(z0, bubble(.(x0, .(z1, z2)))), .(x0, if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))))), bsort(butlast(bubble(.(x0, .(z0, .(z1, z2))))))))) BSORT(.(z0, .(x1, nil))) -> c(LAST(.(if(<=(z0, x1), .(x1, .(z0, nil)), .(z0, bubble(.(x1, nil)))), bsort(butlast(bubble(.(z0, .(x1, nil)))))))) BSORT(.(z0, .(x1, .(z1, z2)))) -> c(LAST(.(if(<=(z0, x1), .(x1, if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2))))), .(z0, bubble(.(x1, .(z1, z2))))), bsort(butlast(bubble(.(z0, .(x1, .(z1, z2))))))))) BSORT(.(z0, .(z1, z2))) -> c3(BUTLAST(bubble(.(z0, .(z1, z2))))) BSORT(.(z0, .(z1, z2))) -> c3(BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, nil))), .(z0, .(z1, nil))), bsort(butlast(bubble(.(z0, .(z1, nil))))))), BSORT(butlast(bubble(.(z0, .(z1, nil))))), BUBBLE(.(z0, .(z1, nil)))) S tuples: LAST(.(z0, .(z1, z2))) -> c9(LAST(.(z1, z2))) BUTLAST(.(z0, .(z1, z2))) -> c12(BUTLAST(.(z1, z2))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) K tuples: BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BSORT(.(z0, .(z1, z2))) -> c2(BSORT(butlast(bubble(.(z0, .(z1, z2))))), BUTLAST(bubble(.(z0, .(z1, z2)))), BUBBLE(.(z0, .(z1, z2)))) BUBBLE(.(z0, .(z1, z2))) -> c5(BUBBLE(.(z0, z2))) BUBBLE(.(z0, .(z1, z2))) -> c6(BUBBLE(.(z1, z2))) Defined Rule Symbols: bubble_1, bsort_1, butlast_1, last_1 Defined Pair Symbols: BUBBLE_1, LAST_1, BUTLAST_1, BSORT_1 Compound Symbols: c5_1, c6_1, c9_1, c12_1, c_1, c2_4, c2_3, c1_1, c3_1 ---------------------------------------- (101) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace BSORT(.(x0, .(z0, .(z1, z2)))) -> c2(LAST(.(if(<=(x0, z0), .(z0, bubble(.(x0, .(z1, z2)))), .(x0, if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))))), bsort(butlast(bubble(.(x0, .(z0, .(z1, z2)))))))), BSORT(butlast(bubble(.(x0, .(z0, .(z1, z2)))))), BUTLAST(bubble(.(x0, .(z0, .(z1, z2))))), BUBBLE(.(x0, .(z0, .(z1, z2))))) by BSORT(.(z0, .(z1, .(z2, z3)))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, .(z2, z3)))), .(z0, if(<=(z1, z2), .(z2, bubble(.(z1, z3))), .(z1, bubble(.(z2, z3)))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3)))))))), BSORT(butlast(bubble(.(z0, .(z1, .(z2, z3)))))), BUTLAST(if(<=(z0, z1), .(z1, bubble(.(z0, .(z2, z3)))), .(z0, bubble(.(z1, .(z2, z3)))))), BUBBLE(.(z0, .(z1, .(z2, z3))))) ---------------------------------------- (102) Obligation: Complexity Dependency Tuples Problem Rules: bubble(.(z0, nil)) -> .(z0, nil) bubble(.(z0, .(z1, z2))) -> if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))) bsort(nil) -> nil bsort(.(z0, z1)) -> last(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1)))))) butlast(nil) -> nil butlast(.(z0, nil)) -> nil butlast(.(z0, .(z1, z2))) -> .(z0, butlast(.(z1, z2))) last(.(z0, nil)) -> z0 last(.(z0, .(z1, z2))) -> last(.(z1, z2)) Tuples: BUBBLE(.(z0, .(z1, z2))) -> c5(BUBBLE(.(z0, z2))) BUBBLE(.(z0, .(z1, z2))) -> c6(BUBBLE(.(z1, z2))) LAST(.(z0, .(z1, z2))) -> c9(LAST(.(z1, z2))) BUTLAST(.(z0, .(z1, z2))) -> c12(BUTLAST(.(z1, z2))) BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BSORT(.(z0, .(x1, nil))) -> c2(LAST(.(if(<=(z0, x1), .(x1, .(z0, nil)), .(z0, bubble(.(x1, nil)))), bsort(butlast(bubble(.(z0, .(x1, nil))))))), BSORT(butlast(bubble(.(z0, .(x1, nil))))), BUTLAST(bubble(.(z0, .(x1, nil)))), BUBBLE(.(z0, .(x1, nil)))) BSORT(.(z0, .(x1, .(z1, z2)))) -> c2(LAST(.(if(<=(z0, x1), .(x1, if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2))))), .(z0, bubble(.(x1, .(z1, z2))))), bsort(butlast(bubble(.(z0, .(x1, .(z1, z2)))))))), BSORT(butlast(bubble(.(z0, .(x1, .(z1, z2)))))), BUTLAST(bubble(.(z0, .(x1, .(z1, z2))))), BUBBLE(.(z0, .(x1, .(z1, z2))))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) BSORT(.(z0, .(z1, z2))) -> c1(BUTLAST(bubble(.(z0, .(z1, z2))))) BSORT(.(z0, .(z1, z2))) -> c1(BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, nil)) -> c(LAST(.(bubble(.(z0, nil)), bsort(nil)))) BSORT(.(z0, nil)) -> c(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil)))))) BSORT(.(x0, .(z0, nil))) -> c(LAST(.(if(<=(x0, z0), .(z0, bubble(.(x0, nil))), .(x0, .(z0, nil))), bsort(butlast(bubble(.(x0, .(z0, nil)))))))) BSORT(.(x0, .(z0, .(z1, z2)))) -> c(LAST(.(if(<=(x0, z0), .(z0, bubble(.(x0, .(z1, z2)))), .(x0, if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))))), bsort(butlast(bubble(.(x0, .(z0, .(z1, z2))))))))) BSORT(.(z0, .(x1, nil))) -> c(LAST(.(if(<=(z0, x1), .(x1, .(z0, nil)), .(z0, bubble(.(x1, nil)))), bsort(butlast(bubble(.(z0, .(x1, nil)))))))) BSORT(.(z0, .(x1, .(z1, z2)))) -> c(LAST(.(if(<=(z0, x1), .(x1, if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2))))), .(z0, bubble(.(x1, .(z1, z2))))), bsort(butlast(bubble(.(z0, .(x1, .(z1, z2))))))))) BSORT(.(z0, .(z1, z2))) -> c3(BUTLAST(bubble(.(z0, .(z1, z2))))) BSORT(.(z0, .(z1, z2))) -> c3(BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, nil))), .(z0, .(z1, nil))), bsort(butlast(bubble(.(z0, .(z1, nil))))))), BSORT(butlast(bubble(.(z0, .(z1, nil))))), BUBBLE(.(z0, .(z1, nil)))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, .(z2, z3)))), .(z0, if(<=(z1, z2), .(z2, bubble(.(z1, z3))), .(z1, bubble(.(z2, z3)))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3)))))))), BSORT(butlast(bubble(.(z0, .(z1, .(z2, z3)))))), BUTLAST(if(<=(z0, z1), .(z1, bubble(.(z0, .(z2, z3)))), .(z0, bubble(.(z1, .(z2, z3)))))), BUBBLE(.(z0, .(z1, .(z2, z3))))) S tuples: LAST(.(z0, .(z1, z2))) -> c9(LAST(.(z1, z2))) BUTLAST(.(z0, .(z1, z2))) -> c12(BUTLAST(.(z1, z2))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) K tuples: BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BSORT(.(z0, .(z1, z2))) -> c2(BSORT(butlast(bubble(.(z0, .(z1, z2))))), BUTLAST(bubble(.(z0, .(z1, z2)))), BUBBLE(.(z0, .(z1, z2)))) BUBBLE(.(z0, .(z1, z2))) -> c5(BUBBLE(.(z0, z2))) BUBBLE(.(z0, .(z1, z2))) -> c6(BUBBLE(.(z1, z2))) Defined Rule Symbols: bubble_1, bsort_1, butlast_1, last_1 Defined Pair Symbols: BUBBLE_1, LAST_1, BUTLAST_1, BSORT_1 Compound Symbols: c5_1, c6_1, c9_1, c12_1, c_1, c2_4, c2_3, c1_1, c3_1 ---------------------------------------- (103) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (104) Obligation: Complexity Dependency Tuples Problem Rules: bubble(.(z0, nil)) -> .(z0, nil) bubble(.(z0, .(z1, z2))) -> if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))) bsort(nil) -> nil bsort(.(z0, z1)) -> last(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1)))))) butlast(nil) -> nil butlast(.(z0, nil)) -> nil butlast(.(z0, .(z1, z2))) -> .(z0, butlast(.(z1, z2))) last(.(z0, nil)) -> z0 last(.(z0, .(z1, z2))) -> last(.(z1, z2)) Tuples: BUBBLE(.(z0, .(z1, z2))) -> c5(BUBBLE(.(z0, z2))) BUBBLE(.(z0, .(z1, z2))) -> c6(BUBBLE(.(z1, z2))) LAST(.(z0, .(z1, z2))) -> c9(LAST(.(z1, z2))) BUTLAST(.(z0, .(z1, z2))) -> c12(BUTLAST(.(z1, z2))) BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BSORT(.(z0, .(x1, nil))) -> c2(LAST(.(if(<=(z0, x1), .(x1, .(z0, nil)), .(z0, bubble(.(x1, nil)))), bsort(butlast(bubble(.(z0, .(x1, nil))))))), BSORT(butlast(bubble(.(z0, .(x1, nil))))), BUTLAST(bubble(.(z0, .(x1, nil)))), BUBBLE(.(z0, .(x1, nil)))) BSORT(.(z0, .(x1, .(z1, z2)))) -> c2(LAST(.(if(<=(z0, x1), .(x1, if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2))))), .(z0, bubble(.(x1, .(z1, z2))))), bsort(butlast(bubble(.(z0, .(x1, .(z1, z2)))))))), BSORT(butlast(bubble(.(z0, .(x1, .(z1, z2)))))), BUTLAST(bubble(.(z0, .(x1, .(z1, z2))))), BUBBLE(.(z0, .(x1, .(z1, z2))))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) BSORT(.(z0, .(z1, z2))) -> c1(BUTLAST(bubble(.(z0, .(z1, z2))))) BSORT(.(z0, .(z1, z2))) -> c1(BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, nil)) -> c(LAST(.(bubble(.(z0, nil)), bsort(nil)))) BSORT(.(z0, nil)) -> c(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil)))))) BSORT(.(x0, .(z0, nil))) -> c(LAST(.(if(<=(x0, z0), .(z0, bubble(.(x0, nil))), .(x0, .(z0, nil))), bsort(butlast(bubble(.(x0, .(z0, nil)))))))) BSORT(.(x0, .(z0, .(z1, z2)))) -> c(LAST(.(if(<=(x0, z0), .(z0, bubble(.(x0, .(z1, z2)))), .(x0, if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))))), bsort(butlast(bubble(.(x0, .(z0, .(z1, z2))))))))) BSORT(.(z0, .(x1, nil))) -> c(LAST(.(if(<=(z0, x1), .(x1, .(z0, nil)), .(z0, bubble(.(x1, nil)))), bsort(butlast(bubble(.(z0, .(x1, nil)))))))) BSORT(.(z0, .(x1, .(z1, z2)))) -> c(LAST(.(if(<=(z0, x1), .(x1, if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2))))), .(z0, bubble(.(x1, .(z1, z2))))), bsort(butlast(bubble(.(z0, .(x1, .(z1, z2))))))))) BSORT(.(z0, .(z1, z2))) -> c3(BUTLAST(bubble(.(z0, .(z1, z2))))) BSORT(.(z0, .(z1, z2))) -> c3(BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, nil))), .(z0, .(z1, nil))), bsort(butlast(bubble(.(z0, .(z1, nil))))))), BSORT(butlast(bubble(.(z0, .(z1, nil))))), BUBBLE(.(z0, .(z1, nil)))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, .(z2, z3)))), .(z0, if(<=(z1, z2), .(z2, bubble(.(z1, z3))), .(z1, bubble(.(z2, z3)))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3)))))))), BSORT(butlast(bubble(.(z0, .(z1, .(z2, z3)))))), BUBBLE(.(z0, .(z1, .(z2, z3))))) S tuples: LAST(.(z0, .(z1, z2))) -> c9(LAST(.(z1, z2))) BUTLAST(.(z0, .(z1, z2))) -> c12(BUTLAST(.(z1, z2))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) K tuples: BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BSORT(.(z0, .(z1, z2))) -> c2(BSORT(butlast(bubble(.(z0, .(z1, z2))))), BUTLAST(bubble(.(z0, .(z1, z2)))), BUBBLE(.(z0, .(z1, z2)))) BUBBLE(.(z0, .(z1, z2))) -> c5(BUBBLE(.(z0, z2))) BUBBLE(.(z0, .(z1, z2))) -> c6(BUBBLE(.(z1, z2))) Defined Rule Symbols: bubble_1, bsort_1, butlast_1, last_1 Defined Pair Symbols: BUBBLE_1, LAST_1, BUTLAST_1, BSORT_1 Compound Symbols: c5_1, c6_1, c9_1, c12_1, c_1, c2_4, c2_3, c1_1, c3_1 ---------------------------------------- (105) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace BSORT(.(z0, .(x1, nil))) -> c2(LAST(.(if(<=(z0, x1), .(x1, .(z0, nil)), .(z0, bubble(.(x1, nil)))), bsort(butlast(bubble(.(z0, .(x1, nil))))))), BSORT(butlast(bubble(.(z0, .(x1, nil))))), BUTLAST(bubble(.(z0, .(x1, nil)))), BUBBLE(.(z0, .(x1, nil)))) by BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, .(z0, nil)), .(z0, bubble(.(z1, nil)))), bsort(butlast(bubble(.(z0, .(z1, nil))))))), BSORT(butlast(bubble(.(z0, .(z1, nil))))), BUTLAST(if(<=(z0, z1), .(z1, bubble(.(z0, nil))), .(z0, bubble(.(z1, nil))))), BUBBLE(.(z0, .(z1, nil)))) ---------------------------------------- (106) Obligation: Complexity Dependency Tuples Problem Rules: bubble(.(z0, nil)) -> .(z0, nil) bubble(.(z0, .(z1, z2))) -> if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))) bsort(nil) -> nil bsort(.(z0, z1)) -> last(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1)))))) butlast(nil) -> nil butlast(.(z0, nil)) -> nil butlast(.(z0, .(z1, z2))) -> .(z0, butlast(.(z1, z2))) last(.(z0, nil)) -> z0 last(.(z0, .(z1, z2))) -> last(.(z1, z2)) Tuples: BUBBLE(.(z0, .(z1, z2))) -> c5(BUBBLE(.(z0, z2))) BUBBLE(.(z0, .(z1, z2))) -> c6(BUBBLE(.(z1, z2))) LAST(.(z0, .(z1, z2))) -> c9(LAST(.(z1, z2))) BUTLAST(.(z0, .(z1, z2))) -> c12(BUTLAST(.(z1, z2))) BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BSORT(.(z0, .(x1, .(z1, z2)))) -> c2(LAST(.(if(<=(z0, x1), .(x1, if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2))))), .(z0, bubble(.(x1, .(z1, z2))))), bsort(butlast(bubble(.(z0, .(x1, .(z1, z2)))))))), BSORT(butlast(bubble(.(z0, .(x1, .(z1, z2)))))), BUTLAST(bubble(.(z0, .(x1, .(z1, z2))))), BUBBLE(.(z0, .(x1, .(z1, z2))))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) BSORT(.(z0, .(z1, z2))) -> c1(BUTLAST(bubble(.(z0, .(z1, z2))))) BSORT(.(z0, .(z1, z2))) -> c1(BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, nil)) -> c(LAST(.(bubble(.(z0, nil)), bsort(nil)))) BSORT(.(z0, nil)) -> c(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil)))))) BSORT(.(x0, .(z0, nil))) -> c(LAST(.(if(<=(x0, z0), .(z0, bubble(.(x0, nil))), .(x0, .(z0, nil))), bsort(butlast(bubble(.(x0, .(z0, nil)))))))) BSORT(.(x0, .(z0, .(z1, z2)))) -> c(LAST(.(if(<=(x0, z0), .(z0, bubble(.(x0, .(z1, z2)))), .(x0, if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))))), bsort(butlast(bubble(.(x0, .(z0, .(z1, z2))))))))) BSORT(.(z0, .(x1, nil))) -> c(LAST(.(if(<=(z0, x1), .(x1, .(z0, nil)), .(z0, bubble(.(x1, nil)))), bsort(butlast(bubble(.(z0, .(x1, nil)))))))) BSORT(.(z0, .(x1, .(z1, z2)))) -> c(LAST(.(if(<=(z0, x1), .(x1, if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2))))), .(z0, bubble(.(x1, .(z1, z2))))), bsort(butlast(bubble(.(z0, .(x1, .(z1, z2))))))))) BSORT(.(z0, .(z1, z2))) -> c3(BUTLAST(bubble(.(z0, .(z1, z2))))) BSORT(.(z0, .(z1, z2))) -> c3(BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, nil))), .(z0, .(z1, nil))), bsort(butlast(bubble(.(z0, .(z1, nil))))))), BSORT(butlast(bubble(.(z0, .(z1, nil))))), BUBBLE(.(z0, .(z1, nil)))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, .(z2, z3)))), .(z0, if(<=(z1, z2), .(z2, bubble(.(z1, z3))), .(z1, bubble(.(z2, z3)))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3)))))))), BSORT(butlast(bubble(.(z0, .(z1, .(z2, z3)))))), BUBBLE(.(z0, .(z1, .(z2, z3))))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, .(z0, nil)), .(z0, bubble(.(z1, nil)))), bsort(butlast(bubble(.(z0, .(z1, nil))))))), BSORT(butlast(bubble(.(z0, .(z1, nil))))), BUTLAST(if(<=(z0, z1), .(z1, bubble(.(z0, nil))), .(z0, bubble(.(z1, nil))))), BUBBLE(.(z0, .(z1, nil)))) S tuples: LAST(.(z0, .(z1, z2))) -> c9(LAST(.(z1, z2))) BUTLAST(.(z0, .(z1, z2))) -> c12(BUTLAST(.(z1, z2))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) K tuples: BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BSORT(.(z0, .(z1, z2))) -> c2(BSORT(butlast(bubble(.(z0, .(z1, z2))))), BUTLAST(bubble(.(z0, .(z1, z2)))), BUBBLE(.(z0, .(z1, z2)))) BUBBLE(.(z0, .(z1, z2))) -> c5(BUBBLE(.(z0, z2))) BUBBLE(.(z0, .(z1, z2))) -> c6(BUBBLE(.(z1, z2))) Defined Rule Symbols: bubble_1, bsort_1, butlast_1, last_1 Defined Pair Symbols: BUBBLE_1, LAST_1, BUTLAST_1, BSORT_1 Compound Symbols: c5_1, c6_1, c9_1, c12_1, c_1, c2_4, c2_3, c1_1, c3_1 ---------------------------------------- (107) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (108) Obligation: Complexity Dependency Tuples Problem Rules: bubble(.(z0, nil)) -> .(z0, nil) bubble(.(z0, .(z1, z2))) -> if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))) bsort(nil) -> nil bsort(.(z0, z1)) -> last(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1)))))) butlast(nil) -> nil butlast(.(z0, nil)) -> nil butlast(.(z0, .(z1, z2))) -> .(z0, butlast(.(z1, z2))) last(.(z0, nil)) -> z0 last(.(z0, .(z1, z2))) -> last(.(z1, z2)) Tuples: BUBBLE(.(z0, .(z1, z2))) -> c5(BUBBLE(.(z0, z2))) BUBBLE(.(z0, .(z1, z2))) -> c6(BUBBLE(.(z1, z2))) LAST(.(z0, .(z1, z2))) -> c9(LAST(.(z1, z2))) BUTLAST(.(z0, .(z1, z2))) -> c12(BUTLAST(.(z1, z2))) BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BSORT(.(z0, .(x1, .(z1, z2)))) -> c2(LAST(.(if(<=(z0, x1), .(x1, if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2))))), .(z0, bubble(.(x1, .(z1, z2))))), bsort(butlast(bubble(.(z0, .(x1, .(z1, z2)))))))), BSORT(butlast(bubble(.(z0, .(x1, .(z1, z2)))))), BUTLAST(bubble(.(z0, .(x1, .(z1, z2))))), BUBBLE(.(z0, .(x1, .(z1, z2))))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) BSORT(.(z0, .(z1, z2))) -> c1(BUTLAST(bubble(.(z0, .(z1, z2))))) BSORT(.(z0, .(z1, z2))) -> c1(BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, nil)) -> c(LAST(.(bubble(.(z0, nil)), bsort(nil)))) BSORT(.(z0, nil)) -> c(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil)))))) BSORT(.(x0, .(z0, nil))) -> c(LAST(.(if(<=(x0, z0), .(z0, bubble(.(x0, nil))), .(x0, .(z0, nil))), bsort(butlast(bubble(.(x0, .(z0, nil)))))))) BSORT(.(x0, .(z0, .(z1, z2)))) -> c(LAST(.(if(<=(x0, z0), .(z0, bubble(.(x0, .(z1, z2)))), .(x0, if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))))), bsort(butlast(bubble(.(x0, .(z0, .(z1, z2))))))))) BSORT(.(z0, .(x1, nil))) -> c(LAST(.(if(<=(z0, x1), .(x1, .(z0, nil)), .(z0, bubble(.(x1, nil)))), bsort(butlast(bubble(.(z0, .(x1, nil)))))))) BSORT(.(z0, .(x1, .(z1, z2)))) -> c(LAST(.(if(<=(z0, x1), .(x1, if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2))))), .(z0, bubble(.(x1, .(z1, z2))))), bsort(butlast(bubble(.(z0, .(x1, .(z1, z2))))))))) BSORT(.(z0, .(z1, z2))) -> c3(BUTLAST(bubble(.(z0, .(z1, z2))))) BSORT(.(z0, .(z1, z2))) -> c3(BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, nil))), .(z0, .(z1, nil))), bsort(butlast(bubble(.(z0, .(z1, nil))))))), BSORT(butlast(bubble(.(z0, .(z1, nil))))), BUBBLE(.(z0, .(z1, nil)))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, .(z2, z3)))), .(z0, if(<=(z1, z2), .(z2, bubble(.(z1, z3))), .(z1, bubble(.(z2, z3)))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3)))))))), BSORT(butlast(bubble(.(z0, .(z1, .(z2, z3)))))), BUBBLE(.(z0, .(z1, .(z2, z3))))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, .(z0, nil)), .(z0, bubble(.(z1, nil)))), bsort(butlast(bubble(.(z0, .(z1, nil))))))), BSORT(butlast(bubble(.(z0, .(z1, nil))))), BUBBLE(.(z0, .(z1, nil)))) S tuples: LAST(.(z0, .(z1, z2))) -> c9(LAST(.(z1, z2))) BUTLAST(.(z0, .(z1, z2))) -> c12(BUTLAST(.(z1, z2))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) K tuples: BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BSORT(.(z0, .(z1, z2))) -> c2(BSORT(butlast(bubble(.(z0, .(z1, z2))))), BUTLAST(bubble(.(z0, .(z1, z2)))), BUBBLE(.(z0, .(z1, z2)))) BUBBLE(.(z0, .(z1, z2))) -> c5(BUBBLE(.(z0, z2))) BUBBLE(.(z0, .(z1, z2))) -> c6(BUBBLE(.(z1, z2))) Defined Rule Symbols: bubble_1, bsort_1, butlast_1, last_1 Defined Pair Symbols: BUBBLE_1, LAST_1, BUTLAST_1, BSORT_1 Compound Symbols: c5_1, c6_1, c9_1, c12_1, c_1, c2_4, c2_3, c1_1, c3_1 ---------------------------------------- (109) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace BSORT(.(z0, .(x1, .(z1, z2)))) -> c2(LAST(.(if(<=(z0, x1), .(x1, if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2))))), .(z0, bubble(.(x1, .(z1, z2))))), bsort(butlast(bubble(.(z0, .(x1, .(z1, z2)))))))), BSORT(butlast(bubble(.(z0, .(x1, .(z1, z2)))))), BUTLAST(bubble(.(z0, .(x1, .(z1, z2))))), BUBBLE(.(z0, .(x1, .(z1, z2))))) by BSORT(.(z0, .(z1, .(z2, z3)))) -> c2(LAST(.(if(<=(z0, z1), .(z1, if(<=(z0, z2), .(z2, bubble(.(z0, z3))), .(z0, bubble(.(z2, z3))))), .(z0, bubble(.(z1, .(z2, z3))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3)))))))), BSORT(butlast(bubble(.(z0, .(z1, .(z2, z3)))))), BUTLAST(if(<=(z0, z1), .(z1, bubble(.(z0, .(z2, z3)))), .(z0, bubble(.(z1, .(z2, z3)))))), BUBBLE(.(z0, .(z1, .(z2, z3))))) ---------------------------------------- (110) Obligation: Complexity Dependency Tuples Problem Rules: bubble(.(z0, nil)) -> .(z0, nil) bubble(.(z0, .(z1, z2))) -> if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))) bsort(nil) -> nil bsort(.(z0, z1)) -> last(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1)))))) butlast(nil) -> nil butlast(.(z0, nil)) -> nil butlast(.(z0, .(z1, z2))) -> .(z0, butlast(.(z1, z2))) last(.(z0, nil)) -> z0 last(.(z0, .(z1, z2))) -> last(.(z1, z2)) Tuples: BUBBLE(.(z0, .(z1, z2))) -> c5(BUBBLE(.(z0, z2))) BUBBLE(.(z0, .(z1, z2))) -> c6(BUBBLE(.(z1, z2))) LAST(.(z0, .(z1, z2))) -> c9(LAST(.(z1, z2))) BUTLAST(.(z0, .(z1, z2))) -> c12(BUTLAST(.(z1, z2))) BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) BSORT(.(z0, .(z1, z2))) -> c1(BUTLAST(bubble(.(z0, .(z1, z2))))) BSORT(.(z0, .(z1, z2))) -> c1(BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, nil)) -> c(LAST(.(bubble(.(z0, nil)), bsort(nil)))) BSORT(.(z0, nil)) -> c(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil)))))) BSORT(.(x0, .(z0, nil))) -> c(LAST(.(if(<=(x0, z0), .(z0, bubble(.(x0, nil))), .(x0, .(z0, nil))), bsort(butlast(bubble(.(x0, .(z0, nil)))))))) BSORT(.(x0, .(z0, .(z1, z2)))) -> c(LAST(.(if(<=(x0, z0), .(z0, bubble(.(x0, .(z1, z2)))), .(x0, if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))))), bsort(butlast(bubble(.(x0, .(z0, .(z1, z2))))))))) BSORT(.(z0, .(x1, nil))) -> c(LAST(.(if(<=(z0, x1), .(x1, .(z0, nil)), .(z0, bubble(.(x1, nil)))), bsort(butlast(bubble(.(z0, .(x1, nil)))))))) BSORT(.(z0, .(x1, .(z1, z2)))) -> c(LAST(.(if(<=(z0, x1), .(x1, if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2))))), .(z0, bubble(.(x1, .(z1, z2))))), bsort(butlast(bubble(.(z0, .(x1, .(z1, z2))))))))) BSORT(.(z0, .(z1, z2))) -> c3(BUTLAST(bubble(.(z0, .(z1, z2))))) BSORT(.(z0, .(z1, z2))) -> c3(BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, nil))), .(z0, .(z1, nil))), bsort(butlast(bubble(.(z0, .(z1, nil))))))), BSORT(butlast(bubble(.(z0, .(z1, nil))))), BUBBLE(.(z0, .(z1, nil)))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, .(z2, z3)))), .(z0, if(<=(z1, z2), .(z2, bubble(.(z1, z3))), .(z1, bubble(.(z2, z3)))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3)))))))), BSORT(butlast(bubble(.(z0, .(z1, .(z2, z3)))))), BUBBLE(.(z0, .(z1, .(z2, z3))))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, .(z0, nil)), .(z0, bubble(.(z1, nil)))), bsort(butlast(bubble(.(z0, .(z1, nil))))))), BSORT(butlast(bubble(.(z0, .(z1, nil))))), BUBBLE(.(z0, .(z1, nil)))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c2(LAST(.(if(<=(z0, z1), .(z1, if(<=(z0, z2), .(z2, bubble(.(z0, z3))), .(z0, bubble(.(z2, z3))))), .(z0, bubble(.(z1, .(z2, z3))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3)))))))), BSORT(butlast(bubble(.(z0, .(z1, .(z2, z3)))))), BUTLAST(if(<=(z0, z1), .(z1, bubble(.(z0, .(z2, z3)))), .(z0, bubble(.(z1, .(z2, z3)))))), BUBBLE(.(z0, .(z1, .(z2, z3))))) S tuples: LAST(.(z0, .(z1, z2))) -> c9(LAST(.(z1, z2))) BUTLAST(.(z0, .(z1, z2))) -> c12(BUTLAST(.(z1, z2))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) K tuples: BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BSORT(.(z0, .(z1, z2))) -> c2(BSORT(butlast(bubble(.(z0, .(z1, z2))))), BUTLAST(bubble(.(z0, .(z1, z2)))), BUBBLE(.(z0, .(z1, z2)))) BUBBLE(.(z0, .(z1, z2))) -> c5(BUBBLE(.(z0, z2))) BUBBLE(.(z0, .(z1, z2))) -> c6(BUBBLE(.(z1, z2))) Defined Rule Symbols: bubble_1, bsort_1, butlast_1, last_1 Defined Pair Symbols: BUBBLE_1, LAST_1, BUTLAST_1, BSORT_1 Compound Symbols: c5_1, c6_1, c9_1, c12_1, c_1, c2_3, c1_1, c3_1, c2_4 ---------------------------------------- (111) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (112) Obligation: Complexity Dependency Tuples Problem Rules: bubble(.(z0, nil)) -> .(z0, nil) bubble(.(z0, .(z1, z2))) -> if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))) bsort(nil) -> nil bsort(.(z0, z1)) -> last(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1)))))) butlast(nil) -> nil butlast(.(z0, nil)) -> nil butlast(.(z0, .(z1, z2))) -> .(z0, butlast(.(z1, z2))) last(.(z0, nil)) -> z0 last(.(z0, .(z1, z2))) -> last(.(z1, z2)) Tuples: BUBBLE(.(z0, .(z1, z2))) -> c5(BUBBLE(.(z0, z2))) BUBBLE(.(z0, .(z1, z2))) -> c6(BUBBLE(.(z1, z2))) LAST(.(z0, .(z1, z2))) -> c9(LAST(.(z1, z2))) BUTLAST(.(z0, .(z1, z2))) -> c12(BUTLAST(.(z1, z2))) BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) BSORT(.(z0, .(z1, z2))) -> c1(BUTLAST(bubble(.(z0, .(z1, z2))))) BSORT(.(z0, .(z1, z2))) -> c1(BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, nil)) -> c(LAST(.(bubble(.(z0, nil)), bsort(nil)))) BSORT(.(z0, nil)) -> c(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil)))))) BSORT(.(x0, .(z0, nil))) -> c(LAST(.(if(<=(x0, z0), .(z0, bubble(.(x0, nil))), .(x0, .(z0, nil))), bsort(butlast(bubble(.(x0, .(z0, nil)))))))) BSORT(.(x0, .(z0, .(z1, z2)))) -> c(LAST(.(if(<=(x0, z0), .(z0, bubble(.(x0, .(z1, z2)))), .(x0, if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))))), bsort(butlast(bubble(.(x0, .(z0, .(z1, z2))))))))) BSORT(.(z0, .(x1, nil))) -> c(LAST(.(if(<=(z0, x1), .(x1, .(z0, nil)), .(z0, bubble(.(x1, nil)))), bsort(butlast(bubble(.(z0, .(x1, nil)))))))) BSORT(.(z0, .(x1, .(z1, z2)))) -> c(LAST(.(if(<=(z0, x1), .(x1, if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2))))), .(z0, bubble(.(x1, .(z1, z2))))), bsort(butlast(bubble(.(z0, .(x1, .(z1, z2))))))))) BSORT(.(z0, .(z1, z2))) -> c3(BUTLAST(bubble(.(z0, .(z1, z2))))) BSORT(.(z0, .(z1, z2))) -> c3(BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, nil))), .(z0, .(z1, nil))), bsort(butlast(bubble(.(z0, .(z1, nil))))))), BSORT(butlast(bubble(.(z0, .(z1, nil))))), BUBBLE(.(z0, .(z1, nil)))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, .(z2, z3)))), .(z0, if(<=(z1, z2), .(z2, bubble(.(z1, z3))), .(z1, bubble(.(z2, z3)))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3)))))))), BSORT(butlast(bubble(.(z0, .(z1, .(z2, z3)))))), BUBBLE(.(z0, .(z1, .(z2, z3))))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, .(z0, nil)), .(z0, bubble(.(z1, nil)))), bsort(butlast(bubble(.(z0, .(z1, nil))))))), BSORT(butlast(bubble(.(z0, .(z1, nil))))), BUBBLE(.(z0, .(z1, nil)))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c2(LAST(.(if(<=(z0, z1), .(z1, if(<=(z0, z2), .(z2, bubble(.(z0, z3))), .(z0, bubble(.(z2, z3))))), .(z0, bubble(.(z1, .(z2, z3))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3)))))))), BSORT(butlast(bubble(.(z0, .(z1, .(z2, z3)))))), BUBBLE(.(z0, .(z1, .(z2, z3))))) S tuples: LAST(.(z0, .(z1, z2))) -> c9(LAST(.(z1, z2))) BUTLAST(.(z0, .(z1, z2))) -> c12(BUTLAST(.(z1, z2))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) K tuples: BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BSORT(.(z0, .(z1, z2))) -> c2(BSORT(butlast(bubble(.(z0, .(z1, z2))))), BUTLAST(bubble(.(z0, .(z1, z2)))), BUBBLE(.(z0, .(z1, z2)))) BUBBLE(.(z0, .(z1, z2))) -> c5(BUBBLE(.(z0, z2))) BUBBLE(.(z0, .(z1, z2))) -> c6(BUBBLE(.(z1, z2))) Defined Rule Symbols: bubble_1, bsort_1, butlast_1, last_1 Defined Pair Symbols: BUBBLE_1, LAST_1, BUTLAST_1, BSORT_1 Compound Symbols: c5_1, c6_1, c9_1, c12_1, c_1, c2_3, c1_1, c3_1 ---------------------------------------- (113) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) by BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(.(z0, nil))) ---------------------------------------- (114) Obligation: Complexity Dependency Tuples Problem Rules: bubble(.(z0, nil)) -> .(z0, nil) bubble(.(z0, .(z1, z2))) -> if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))) bsort(nil) -> nil bsort(.(z0, z1)) -> last(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1)))))) butlast(nil) -> nil butlast(.(z0, nil)) -> nil butlast(.(z0, .(z1, z2))) -> .(z0, butlast(.(z1, z2))) last(.(z0, nil)) -> z0 last(.(z0, .(z1, z2))) -> last(.(z1, z2)) Tuples: BUBBLE(.(z0, .(z1, z2))) -> c5(BUBBLE(.(z0, z2))) BUBBLE(.(z0, .(z1, z2))) -> c6(BUBBLE(.(z1, z2))) LAST(.(z0, .(z1, z2))) -> c9(LAST(.(z1, z2))) BUTLAST(.(z0, .(z1, z2))) -> c12(BUTLAST(.(z1, z2))) BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) BSORT(.(z0, .(z1, z2))) -> c1(BUTLAST(bubble(.(z0, .(z1, z2))))) BSORT(.(z0, .(z1, z2))) -> c1(BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, nil)) -> c(LAST(.(bubble(.(z0, nil)), bsort(nil)))) BSORT(.(z0, nil)) -> c(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil)))))) BSORT(.(x0, .(z0, nil))) -> c(LAST(.(if(<=(x0, z0), .(z0, bubble(.(x0, nil))), .(x0, .(z0, nil))), bsort(butlast(bubble(.(x0, .(z0, nil)))))))) BSORT(.(x0, .(z0, .(z1, z2)))) -> c(LAST(.(if(<=(x0, z0), .(z0, bubble(.(x0, .(z1, z2)))), .(x0, if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))))), bsort(butlast(bubble(.(x0, .(z0, .(z1, z2))))))))) BSORT(.(z0, .(x1, nil))) -> c(LAST(.(if(<=(z0, x1), .(x1, .(z0, nil)), .(z0, bubble(.(x1, nil)))), bsort(butlast(bubble(.(z0, .(x1, nil)))))))) BSORT(.(z0, .(x1, .(z1, z2)))) -> c(LAST(.(if(<=(z0, x1), .(x1, if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2))))), .(z0, bubble(.(x1, .(z1, z2))))), bsort(butlast(bubble(.(z0, .(x1, .(z1, z2))))))))) BSORT(.(z0, .(z1, z2))) -> c3(BUTLAST(bubble(.(z0, .(z1, z2))))) BSORT(.(z0, .(z1, z2))) -> c3(BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, nil))), .(z0, .(z1, nil))), bsort(butlast(bubble(.(z0, .(z1, nil))))))), BSORT(butlast(bubble(.(z0, .(z1, nil))))), BUBBLE(.(z0, .(z1, nil)))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, .(z2, z3)))), .(z0, if(<=(z1, z2), .(z2, bubble(.(z1, z3))), .(z1, bubble(.(z2, z3)))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3)))))))), BSORT(butlast(bubble(.(z0, .(z1, .(z2, z3)))))), BUBBLE(.(z0, .(z1, .(z2, z3))))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, .(z0, nil)), .(z0, bubble(.(z1, nil)))), bsort(butlast(bubble(.(z0, .(z1, nil))))))), BSORT(butlast(bubble(.(z0, .(z1, nil))))), BUBBLE(.(z0, .(z1, nil)))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c2(LAST(.(if(<=(z0, z1), .(z1, if(<=(z0, z2), .(z2, bubble(.(z0, z3))), .(z0, bubble(.(z2, z3))))), .(z0, bubble(.(z1, .(z2, z3))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3)))))))), BSORT(butlast(bubble(.(z0, .(z1, .(z2, z3)))))), BUBBLE(.(z0, .(z1, .(z2, z3))))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(.(z0, nil))) S tuples: LAST(.(z0, .(z1, z2))) -> c9(LAST(.(z1, z2))) BUTLAST(.(z0, .(z1, z2))) -> c12(BUTLAST(.(z1, z2))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(.(z0, nil))) K tuples: BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BSORT(.(z0, .(z1, z2))) -> c2(BSORT(butlast(bubble(.(z0, .(z1, z2))))), BUTLAST(bubble(.(z0, .(z1, z2)))), BUBBLE(.(z0, .(z1, z2)))) BUBBLE(.(z0, .(z1, z2))) -> c5(BUBBLE(.(z0, z2))) BUBBLE(.(z0, .(z1, z2))) -> c6(BUBBLE(.(z1, z2))) Defined Rule Symbols: bubble_1, bsort_1, butlast_1, last_1 Defined Pair Symbols: BUBBLE_1, LAST_1, BUTLAST_1, BSORT_1 Compound Symbols: c5_1, c6_1, c9_1, c12_1, c_1, c2_3, c1_1, c3_1 ---------------------------------------- (115) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (116) Obligation: Complexity Dependency Tuples Problem Rules: bubble(.(z0, nil)) -> .(z0, nil) bubble(.(z0, .(z1, z2))) -> if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))) bsort(nil) -> nil bsort(.(z0, z1)) -> last(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1)))))) butlast(nil) -> nil butlast(.(z0, nil)) -> nil butlast(.(z0, .(z1, z2))) -> .(z0, butlast(.(z1, z2))) last(.(z0, nil)) -> z0 last(.(z0, .(z1, z2))) -> last(.(z1, z2)) Tuples: BUBBLE(.(z0, .(z1, z2))) -> c5(BUBBLE(.(z0, z2))) BUBBLE(.(z0, .(z1, z2))) -> c6(BUBBLE(.(z1, z2))) LAST(.(z0, .(z1, z2))) -> c9(LAST(.(z1, z2))) BUTLAST(.(z0, .(z1, z2))) -> c12(BUTLAST(.(z1, z2))) BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) BSORT(.(z0, .(z1, z2))) -> c1(BUTLAST(bubble(.(z0, .(z1, z2))))) BSORT(.(z0, .(z1, z2))) -> c1(BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, nil)) -> c(LAST(.(bubble(.(z0, nil)), bsort(nil)))) BSORT(.(z0, nil)) -> c(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil)))))) BSORT(.(x0, .(z0, nil))) -> c(LAST(.(if(<=(x0, z0), .(z0, bubble(.(x0, nil))), .(x0, .(z0, nil))), bsort(butlast(bubble(.(x0, .(z0, nil)))))))) BSORT(.(x0, .(z0, .(z1, z2)))) -> c(LAST(.(if(<=(x0, z0), .(z0, bubble(.(x0, .(z1, z2)))), .(x0, if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))))), bsort(butlast(bubble(.(x0, .(z0, .(z1, z2))))))))) BSORT(.(z0, .(x1, nil))) -> c(LAST(.(if(<=(z0, x1), .(x1, .(z0, nil)), .(z0, bubble(.(x1, nil)))), bsort(butlast(bubble(.(z0, .(x1, nil)))))))) BSORT(.(z0, .(x1, .(z1, z2)))) -> c(LAST(.(if(<=(z0, x1), .(x1, if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2))))), .(z0, bubble(.(x1, .(z1, z2))))), bsort(butlast(bubble(.(z0, .(x1, .(z1, z2))))))))) BSORT(.(z0, .(z1, z2))) -> c3(BUTLAST(bubble(.(z0, .(z1, z2))))) BSORT(.(z0, .(z1, z2))) -> c3(BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, nil))), .(z0, .(z1, nil))), bsort(butlast(bubble(.(z0, .(z1, nil))))))), BSORT(butlast(bubble(.(z0, .(z1, nil))))), BUBBLE(.(z0, .(z1, nil)))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, .(z2, z3)))), .(z0, if(<=(z1, z2), .(z2, bubble(.(z1, z3))), .(z1, bubble(.(z2, z3)))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3)))))))), BSORT(butlast(bubble(.(z0, .(z1, .(z2, z3)))))), BUBBLE(.(z0, .(z1, .(z2, z3))))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, .(z0, nil)), .(z0, bubble(.(z1, nil)))), bsort(butlast(bubble(.(z0, .(z1, nil))))))), BSORT(butlast(bubble(.(z0, .(z1, nil))))), BUBBLE(.(z0, .(z1, nil)))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c2(LAST(.(if(<=(z0, z1), .(z1, if(<=(z0, z2), .(z2, bubble(.(z0, z3))), .(z0, bubble(.(z2, z3))))), .(z0, bubble(.(z1, .(z2, z3))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3)))))))), BSORT(butlast(bubble(.(z0, .(z1, .(z2, z3)))))), BUBBLE(.(z0, .(z1, .(z2, z3))))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil))))) S tuples: LAST(.(z0, .(z1, z2))) -> c9(LAST(.(z1, z2))) BUTLAST(.(z0, .(z1, z2))) -> c12(BUTLAST(.(z1, z2))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil))))) K tuples: BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BSORT(.(z0, .(z1, z2))) -> c2(BSORT(butlast(bubble(.(z0, .(z1, z2))))), BUTLAST(bubble(.(z0, .(z1, z2)))), BUBBLE(.(z0, .(z1, z2)))) BUBBLE(.(z0, .(z1, z2))) -> c5(BUBBLE(.(z0, z2))) BUBBLE(.(z0, .(z1, z2))) -> c6(BUBBLE(.(z1, z2))) Defined Rule Symbols: bubble_1, bsort_1, butlast_1, last_1 Defined Pair Symbols: BUBBLE_1, LAST_1, BUTLAST_1, BSORT_1 Compound Symbols: c5_1, c6_1, c9_1, c12_1, c_1, c2_3, c1_1, c3_1, c2_2 ---------------------------------------- (117) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) by BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(.(z0, nil))) ---------------------------------------- (118) Obligation: Complexity Dependency Tuples Problem Rules: bubble(.(z0, nil)) -> .(z0, nil) bubble(.(z0, .(z1, z2))) -> if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))) bsort(nil) -> nil bsort(.(z0, z1)) -> last(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1)))))) butlast(nil) -> nil butlast(.(z0, nil)) -> nil butlast(.(z0, .(z1, z2))) -> .(z0, butlast(.(z1, z2))) last(.(z0, nil)) -> z0 last(.(z0, .(z1, z2))) -> last(.(z1, z2)) Tuples: BUBBLE(.(z0, .(z1, z2))) -> c5(BUBBLE(.(z0, z2))) BUBBLE(.(z0, .(z1, z2))) -> c6(BUBBLE(.(z1, z2))) LAST(.(z0, .(z1, z2))) -> c9(LAST(.(z1, z2))) BUTLAST(.(z0, .(z1, z2))) -> c12(BUTLAST(.(z1, z2))) BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) BSORT(.(z0, .(z1, z2))) -> c1(BUTLAST(bubble(.(z0, .(z1, z2))))) BSORT(.(z0, .(z1, z2))) -> c1(BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, nil)) -> c(LAST(.(bubble(.(z0, nil)), bsort(nil)))) BSORT(.(z0, nil)) -> c(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil)))))) BSORT(.(x0, .(z0, nil))) -> c(LAST(.(if(<=(x0, z0), .(z0, bubble(.(x0, nil))), .(x0, .(z0, nil))), bsort(butlast(bubble(.(x0, .(z0, nil)))))))) BSORT(.(x0, .(z0, .(z1, z2)))) -> c(LAST(.(if(<=(x0, z0), .(z0, bubble(.(x0, .(z1, z2)))), .(x0, if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))))), bsort(butlast(bubble(.(x0, .(z0, .(z1, z2))))))))) BSORT(.(z0, .(x1, nil))) -> c(LAST(.(if(<=(z0, x1), .(x1, .(z0, nil)), .(z0, bubble(.(x1, nil)))), bsort(butlast(bubble(.(z0, .(x1, nil)))))))) BSORT(.(z0, .(x1, .(z1, z2)))) -> c(LAST(.(if(<=(z0, x1), .(x1, if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2))))), .(z0, bubble(.(x1, .(z1, z2))))), bsort(butlast(bubble(.(z0, .(x1, .(z1, z2))))))))) BSORT(.(z0, .(z1, z2))) -> c3(BUTLAST(bubble(.(z0, .(z1, z2))))) BSORT(.(z0, .(z1, z2))) -> c3(BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, nil))), .(z0, .(z1, nil))), bsort(butlast(bubble(.(z0, .(z1, nil))))))), BSORT(butlast(bubble(.(z0, .(z1, nil))))), BUBBLE(.(z0, .(z1, nil)))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, .(z2, z3)))), .(z0, if(<=(z1, z2), .(z2, bubble(.(z1, z3))), .(z1, bubble(.(z2, z3)))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3)))))))), BSORT(butlast(bubble(.(z0, .(z1, .(z2, z3)))))), BUBBLE(.(z0, .(z1, .(z2, z3))))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, .(z0, nil)), .(z0, bubble(.(z1, nil)))), bsort(butlast(bubble(.(z0, .(z1, nil))))))), BSORT(butlast(bubble(.(z0, .(z1, nil))))), BUBBLE(.(z0, .(z1, nil)))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c2(LAST(.(if(<=(z0, z1), .(z1, if(<=(z0, z2), .(z2, bubble(.(z0, z3))), .(z0, bubble(.(z2, z3))))), .(z0, bubble(.(z1, .(z2, z3))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3)))))))), BSORT(butlast(bubble(.(z0, .(z1, .(z2, z3)))))), BUBBLE(.(z0, .(z1, .(z2, z3))))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil))))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(.(z0, nil))) S tuples: LAST(.(z0, .(z1, z2))) -> c9(LAST(.(z1, z2))) BUTLAST(.(z0, .(z1, z2))) -> c12(BUTLAST(.(z1, z2))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil))))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(.(z0, nil))) K tuples: BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BSORT(.(z0, .(z1, z2))) -> c2(BSORT(butlast(bubble(.(z0, .(z1, z2))))), BUTLAST(bubble(.(z0, .(z1, z2)))), BUBBLE(.(z0, .(z1, z2)))) BUBBLE(.(z0, .(z1, z2))) -> c5(BUBBLE(.(z0, z2))) BUBBLE(.(z0, .(z1, z2))) -> c6(BUBBLE(.(z1, z2))) Defined Rule Symbols: bubble_1, bsort_1, butlast_1, last_1 Defined Pair Symbols: BUBBLE_1, LAST_1, BUTLAST_1, BSORT_1 Compound Symbols: c5_1, c6_1, c9_1, c12_1, c_1, c2_3, c1_1, c3_1, c2_2 ---------------------------------------- (119) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (120) Obligation: Complexity Dependency Tuples Problem Rules: bubble(.(z0, nil)) -> .(z0, nil) bubble(.(z0, .(z1, z2))) -> if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))) bsort(nil) -> nil bsort(.(z0, z1)) -> last(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1)))))) butlast(nil) -> nil butlast(.(z0, nil)) -> nil butlast(.(z0, .(z1, z2))) -> .(z0, butlast(.(z1, z2))) last(.(z0, nil)) -> z0 last(.(z0, .(z1, z2))) -> last(.(z1, z2)) Tuples: BUBBLE(.(z0, .(z1, z2))) -> c5(BUBBLE(.(z0, z2))) BUBBLE(.(z0, .(z1, z2))) -> c6(BUBBLE(.(z1, z2))) LAST(.(z0, .(z1, z2))) -> c9(LAST(.(z1, z2))) BUTLAST(.(z0, .(z1, z2))) -> c12(BUTLAST(.(z1, z2))) BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) BSORT(.(z0, .(z1, z2))) -> c1(BUTLAST(bubble(.(z0, .(z1, z2))))) BSORT(.(z0, .(z1, z2))) -> c1(BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, nil)) -> c(LAST(.(bubble(.(z0, nil)), bsort(nil)))) BSORT(.(z0, nil)) -> c(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil)))))) BSORT(.(x0, .(z0, nil))) -> c(LAST(.(if(<=(x0, z0), .(z0, bubble(.(x0, nil))), .(x0, .(z0, nil))), bsort(butlast(bubble(.(x0, .(z0, nil)))))))) BSORT(.(x0, .(z0, .(z1, z2)))) -> c(LAST(.(if(<=(x0, z0), .(z0, bubble(.(x0, .(z1, z2)))), .(x0, if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))))), bsort(butlast(bubble(.(x0, .(z0, .(z1, z2))))))))) BSORT(.(z0, .(x1, nil))) -> c(LAST(.(if(<=(z0, x1), .(x1, .(z0, nil)), .(z0, bubble(.(x1, nil)))), bsort(butlast(bubble(.(z0, .(x1, nil)))))))) BSORT(.(z0, .(x1, .(z1, z2)))) -> c(LAST(.(if(<=(z0, x1), .(x1, if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2))))), .(z0, bubble(.(x1, .(z1, z2))))), bsort(butlast(bubble(.(z0, .(x1, .(z1, z2))))))))) BSORT(.(z0, .(z1, z2))) -> c3(BUTLAST(bubble(.(z0, .(z1, z2))))) BSORT(.(z0, .(z1, z2))) -> c3(BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, nil))), .(z0, .(z1, nil))), bsort(butlast(bubble(.(z0, .(z1, nil))))))), BSORT(butlast(bubble(.(z0, .(z1, nil))))), BUBBLE(.(z0, .(z1, nil)))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, .(z2, z3)))), .(z0, if(<=(z1, z2), .(z2, bubble(.(z1, z3))), .(z1, bubble(.(z2, z3)))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3)))))))), BSORT(butlast(bubble(.(z0, .(z1, .(z2, z3)))))), BUBBLE(.(z0, .(z1, .(z2, z3))))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, .(z0, nil)), .(z0, bubble(.(z1, nil)))), bsort(butlast(bubble(.(z0, .(z1, nil))))))), BSORT(butlast(bubble(.(z0, .(z1, nil))))), BUBBLE(.(z0, .(z1, nil)))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c2(LAST(.(if(<=(z0, z1), .(z1, if(<=(z0, z2), .(z2, bubble(.(z0, z3))), .(z0, bubble(.(z2, z3))))), .(z0, bubble(.(z1, .(z2, z3))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3)))))))), BSORT(butlast(bubble(.(z0, .(z1, .(z2, z3)))))), BUBBLE(.(z0, .(z1, .(z2, z3))))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil))))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil))))) S tuples: LAST(.(z0, .(z1, z2))) -> c9(LAST(.(z1, z2))) BUTLAST(.(z0, .(z1, z2))) -> c12(BUTLAST(.(z1, z2))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil))))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil))))) K tuples: BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BSORT(.(z0, .(z1, z2))) -> c2(BSORT(butlast(bubble(.(z0, .(z1, z2))))), BUTLAST(bubble(.(z0, .(z1, z2)))), BUBBLE(.(z0, .(z1, z2)))) BUBBLE(.(z0, .(z1, z2))) -> c5(BUBBLE(.(z0, z2))) BUBBLE(.(z0, .(z1, z2))) -> c6(BUBBLE(.(z1, z2))) Defined Rule Symbols: bubble_1, bsort_1, butlast_1, last_1 Defined Pair Symbols: BUBBLE_1, LAST_1, BUTLAST_1, BSORT_1 Compound Symbols: c5_1, c6_1, c9_1, c12_1, c_1, c2_3, c1_1, c3_1, c2_2 ---------------------------------------- (121) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace BSORT(.(z0, .(z1, z2))) -> c1(BUTLAST(bubble(.(z0, .(z1, z2))))) by BSORT(.(z0, .(z1, z2))) -> c1(BUTLAST(if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))))) ---------------------------------------- (122) Obligation: Complexity Dependency Tuples Problem Rules: bubble(.(z0, nil)) -> .(z0, nil) bubble(.(z0, .(z1, z2))) -> if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))) bsort(nil) -> nil bsort(.(z0, z1)) -> last(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1)))))) butlast(nil) -> nil butlast(.(z0, nil)) -> nil butlast(.(z0, .(z1, z2))) -> .(z0, butlast(.(z1, z2))) last(.(z0, nil)) -> z0 last(.(z0, .(z1, z2))) -> last(.(z1, z2)) Tuples: BUBBLE(.(z0, .(z1, z2))) -> c5(BUBBLE(.(z0, z2))) BUBBLE(.(z0, .(z1, z2))) -> c6(BUBBLE(.(z1, z2))) LAST(.(z0, .(z1, z2))) -> c9(LAST(.(z1, z2))) BUTLAST(.(z0, .(z1, z2))) -> c12(BUTLAST(.(z1, z2))) BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) BSORT(.(z0, .(z1, z2))) -> c1(BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, nil)) -> c(LAST(.(bubble(.(z0, nil)), bsort(nil)))) BSORT(.(z0, nil)) -> c(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil)))))) BSORT(.(x0, .(z0, nil))) -> c(LAST(.(if(<=(x0, z0), .(z0, bubble(.(x0, nil))), .(x0, .(z0, nil))), bsort(butlast(bubble(.(x0, .(z0, nil)))))))) BSORT(.(x0, .(z0, .(z1, z2)))) -> c(LAST(.(if(<=(x0, z0), .(z0, bubble(.(x0, .(z1, z2)))), .(x0, if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))))), bsort(butlast(bubble(.(x0, .(z0, .(z1, z2))))))))) BSORT(.(z0, .(x1, nil))) -> c(LAST(.(if(<=(z0, x1), .(x1, .(z0, nil)), .(z0, bubble(.(x1, nil)))), bsort(butlast(bubble(.(z0, .(x1, nil)))))))) BSORT(.(z0, .(x1, .(z1, z2)))) -> c(LAST(.(if(<=(z0, x1), .(x1, if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2))))), .(z0, bubble(.(x1, .(z1, z2))))), bsort(butlast(bubble(.(z0, .(x1, .(z1, z2))))))))) BSORT(.(z0, .(z1, z2))) -> c3(BUTLAST(bubble(.(z0, .(z1, z2))))) BSORT(.(z0, .(z1, z2))) -> c3(BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, nil))), .(z0, .(z1, nil))), bsort(butlast(bubble(.(z0, .(z1, nil))))))), BSORT(butlast(bubble(.(z0, .(z1, nil))))), BUBBLE(.(z0, .(z1, nil)))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, .(z2, z3)))), .(z0, if(<=(z1, z2), .(z2, bubble(.(z1, z3))), .(z1, bubble(.(z2, z3)))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3)))))))), BSORT(butlast(bubble(.(z0, .(z1, .(z2, z3)))))), BUBBLE(.(z0, .(z1, .(z2, z3))))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, .(z0, nil)), .(z0, bubble(.(z1, nil)))), bsort(butlast(bubble(.(z0, .(z1, nil))))))), BSORT(butlast(bubble(.(z0, .(z1, nil))))), BUBBLE(.(z0, .(z1, nil)))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c2(LAST(.(if(<=(z0, z1), .(z1, if(<=(z0, z2), .(z2, bubble(.(z0, z3))), .(z0, bubble(.(z2, z3))))), .(z0, bubble(.(z1, .(z2, z3))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3)))))))), BSORT(butlast(bubble(.(z0, .(z1, .(z2, z3)))))), BUBBLE(.(z0, .(z1, .(z2, z3))))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil))))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil))))) BSORT(.(z0, .(z1, z2))) -> c1(BUTLAST(if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))))) S tuples: LAST(.(z0, .(z1, z2))) -> c9(LAST(.(z1, z2))) BUTLAST(.(z0, .(z1, z2))) -> c12(BUTLAST(.(z1, z2))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil))))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil))))) K tuples: BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BSORT(.(z0, .(z1, z2))) -> c2(BSORT(butlast(bubble(.(z0, .(z1, z2))))), BUTLAST(bubble(.(z0, .(z1, z2)))), BUBBLE(.(z0, .(z1, z2)))) BUBBLE(.(z0, .(z1, z2))) -> c5(BUBBLE(.(z0, z2))) BUBBLE(.(z0, .(z1, z2))) -> c6(BUBBLE(.(z1, z2))) Defined Rule Symbols: bubble_1, bsort_1, butlast_1, last_1 Defined Pair Symbols: BUBBLE_1, LAST_1, BUTLAST_1, BSORT_1 Compound Symbols: c5_1, c6_1, c9_1, c12_1, c_1, c2_3, c1_1, c3_1, c2_2 ---------------------------------------- (123) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: BSORT(.(z0, .(z1, z2))) -> c1(BUTLAST(if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))))) ---------------------------------------- (124) Obligation: Complexity Dependency Tuples Problem Rules: bubble(.(z0, nil)) -> .(z0, nil) bubble(.(z0, .(z1, z2))) -> if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))) bsort(nil) -> nil bsort(.(z0, z1)) -> last(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1)))))) butlast(nil) -> nil butlast(.(z0, nil)) -> nil butlast(.(z0, .(z1, z2))) -> .(z0, butlast(.(z1, z2))) last(.(z0, nil)) -> z0 last(.(z0, .(z1, z2))) -> last(.(z1, z2)) Tuples: BUBBLE(.(z0, .(z1, z2))) -> c5(BUBBLE(.(z0, z2))) BUBBLE(.(z0, .(z1, z2))) -> c6(BUBBLE(.(z1, z2))) LAST(.(z0, .(z1, z2))) -> c9(LAST(.(z1, z2))) BUTLAST(.(z0, .(z1, z2))) -> c12(BUTLAST(.(z1, z2))) BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BSORT(.(z0, .(z1, z2))) -> c1(BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) BSORT(.(z0, nil)) -> c(LAST(.(bubble(.(z0, nil)), bsort(nil)))) BSORT(.(z0, nil)) -> c(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil)))))) BSORT(.(x0, .(z0, nil))) -> c(LAST(.(if(<=(x0, z0), .(z0, bubble(.(x0, nil))), .(x0, .(z0, nil))), bsort(butlast(bubble(.(x0, .(z0, nil)))))))) BSORT(.(x0, .(z0, .(z1, z2)))) -> c(LAST(.(if(<=(x0, z0), .(z0, bubble(.(x0, .(z1, z2)))), .(x0, if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))))), bsort(butlast(bubble(.(x0, .(z0, .(z1, z2))))))))) BSORT(.(z0, .(x1, nil))) -> c(LAST(.(if(<=(z0, x1), .(x1, .(z0, nil)), .(z0, bubble(.(x1, nil)))), bsort(butlast(bubble(.(z0, .(x1, nil)))))))) BSORT(.(z0, .(x1, .(z1, z2)))) -> c(LAST(.(if(<=(z0, x1), .(x1, if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2))))), .(z0, bubble(.(x1, .(z1, z2))))), bsort(butlast(bubble(.(z0, .(x1, .(z1, z2))))))))) BSORT(.(z0, .(z1, z2))) -> c3(BUTLAST(bubble(.(z0, .(z1, z2))))) BSORT(.(z0, .(z1, z2))) -> c3(BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, nil))), .(z0, .(z1, nil))), bsort(butlast(bubble(.(z0, .(z1, nil))))))), BSORT(butlast(bubble(.(z0, .(z1, nil))))), BUBBLE(.(z0, .(z1, nil)))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, .(z2, z3)))), .(z0, if(<=(z1, z2), .(z2, bubble(.(z1, z3))), .(z1, bubble(.(z2, z3)))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3)))))))), BSORT(butlast(bubble(.(z0, .(z1, .(z2, z3)))))), BUBBLE(.(z0, .(z1, .(z2, z3))))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, .(z0, nil)), .(z0, bubble(.(z1, nil)))), bsort(butlast(bubble(.(z0, .(z1, nil))))))), BSORT(butlast(bubble(.(z0, .(z1, nil))))), BUBBLE(.(z0, .(z1, nil)))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c2(LAST(.(if(<=(z0, z1), .(z1, if(<=(z0, z2), .(z2, bubble(.(z0, z3))), .(z0, bubble(.(z2, z3))))), .(z0, bubble(.(z1, .(z2, z3))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3)))))))), BSORT(butlast(bubble(.(z0, .(z1, .(z2, z3)))))), BUBBLE(.(z0, .(z1, .(z2, z3))))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil))))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil))))) S tuples: LAST(.(z0, .(z1, z2))) -> c9(LAST(.(z1, z2))) BUTLAST(.(z0, .(z1, z2))) -> c12(BUTLAST(.(z1, z2))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil))))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil))))) K tuples: BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BUBBLE(.(z0, .(z1, z2))) -> c5(BUBBLE(.(z0, z2))) BUBBLE(.(z0, .(z1, z2))) -> c6(BUBBLE(.(z1, z2))) Defined Rule Symbols: bubble_1, bsort_1, butlast_1, last_1 Defined Pair Symbols: BUBBLE_1, LAST_1, BUTLAST_1, BSORT_1 Compound Symbols: c5_1, c6_1, c9_1, c12_1, c_1, c1_1, c2_3, c3_1, c2_2 ---------------------------------------- (125) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace BUBBLE(.(z0, .(z1, z2))) -> c5(BUBBLE(.(z0, z2))) by BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c5(BUBBLE(.(z0, .(y1, y2)))) ---------------------------------------- (126) Obligation: Complexity Dependency Tuples Problem Rules: bubble(.(z0, nil)) -> .(z0, nil) bubble(.(z0, .(z1, z2))) -> if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))) bsort(nil) -> nil bsort(.(z0, z1)) -> last(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1)))))) butlast(nil) -> nil butlast(.(z0, nil)) -> nil butlast(.(z0, .(z1, z2))) -> .(z0, butlast(.(z1, z2))) last(.(z0, nil)) -> z0 last(.(z0, .(z1, z2))) -> last(.(z1, z2)) Tuples: BUBBLE(.(z0, .(z1, z2))) -> c6(BUBBLE(.(z1, z2))) LAST(.(z0, .(z1, z2))) -> c9(LAST(.(z1, z2))) BUTLAST(.(z0, .(z1, z2))) -> c12(BUTLAST(.(z1, z2))) BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BSORT(.(z0, .(z1, z2))) -> c1(BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) BSORT(.(z0, nil)) -> c(LAST(.(bubble(.(z0, nil)), bsort(nil)))) BSORT(.(z0, nil)) -> c(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil)))))) BSORT(.(x0, .(z0, nil))) -> c(LAST(.(if(<=(x0, z0), .(z0, bubble(.(x0, nil))), .(x0, .(z0, nil))), bsort(butlast(bubble(.(x0, .(z0, nil)))))))) BSORT(.(x0, .(z0, .(z1, z2)))) -> c(LAST(.(if(<=(x0, z0), .(z0, bubble(.(x0, .(z1, z2)))), .(x0, if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))))), bsort(butlast(bubble(.(x0, .(z0, .(z1, z2))))))))) BSORT(.(z0, .(x1, nil))) -> c(LAST(.(if(<=(z0, x1), .(x1, .(z0, nil)), .(z0, bubble(.(x1, nil)))), bsort(butlast(bubble(.(z0, .(x1, nil)))))))) BSORT(.(z0, .(x1, .(z1, z2)))) -> c(LAST(.(if(<=(z0, x1), .(x1, if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2))))), .(z0, bubble(.(x1, .(z1, z2))))), bsort(butlast(bubble(.(z0, .(x1, .(z1, z2))))))))) BSORT(.(z0, .(z1, z2))) -> c3(BUTLAST(bubble(.(z0, .(z1, z2))))) BSORT(.(z0, .(z1, z2))) -> c3(BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, nil))), .(z0, .(z1, nil))), bsort(butlast(bubble(.(z0, .(z1, nil))))))), BSORT(butlast(bubble(.(z0, .(z1, nil))))), BUBBLE(.(z0, .(z1, nil)))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, .(z2, z3)))), .(z0, if(<=(z1, z2), .(z2, bubble(.(z1, z3))), .(z1, bubble(.(z2, z3)))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3)))))))), BSORT(butlast(bubble(.(z0, .(z1, .(z2, z3)))))), BUBBLE(.(z0, .(z1, .(z2, z3))))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, .(z0, nil)), .(z0, bubble(.(z1, nil)))), bsort(butlast(bubble(.(z0, .(z1, nil))))))), BSORT(butlast(bubble(.(z0, .(z1, nil))))), BUBBLE(.(z0, .(z1, nil)))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c2(LAST(.(if(<=(z0, z1), .(z1, if(<=(z0, z2), .(z2, bubble(.(z0, z3))), .(z0, bubble(.(z2, z3))))), .(z0, bubble(.(z1, .(z2, z3))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3)))))))), BSORT(butlast(bubble(.(z0, .(z1, .(z2, z3)))))), BUBBLE(.(z0, .(z1, .(z2, z3))))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil))))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil))))) BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c5(BUBBLE(.(z0, .(y1, y2)))) S tuples: LAST(.(z0, .(z1, z2))) -> c9(LAST(.(z1, z2))) BUTLAST(.(z0, .(z1, z2))) -> c12(BUTLAST(.(z1, z2))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil))))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil))))) K tuples: BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BUBBLE(.(z0, .(z1, z2))) -> c6(BUBBLE(.(z1, z2))) BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c5(BUBBLE(.(z0, .(y1, y2)))) Defined Rule Symbols: bubble_1, bsort_1, butlast_1, last_1 Defined Pair Symbols: BUBBLE_1, LAST_1, BUTLAST_1, BSORT_1 Compound Symbols: c6_1, c9_1, c12_1, c_1, c1_1, c2_3, c3_1, c2_2, c5_1 ---------------------------------------- (127) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(bubble(.(z0, nil)))) by BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(.(z0, nil))) ---------------------------------------- (128) Obligation: Complexity Dependency Tuples Problem Rules: bubble(.(z0, nil)) -> .(z0, nil) bubble(.(z0, .(z1, z2))) -> if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))) bsort(nil) -> nil bsort(.(z0, z1)) -> last(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1)))))) butlast(nil) -> nil butlast(.(z0, nil)) -> nil butlast(.(z0, .(z1, z2))) -> .(z0, butlast(.(z1, z2))) last(.(z0, nil)) -> z0 last(.(z0, .(z1, z2))) -> last(.(z1, z2)) Tuples: BUBBLE(.(z0, .(z1, z2))) -> c6(BUBBLE(.(z1, z2))) LAST(.(z0, .(z1, z2))) -> c9(LAST(.(z1, z2))) BUTLAST(.(z0, .(z1, z2))) -> c12(BUTLAST(.(z1, z2))) BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BSORT(.(z0, .(z1, z2))) -> c1(BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, nil)) -> c(LAST(.(bubble(.(z0, nil)), bsort(nil)))) BSORT(.(z0, nil)) -> c(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil)))))) BSORT(.(x0, .(z0, nil))) -> c(LAST(.(if(<=(x0, z0), .(z0, bubble(.(x0, nil))), .(x0, .(z0, nil))), bsort(butlast(bubble(.(x0, .(z0, nil)))))))) BSORT(.(x0, .(z0, .(z1, z2)))) -> c(LAST(.(if(<=(x0, z0), .(z0, bubble(.(x0, .(z1, z2)))), .(x0, if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))))), bsort(butlast(bubble(.(x0, .(z0, .(z1, z2))))))))) BSORT(.(z0, .(x1, nil))) -> c(LAST(.(if(<=(z0, x1), .(x1, .(z0, nil)), .(z0, bubble(.(x1, nil)))), bsort(butlast(bubble(.(z0, .(x1, nil)))))))) BSORT(.(z0, .(x1, .(z1, z2)))) -> c(LAST(.(if(<=(z0, x1), .(x1, if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2))))), .(z0, bubble(.(x1, .(z1, z2))))), bsort(butlast(bubble(.(z0, .(x1, .(z1, z2))))))))) BSORT(.(z0, .(z1, z2))) -> c3(BUTLAST(bubble(.(z0, .(z1, z2))))) BSORT(.(z0, .(z1, z2))) -> c3(BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, nil))), .(z0, .(z1, nil))), bsort(butlast(bubble(.(z0, .(z1, nil))))))), BSORT(butlast(bubble(.(z0, .(z1, nil))))), BUBBLE(.(z0, .(z1, nil)))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, .(z2, z3)))), .(z0, if(<=(z1, z2), .(z2, bubble(.(z1, z3))), .(z1, bubble(.(z2, z3)))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3)))))))), BSORT(butlast(bubble(.(z0, .(z1, .(z2, z3)))))), BUBBLE(.(z0, .(z1, .(z2, z3))))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, .(z0, nil)), .(z0, bubble(.(z1, nil)))), bsort(butlast(bubble(.(z0, .(z1, nil))))))), BSORT(butlast(bubble(.(z0, .(z1, nil))))), BUBBLE(.(z0, .(z1, nil)))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c2(LAST(.(if(<=(z0, z1), .(z1, if(<=(z0, z2), .(z2, bubble(.(z0, z3))), .(z0, bubble(.(z2, z3))))), .(z0, bubble(.(z1, .(z2, z3))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3)))))))), BSORT(butlast(bubble(.(z0, .(z1, .(z2, z3)))))), BUBBLE(.(z0, .(z1, .(z2, z3))))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil))))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil))))) BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c5(BUBBLE(.(z0, .(y1, y2)))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil)))), BUTLAST(.(z0, nil))) S tuples: LAST(.(z0, .(z1, z2))) -> c9(LAST(.(z1, z2))) BUTLAST(.(z0, .(z1, z2))) -> c12(BUTLAST(.(z1, z2))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil))))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil))))) K tuples: BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BUBBLE(.(z0, .(z1, z2))) -> c6(BUBBLE(.(z1, z2))) BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c5(BUBBLE(.(z0, .(y1, y2)))) Defined Rule Symbols: bubble_1, bsort_1, butlast_1, last_1 Defined Pair Symbols: BUBBLE_1, LAST_1, BUTLAST_1, BSORT_1 Compound Symbols: c6_1, c9_1, c12_1, c_1, c1_1, c3_1, c2_3, c2_2, c5_1 ---------------------------------------- (129) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (130) Obligation: Complexity Dependency Tuples Problem Rules: bubble(.(z0, nil)) -> .(z0, nil) bubble(.(z0, .(z1, z2))) -> if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))) bsort(nil) -> nil bsort(.(z0, z1)) -> last(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1)))))) butlast(nil) -> nil butlast(.(z0, nil)) -> nil butlast(.(z0, .(z1, z2))) -> .(z0, butlast(.(z1, z2))) last(.(z0, nil)) -> z0 last(.(z0, .(z1, z2))) -> last(.(z1, z2)) Tuples: BUBBLE(.(z0, .(z1, z2))) -> c6(BUBBLE(.(z1, z2))) LAST(.(z0, .(z1, z2))) -> c9(LAST(.(z1, z2))) BUTLAST(.(z0, .(z1, z2))) -> c12(BUTLAST(.(z1, z2))) BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BSORT(.(z0, .(z1, z2))) -> c1(BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, nil)) -> c(LAST(.(bubble(.(z0, nil)), bsort(nil)))) BSORT(.(z0, nil)) -> c(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil)))))) BSORT(.(x0, .(z0, nil))) -> c(LAST(.(if(<=(x0, z0), .(z0, bubble(.(x0, nil))), .(x0, .(z0, nil))), bsort(butlast(bubble(.(x0, .(z0, nil)))))))) BSORT(.(x0, .(z0, .(z1, z2)))) -> c(LAST(.(if(<=(x0, z0), .(z0, bubble(.(x0, .(z1, z2)))), .(x0, if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))))), bsort(butlast(bubble(.(x0, .(z0, .(z1, z2))))))))) BSORT(.(z0, .(x1, nil))) -> c(LAST(.(if(<=(z0, x1), .(x1, .(z0, nil)), .(z0, bubble(.(x1, nil)))), bsort(butlast(bubble(.(z0, .(x1, nil)))))))) BSORT(.(z0, .(x1, .(z1, z2)))) -> c(LAST(.(if(<=(z0, x1), .(x1, if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2))))), .(z0, bubble(.(x1, .(z1, z2))))), bsort(butlast(bubble(.(z0, .(x1, .(z1, z2))))))))) BSORT(.(z0, .(z1, z2))) -> c3(BUTLAST(bubble(.(z0, .(z1, z2))))) BSORT(.(z0, .(z1, z2))) -> c3(BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, nil))), .(z0, .(z1, nil))), bsort(butlast(bubble(.(z0, .(z1, nil))))))), BSORT(butlast(bubble(.(z0, .(z1, nil))))), BUBBLE(.(z0, .(z1, nil)))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, .(z2, z3)))), .(z0, if(<=(z1, z2), .(z2, bubble(.(z1, z3))), .(z1, bubble(.(z2, z3)))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3)))))))), BSORT(butlast(bubble(.(z0, .(z1, .(z2, z3)))))), BUBBLE(.(z0, .(z1, .(z2, z3))))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, .(z0, nil)), .(z0, bubble(.(z1, nil)))), bsort(butlast(bubble(.(z0, .(z1, nil))))))), BSORT(butlast(bubble(.(z0, .(z1, nil))))), BUBBLE(.(z0, .(z1, nil)))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c2(LAST(.(if(<=(z0, z1), .(z1, if(<=(z0, z2), .(z2, bubble(.(z0, z3))), .(z0, bubble(.(z2, z3))))), .(z0, bubble(.(z1, .(z2, z3))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3)))))))), BSORT(butlast(bubble(.(z0, .(z1, .(z2, z3)))))), BUBBLE(.(z0, .(z1, .(z2, z3))))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil))))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil))))) BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c5(BUBBLE(.(z0, .(y1, y2)))) S tuples: LAST(.(z0, .(z1, z2))) -> c9(LAST(.(z1, z2))) BUTLAST(.(z0, .(z1, z2))) -> c12(BUTLAST(.(z1, z2))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil))))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil))))) K tuples: BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BUBBLE(.(z0, .(z1, z2))) -> c6(BUBBLE(.(z1, z2))) BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c5(BUBBLE(.(z0, .(y1, y2)))) Defined Rule Symbols: bubble_1, bsort_1, butlast_1, last_1 Defined Pair Symbols: BUBBLE_1, LAST_1, BUTLAST_1, BSORT_1 Compound Symbols: c6_1, c9_1, c12_1, c_1, c1_1, c3_1, c2_3, c2_2, c5_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. BUTLAST(.(z0, .(z1, z2))) -> c12(BUTLAST(.(z1, z2))) We considered the (Usable) Rules: bubble(.(z0, .(z1, z2))) -> if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))) And the Tuples: BUBBLE(.(z0, .(z1, z2))) -> c6(BUBBLE(.(z1, z2))) LAST(.(z0, .(z1, z2))) -> c9(LAST(.(z1, z2))) BUTLAST(.(z0, .(z1, z2))) -> c12(BUTLAST(.(z1, z2))) BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BSORT(.(z0, .(z1, z2))) -> c1(BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, nil)) -> c(LAST(.(bubble(.(z0, nil)), bsort(nil)))) BSORT(.(z0, nil)) -> c(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil)))))) BSORT(.(x0, .(z0, nil))) -> c(LAST(.(if(<=(x0, z0), .(z0, bubble(.(x0, nil))), .(x0, .(z0, nil))), bsort(butlast(bubble(.(x0, .(z0, nil)))))))) BSORT(.(x0, .(z0, .(z1, z2)))) -> c(LAST(.(if(<=(x0, z0), .(z0, bubble(.(x0, .(z1, z2)))), .(x0, if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))))), bsort(butlast(bubble(.(x0, .(z0, .(z1, z2))))))))) BSORT(.(z0, .(x1, nil))) -> c(LAST(.(if(<=(z0, x1), .(x1, .(z0, nil)), .(z0, bubble(.(x1, nil)))), bsort(butlast(bubble(.(z0, .(x1, nil)))))))) BSORT(.(z0, .(x1, .(z1, z2)))) -> c(LAST(.(if(<=(z0, x1), .(x1, if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2))))), .(z0, bubble(.(x1, .(z1, z2))))), bsort(butlast(bubble(.(z0, .(x1, .(z1, z2))))))))) BSORT(.(z0, .(z1, z2))) -> c3(BUTLAST(bubble(.(z0, .(z1, z2))))) BSORT(.(z0, .(z1, z2))) -> c3(BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, nil))), .(z0, .(z1, nil))), bsort(butlast(bubble(.(z0, .(z1, nil))))))), BSORT(butlast(bubble(.(z0, .(z1, nil))))), BUBBLE(.(z0, .(z1, nil)))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, .(z2, z3)))), .(z0, if(<=(z1, z2), .(z2, bubble(.(z1, z3))), .(z1, bubble(.(z2, z3)))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3)))))))), BSORT(butlast(bubble(.(z0, .(z1, .(z2, z3)))))), BUBBLE(.(z0, .(z1, .(z2, z3))))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, .(z0, nil)), .(z0, bubble(.(z1, nil)))), bsort(butlast(bubble(.(z0, .(z1, nil))))))), BSORT(butlast(bubble(.(z0, .(z1, nil))))), BUBBLE(.(z0, .(z1, nil)))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c2(LAST(.(if(<=(z0, z1), .(z1, if(<=(z0, z2), .(z2, bubble(.(z0, z3))), .(z0, bubble(.(z2, z3))))), .(z0, bubble(.(z1, .(z2, z3))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3)))))))), BSORT(butlast(bubble(.(z0, .(z1, .(z2, z3)))))), BUBBLE(.(z0, .(z1, .(z2, z3))))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil))))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil))))) BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c5(BUBBLE(.(z0, .(y1, y2)))) The order we found is given by the following interpretation: Polynomial interpretation : POL(.(x_1, x_2)) = [1] + x_2 POL(<=(x_1, x_2)) = 0 POL(BSORT(x_1)) = 0 POL(BUBBLE(x_1)) = 0 POL(BUTLAST(x_1)) = x_1 POL(LAST(x_1)) = 0 POL(bsort(x_1)) = [1] POL(bubble(x_1)) = 0 POL(butlast(x_1)) = 0 POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c2(x_1, x_2)) = x_1 + x_2 POL(c2(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c3(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c9(x_1)) = x_1 POL(if(x_1, x_2, x_3)) = 0 POL(last(x_1)) = [1] POL(nil) = 0 ---------------------------------------- (132) Obligation: Complexity Dependency Tuples Problem Rules: bubble(.(z0, nil)) -> .(z0, nil) bubble(.(z0, .(z1, z2))) -> if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))) bsort(nil) -> nil bsort(.(z0, z1)) -> last(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1)))))) butlast(nil) -> nil butlast(.(z0, nil)) -> nil butlast(.(z0, .(z1, z2))) -> .(z0, butlast(.(z1, z2))) last(.(z0, nil)) -> z0 last(.(z0, .(z1, z2))) -> last(.(z1, z2)) Tuples: BUBBLE(.(z0, .(z1, z2))) -> c6(BUBBLE(.(z1, z2))) LAST(.(z0, .(z1, z2))) -> c9(LAST(.(z1, z2))) BUTLAST(.(z0, .(z1, z2))) -> c12(BUTLAST(.(z1, z2))) BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BSORT(.(z0, .(z1, z2))) -> c1(BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, nil)) -> c(LAST(.(bubble(.(z0, nil)), bsort(nil)))) BSORT(.(z0, nil)) -> c(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil)))))) BSORT(.(x0, .(z0, nil))) -> c(LAST(.(if(<=(x0, z0), .(z0, bubble(.(x0, nil))), .(x0, .(z0, nil))), bsort(butlast(bubble(.(x0, .(z0, nil)))))))) BSORT(.(x0, .(z0, .(z1, z2)))) -> c(LAST(.(if(<=(x0, z0), .(z0, bubble(.(x0, .(z1, z2)))), .(x0, if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))))), bsort(butlast(bubble(.(x0, .(z0, .(z1, z2))))))))) BSORT(.(z0, .(x1, nil))) -> c(LAST(.(if(<=(z0, x1), .(x1, .(z0, nil)), .(z0, bubble(.(x1, nil)))), bsort(butlast(bubble(.(z0, .(x1, nil)))))))) BSORT(.(z0, .(x1, .(z1, z2)))) -> c(LAST(.(if(<=(z0, x1), .(x1, if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2))))), .(z0, bubble(.(x1, .(z1, z2))))), bsort(butlast(bubble(.(z0, .(x1, .(z1, z2))))))))) BSORT(.(z0, .(z1, z2))) -> c3(BUTLAST(bubble(.(z0, .(z1, z2))))) BSORT(.(z0, .(z1, z2))) -> c3(BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, nil))), .(z0, .(z1, nil))), bsort(butlast(bubble(.(z0, .(z1, nil))))))), BSORT(butlast(bubble(.(z0, .(z1, nil))))), BUBBLE(.(z0, .(z1, nil)))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, .(z2, z3)))), .(z0, if(<=(z1, z2), .(z2, bubble(.(z1, z3))), .(z1, bubble(.(z2, z3)))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3)))))))), BSORT(butlast(bubble(.(z0, .(z1, .(z2, z3)))))), BUBBLE(.(z0, .(z1, .(z2, z3))))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, .(z0, nil)), .(z0, bubble(.(z1, nil)))), bsort(butlast(bubble(.(z0, .(z1, nil))))))), BSORT(butlast(bubble(.(z0, .(z1, nil))))), BUBBLE(.(z0, .(z1, nil)))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c2(LAST(.(if(<=(z0, z1), .(z1, if(<=(z0, z2), .(z2, bubble(.(z0, z3))), .(z0, bubble(.(z2, z3))))), .(z0, bubble(.(z1, .(z2, z3))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3)))))))), BSORT(butlast(bubble(.(z0, .(z1, .(z2, z3)))))), BUBBLE(.(z0, .(z1, .(z2, z3))))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil))))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil))))) BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c5(BUBBLE(.(z0, .(y1, y2)))) S tuples: LAST(.(z0, .(z1, z2))) -> c9(LAST(.(z1, z2))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil))))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil))))) K tuples: BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BUBBLE(.(z0, .(z1, z2))) -> c6(BUBBLE(.(z1, z2))) BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c5(BUBBLE(.(z0, .(y1, y2)))) BUTLAST(.(z0, .(z1, z2))) -> c12(BUTLAST(.(z1, z2))) Defined Rule Symbols: bubble_1, bsort_1, butlast_1, last_1 Defined Pair Symbols: BUBBLE_1, LAST_1, BUTLAST_1, BSORT_1 Compound Symbols: c6_1, c9_1, c12_1, c_1, c1_1, c3_1, c2_3, c2_2, c5_1 ---------------------------------------- (133) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace BSORT(.(z0, nil)) -> c(LAST(.(bubble(.(z0, nil)), bsort(nil)))) by BSORT(.(z0, nil)) -> c(LAST(.(bubble(.(z0, nil)), nil))) ---------------------------------------- (134) Obligation: Complexity Dependency Tuples Problem Rules: bubble(.(z0, nil)) -> .(z0, nil) bubble(.(z0, .(z1, z2))) -> if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))) bsort(nil) -> nil bsort(.(z0, z1)) -> last(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1)))))) butlast(nil) -> nil butlast(.(z0, nil)) -> nil butlast(.(z0, .(z1, z2))) -> .(z0, butlast(.(z1, z2))) last(.(z0, nil)) -> z0 last(.(z0, .(z1, z2))) -> last(.(z1, z2)) Tuples: BUBBLE(.(z0, .(z1, z2))) -> c6(BUBBLE(.(z1, z2))) LAST(.(z0, .(z1, z2))) -> c9(LAST(.(z1, z2))) BUTLAST(.(z0, .(z1, z2))) -> c12(BUTLAST(.(z1, z2))) BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BSORT(.(z0, .(z1, z2))) -> c1(BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, nil)) -> c(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil)))))) BSORT(.(x0, .(z0, nil))) -> c(LAST(.(if(<=(x0, z0), .(z0, bubble(.(x0, nil))), .(x0, .(z0, nil))), bsort(butlast(bubble(.(x0, .(z0, nil)))))))) BSORT(.(x0, .(z0, .(z1, z2)))) -> c(LAST(.(if(<=(x0, z0), .(z0, bubble(.(x0, .(z1, z2)))), .(x0, if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))))), bsort(butlast(bubble(.(x0, .(z0, .(z1, z2))))))))) BSORT(.(z0, .(x1, nil))) -> c(LAST(.(if(<=(z0, x1), .(x1, .(z0, nil)), .(z0, bubble(.(x1, nil)))), bsort(butlast(bubble(.(z0, .(x1, nil)))))))) BSORT(.(z0, .(x1, .(z1, z2)))) -> c(LAST(.(if(<=(z0, x1), .(x1, if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2))))), .(z0, bubble(.(x1, .(z1, z2))))), bsort(butlast(bubble(.(z0, .(x1, .(z1, z2))))))))) BSORT(.(z0, .(z1, z2))) -> c3(BUTLAST(bubble(.(z0, .(z1, z2))))) BSORT(.(z0, .(z1, z2))) -> c3(BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, nil))), .(z0, .(z1, nil))), bsort(butlast(bubble(.(z0, .(z1, nil))))))), BSORT(butlast(bubble(.(z0, .(z1, nil))))), BUBBLE(.(z0, .(z1, nil)))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, .(z2, z3)))), .(z0, if(<=(z1, z2), .(z2, bubble(.(z1, z3))), .(z1, bubble(.(z2, z3)))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3)))))))), BSORT(butlast(bubble(.(z0, .(z1, .(z2, z3)))))), BUBBLE(.(z0, .(z1, .(z2, z3))))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, .(z0, nil)), .(z0, bubble(.(z1, nil)))), bsort(butlast(bubble(.(z0, .(z1, nil))))))), BSORT(butlast(bubble(.(z0, .(z1, nil))))), BUBBLE(.(z0, .(z1, nil)))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c2(LAST(.(if(<=(z0, z1), .(z1, if(<=(z0, z2), .(z2, bubble(.(z0, z3))), .(z0, bubble(.(z2, z3))))), .(z0, bubble(.(z1, .(z2, z3))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3)))))))), BSORT(butlast(bubble(.(z0, .(z1, .(z2, z3)))))), BUBBLE(.(z0, .(z1, .(z2, z3))))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil))))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil))))) BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c5(BUBBLE(.(z0, .(y1, y2)))) BSORT(.(z0, nil)) -> c(LAST(.(bubble(.(z0, nil)), nil))) S tuples: LAST(.(z0, .(z1, z2))) -> c9(LAST(.(z1, z2))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil))))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil))))) K tuples: BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BUBBLE(.(z0, .(z1, z2))) -> c6(BUBBLE(.(z1, z2))) BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c5(BUBBLE(.(z0, .(y1, y2)))) BUTLAST(.(z0, .(z1, z2))) -> c12(BUTLAST(.(z1, z2))) Defined Rule Symbols: bubble_1, bsort_1, butlast_1, last_1 Defined Pair Symbols: BUBBLE_1, LAST_1, BUTLAST_1, BSORT_1 Compound Symbols: c6_1, c9_1, c12_1, c_1, c1_1, c3_1, c2_3, c2_2, c5_1 ---------------------------------------- (135) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: BSORT(.(z0, nil)) -> c(LAST(.(bubble(.(z0, nil)), nil))) ---------------------------------------- (136) Obligation: Complexity Dependency Tuples Problem Rules: bubble(.(z0, nil)) -> .(z0, nil) bubble(.(z0, .(z1, z2))) -> if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))) bsort(nil) -> nil bsort(.(z0, z1)) -> last(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1)))))) butlast(nil) -> nil butlast(.(z0, nil)) -> nil butlast(.(z0, .(z1, z2))) -> .(z0, butlast(.(z1, z2))) last(.(z0, nil)) -> z0 last(.(z0, .(z1, z2))) -> last(.(z1, z2)) Tuples: BUBBLE(.(z0, .(z1, z2))) -> c6(BUBBLE(.(z1, z2))) LAST(.(z0, .(z1, z2))) -> c9(LAST(.(z1, z2))) BUTLAST(.(z0, .(z1, z2))) -> c12(BUTLAST(.(z1, z2))) BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BSORT(.(z0, .(z1, z2))) -> c1(BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, nil)) -> c(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil)))))) BSORT(.(x0, .(z0, nil))) -> c(LAST(.(if(<=(x0, z0), .(z0, bubble(.(x0, nil))), .(x0, .(z0, nil))), bsort(butlast(bubble(.(x0, .(z0, nil)))))))) BSORT(.(x0, .(z0, .(z1, z2)))) -> c(LAST(.(if(<=(x0, z0), .(z0, bubble(.(x0, .(z1, z2)))), .(x0, if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))))), bsort(butlast(bubble(.(x0, .(z0, .(z1, z2))))))))) BSORT(.(z0, .(x1, nil))) -> c(LAST(.(if(<=(z0, x1), .(x1, .(z0, nil)), .(z0, bubble(.(x1, nil)))), bsort(butlast(bubble(.(z0, .(x1, nil)))))))) BSORT(.(z0, .(x1, .(z1, z2)))) -> c(LAST(.(if(<=(z0, x1), .(x1, if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2))))), .(z0, bubble(.(x1, .(z1, z2))))), bsort(butlast(bubble(.(z0, .(x1, .(z1, z2))))))))) BSORT(.(z0, .(z1, z2))) -> c3(BUTLAST(bubble(.(z0, .(z1, z2))))) BSORT(.(z0, .(z1, z2))) -> c3(BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, nil))), .(z0, .(z1, nil))), bsort(butlast(bubble(.(z0, .(z1, nil))))))), BSORT(butlast(bubble(.(z0, .(z1, nil))))), BUBBLE(.(z0, .(z1, nil)))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, .(z2, z3)))), .(z0, if(<=(z1, z2), .(z2, bubble(.(z1, z3))), .(z1, bubble(.(z2, z3)))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3)))))))), BSORT(butlast(bubble(.(z0, .(z1, .(z2, z3)))))), BUBBLE(.(z0, .(z1, .(z2, z3))))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, .(z0, nil)), .(z0, bubble(.(z1, nil)))), bsort(butlast(bubble(.(z0, .(z1, nil))))))), BSORT(butlast(bubble(.(z0, .(z1, nil))))), BUBBLE(.(z0, .(z1, nil)))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c2(LAST(.(if(<=(z0, z1), .(z1, if(<=(z0, z2), .(z2, bubble(.(z0, z3))), .(z0, bubble(.(z2, z3))))), .(z0, bubble(.(z1, .(z2, z3))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3)))))))), BSORT(butlast(bubble(.(z0, .(z1, .(z2, z3)))))), BUBBLE(.(z0, .(z1, .(z2, z3))))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil))))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil))))) BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c5(BUBBLE(.(z0, .(y1, y2)))) S tuples: LAST(.(z0, .(z1, z2))) -> c9(LAST(.(z1, z2))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil))))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil))))) K tuples: BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BUBBLE(.(z0, .(z1, z2))) -> c6(BUBBLE(.(z1, z2))) BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c5(BUBBLE(.(z0, .(y1, y2)))) BUTLAST(.(z0, .(z1, z2))) -> c12(BUTLAST(.(z1, z2))) Defined Rule Symbols: bubble_1, bsort_1, butlast_1, last_1 Defined Pair Symbols: BUBBLE_1, LAST_1, BUTLAST_1, BSORT_1 Compound Symbols: c6_1, c9_1, c12_1, c_1, c1_1, c3_1, c2_3, c2_2, c5_1 ---------------------------------------- (137) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace BSORT(.(z0, nil)) -> c(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil)))))) by BSORT(.(z0, nil)) -> c(LAST(.(.(z0, nil), bsort(nil)))) ---------------------------------------- (138) Obligation: Complexity Dependency Tuples Problem Rules: bubble(.(z0, nil)) -> .(z0, nil) bubble(.(z0, .(z1, z2))) -> if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))) bsort(nil) -> nil bsort(.(z0, z1)) -> last(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1)))))) butlast(nil) -> nil butlast(.(z0, nil)) -> nil butlast(.(z0, .(z1, z2))) -> .(z0, butlast(.(z1, z2))) last(.(z0, nil)) -> z0 last(.(z0, .(z1, z2))) -> last(.(z1, z2)) Tuples: BUBBLE(.(z0, .(z1, z2))) -> c6(BUBBLE(.(z1, z2))) LAST(.(z0, .(z1, z2))) -> c9(LAST(.(z1, z2))) BUTLAST(.(z0, .(z1, z2))) -> c12(BUTLAST(.(z1, z2))) BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BSORT(.(z0, .(z1, z2))) -> c1(BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, nil)) -> c(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil)))))) BSORT(.(x0, .(z0, nil))) -> c(LAST(.(if(<=(x0, z0), .(z0, bubble(.(x0, nil))), .(x0, .(z0, nil))), bsort(butlast(bubble(.(x0, .(z0, nil)))))))) BSORT(.(x0, .(z0, .(z1, z2)))) -> c(LAST(.(if(<=(x0, z0), .(z0, bubble(.(x0, .(z1, z2)))), .(x0, if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))))), bsort(butlast(bubble(.(x0, .(z0, .(z1, z2))))))))) BSORT(.(z0, .(x1, nil))) -> c(LAST(.(if(<=(z0, x1), .(x1, .(z0, nil)), .(z0, bubble(.(x1, nil)))), bsort(butlast(bubble(.(z0, .(x1, nil)))))))) BSORT(.(z0, .(x1, .(z1, z2)))) -> c(LAST(.(if(<=(z0, x1), .(x1, if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2))))), .(z0, bubble(.(x1, .(z1, z2))))), bsort(butlast(bubble(.(z0, .(x1, .(z1, z2))))))))) BSORT(.(z0, .(z1, z2))) -> c3(BUTLAST(bubble(.(z0, .(z1, z2))))) BSORT(.(z0, .(z1, z2))) -> c3(BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, nil))), .(z0, .(z1, nil))), bsort(butlast(bubble(.(z0, .(z1, nil))))))), BSORT(butlast(bubble(.(z0, .(z1, nil))))), BUBBLE(.(z0, .(z1, nil)))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, .(z2, z3)))), .(z0, if(<=(z1, z2), .(z2, bubble(.(z1, z3))), .(z1, bubble(.(z2, z3)))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3)))))))), BSORT(butlast(bubble(.(z0, .(z1, .(z2, z3)))))), BUBBLE(.(z0, .(z1, .(z2, z3))))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, .(z0, nil)), .(z0, bubble(.(z1, nil)))), bsort(butlast(bubble(.(z0, .(z1, nil))))))), BSORT(butlast(bubble(.(z0, .(z1, nil))))), BUBBLE(.(z0, .(z1, nil)))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c2(LAST(.(if(<=(z0, z1), .(z1, if(<=(z0, z2), .(z2, bubble(.(z0, z3))), .(z0, bubble(.(z2, z3))))), .(z0, bubble(.(z1, .(z2, z3))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3)))))))), BSORT(butlast(bubble(.(z0, .(z1, .(z2, z3)))))), BUBBLE(.(z0, .(z1, .(z2, z3))))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil))))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil))))) BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c5(BUBBLE(.(z0, .(y1, y2)))) BSORT(.(z0, nil)) -> c(LAST(.(.(z0, nil), bsort(nil)))) S tuples: LAST(.(z0, .(z1, z2))) -> c9(LAST(.(z1, z2))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil))))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil))))) K tuples: BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BUBBLE(.(z0, .(z1, z2))) -> c6(BUBBLE(.(z1, z2))) BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c5(BUBBLE(.(z0, .(y1, y2)))) BUTLAST(.(z0, .(z1, z2))) -> c12(BUTLAST(.(z1, z2))) Defined Rule Symbols: bubble_1, bsort_1, butlast_1, last_1 Defined Pair Symbols: BUBBLE_1, LAST_1, BUTLAST_1, BSORT_1 Compound Symbols: c6_1, c9_1, c12_1, c_1, c1_1, c3_1, c2_3, c2_2, c5_1 ---------------------------------------- (139) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace BSORT(.(z0, nil)) -> c(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil)))))) by BSORT(.(z0, nil)) -> c(LAST(.(.(z0, nil), bsort(nil)))) ---------------------------------------- (140) Obligation: Complexity Dependency Tuples Problem Rules: bubble(.(z0, nil)) -> .(z0, nil) bubble(.(z0, .(z1, z2))) -> if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))) bsort(nil) -> nil bsort(.(z0, z1)) -> last(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1)))))) butlast(nil) -> nil butlast(.(z0, nil)) -> nil butlast(.(z0, .(z1, z2))) -> .(z0, butlast(.(z1, z2))) last(.(z0, nil)) -> z0 last(.(z0, .(z1, z2))) -> last(.(z1, z2)) Tuples: BUBBLE(.(z0, .(z1, z2))) -> c6(BUBBLE(.(z1, z2))) LAST(.(z0, .(z1, z2))) -> c9(LAST(.(z1, z2))) BUTLAST(.(z0, .(z1, z2))) -> c12(BUTLAST(.(z1, z2))) BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BSORT(.(z0, .(z1, z2))) -> c1(BUBBLE(.(z0, .(z1, z2)))) BSORT(.(x0, .(z0, nil))) -> c(LAST(.(if(<=(x0, z0), .(z0, bubble(.(x0, nil))), .(x0, .(z0, nil))), bsort(butlast(bubble(.(x0, .(z0, nil)))))))) BSORT(.(x0, .(z0, .(z1, z2)))) -> c(LAST(.(if(<=(x0, z0), .(z0, bubble(.(x0, .(z1, z2)))), .(x0, if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))))), bsort(butlast(bubble(.(x0, .(z0, .(z1, z2))))))))) BSORT(.(z0, .(x1, nil))) -> c(LAST(.(if(<=(z0, x1), .(x1, .(z0, nil)), .(z0, bubble(.(x1, nil)))), bsort(butlast(bubble(.(z0, .(x1, nil)))))))) BSORT(.(z0, .(x1, .(z1, z2)))) -> c(LAST(.(if(<=(z0, x1), .(x1, if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2))))), .(z0, bubble(.(x1, .(z1, z2))))), bsort(butlast(bubble(.(z0, .(x1, .(z1, z2))))))))) BSORT(.(z0, .(z1, z2))) -> c3(BUTLAST(bubble(.(z0, .(z1, z2))))) BSORT(.(z0, .(z1, z2))) -> c3(BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, nil))), .(z0, .(z1, nil))), bsort(butlast(bubble(.(z0, .(z1, nil))))))), BSORT(butlast(bubble(.(z0, .(z1, nil))))), BUBBLE(.(z0, .(z1, nil)))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, .(z2, z3)))), .(z0, if(<=(z1, z2), .(z2, bubble(.(z1, z3))), .(z1, bubble(.(z2, z3)))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3)))))))), BSORT(butlast(bubble(.(z0, .(z1, .(z2, z3)))))), BUBBLE(.(z0, .(z1, .(z2, z3))))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, .(z0, nil)), .(z0, bubble(.(z1, nil)))), bsort(butlast(bubble(.(z0, .(z1, nil))))))), BSORT(butlast(bubble(.(z0, .(z1, nil))))), BUBBLE(.(z0, .(z1, nil)))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c2(LAST(.(if(<=(z0, z1), .(z1, if(<=(z0, z2), .(z2, bubble(.(z0, z3))), .(z0, bubble(.(z2, z3))))), .(z0, bubble(.(z1, .(z2, z3))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3)))))))), BSORT(butlast(bubble(.(z0, .(z1, .(z2, z3)))))), BUBBLE(.(z0, .(z1, .(z2, z3))))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil))))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil))))) BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c5(BUBBLE(.(z0, .(y1, y2)))) BSORT(.(z0, nil)) -> c(LAST(.(.(z0, nil), bsort(nil)))) S tuples: LAST(.(z0, .(z1, z2))) -> c9(LAST(.(z1, z2))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil))))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil))))) K tuples: BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BUBBLE(.(z0, .(z1, z2))) -> c6(BUBBLE(.(z1, z2))) BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c5(BUBBLE(.(z0, .(y1, y2)))) BUTLAST(.(z0, .(z1, z2))) -> c12(BUTLAST(.(z1, z2))) Defined Rule Symbols: bubble_1, bsort_1, butlast_1, last_1 Defined Pair Symbols: BUBBLE_1, LAST_1, BUTLAST_1, BSORT_1 Compound Symbols: c6_1, c9_1, c12_1, c_1, c1_1, c3_1, c2_3, c2_2, c5_1 ---------------------------------------- (141) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace BUBBLE(.(z0, .(z1, z2))) -> c6(BUBBLE(.(z1, z2))) by BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c6(BUBBLE(.(z1, .(y1, y2)))) BUBBLE(.(z0, .(z1, .(y1, .(y2, y3))))) -> c6(BUBBLE(.(z1, .(y1, .(y2, y3))))) ---------------------------------------- (142) Obligation: Complexity Dependency Tuples Problem Rules: bubble(.(z0, nil)) -> .(z0, nil) bubble(.(z0, .(z1, z2))) -> if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))) bsort(nil) -> nil bsort(.(z0, z1)) -> last(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1)))))) butlast(nil) -> nil butlast(.(z0, nil)) -> nil butlast(.(z0, .(z1, z2))) -> .(z0, butlast(.(z1, z2))) last(.(z0, nil)) -> z0 last(.(z0, .(z1, z2))) -> last(.(z1, z2)) Tuples: LAST(.(z0, .(z1, z2))) -> c9(LAST(.(z1, z2))) BUTLAST(.(z0, .(z1, z2))) -> c12(BUTLAST(.(z1, z2))) BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BSORT(.(z0, .(z1, z2))) -> c1(BUBBLE(.(z0, .(z1, z2)))) BSORT(.(x0, .(z0, nil))) -> c(LAST(.(if(<=(x0, z0), .(z0, bubble(.(x0, nil))), .(x0, .(z0, nil))), bsort(butlast(bubble(.(x0, .(z0, nil)))))))) BSORT(.(x0, .(z0, .(z1, z2)))) -> c(LAST(.(if(<=(x0, z0), .(z0, bubble(.(x0, .(z1, z2)))), .(x0, if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))))), bsort(butlast(bubble(.(x0, .(z0, .(z1, z2))))))))) BSORT(.(z0, .(x1, nil))) -> c(LAST(.(if(<=(z0, x1), .(x1, .(z0, nil)), .(z0, bubble(.(x1, nil)))), bsort(butlast(bubble(.(z0, .(x1, nil)))))))) BSORT(.(z0, .(x1, .(z1, z2)))) -> c(LAST(.(if(<=(z0, x1), .(x1, if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2))))), .(z0, bubble(.(x1, .(z1, z2))))), bsort(butlast(bubble(.(z0, .(x1, .(z1, z2))))))))) BSORT(.(z0, .(z1, z2))) -> c3(BUTLAST(bubble(.(z0, .(z1, z2))))) BSORT(.(z0, .(z1, z2))) -> c3(BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, nil))), .(z0, .(z1, nil))), bsort(butlast(bubble(.(z0, .(z1, nil))))))), BSORT(butlast(bubble(.(z0, .(z1, nil))))), BUBBLE(.(z0, .(z1, nil)))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, .(z2, z3)))), .(z0, if(<=(z1, z2), .(z2, bubble(.(z1, z3))), .(z1, bubble(.(z2, z3)))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3)))))))), BSORT(butlast(bubble(.(z0, .(z1, .(z2, z3)))))), BUBBLE(.(z0, .(z1, .(z2, z3))))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, .(z0, nil)), .(z0, bubble(.(z1, nil)))), bsort(butlast(bubble(.(z0, .(z1, nil))))))), BSORT(butlast(bubble(.(z0, .(z1, nil))))), BUBBLE(.(z0, .(z1, nil)))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c2(LAST(.(if(<=(z0, z1), .(z1, if(<=(z0, z2), .(z2, bubble(.(z0, z3))), .(z0, bubble(.(z2, z3))))), .(z0, bubble(.(z1, .(z2, z3))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3)))))))), BSORT(butlast(bubble(.(z0, .(z1, .(z2, z3)))))), BUBBLE(.(z0, .(z1, .(z2, z3))))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil))))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil))))) BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c5(BUBBLE(.(z0, .(y1, y2)))) BSORT(.(z0, nil)) -> c(LAST(.(.(z0, nil), bsort(nil)))) BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c6(BUBBLE(.(z1, .(y1, y2)))) BUBBLE(.(z0, .(z1, .(y1, .(y2, y3))))) -> c6(BUBBLE(.(z1, .(y1, .(y2, y3))))) S tuples: LAST(.(z0, .(z1, z2))) -> c9(LAST(.(z1, z2))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil))))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil))))) K tuples: BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c5(BUBBLE(.(z0, .(y1, y2)))) BUTLAST(.(z0, .(z1, z2))) -> c12(BUTLAST(.(z1, z2))) BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c6(BUBBLE(.(z1, .(y1, y2)))) BUBBLE(.(z0, .(z1, .(y1, .(y2, y3))))) -> c6(BUBBLE(.(z1, .(y1, .(y2, y3))))) Defined Rule Symbols: bubble_1, bsort_1, butlast_1, last_1 Defined Pair Symbols: LAST_1, BUTLAST_1, BSORT_1, BUBBLE_1 Compound Symbols: c9_1, c12_1, c_1, c1_1, c3_1, c2_3, c2_2, c5_1, c6_1 ---------------------------------------- (143) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (144) Obligation: Complexity Dependency Tuples Problem Rules: bubble(.(z0, nil)) -> .(z0, nil) bubble(.(z0, .(z1, z2))) -> if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))) bsort(nil) -> nil bsort(.(z0, z1)) -> last(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1)))))) butlast(nil) -> nil butlast(.(z0, nil)) -> nil butlast(.(z0, .(z1, z2))) -> .(z0, butlast(.(z1, z2))) last(.(z0, nil)) -> z0 last(.(z0, .(z1, z2))) -> last(.(z1, z2)) Tuples: LAST(.(z0, .(z1, z2))) -> c9(LAST(.(z1, z2))) BUTLAST(.(z0, .(z1, z2))) -> c12(BUTLAST(.(z1, z2))) BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BSORT(.(z0, .(z1, z2))) -> c1(BUBBLE(.(z0, .(z1, z2)))) BSORT(.(x0, .(z0, nil))) -> c(LAST(.(if(<=(x0, z0), .(z0, bubble(.(x0, nil))), .(x0, .(z0, nil))), bsort(butlast(bubble(.(x0, .(z0, nil)))))))) BSORT(.(x0, .(z0, .(z1, z2)))) -> c(LAST(.(if(<=(x0, z0), .(z0, bubble(.(x0, .(z1, z2)))), .(x0, if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))))), bsort(butlast(bubble(.(x0, .(z0, .(z1, z2))))))))) BSORT(.(z0, .(x1, nil))) -> c(LAST(.(if(<=(z0, x1), .(x1, .(z0, nil)), .(z0, bubble(.(x1, nil)))), bsort(butlast(bubble(.(z0, .(x1, nil)))))))) BSORT(.(z0, .(x1, .(z1, z2)))) -> c(LAST(.(if(<=(z0, x1), .(x1, if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2))))), .(z0, bubble(.(x1, .(z1, z2))))), bsort(butlast(bubble(.(z0, .(x1, .(z1, z2))))))))) BSORT(.(z0, .(z1, z2))) -> c3(BUTLAST(bubble(.(z0, .(z1, z2))))) BSORT(.(z0, .(z1, z2))) -> c3(BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, .(z2, z3)))), .(z0, if(<=(z1, z2), .(z2, bubble(.(z1, z3))), .(z1, bubble(.(z2, z3)))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3)))))))), BSORT(butlast(bubble(.(z0, .(z1, .(z2, z3)))))), BUBBLE(.(z0, .(z1, .(z2, z3))))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c2(LAST(.(if(<=(z0, z1), .(z1, if(<=(z0, z2), .(z2, bubble(.(z0, z3))), .(z0, bubble(.(z2, z3))))), .(z0, bubble(.(z1, .(z2, z3))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3)))))))), BSORT(butlast(bubble(.(z0, .(z1, .(z2, z3)))))), BUBBLE(.(z0, .(z1, .(z2, z3))))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil))))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil))))) BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c5(BUBBLE(.(z0, .(y1, y2)))) BSORT(.(z0, nil)) -> c(LAST(.(.(z0, nil), bsort(nil)))) BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c6(BUBBLE(.(z1, .(y1, y2)))) BUBBLE(.(z0, .(z1, .(y1, .(y2, y3))))) -> c6(BUBBLE(.(z1, .(y1, .(y2, y3))))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, nil))), .(z0, .(z1, nil))), bsort(butlast(bubble(.(z0, .(z1, nil))))))), BSORT(butlast(bubble(.(z0, .(z1, nil)))))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, .(z0, nil)), .(z0, bubble(.(z1, nil)))), bsort(butlast(bubble(.(z0, .(z1, nil))))))), BSORT(butlast(bubble(.(z0, .(z1, nil)))))) S tuples: LAST(.(z0, .(z1, z2))) -> c9(LAST(.(z1, z2))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil))))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil))))) K tuples: BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c5(BUBBLE(.(z0, .(y1, y2)))) BUTLAST(.(z0, .(z1, z2))) -> c12(BUTLAST(.(z1, z2))) BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c6(BUBBLE(.(z1, .(y1, y2)))) BUBBLE(.(z0, .(z1, .(y1, .(y2, y3))))) -> c6(BUBBLE(.(z1, .(y1, .(y2, y3))))) Defined Rule Symbols: bubble_1, bsort_1, butlast_1, last_1 Defined Pair Symbols: LAST_1, BUTLAST_1, BSORT_1, BUBBLE_1 Compound Symbols: c9_1, c12_1, c_1, c1_1, c3_1, c2_3, c2_2, c5_1, c6_1 ---------------------------------------- (145) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace BSORT(.(x0, .(z0, nil))) -> c(LAST(.(if(<=(x0, z0), .(z0, bubble(.(x0, nil))), .(x0, .(z0, nil))), bsort(butlast(bubble(.(x0, .(z0, nil)))))))) by BSORT(.(z0, .(z1, nil))) -> c(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, nil))), .(z0, .(z1, nil))), bsort(butlast(if(<=(z0, z1), .(z1, bubble(.(z0, nil))), .(z0, bubble(.(z1, nil))))))))) ---------------------------------------- (146) Obligation: Complexity Dependency Tuples Problem Rules: bubble(.(z0, nil)) -> .(z0, nil) bubble(.(z0, .(z1, z2))) -> if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))) bsort(nil) -> nil bsort(.(z0, z1)) -> last(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1)))))) butlast(nil) -> nil butlast(.(z0, nil)) -> nil butlast(.(z0, .(z1, z2))) -> .(z0, butlast(.(z1, z2))) last(.(z0, nil)) -> z0 last(.(z0, .(z1, z2))) -> last(.(z1, z2)) Tuples: LAST(.(z0, .(z1, z2))) -> c9(LAST(.(z1, z2))) BUTLAST(.(z0, .(z1, z2))) -> c12(BUTLAST(.(z1, z2))) BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BSORT(.(z0, .(z1, z2))) -> c1(BUBBLE(.(z0, .(z1, z2)))) BSORT(.(x0, .(z0, .(z1, z2)))) -> c(LAST(.(if(<=(x0, z0), .(z0, bubble(.(x0, .(z1, z2)))), .(x0, if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))))), bsort(butlast(bubble(.(x0, .(z0, .(z1, z2))))))))) BSORT(.(z0, .(x1, nil))) -> c(LAST(.(if(<=(z0, x1), .(x1, .(z0, nil)), .(z0, bubble(.(x1, nil)))), bsort(butlast(bubble(.(z0, .(x1, nil)))))))) BSORT(.(z0, .(x1, .(z1, z2)))) -> c(LAST(.(if(<=(z0, x1), .(x1, if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2))))), .(z0, bubble(.(x1, .(z1, z2))))), bsort(butlast(bubble(.(z0, .(x1, .(z1, z2))))))))) BSORT(.(z0, .(z1, z2))) -> c3(BUTLAST(bubble(.(z0, .(z1, z2))))) BSORT(.(z0, .(z1, z2))) -> c3(BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, .(z2, z3)))), .(z0, if(<=(z1, z2), .(z2, bubble(.(z1, z3))), .(z1, bubble(.(z2, z3)))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3)))))))), BSORT(butlast(bubble(.(z0, .(z1, .(z2, z3)))))), BUBBLE(.(z0, .(z1, .(z2, z3))))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c2(LAST(.(if(<=(z0, z1), .(z1, if(<=(z0, z2), .(z2, bubble(.(z0, z3))), .(z0, bubble(.(z2, z3))))), .(z0, bubble(.(z1, .(z2, z3))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3)))))))), BSORT(butlast(bubble(.(z0, .(z1, .(z2, z3)))))), BUBBLE(.(z0, .(z1, .(z2, z3))))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil))))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil))))) BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c5(BUBBLE(.(z0, .(y1, y2)))) BSORT(.(z0, nil)) -> c(LAST(.(.(z0, nil), bsort(nil)))) BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c6(BUBBLE(.(z1, .(y1, y2)))) BUBBLE(.(z0, .(z1, .(y1, .(y2, y3))))) -> c6(BUBBLE(.(z1, .(y1, .(y2, y3))))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, nil))), .(z0, .(z1, nil))), bsort(butlast(bubble(.(z0, .(z1, nil))))))), BSORT(butlast(bubble(.(z0, .(z1, nil)))))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, .(z0, nil)), .(z0, bubble(.(z1, nil)))), bsort(butlast(bubble(.(z0, .(z1, nil))))))), BSORT(butlast(bubble(.(z0, .(z1, nil)))))) BSORT(.(z0, .(z1, nil))) -> c(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, nil))), .(z0, .(z1, nil))), bsort(butlast(if(<=(z0, z1), .(z1, bubble(.(z0, nil))), .(z0, bubble(.(z1, nil))))))))) S tuples: LAST(.(z0, .(z1, z2))) -> c9(LAST(.(z1, z2))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil))))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil))))) K tuples: BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c5(BUBBLE(.(z0, .(y1, y2)))) BUTLAST(.(z0, .(z1, z2))) -> c12(BUTLAST(.(z1, z2))) BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c6(BUBBLE(.(z1, .(y1, y2)))) BUBBLE(.(z0, .(z1, .(y1, .(y2, y3))))) -> c6(BUBBLE(.(z1, .(y1, .(y2, y3))))) Defined Rule Symbols: bubble_1, bsort_1, butlast_1, last_1 Defined Pair Symbols: LAST_1, BUTLAST_1, BSORT_1, BUBBLE_1 Compound Symbols: c9_1, c12_1, c_1, c1_1, c3_1, c2_3, c2_2, c5_1, c6_1 ---------------------------------------- (147) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: BSORT(.(z0, .(z1, nil))) -> c(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, nil))), .(z0, .(z1, nil))), bsort(butlast(if(<=(z0, z1), .(z1, bubble(.(z0, nil))), .(z0, bubble(.(z1, nil))))))))) ---------------------------------------- (148) Obligation: Complexity Dependency Tuples Problem Rules: bubble(.(z0, nil)) -> .(z0, nil) bubble(.(z0, .(z1, z2))) -> if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))) bsort(nil) -> nil bsort(.(z0, z1)) -> last(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1)))))) butlast(nil) -> nil butlast(.(z0, nil)) -> nil butlast(.(z0, .(z1, z2))) -> .(z0, butlast(.(z1, z2))) last(.(z0, nil)) -> z0 last(.(z0, .(z1, z2))) -> last(.(z1, z2)) Tuples: LAST(.(z0, .(z1, z2))) -> c9(LAST(.(z1, z2))) BUTLAST(.(z0, .(z1, z2))) -> c12(BUTLAST(.(z1, z2))) BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BSORT(.(z0, .(z1, z2))) -> c1(BUBBLE(.(z0, .(z1, z2)))) BSORT(.(x0, .(z0, .(z1, z2)))) -> c(LAST(.(if(<=(x0, z0), .(z0, bubble(.(x0, .(z1, z2)))), .(x0, if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))))), bsort(butlast(bubble(.(x0, .(z0, .(z1, z2))))))))) BSORT(.(z0, .(x1, nil))) -> c(LAST(.(if(<=(z0, x1), .(x1, .(z0, nil)), .(z0, bubble(.(x1, nil)))), bsort(butlast(bubble(.(z0, .(x1, nil)))))))) BSORT(.(z0, .(x1, .(z1, z2)))) -> c(LAST(.(if(<=(z0, x1), .(x1, if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2))))), .(z0, bubble(.(x1, .(z1, z2))))), bsort(butlast(bubble(.(z0, .(x1, .(z1, z2))))))))) BSORT(.(z0, .(z1, z2))) -> c3(BUTLAST(bubble(.(z0, .(z1, z2))))) BSORT(.(z0, .(z1, z2))) -> c3(BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, .(z2, z3)))), .(z0, if(<=(z1, z2), .(z2, bubble(.(z1, z3))), .(z1, bubble(.(z2, z3)))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3)))))))), BSORT(butlast(bubble(.(z0, .(z1, .(z2, z3)))))), BUBBLE(.(z0, .(z1, .(z2, z3))))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c2(LAST(.(if(<=(z0, z1), .(z1, if(<=(z0, z2), .(z2, bubble(.(z0, z3))), .(z0, bubble(.(z2, z3))))), .(z0, bubble(.(z1, .(z2, z3))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3)))))))), BSORT(butlast(bubble(.(z0, .(z1, .(z2, z3)))))), BUBBLE(.(z0, .(z1, .(z2, z3))))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil))))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil))))) BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c5(BUBBLE(.(z0, .(y1, y2)))) BSORT(.(z0, nil)) -> c(LAST(.(.(z0, nil), bsort(nil)))) BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c6(BUBBLE(.(z1, .(y1, y2)))) BUBBLE(.(z0, .(z1, .(y1, .(y2, y3))))) -> c6(BUBBLE(.(z1, .(y1, .(y2, y3))))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, nil))), .(z0, .(z1, nil))), bsort(butlast(bubble(.(z0, .(z1, nil))))))), BSORT(butlast(bubble(.(z0, .(z1, nil)))))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, .(z0, nil)), .(z0, bubble(.(z1, nil)))), bsort(butlast(bubble(.(z0, .(z1, nil))))))), BSORT(butlast(bubble(.(z0, .(z1, nil)))))) S tuples: LAST(.(z0, .(z1, z2))) -> c9(LAST(.(z1, z2))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil))))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil))))) K tuples: BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c5(BUBBLE(.(z0, .(y1, y2)))) BUTLAST(.(z0, .(z1, z2))) -> c12(BUTLAST(.(z1, z2))) BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c6(BUBBLE(.(z1, .(y1, y2)))) BUBBLE(.(z0, .(z1, .(y1, .(y2, y3))))) -> c6(BUBBLE(.(z1, .(y1, .(y2, y3))))) Defined Rule Symbols: bubble_1, bsort_1, butlast_1, last_1 Defined Pair Symbols: LAST_1, BUTLAST_1, BSORT_1, BUBBLE_1 Compound Symbols: c9_1, c12_1, c_1, c1_1, c3_1, c2_3, c2_2, c5_1, c6_1 ---------------------------------------- (149) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace BSORT(.(x0, .(z0, .(z1, z2)))) -> c(LAST(.(if(<=(x0, z0), .(z0, bubble(.(x0, .(z1, z2)))), .(x0, if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))))), bsort(butlast(bubble(.(x0, .(z0, .(z1, z2))))))))) by BSORT(.(z0, .(z1, .(z2, z3)))) -> c(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, .(z2, z3)))), .(z0, if(<=(z1, z2), .(z2, bubble(.(z1, z3))), .(z1, bubble(.(z2, z3)))))), bsort(butlast(if(<=(z0, z1), .(z1, bubble(.(z0, .(z2, z3)))), .(z0, bubble(.(z1, .(z2, z3)))))))))) ---------------------------------------- (150) Obligation: Complexity Dependency Tuples Problem Rules: bubble(.(z0, nil)) -> .(z0, nil) bubble(.(z0, .(z1, z2))) -> if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))) bsort(nil) -> nil bsort(.(z0, z1)) -> last(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1)))))) butlast(nil) -> nil butlast(.(z0, nil)) -> nil butlast(.(z0, .(z1, z2))) -> .(z0, butlast(.(z1, z2))) last(.(z0, nil)) -> z0 last(.(z0, .(z1, z2))) -> last(.(z1, z2)) Tuples: LAST(.(z0, .(z1, z2))) -> c9(LAST(.(z1, z2))) BUTLAST(.(z0, .(z1, z2))) -> c12(BUTLAST(.(z1, z2))) BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BSORT(.(z0, .(z1, z2))) -> c1(BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, .(x1, nil))) -> c(LAST(.(if(<=(z0, x1), .(x1, .(z0, nil)), .(z0, bubble(.(x1, nil)))), bsort(butlast(bubble(.(z0, .(x1, nil)))))))) BSORT(.(z0, .(x1, .(z1, z2)))) -> c(LAST(.(if(<=(z0, x1), .(x1, if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2))))), .(z0, bubble(.(x1, .(z1, z2))))), bsort(butlast(bubble(.(z0, .(x1, .(z1, z2))))))))) BSORT(.(z0, .(z1, z2))) -> c3(BUTLAST(bubble(.(z0, .(z1, z2))))) BSORT(.(z0, .(z1, z2))) -> c3(BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, .(z2, z3)))), .(z0, if(<=(z1, z2), .(z2, bubble(.(z1, z3))), .(z1, bubble(.(z2, z3)))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3)))))))), BSORT(butlast(bubble(.(z0, .(z1, .(z2, z3)))))), BUBBLE(.(z0, .(z1, .(z2, z3))))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c2(LAST(.(if(<=(z0, z1), .(z1, if(<=(z0, z2), .(z2, bubble(.(z0, z3))), .(z0, bubble(.(z2, z3))))), .(z0, bubble(.(z1, .(z2, z3))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3)))))))), BSORT(butlast(bubble(.(z0, .(z1, .(z2, z3)))))), BUBBLE(.(z0, .(z1, .(z2, z3))))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil))))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil))))) BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c5(BUBBLE(.(z0, .(y1, y2)))) BSORT(.(z0, nil)) -> c(LAST(.(.(z0, nil), bsort(nil)))) BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c6(BUBBLE(.(z1, .(y1, y2)))) BUBBLE(.(z0, .(z1, .(y1, .(y2, y3))))) -> c6(BUBBLE(.(z1, .(y1, .(y2, y3))))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, nil))), .(z0, .(z1, nil))), bsort(butlast(bubble(.(z0, .(z1, nil))))))), BSORT(butlast(bubble(.(z0, .(z1, nil)))))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, .(z0, nil)), .(z0, bubble(.(z1, nil)))), bsort(butlast(bubble(.(z0, .(z1, nil))))))), BSORT(butlast(bubble(.(z0, .(z1, nil)))))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, .(z2, z3)))), .(z0, if(<=(z1, z2), .(z2, bubble(.(z1, z3))), .(z1, bubble(.(z2, z3)))))), bsort(butlast(if(<=(z0, z1), .(z1, bubble(.(z0, .(z2, z3)))), .(z0, bubble(.(z1, .(z2, z3)))))))))) S tuples: LAST(.(z0, .(z1, z2))) -> c9(LAST(.(z1, z2))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil))))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil))))) K tuples: BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c5(BUBBLE(.(z0, .(y1, y2)))) BUTLAST(.(z0, .(z1, z2))) -> c12(BUTLAST(.(z1, z2))) BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c6(BUBBLE(.(z1, .(y1, y2)))) BUBBLE(.(z0, .(z1, .(y1, .(y2, y3))))) -> c6(BUBBLE(.(z1, .(y1, .(y2, y3))))) Defined Rule Symbols: bubble_1, bsort_1, butlast_1, last_1 Defined Pair Symbols: LAST_1, BUTLAST_1, BSORT_1, BUBBLE_1 Compound Symbols: c9_1, c12_1, c_1, c1_1, c3_1, c2_3, c2_2, c5_1, c6_1 ---------------------------------------- (151) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: BSORT(.(z0, .(z1, .(z2, z3)))) -> c(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, .(z2, z3)))), .(z0, if(<=(z1, z2), .(z2, bubble(.(z1, z3))), .(z1, bubble(.(z2, z3)))))), bsort(butlast(if(<=(z0, z1), .(z1, bubble(.(z0, .(z2, z3)))), .(z0, bubble(.(z1, .(z2, z3)))))))))) ---------------------------------------- (152) Obligation: Complexity Dependency Tuples Problem Rules: bubble(.(z0, nil)) -> .(z0, nil) bubble(.(z0, .(z1, z2))) -> if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))) bsort(nil) -> nil bsort(.(z0, z1)) -> last(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1)))))) butlast(nil) -> nil butlast(.(z0, nil)) -> nil butlast(.(z0, .(z1, z2))) -> .(z0, butlast(.(z1, z2))) last(.(z0, nil)) -> z0 last(.(z0, .(z1, z2))) -> last(.(z1, z2)) Tuples: LAST(.(z0, .(z1, z2))) -> c9(LAST(.(z1, z2))) BUTLAST(.(z0, .(z1, z2))) -> c12(BUTLAST(.(z1, z2))) BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BSORT(.(z0, .(z1, z2))) -> c1(BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, .(x1, nil))) -> c(LAST(.(if(<=(z0, x1), .(x1, .(z0, nil)), .(z0, bubble(.(x1, nil)))), bsort(butlast(bubble(.(z0, .(x1, nil)))))))) BSORT(.(z0, .(x1, .(z1, z2)))) -> c(LAST(.(if(<=(z0, x1), .(x1, if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2))))), .(z0, bubble(.(x1, .(z1, z2))))), bsort(butlast(bubble(.(z0, .(x1, .(z1, z2))))))))) BSORT(.(z0, .(z1, z2))) -> c3(BUTLAST(bubble(.(z0, .(z1, z2))))) BSORT(.(z0, .(z1, z2))) -> c3(BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, .(z2, z3)))), .(z0, if(<=(z1, z2), .(z2, bubble(.(z1, z3))), .(z1, bubble(.(z2, z3)))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3)))))))), BSORT(butlast(bubble(.(z0, .(z1, .(z2, z3)))))), BUBBLE(.(z0, .(z1, .(z2, z3))))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c2(LAST(.(if(<=(z0, z1), .(z1, if(<=(z0, z2), .(z2, bubble(.(z0, z3))), .(z0, bubble(.(z2, z3))))), .(z0, bubble(.(z1, .(z2, z3))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3)))))))), BSORT(butlast(bubble(.(z0, .(z1, .(z2, z3)))))), BUBBLE(.(z0, .(z1, .(z2, z3))))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil))))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil))))) BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c5(BUBBLE(.(z0, .(y1, y2)))) BSORT(.(z0, nil)) -> c(LAST(.(.(z0, nil), bsort(nil)))) BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c6(BUBBLE(.(z1, .(y1, y2)))) BUBBLE(.(z0, .(z1, .(y1, .(y2, y3))))) -> c6(BUBBLE(.(z1, .(y1, .(y2, y3))))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, nil))), .(z0, .(z1, nil))), bsort(butlast(bubble(.(z0, .(z1, nil))))))), BSORT(butlast(bubble(.(z0, .(z1, nil)))))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, .(z0, nil)), .(z0, bubble(.(z1, nil)))), bsort(butlast(bubble(.(z0, .(z1, nil))))))), BSORT(butlast(bubble(.(z0, .(z1, nil)))))) S tuples: LAST(.(z0, .(z1, z2))) -> c9(LAST(.(z1, z2))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil))))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil))))) K tuples: BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c5(BUBBLE(.(z0, .(y1, y2)))) BUTLAST(.(z0, .(z1, z2))) -> c12(BUTLAST(.(z1, z2))) BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c6(BUBBLE(.(z1, .(y1, y2)))) BUBBLE(.(z0, .(z1, .(y1, .(y2, y3))))) -> c6(BUBBLE(.(z1, .(y1, .(y2, y3))))) Defined Rule Symbols: bubble_1, bsort_1, butlast_1, last_1 Defined Pair Symbols: LAST_1, BUTLAST_1, BSORT_1, BUBBLE_1 Compound Symbols: c9_1, c12_1, c_1, c1_1, c3_1, c2_3, c2_2, c5_1, c6_1 ---------------------------------------- (153) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace LAST(.(z0, .(z1, z2))) -> c9(LAST(.(z1, z2))) by LAST(.(z0, .(z1, .(y1, y2)))) -> c9(LAST(.(z1, .(y1, y2)))) ---------------------------------------- (154) Obligation: Complexity Dependency Tuples Problem Rules: bubble(.(z0, nil)) -> .(z0, nil) bubble(.(z0, .(z1, z2))) -> if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))) bsort(nil) -> nil bsort(.(z0, z1)) -> last(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1)))))) butlast(nil) -> nil butlast(.(z0, nil)) -> nil butlast(.(z0, .(z1, z2))) -> .(z0, butlast(.(z1, z2))) last(.(z0, nil)) -> z0 last(.(z0, .(z1, z2))) -> last(.(z1, z2)) Tuples: BUTLAST(.(z0, .(z1, z2))) -> c12(BUTLAST(.(z1, z2))) BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BSORT(.(z0, .(z1, z2))) -> c1(BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, .(x1, nil))) -> c(LAST(.(if(<=(z0, x1), .(x1, .(z0, nil)), .(z0, bubble(.(x1, nil)))), bsort(butlast(bubble(.(z0, .(x1, nil)))))))) BSORT(.(z0, .(x1, .(z1, z2)))) -> c(LAST(.(if(<=(z0, x1), .(x1, if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2))))), .(z0, bubble(.(x1, .(z1, z2))))), bsort(butlast(bubble(.(z0, .(x1, .(z1, z2))))))))) BSORT(.(z0, .(z1, z2))) -> c3(BUTLAST(bubble(.(z0, .(z1, z2))))) BSORT(.(z0, .(z1, z2))) -> c3(BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, .(z2, z3)))), .(z0, if(<=(z1, z2), .(z2, bubble(.(z1, z3))), .(z1, bubble(.(z2, z3)))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3)))))))), BSORT(butlast(bubble(.(z0, .(z1, .(z2, z3)))))), BUBBLE(.(z0, .(z1, .(z2, z3))))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c2(LAST(.(if(<=(z0, z1), .(z1, if(<=(z0, z2), .(z2, bubble(.(z0, z3))), .(z0, bubble(.(z2, z3))))), .(z0, bubble(.(z1, .(z2, z3))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3)))))))), BSORT(butlast(bubble(.(z0, .(z1, .(z2, z3)))))), BUBBLE(.(z0, .(z1, .(z2, z3))))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil))))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil))))) BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c5(BUBBLE(.(z0, .(y1, y2)))) BSORT(.(z0, nil)) -> c(LAST(.(.(z0, nil), bsort(nil)))) BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c6(BUBBLE(.(z1, .(y1, y2)))) BUBBLE(.(z0, .(z1, .(y1, .(y2, y3))))) -> c6(BUBBLE(.(z1, .(y1, .(y2, y3))))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, nil))), .(z0, .(z1, nil))), bsort(butlast(bubble(.(z0, .(z1, nil))))))), BSORT(butlast(bubble(.(z0, .(z1, nil)))))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, .(z0, nil)), .(z0, bubble(.(z1, nil)))), bsort(butlast(bubble(.(z0, .(z1, nil))))))), BSORT(butlast(bubble(.(z0, .(z1, nil)))))) LAST(.(z0, .(z1, .(y1, y2)))) -> c9(LAST(.(z1, .(y1, y2)))) S tuples: BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil))))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil))))) LAST(.(z0, .(z1, .(y1, y2)))) -> c9(LAST(.(z1, .(y1, y2)))) K tuples: BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c5(BUBBLE(.(z0, .(y1, y2)))) BUTLAST(.(z0, .(z1, z2))) -> c12(BUTLAST(.(z1, z2))) BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c6(BUBBLE(.(z1, .(y1, y2)))) BUBBLE(.(z0, .(z1, .(y1, .(y2, y3))))) -> c6(BUBBLE(.(z1, .(y1, .(y2, y3))))) Defined Rule Symbols: bubble_1, bsort_1, butlast_1, last_1 Defined Pair Symbols: BUTLAST_1, BSORT_1, BUBBLE_1, LAST_1 Compound Symbols: c12_1, c_1, c1_1, c3_1, c2_3, c2_2, c5_1, c6_1, c9_1 ---------------------------------------- (155) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace BUTLAST(.(z0, .(z1, z2))) -> c12(BUTLAST(.(z1, z2))) by BUTLAST(.(z0, .(z1, .(y1, y2)))) -> c12(BUTLAST(.(z1, .(y1, y2)))) ---------------------------------------- (156) Obligation: Complexity Dependency Tuples Problem Rules: bubble(.(z0, nil)) -> .(z0, nil) bubble(.(z0, .(z1, z2))) -> if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))) bsort(nil) -> nil bsort(.(z0, z1)) -> last(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1)))))) butlast(nil) -> nil butlast(.(z0, nil)) -> nil butlast(.(z0, .(z1, z2))) -> .(z0, butlast(.(z1, z2))) last(.(z0, nil)) -> z0 last(.(z0, .(z1, z2))) -> last(.(z1, z2)) Tuples: BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BSORT(.(z0, .(z1, z2))) -> c1(BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, .(x1, nil))) -> c(LAST(.(if(<=(z0, x1), .(x1, .(z0, nil)), .(z0, bubble(.(x1, nil)))), bsort(butlast(bubble(.(z0, .(x1, nil)))))))) BSORT(.(z0, .(x1, .(z1, z2)))) -> c(LAST(.(if(<=(z0, x1), .(x1, if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2))))), .(z0, bubble(.(x1, .(z1, z2))))), bsort(butlast(bubble(.(z0, .(x1, .(z1, z2))))))))) BSORT(.(z0, .(z1, z2))) -> c3(BUTLAST(bubble(.(z0, .(z1, z2))))) BSORT(.(z0, .(z1, z2))) -> c3(BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, .(z2, z3)))), .(z0, if(<=(z1, z2), .(z2, bubble(.(z1, z3))), .(z1, bubble(.(z2, z3)))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3)))))))), BSORT(butlast(bubble(.(z0, .(z1, .(z2, z3)))))), BUBBLE(.(z0, .(z1, .(z2, z3))))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c2(LAST(.(if(<=(z0, z1), .(z1, if(<=(z0, z2), .(z2, bubble(.(z0, z3))), .(z0, bubble(.(z2, z3))))), .(z0, bubble(.(z1, .(z2, z3))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3)))))))), BSORT(butlast(bubble(.(z0, .(z1, .(z2, z3)))))), BUBBLE(.(z0, .(z1, .(z2, z3))))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil))))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil))))) BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c5(BUBBLE(.(z0, .(y1, y2)))) BSORT(.(z0, nil)) -> c(LAST(.(.(z0, nil), bsort(nil)))) BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c6(BUBBLE(.(z1, .(y1, y2)))) BUBBLE(.(z0, .(z1, .(y1, .(y2, y3))))) -> c6(BUBBLE(.(z1, .(y1, .(y2, y3))))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, nil))), .(z0, .(z1, nil))), bsort(butlast(bubble(.(z0, .(z1, nil))))))), BSORT(butlast(bubble(.(z0, .(z1, nil)))))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, .(z0, nil)), .(z0, bubble(.(z1, nil)))), bsort(butlast(bubble(.(z0, .(z1, nil))))))), BSORT(butlast(bubble(.(z0, .(z1, nil)))))) LAST(.(z0, .(z1, .(y1, y2)))) -> c9(LAST(.(z1, .(y1, y2)))) BUTLAST(.(z0, .(z1, .(y1, y2)))) -> c12(BUTLAST(.(z1, .(y1, y2)))) S tuples: BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil))))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil))))) LAST(.(z0, .(z1, .(y1, y2)))) -> c9(LAST(.(z1, .(y1, y2)))) K tuples: BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c5(BUBBLE(.(z0, .(y1, y2)))) BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c6(BUBBLE(.(z1, .(y1, y2)))) BUBBLE(.(z0, .(z1, .(y1, .(y2, y3))))) -> c6(BUBBLE(.(z1, .(y1, .(y2, y3))))) BUTLAST(.(z0, .(z1, .(y1, y2)))) -> c12(BUTLAST(.(z1, .(y1, y2)))) Defined Rule Symbols: bubble_1, bsort_1, butlast_1, last_1 Defined Pair Symbols: BSORT_1, BUBBLE_1, LAST_1, BUTLAST_1 Compound Symbols: c_1, c1_1, c3_1, c2_3, c2_2, c5_1, c6_1, c9_1, c12_1 ---------------------------------------- (157) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace BSORT(.(z0, .(x1, nil))) -> c(LAST(.(if(<=(z0, x1), .(x1, .(z0, nil)), .(z0, bubble(.(x1, nil)))), bsort(butlast(bubble(.(z0, .(x1, nil)))))))) by BSORT(.(z0, .(z1, nil))) -> c(LAST(.(if(<=(z0, z1), .(z1, .(z0, nil)), .(z0, bubble(.(z1, nil)))), bsort(butlast(if(<=(z0, z1), .(z1, bubble(.(z0, nil))), .(z0, bubble(.(z1, nil))))))))) ---------------------------------------- (158) Obligation: Complexity Dependency Tuples Problem Rules: bubble(.(z0, nil)) -> .(z0, nil) bubble(.(z0, .(z1, z2))) -> if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))) bsort(nil) -> nil bsort(.(z0, z1)) -> last(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1)))))) butlast(nil) -> nil butlast(.(z0, nil)) -> nil butlast(.(z0, .(z1, z2))) -> .(z0, butlast(.(z1, z2))) last(.(z0, nil)) -> z0 last(.(z0, .(z1, z2))) -> last(.(z1, z2)) Tuples: BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BSORT(.(z0, .(z1, z2))) -> c1(BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, .(x1, .(z1, z2)))) -> c(LAST(.(if(<=(z0, x1), .(x1, if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2))))), .(z0, bubble(.(x1, .(z1, z2))))), bsort(butlast(bubble(.(z0, .(x1, .(z1, z2))))))))) BSORT(.(z0, .(z1, z2))) -> c3(BUTLAST(bubble(.(z0, .(z1, z2))))) BSORT(.(z0, .(z1, z2))) -> c3(BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, .(z2, z3)))), .(z0, if(<=(z1, z2), .(z2, bubble(.(z1, z3))), .(z1, bubble(.(z2, z3)))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3)))))))), BSORT(butlast(bubble(.(z0, .(z1, .(z2, z3)))))), BUBBLE(.(z0, .(z1, .(z2, z3))))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c2(LAST(.(if(<=(z0, z1), .(z1, if(<=(z0, z2), .(z2, bubble(.(z0, z3))), .(z0, bubble(.(z2, z3))))), .(z0, bubble(.(z1, .(z2, z3))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3)))))))), BSORT(butlast(bubble(.(z0, .(z1, .(z2, z3)))))), BUBBLE(.(z0, .(z1, .(z2, z3))))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil))))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil))))) BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c5(BUBBLE(.(z0, .(y1, y2)))) BSORT(.(z0, nil)) -> c(LAST(.(.(z0, nil), bsort(nil)))) BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c6(BUBBLE(.(z1, .(y1, y2)))) BUBBLE(.(z0, .(z1, .(y1, .(y2, y3))))) -> c6(BUBBLE(.(z1, .(y1, .(y2, y3))))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, nil))), .(z0, .(z1, nil))), bsort(butlast(bubble(.(z0, .(z1, nil))))))), BSORT(butlast(bubble(.(z0, .(z1, nil)))))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, .(z0, nil)), .(z0, bubble(.(z1, nil)))), bsort(butlast(bubble(.(z0, .(z1, nil))))))), BSORT(butlast(bubble(.(z0, .(z1, nil)))))) LAST(.(z0, .(z1, .(y1, y2)))) -> c9(LAST(.(z1, .(y1, y2)))) BUTLAST(.(z0, .(z1, .(y1, y2)))) -> c12(BUTLAST(.(z1, .(y1, y2)))) BSORT(.(z0, .(z1, nil))) -> c(LAST(.(if(<=(z0, z1), .(z1, .(z0, nil)), .(z0, bubble(.(z1, nil)))), bsort(butlast(if(<=(z0, z1), .(z1, bubble(.(z0, nil))), .(z0, bubble(.(z1, nil))))))))) S tuples: BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil))))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil))))) LAST(.(z0, .(z1, .(y1, y2)))) -> c9(LAST(.(z1, .(y1, y2)))) K tuples: BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c5(BUBBLE(.(z0, .(y1, y2)))) BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c6(BUBBLE(.(z1, .(y1, y2)))) BUBBLE(.(z0, .(z1, .(y1, .(y2, y3))))) -> c6(BUBBLE(.(z1, .(y1, .(y2, y3))))) BUTLAST(.(z0, .(z1, .(y1, y2)))) -> c12(BUTLAST(.(z1, .(y1, y2)))) Defined Rule Symbols: bubble_1, bsort_1, butlast_1, last_1 Defined Pair Symbols: BSORT_1, BUBBLE_1, LAST_1, BUTLAST_1 Compound Symbols: c_1, c1_1, c3_1, c2_3, c2_2, c5_1, c6_1, c9_1, c12_1 ---------------------------------------- (159) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: BSORT(.(z0, .(z1, nil))) -> c(LAST(.(if(<=(z0, z1), .(z1, .(z0, nil)), .(z0, bubble(.(z1, nil)))), bsort(butlast(if(<=(z0, z1), .(z1, bubble(.(z0, nil))), .(z0, bubble(.(z1, nil))))))))) ---------------------------------------- (160) Obligation: Complexity Dependency Tuples Problem Rules: bubble(.(z0, nil)) -> .(z0, nil) bubble(.(z0, .(z1, z2))) -> if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))) bsort(nil) -> nil bsort(.(z0, z1)) -> last(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1)))))) butlast(nil) -> nil butlast(.(z0, nil)) -> nil butlast(.(z0, .(z1, z2))) -> .(z0, butlast(.(z1, z2))) last(.(z0, nil)) -> z0 last(.(z0, .(z1, z2))) -> last(.(z1, z2)) Tuples: BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BSORT(.(z0, .(z1, z2))) -> c1(BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, .(x1, .(z1, z2)))) -> c(LAST(.(if(<=(z0, x1), .(x1, if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2))))), .(z0, bubble(.(x1, .(z1, z2))))), bsort(butlast(bubble(.(z0, .(x1, .(z1, z2))))))))) BSORT(.(z0, .(z1, z2))) -> c3(BUTLAST(bubble(.(z0, .(z1, z2))))) BSORT(.(z0, .(z1, z2))) -> c3(BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, .(z2, z3)))), .(z0, if(<=(z1, z2), .(z2, bubble(.(z1, z3))), .(z1, bubble(.(z2, z3)))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3)))))))), BSORT(butlast(bubble(.(z0, .(z1, .(z2, z3)))))), BUBBLE(.(z0, .(z1, .(z2, z3))))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c2(LAST(.(if(<=(z0, z1), .(z1, if(<=(z0, z2), .(z2, bubble(.(z0, z3))), .(z0, bubble(.(z2, z3))))), .(z0, bubble(.(z1, .(z2, z3))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3)))))))), BSORT(butlast(bubble(.(z0, .(z1, .(z2, z3)))))), BUBBLE(.(z0, .(z1, .(z2, z3))))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil))))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil))))) BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c5(BUBBLE(.(z0, .(y1, y2)))) BSORT(.(z0, nil)) -> c(LAST(.(.(z0, nil), bsort(nil)))) BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c6(BUBBLE(.(z1, .(y1, y2)))) BUBBLE(.(z0, .(z1, .(y1, .(y2, y3))))) -> c6(BUBBLE(.(z1, .(y1, .(y2, y3))))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, nil))), .(z0, .(z1, nil))), bsort(butlast(bubble(.(z0, .(z1, nil))))))), BSORT(butlast(bubble(.(z0, .(z1, nil)))))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, .(z0, nil)), .(z0, bubble(.(z1, nil)))), bsort(butlast(bubble(.(z0, .(z1, nil))))))), BSORT(butlast(bubble(.(z0, .(z1, nil)))))) LAST(.(z0, .(z1, .(y1, y2)))) -> c9(LAST(.(z1, .(y1, y2)))) BUTLAST(.(z0, .(z1, .(y1, y2)))) -> c12(BUTLAST(.(z1, .(y1, y2)))) S tuples: BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil))))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil))))) LAST(.(z0, .(z1, .(y1, y2)))) -> c9(LAST(.(z1, .(y1, y2)))) K tuples: BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c5(BUBBLE(.(z0, .(y1, y2)))) BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c6(BUBBLE(.(z1, .(y1, y2)))) BUBBLE(.(z0, .(z1, .(y1, .(y2, y3))))) -> c6(BUBBLE(.(z1, .(y1, .(y2, y3))))) BUTLAST(.(z0, .(z1, .(y1, y2)))) -> c12(BUTLAST(.(z1, .(y1, y2)))) Defined Rule Symbols: bubble_1, bsort_1, butlast_1, last_1 Defined Pair Symbols: BSORT_1, BUBBLE_1, LAST_1, BUTLAST_1 Compound Symbols: c_1, c1_1, c3_1, c2_3, c2_2, c5_1, c6_1, c9_1, c12_1 ---------------------------------------- (161) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace BSORT(.(z0, .(x1, .(z1, z2)))) -> c(LAST(.(if(<=(z0, x1), .(x1, if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2))))), .(z0, bubble(.(x1, .(z1, z2))))), bsort(butlast(bubble(.(z0, .(x1, .(z1, z2))))))))) by BSORT(.(z0, .(z1, .(z2, z3)))) -> c(LAST(.(if(<=(z0, z1), .(z1, if(<=(z0, z2), .(z2, bubble(.(z0, z3))), .(z0, bubble(.(z2, z3))))), .(z0, bubble(.(z1, .(z2, z3))))), bsort(butlast(if(<=(z0, z1), .(z1, bubble(.(z0, .(z2, z3)))), .(z0, bubble(.(z1, .(z2, z3)))))))))) ---------------------------------------- (162) Obligation: Complexity Dependency Tuples Problem Rules: bubble(.(z0, nil)) -> .(z0, nil) bubble(.(z0, .(z1, z2))) -> if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))) bsort(nil) -> nil bsort(.(z0, z1)) -> last(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1)))))) butlast(nil) -> nil butlast(.(z0, nil)) -> nil butlast(.(z0, .(z1, z2))) -> .(z0, butlast(.(z1, z2))) last(.(z0, nil)) -> z0 last(.(z0, .(z1, z2))) -> last(.(z1, z2)) Tuples: BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BSORT(.(z0, .(z1, z2))) -> c1(BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, .(z1, z2))) -> c3(BUTLAST(bubble(.(z0, .(z1, z2))))) BSORT(.(z0, .(z1, z2))) -> c3(BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, .(z2, z3)))), .(z0, if(<=(z1, z2), .(z2, bubble(.(z1, z3))), .(z1, bubble(.(z2, z3)))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3)))))))), BSORT(butlast(bubble(.(z0, .(z1, .(z2, z3)))))), BUBBLE(.(z0, .(z1, .(z2, z3))))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c2(LAST(.(if(<=(z0, z1), .(z1, if(<=(z0, z2), .(z2, bubble(.(z0, z3))), .(z0, bubble(.(z2, z3))))), .(z0, bubble(.(z1, .(z2, z3))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3)))))))), BSORT(butlast(bubble(.(z0, .(z1, .(z2, z3)))))), BUBBLE(.(z0, .(z1, .(z2, z3))))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil))))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil))))) BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c5(BUBBLE(.(z0, .(y1, y2)))) BSORT(.(z0, nil)) -> c(LAST(.(.(z0, nil), bsort(nil)))) BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c6(BUBBLE(.(z1, .(y1, y2)))) BUBBLE(.(z0, .(z1, .(y1, .(y2, y3))))) -> c6(BUBBLE(.(z1, .(y1, .(y2, y3))))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, nil))), .(z0, .(z1, nil))), bsort(butlast(bubble(.(z0, .(z1, nil))))))), BSORT(butlast(bubble(.(z0, .(z1, nil)))))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, .(z0, nil)), .(z0, bubble(.(z1, nil)))), bsort(butlast(bubble(.(z0, .(z1, nil))))))), BSORT(butlast(bubble(.(z0, .(z1, nil)))))) LAST(.(z0, .(z1, .(y1, y2)))) -> c9(LAST(.(z1, .(y1, y2)))) BUTLAST(.(z0, .(z1, .(y1, y2)))) -> c12(BUTLAST(.(z1, .(y1, y2)))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c(LAST(.(if(<=(z0, z1), .(z1, if(<=(z0, z2), .(z2, bubble(.(z0, z3))), .(z0, bubble(.(z2, z3))))), .(z0, bubble(.(z1, .(z2, z3))))), bsort(butlast(if(<=(z0, z1), .(z1, bubble(.(z0, .(z2, z3)))), .(z0, bubble(.(z1, .(z2, z3)))))))))) S tuples: BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil))))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil))))) LAST(.(z0, .(z1, .(y1, y2)))) -> c9(LAST(.(z1, .(y1, y2)))) K tuples: BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c5(BUBBLE(.(z0, .(y1, y2)))) BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c6(BUBBLE(.(z1, .(y1, y2)))) BUBBLE(.(z0, .(z1, .(y1, .(y2, y3))))) -> c6(BUBBLE(.(z1, .(y1, .(y2, y3))))) BUTLAST(.(z0, .(z1, .(y1, y2)))) -> c12(BUTLAST(.(z1, .(y1, y2)))) Defined Rule Symbols: bubble_1, bsort_1, butlast_1, last_1 Defined Pair Symbols: BSORT_1, BUBBLE_1, LAST_1, BUTLAST_1 Compound Symbols: c_1, c1_1, c3_1, c2_3, c2_2, c5_1, c6_1, c9_1, c12_1 ---------------------------------------- (163) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: BSORT(.(z0, .(z1, .(z2, z3)))) -> c(LAST(.(if(<=(z0, z1), .(z1, if(<=(z0, z2), .(z2, bubble(.(z0, z3))), .(z0, bubble(.(z2, z3))))), .(z0, bubble(.(z1, .(z2, z3))))), bsort(butlast(if(<=(z0, z1), .(z1, bubble(.(z0, .(z2, z3)))), .(z0, bubble(.(z1, .(z2, z3)))))))))) ---------------------------------------- (164) Obligation: Complexity Dependency Tuples Problem Rules: bubble(.(z0, nil)) -> .(z0, nil) bubble(.(z0, .(z1, z2))) -> if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))) bsort(nil) -> nil bsort(.(z0, z1)) -> last(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1)))))) butlast(nil) -> nil butlast(.(z0, nil)) -> nil butlast(.(z0, .(z1, z2))) -> .(z0, butlast(.(z1, z2))) last(.(z0, nil)) -> z0 last(.(z0, .(z1, z2))) -> last(.(z1, z2)) Tuples: BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BSORT(.(z0, .(z1, z2))) -> c1(BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, .(z1, z2))) -> c3(BUTLAST(bubble(.(z0, .(z1, z2))))) BSORT(.(z0, .(z1, z2))) -> c3(BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, .(z2, z3)))), .(z0, if(<=(z1, z2), .(z2, bubble(.(z1, z3))), .(z1, bubble(.(z2, z3)))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3)))))))), BSORT(butlast(bubble(.(z0, .(z1, .(z2, z3)))))), BUBBLE(.(z0, .(z1, .(z2, z3))))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c2(LAST(.(if(<=(z0, z1), .(z1, if(<=(z0, z2), .(z2, bubble(.(z0, z3))), .(z0, bubble(.(z2, z3))))), .(z0, bubble(.(z1, .(z2, z3))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3)))))))), BSORT(butlast(bubble(.(z0, .(z1, .(z2, z3)))))), BUBBLE(.(z0, .(z1, .(z2, z3))))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil))))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil))))) BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c5(BUBBLE(.(z0, .(y1, y2)))) BSORT(.(z0, nil)) -> c(LAST(.(.(z0, nil), bsort(nil)))) BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c6(BUBBLE(.(z1, .(y1, y2)))) BUBBLE(.(z0, .(z1, .(y1, .(y2, y3))))) -> c6(BUBBLE(.(z1, .(y1, .(y2, y3))))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, nil))), .(z0, .(z1, nil))), bsort(butlast(bubble(.(z0, .(z1, nil))))))), BSORT(butlast(bubble(.(z0, .(z1, nil)))))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, .(z0, nil)), .(z0, bubble(.(z1, nil)))), bsort(butlast(bubble(.(z0, .(z1, nil))))))), BSORT(butlast(bubble(.(z0, .(z1, nil)))))) LAST(.(z0, .(z1, .(y1, y2)))) -> c9(LAST(.(z1, .(y1, y2)))) BUTLAST(.(z0, .(z1, .(y1, y2)))) -> c12(BUTLAST(.(z1, .(y1, y2)))) S tuples: BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil))))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil))))) LAST(.(z0, .(z1, .(y1, y2)))) -> c9(LAST(.(z1, .(y1, y2)))) K tuples: BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c5(BUBBLE(.(z0, .(y1, y2)))) BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c6(BUBBLE(.(z1, .(y1, y2)))) BUBBLE(.(z0, .(z1, .(y1, .(y2, y3))))) -> c6(BUBBLE(.(z1, .(y1, .(y2, y3))))) BUTLAST(.(z0, .(z1, .(y1, y2)))) -> c12(BUTLAST(.(z1, .(y1, y2)))) Defined Rule Symbols: bubble_1, bsort_1, butlast_1, last_1 Defined Pair Symbols: BSORT_1, BUBBLE_1, LAST_1, BUTLAST_1 Compound Symbols: c_1, c1_1, c3_1, c2_3, c2_2, c5_1, c6_1, c9_1, c12_1 ---------------------------------------- (165) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace BSORT(.(z0, .(z1, z2))) -> c3(BUTLAST(bubble(.(z0, .(z1, z2))))) by BSORT(.(z0, .(z1, z2))) -> c3(BUTLAST(if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))))) ---------------------------------------- (166) Obligation: Complexity Dependency Tuples Problem Rules: bubble(.(z0, nil)) -> .(z0, nil) bubble(.(z0, .(z1, z2))) -> if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))) bsort(nil) -> nil bsort(.(z0, z1)) -> last(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1)))))) butlast(nil) -> nil butlast(.(z0, nil)) -> nil butlast(.(z0, .(z1, z2))) -> .(z0, butlast(.(z1, z2))) last(.(z0, nil)) -> z0 last(.(z0, .(z1, z2))) -> last(.(z1, z2)) Tuples: BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BSORT(.(z0, .(z1, z2))) -> c1(BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, .(z1, z2))) -> c3(BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, .(z2, z3)))), .(z0, if(<=(z1, z2), .(z2, bubble(.(z1, z3))), .(z1, bubble(.(z2, z3)))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3)))))))), BSORT(butlast(bubble(.(z0, .(z1, .(z2, z3)))))), BUBBLE(.(z0, .(z1, .(z2, z3))))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c2(LAST(.(if(<=(z0, z1), .(z1, if(<=(z0, z2), .(z2, bubble(.(z0, z3))), .(z0, bubble(.(z2, z3))))), .(z0, bubble(.(z1, .(z2, z3))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3)))))))), BSORT(butlast(bubble(.(z0, .(z1, .(z2, z3)))))), BUBBLE(.(z0, .(z1, .(z2, z3))))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil))))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil))))) BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c5(BUBBLE(.(z0, .(y1, y2)))) BSORT(.(z0, nil)) -> c(LAST(.(.(z0, nil), bsort(nil)))) BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c6(BUBBLE(.(z1, .(y1, y2)))) BUBBLE(.(z0, .(z1, .(y1, .(y2, y3))))) -> c6(BUBBLE(.(z1, .(y1, .(y2, y3))))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, nil))), .(z0, .(z1, nil))), bsort(butlast(bubble(.(z0, .(z1, nil))))))), BSORT(butlast(bubble(.(z0, .(z1, nil)))))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, .(z0, nil)), .(z0, bubble(.(z1, nil)))), bsort(butlast(bubble(.(z0, .(z1, nil))))))), BSORT(butlast(bubble(.(z0, .(z1, nil)))))) LAST(.(z0, .(z1, .(y1, y2)))) -> c9(LAST(.(z1, .(y1, y2)))) BUTLAST(.(z0, .(z1, .(y1, y2)))) -> c12(BUTLAST(.(z1, .(y1, y2)))) BSORT(.(z0, .(z1, z2))) -> c3(BUTLAST(if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))))) S tuples: BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil))))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil))))) LAST(.(z0, .(z1, .(y1, y2)))) -> c9(LAST(.(z1, .(y1, y2)))) K tuples: BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c5(BUBBLE(.(z0, .(y1, y2)))) BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c6(BUBBLE(.(z1, .(y1, y2)))) BUBBLE(.(z0, .(z1, .(y1, .(y2, y3))))) -> c6(BUBBLE(.(z1, .(y1, .(y2, y3))))) BUTLAST(.(z0, .(z1, .(y1, y2)))) -> c12(BUTLAST(.(z1, .(y1, y2)))) Defined Rule Symbols: bubble_1, bsort_1, butlast_1, last_1 Defined Pair Symbols: BSORT_1, BUBBLE_1, LAST_1, BUTLAST_1 Compound Symbols: c_1, c1_1, c3_1, c2_3, c2_2, c5_1, c6_1, c9_1, c12_1 ---------------------------------------- (167) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: BSORT(.(z0, .(z1, z2))) -> c3(BUTLAST(if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))))) ---------------------------------------- (168) Obligation: Complexity Dependency Tuples Problem Rules: bubble(.(z0, nil)) -> .(z0, nil) bubble(.(z0, .(z1, z2))) -> if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))) bsort(nil) -> nil bsort(.(z0, z1)) -> last(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1)))))) butlast(nil) -> nil butlast(.(z0, nil)) -> nil butlast(.(z0, .(z1, z2))) -> .(z0, butlast(.(z1, z2))) last(.(z0, nil)) -> z0 last(.(z0, .(z1, z2))) -> last(.(z1, z2)) Tuples: BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BSORT(.(z0, .(z1, z2))) -> c1(BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, .(z1, z2))) -> c3(BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, .(z2, z3)))), .(z0, if(<=(z1, z2), .(z2, bubble(.(z1, z3))), .(z1, bubble(.(z2, z3)))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3)))))))), BSORT(butlast(bubble(.(z0, .(z1, .(z2, z3)))))), BUBBLE(.(z0, .(z1, .(z2, z3))))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c2(LAST(.(if(<=(z0, z1), .(z1, if(<=(z0, z2), .(z2, bubble(.(z0, z3))), .(z0, bubble(.(z2, z3))))), .(z0, bubble(.(z1, .(z2, z3))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3)))))))), BSORT(butlast(bubble(.(z0, .(z1, .(z2, z3)))))), BUBBLE(.(z0, .(z1, .(z2, z3))))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil))))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil))))) BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c5(BUBBLE(.(z0, .(y1, y2)))) BSORT(.(z0, nil)) -> c(LAST(.(.(z0, nil), bsort(nil)))) BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c6(BUBBLE(.(z1, .(y1, y2)))) BUBBLE(.(z0, .(z1, .(y1, .(y2, y3))))) -> c6(BUBBLE(.(z1, .(y1, .(y2, y3))))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, nil))), .(z0, .(z1, nil))), bsort(butlast(bubble(.(z0, .(z1, nil))))))), BSORT(butlast(bubble(.(z0, .(z1, nil)))))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, .(z0, nil)), .(z0, bubble(.(z1, nil)))), bsort(butlast(bubble(.(z0, .(z1, nil))))))), BSORT(butlast(bubble(.(z0, .(z1, nil)))))) LAST(.(z0, .(z1, .(y1, y2)))) -> c9(LAST(.(z1, .(y1, y2)))) BUTLAST(.(z0, .(z1, .(y1, y2)))) -> c12(BUTLAST(.(z1, .(y1, y2)))) S tuples: BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil))))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil))))) LAST(.(z0, .(z1, .(y1, y2)))) -> c9(LAST(.(z1, .(y1, y2)))) K tuples: BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c5(BUBBLE(.(z0, .(y1, y2)))) BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c6(BUBBLE(.(z1, .(y1, y2)))) BUBBLE(.(z0, .(z1, .(y1, .(y2, y3))))) -> c6(BUBBLE(.(z1, .(y1, .(y2, y3))))) BUTLAST(.(z0, .(z1, .(y1, y2)))) -> c12(BUTLAST(.(z1, .(y1, y2)))) Defined Rule Symbols: bubble_1, bsort_1, butlast_1, last_1 Defined Pair Symbols: BSORT_1, BUBBLE_1, LAST_1, BUTLAST_1 Compound Symbols: c_1, c1_1, c3_1, c2_3, c2_2, c5_1, c6_1, c9_1, c12_1 ---------------------------------------- (169) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace BSORT(.(z0, .(z1, .(z2, z3)))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, .(z2, z3)))), .(z0, if(<=(z1, z2), .(z2, bubble(.(z1, z3))), .(z1, bubble(.(z2, z3)))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3)))))))), BSORT(butlast(bubble(.(z0, .(z1, .(z2, z3)))))), BUBBLE(.(z0, .(z1, .(z2, z3))))) by BSORT(.(z0, .(z1, .(z2, z3)))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, .(z2, z3)))), .(z0, if(<=(z1, z2), .(z2, bubble(.(z1, z3))), .(z1, bubble(.(z2, z3)))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3)))))))), BSORT(butlast(if(<=(z0, z1), .(z1, bubble(.(z0, .(z2, z3)))), .(z0, bubble(.(z1, .(z2, z3))))))), BUBBLE(.(z0, .(z1, .(z2, z3))))) ---------------------------------------- (170) Obligation: Complexity Dependency Tuples Problem Rules: bubble(.(z0, nil)) -> .(z0, nil) bubble(.(z0, .(z1, z2))) -> if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))) bsort(nil) -> nil bsort(.(z0, z1)) -> last(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1)))))) butlast(nil) -> nil butlast(.(z0, nil)) -> nil butlast(.(z0, .(z1, z2))) -> .(z0, butlast(.(z1, z2))) last(.(z0, nil)) -> z0 last(.(z0, .(z1, z2))) -> last(.(z1, z2)) Tuples: BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BSORT(.(z0, .(z1, z2))) -> c1(BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, .(z1, z2))) -> c3(BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c2(LAST(.(if(<=(z0, z1), .(z1, if(<=(z0, z2), .(z2, bubble(.(z0, z3))), .(z0, bubble(.(z2, z3))))), .(z0, bubble(.(z1, .(z2, z3))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3)))))))), BSORT(butlast(bubble(.(z0, .(z1, .(z2, z3)))))), BUBBLE(.(z0, .(z1, .(z2, z3))))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil))))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil))))) BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c5(BUBBLE(.(z0, .(y1, y2)))) BSORT(.(z0, nil)) -> c(LAST(.(.(z0, nil), bsort(nil)))) BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c6(BUBBLE(.(z1, .(y1, y2)))) BUBBLE(.(z0, .(z1, .(y1, .(y2, y3))))) -> c6(BUBBLE(.(z1, .(y1, .(y2, y3))))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, nil))), .(z0, .(z1, nil))), bsort(butlast(bubble(.(z0, .(z1, nil))))))), BSORT(butlast(bubble(.(z0, .(z1, nil)))))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, .(z0, nil)), .(z0, bubble(.(z1, nil)))), bsort(butlast(bubble(.(z0, .(z1, nil))))))), BSORT(butlast(bubble(.(z0, .(z1, nil)))))) LAST(.(z0, .(z1, .(y1, y2)))) -> c9(LAST(.(z1, .(y1, y2)))) BUTLAST(.(z0, .(z1, .(y1, y2)))) -> c12(BUTLAST(.(z1, .(y1, y2)))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, .(z2, z3)))), .(z0, if(<=(z1, z2), .(z2, bubble(.(z1, z3))), .(z1, bubble(.(z2, z3)))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3)))))))), BSORT(butlast(if(<=(z0, z1), .(z1, bubble(.(z0, .(z2, z3)))), .(z0, bubble(.(z1, .(z2, z3))))))), BUBBLE(.(z0, .(z1, .(z2, z3))))) S tuples: BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil))))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil))))) LAST(.(z0, .(z1, .(y1, y2)))) -> c9(LAST(.(z1, .(y1, y2)))) K tuples: BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c5(BUBBLE(.(z0, .(y1, y2)))) BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c6(BUBBLE(.(z1, .(y1, y2)))) BUBBLE(.(z0, .(z1, .(y1, .(y2, y3))))) -> c6(BUBBLE(.(z1, .(y1, .(y2, y3))))) BUTLAST(.(z0, .(z1, .(y1, y2)))) -> c12(BUTLAST(.(z1, .(y1, y2)))) Defined Rule Symbols: bubble_1, bsort_1, butlast_1, last_1 Defined Pair Symbols: BSORT_1, BUBBLE_1, LAST_1, BUTLAST_1 Compound Symbols: c_1, c1_1, c3_1, c2_3, c2_2, c5_1, c6_1, c9_1, c12_1 ---------------------------------------- (171) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (172) Obligation: Complexity Dependency Tuples Problem Rules: bubble(.(z0, nil)) -> .(z0, nil) bubble(.(z0, .(z1, z2))) -> if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))) bsort(nil) -> nil bsort(.(z0, z1)) -> last(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1)))))) butlast(nil) -> nil butlast(.(z0, nil)) -> nil butlast(.(z0, .(z1, z2))) -> .(z0, butlast(.(z1, z2))) last(.(z0, nil)) -> z0 last(.(z0, .(z1, z2))) -> last(.(z1, z2)) Tuples: BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BSORT(.(z0, .(z1, z2))) -> c1(BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, .(z1, z2))) -> c3(BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c2(LAST(.(if(<=(z0, z1), .(z1, if(<=(z0, z2), .(z2, bubble(.(z0, z3))), .(z0, bubble(.(z2, z3))))), .(z0, bubble(.(z1, .(z2, z3))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3)))))))), BSORT(butlast(bubble(.(z0, .(z1, .(z2, z3)))))), BUBBLE(.(z0, .(z1, .(z2, z3))))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil))))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil))))) BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c5(BUBBLE(.(z0, .(y1, y2)))) BSORT(.(z0, nil)) -> c(LAST(.(.(z0, nil), bsort(nil)))) BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c6(BUBBLE(.(z1, .(y1, y2)))) BUBBLE(.(z0, .(z1, .(y1, .(y2, y3))))) -> c6(BUBBLE(.(z1, .(y1, .(y2, y3))))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, nil))), .(z0, .(z1, nil))), bsort(butlast(bubble(.(z0, .(z1, nil))))))), BSORT(butlast(bubble(.(z0, .(z1, nil)))))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, .(z0, nil)), .(z0, bubble(.(z1, nil)))), bsort(butlast(bubble(.(z0, .(z1, nil))))))), BSORT(butlast(bubble(.(z0, .(z1, nil)))))) LAST(.(z0, .(z1, .(y1, y2)))) -> c9(LAST(.(z1, .(y1, y2)))) BUTLAST(.(z0, .(z1, .(y1, y2)))) -> c12(BUTLAST(.(z1, .(y1, y2)))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, .(z2, z3)))), .(z0, if(<=(z1, z2), .(z2, bubble(.(z1, z3))), .(z1, bubble(.(z2, z3)))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3)))))))), BUBBLE(.(z0, .(z1, .(z2, z3))))) S tuples: BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil))))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil))))) LAST(.(z0, .(z1, .(y1, y2)))) -> c9(LAST(.(z1, .(y1, y2)))) K tuples: BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c5(BUBBLE(.(z0, .(y1, y2)))) BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c6(BUBBLE(.(z1, .(y1, y2)))) BUBBLE(.(z0, .(z1, .(y1, .(y2, y3))))) -> c6(BUBBLE(.(z1, .(y1, .(y2, y3))))) BUTLAST(.(z0, .(z1, .(y1, y2)))) -> c12(BUTLAST(.(z1, .(y1, y2)))) Defined Rule Symbols: bubble_1, bsort_1, butlast_1, last_1 Defined Pair Symbols: BSORT_1, BUBBLE_1, LAST_1, BUTLAST_1 Compound Symbols: c_1, c1_1, c3_1, c2_3, c2_2, c5_1, c6_1, c9_1, c12_1 ---------------------------------------- (173) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (174) Obligation: Complexity Dependency Tuples Problem Rules: bubble(.(z0, nil)) -> .(z0, nil) bubble(.(z0, .(z1, z2))) -> if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))) bsort(nil) -> nil bsort(.(z0, z1)) -> last(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1)))))) butlast(nil) -> nil butlast(.(z0, nil)) -> nil butlast(.(z0, .(z1, z2))) -> .(z0, butlast(.(z1, z2))) last(.(z0, nil)) -> z0 last(.(z0, .(z1, z2))) -> last(.(z1, z2)) Tuples: BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BSORT(.(z0, .(z1, z2))) -> c1(BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, .(z1, z2))) -> c3(BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c2(LAST(.(if(<=(z0, z1), .(z1, if(<=(z0, z2), .(z2, bubble(.(z0, z3))), .(z0, bubble(.(z2, z3))))), .(z0, bubble(.(z1, .(z2, z3))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3)))))))), BSORT(butlast(bubble(.(z0, .(z1, .(z2, z3)))))), BUBBLE(.(z0, .(z1, .(z2, z3))))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil))))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil))))) BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c5(BUBBLE(.(z0, .(y1, y2)))) BSORT(.(z0, nil)) -> c(LAST(.(.(z0, nil), bsort(nil)))) BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c6(BUBBLE(.(z1, .(y1, y2)))) BUBBLE(.(z0, .(z1, .(y1, .(y2, y3))))) -> c6(BUBBLE(.(z1, .(y1, .(y2, y3))))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, nil))), .(z0, .(z1, nil))), bsort(butlast(bubble(.(z0, .(z1, nil))))))), BSORT(butlast(bubble(.(z0, .(z1, nil)))))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, .(z0, nil)), .(z0, bubble(.(z1, nil)))), bsort(butlast(bubble(.(z0, .(z1, nil))))))), BSORT(butlast(bubble(.(z0, .(z1, nil)))))) LAST(.(z0, .(z1, .(y1, y2)))) -> c9(LAST(.(z1, .(y1, y2)))) BUTLAST(.(z0, .(z1, .(y1, y2)))) -> c12(BUTLAST(.(z1, .(y1, y2)))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c4(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, .(z2, z3)))), .(z0, if(<=(z1, z2), .(z2, bubble(.(z1, z3))), .(z1, bubble(.(z2, z3)))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3))))))))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c4(BUBBLE(.(z0, .(z1, .(z2, z3))))) S tuples: BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil))))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil))))) LAST(.(z0, .(z1, .(y1, y2)))) -> c9(LAST(.(z1, .(y1, y2)))) K tuples: BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c5(BUBBLE(.(z0, .(y1, y2)))) BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c6(BUBBLE(.(z1, .(y1, y2)))) BUBBLE(.(z0, .(z1, .(y1, .(y2, y3))))) -> c6(BUBBLE(.(z1, .(y1, .(y2, y3))))) BUTLAST(.(z0, .(z1, .(y1, y2)))) -> c12(BUTLAST(.(z1, .(y1, y2)))) Defined Rule Symbols: bubble_1, bsort_1, butlast_1, last_1 Defined Pair Symbols: BSORT_1, BUBBLE_1, LAST_1, BUTLAST_1 Compound Symbols: c_1, c1_1, c3_1, c2_3, c2_2, c5_1, c6_1, c9_1, c12_1, c4_1 ---------------------------------------- (175) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace BSORT(.(z0, z1)) -> c(BUBBLE(.(z0, z1))) by BSORT(.(z0, .(y1, .(y2, y3)))) -> c(BUBBLE(.(z0, .(y1, .(y2, y3))))) BSORT(.(z0, .(y1, .(y2, .(y3, y4))))) -> c(BUBBLE(.(z0, .(y1, .(y2, .(y3, y4)))))) ---------------------------------------- (176) Obligation: Complexity Dependency Tuples Problem Rules: bubble(.(z0, nil)) -> .(z0, nil) bubble(.(z0, .(z1, z2))) -> if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))) bsort(nil) -> nil bsort(.(z0, z1)) -> last(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1)))))) butlast(nil) -> nil butlast(.(z0, nil)) -> nil butlast(.(z0, .(z1, z2))) -> .(z0, butlast(.(z1, z2))) last(.(z0, nil)) -> z0 last(.(z0, .(z1, z2))) -> last(.(z1, z2)) Tuples: BSORT(.(z0, .(z1, z2))) -> c1(BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, .(z1, z2))) -> c3(BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c2(LAST(.(if(<=(z0, z1), .(z1, if(<=(z0, z2), .(z2, bubble(.(z0, z3))), .(z0, bubble(.(z2, z3))))), .(z0, bubble(.(z1, .(z2, z3))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3)))))))), BSORT(butlast(bubble(.(z0, .(z1, .(z2, z3)))))), BUBBLE(.(z0, .(z1, .(z2, z3))))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil))))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil))))) BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c5(BUBBLE(.(z0, .(y1, y2)))) BSORT(.(z0, nil)) -> c(LAST(.(.(z0, nil), bsort(nil)))) BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c6(BUBBLE(.(z1, .(y1, y2)))) BUBBLE(.(z0, .(z1, .(y1, .(y2, y3))))) -> c6(BUBBLE(.(z1, .(y1, .(y2, y3))))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, nil))), .(z0, .(z1, nil))), bsort(butlast(bubble(.(z0, .(z1, nil))))))), BSORT(butlast(bubble(.(z0, .(z1, nil)))))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, .(z0, nil)), .(z0, bubble(.(z1, nil)))), bsort(butlast(bubble(.(z0, .(z1, nil))))))), BSORT(butlast(bubble(.(z0, .(z1, nil)))))) LAST(.(z0, .(z1, .(y1, y2)))) -> c9(LAST(.(z1, .(y1, y2)))) BUTLAST(.(z0, .(z1, .(y1, y2)))) -> c12(BUTLAST(.(z1, .(y1, y2)))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c4(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, .(z2, z3)))), .(z0, if(<=(z1, z2), .(z2, bubble(.(z1, z3))), .(z1, bubble(.(z2, z3)))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3))))))))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c4(BUBBLE(.(z0, .(z1, .(z2, z3))))) BSORT(.(z0, .(y1, .(y2, y3)))) -> c(BUBBLE(.(z0, .(y1, .(y2, y3))))) BSORT(.(z0, .(y1, .(y2, .(y3, y4))))) -> c(BUBBLE(.(z0, .(y1, .(y2, .(y3, y4)))))) S tuples: BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil))))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil))))) LAST(.(z0, .(z1, .(y1, y2)))) -> c9(LAST(.(z1, .(y1, y2)))) K tuples: BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c5(BUBBLE(.(z0, .(y1, y2)))) BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c6(BUBBLE(.(z1, .(y1, y2)))) BUBBLE(.(z0, .(z1, .(y1, .(y2, y3))))) -> c6(BUBBLE(.(z1, .(y1, .(y2, y3))))) BUTLAST(.(z0, .(z1, .(y1, y2)))) -> c12(BUTLAST(.(z1, .(y1, y2)))) BSORT(.(z0, .(y1, .(y2, y3)))) -> c(BUBBLE(.(z0, .(y1, .(y2, y3))))) BSORT(.(z0, .(y1, .(y2, .(y3, y4))))) -> c(BUBBLE(.(z0, .(y1, .(y2, .(y3, y4)))))) Defined Rule Symbols: bubble_1, bsort_1, butlast_1, last_1 Defined Pair Symbols: BSORT_1, BUBBLE_1, LAST_1, BUTLAST_1 Compound Symbols: c1_1, c3_1, c2_3, c2_2, c5_1, c_1, c6_1, c9_1, c12_1, c4_1 ---------------------------------------- (177) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace BSORT(.(z0, .(z1, z2))) -> c1(BUBBLE(.(z0, .(z1, z2)))) by BSORT(.(z0, .(z1, .(y2, y3)))) -> c1(BUBBLE(.(z0, .(z1, .(y2, y3))))) BSORT(.(z0, .(z1, .(y2, .(y3, y4))))) -> c1(BUBBLE(.(z0, .(z1, .(y2, .(y3, y4)))))) ---------------------------------------- (178) Obligation: Complexity Dependency Tuples Problem Rules: bubble(.(z0, nil)) -> .(z0, nil) bubble(.(z0, .(z1, z2))) -> if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))) bsort(nil) -> nil bsort(.(z0, z1)) -> last(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1)))))) butlast(nil) -> nil butlast(.(z0, nil)) -> nil butlast(.(z0, .(z1, z2))) -> .(z0, butlast(.(z1, z2))) last(.(z0, nil)) -> z0 last(.(z0, .(z1, z2))) -> last(.(z1, z2)) Tuples: BSORT(.(z0, .(z1, z2))) -> c3(BUBBLE(.(z0, .(z1, z2)))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c2(LAST(.(if(<=(z0, z1), .(z1, if(<=(z0, z2), .(z2, bubble(.(z0, z3))), .(z0, bubble(.(z2, z3))))), .(z0, bubble(.(z1, .(z2, z3))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3)))))))), BSORT(butlast(bubble(.(z0, .(z1, .(z2, z3)))))), BUBBLE(.(z0, .(z1, .(z2, z3))))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil))))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil))))) BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c5(BUBBLE(.(z0, .(y1, y2)))) BSORT(.(z0, nil)) -> c(LAST(.(.(z0, nil), bsort(nil)))) BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c6(BUBBLE(.(z1, .(y1, y2)))) BUBBLE(.(z0, .(z1, .(y1, .(y2, y3))))) -> c6(BUBBLE(.(z1, .(y1, .(y2, y3))))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, nil))), .(z0, .(z1, nil))), bsort(butlast(bubble(.(z0, .(z1, nil))))))), BSORT(butlast(bubble(.(z0, .(z1, nil)))))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, .(z0, nil)), .(z0, bubble(.(z1, nil)))), bsort(butlast(bubble(.(z0, .(z1, nil))))))), BSORT(butlast(bubble(.(z0, .(z1, nil)))))) LAST(.(z0, .(z1, .(y1, y2)))) -> c9(LAST(.(z1, .(y1, y2)))) BUTLAST(.(z0, .(z1, .(y1, y2)))) -> c12(BUTLAST(.(z1, .(y1, y2)))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c4(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, .(z2, z3)))), .(z0, if(<=(z1, z2), .(z2, bubble(.(z1, z3))), .(z1, bubble(.(z2, z3)))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3))))))))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c4(BUBBLE(.(z0, .(z1, .(z2, z3))))) BSORT(.(z0, .(y1, .(y2, y3)))) -> c(BUBBLE(.(z0, .(y1, .(y2, y3))))) BSORT(.(z0, .(y1, .(y2, .(y3, y4))))) -> c(BUBBLE(.(z0, .(y1, .(y2, .(y3, y4)))))) BSORT(.(z0, .(z1, .(y2, y3)))) -> c1(BUBBLE(.(z0, .(z1, .(y2, y3))))) BSORT(.(z0, .(z1, .(y2, .(y3, y4))))) -> c1(BUBBLE(.(z0, .(z1, .(y2, .(y3, y4)))))) S tuples: BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil))))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil))))) LAST(.(z0, .(z1, .(y1, y2)))) -> c9(LAST(.(z1, .(y1, y2)))) K tuples: BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c5(BUBBLE(.(z0, .(y1, y2)))) BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c6(BUBBLE(.(z1, .(y1, y2)))) BUBBLE(.(z0, .(z1, .(y1, .(y2, y3))))) -> c6(BUBBLE(.(z1, .(y1, .(y2, y3))))) BUTLAST(.(z0, .(z1, .(y1, y2)))) -> c12(BUTLAST(.(z1, .(y1, y2)))) BSORT(.(z0, .(y1, .(y2, y3)))) -> c(BUBBLE(.(z0, .(y1, .(y2, y3))))) BSORT(.(z0, .(y1, .(y2, .(y3, y4))))) -> c(BUBBLE(.(z0, .(y1, .(y2, .(y3, y4)))))) Defined Rule Symbols: bubble_1, bsort_1, butlast_1, last_1 Defined Pair Symbols: BSORT_1, BUBBLE_1, LAST_1, BUTLAST_1 Compound Symbols: c3_1, c2_3, c2_2, c5_1, c_1, c6_1, c9_1, c12_1, c4_1, c1_1 ---------------------------------------- (179) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace BSORT(.(z0, .(z1, z2))) -> c3(BUBBLE(.(z0, .(z1, z2)))) by BSORT(.(z0, .(z1, .(y2, y3)))) -> c3(BUBBLE(.(z0, .(z1, .(y2, y3))))) BSORT(.(z0, .(z1, .(y2, .(y3, y4))))) -> c3(BUBBLE(.(z0, .(z1, .(y2, .(y3, y4)))))) ---------------------------------------- (180) Obligation: Complexity Dependency Tuples Problem Rules: bubble(.(z0, nil)) -> .(z0, nil) bubble(.(z0, .(z1, z2))) -> if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))) bsort(nil) -> nil bsort(.(z0, z1)) -> last(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1)))))) butlast(nil) -> nil butlast(.(z0, nil)) -> nil butlast(.(z0, .(z1, z2))) -> .(z0, butlast(.(z1, z2))) last(.(z0, nil)) -> z0 last(.(z0, .(z1, z2))) -> last(.(z1, z2)) Tuples: BSORT(.(z0, .(z1, .(z2, z3)))) -> c2(LAST(.(if(<=(z0, z1), .(z1, if(<=(z0, z2), .(z2, bubble(.(z0, z3))), .(z0, bubble(.(z2, z3))))), .(z0, bubble(.(z1, .(z2, z3))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3)))))))), BSORT(butlast(bubble(.(z0, .(z1, .(z2, z3)))))), BUBBLE(.(z0, .(z1, .(z2, z3))))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil))))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil))))) BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c5(BUBBLE(.(z0, .(y1, y2)))) BSORT(.(z0, nil)) -> c(LAST(.(.(z0, nil), bsort(nil)))) BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c6(BUBBLE(.(z1, .(y1, y2)))) BUBBLE(.(z0, .(z1, .(y1, .(y2, y3))))) -> c6(BUBBLE(.(z1, .(y1, .(y2, y3))))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, nil))), .(z0, .(z1, nil))), bsort(butlast(bubble(.(z0, .(z1, nil))))))), BSORT(butlast(bubble(.(z0, .(z1, nil)))))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, .(z0, nil)), .(z0, bubble(.(z1, nil)))), bsort(butlast(bubble(.(z0, .(z1, nil))))))), BSORT(butlast(bubble(.(z0, .(z1, nil)))))) LAST(.(z0, .(z1, .(y1, y2)))) -> c9(LAST(.(z1, .(y1, y2)))) BUTLAST(.(z0, .(z1, .(y1, y2)))) -> c12(BUTLAST(.(z1, .(y1, y2)))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c4(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, .(z2, z3)))), .(z0, if(<=(z1, z2), .(z2, bubble(.(z1, z3))), .(z1, bubble(.(z2, z3)))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3))))))))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c4(BUBBLE(.(z0, .(z1, .(z2, z3))))) BSORT(.(z0, .(y1, .(y2, y3)))) -> c(BUBBLE(.(z0, .(y1, .(y2, y3))))) BSORT(.(z0, .(y1, .(y2, .(y3, y4))))) -> c(BUBBLE(.(z0, .(y1, .(y2, .(y3, y4)))))) BSORT(.(z0, .(z1, .(y2, y3)))) -> c1(BUBBLE(.(z0, .(z1, .(y2, y3))))) BSORT(.(z0, .(z1, .(y2, .(y3, y4))))) -> c1(BUBBLE(.(z0, .(z1, .(y2, .(y3, y4)))))) BSORT(.(z0, .(z1, .(y2, y3)))) -> c3(BUBBLE(.(z0, .(z1, .(y2, y3))))) BSORT(.(z0, .(z1, .(y2, .(y3, y4))))) -> c3(BUBBLE(.(z0, .(z1, .(y2, .(y3, y4)))))) S tuples: BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil))))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil))))) LAST(.(z0, .(z1, .(y1, y2)))) -> c9(LAST(.(z1, .(y1, y2)))) K tuples: BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c5(BUBBLE(.(z0, .(y1, y2)))) BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c6(BUBBLE(.(z1, .(y1, y2)))) BUBBLE(.(z0, .(z1, .(y1, .(y2, y3))))) -> c6(BUBBLE(.(z1, .(y1, .(y2, y3))))) BUTLAST(.(z0, .(z1, .(y1, y2)))) -> c12(BUTLAST(.(z1, .(y1, y2)))) BSORT(.(z0, .(y1, .(y2, y3)))) -> c(BUBBLE(.(z0, .(y1, .(y2, y3))))) BSORT(.(z0, .(y1, .(y2, .(y3, y4))))) -> c(BUBBLE(.(z0, .(y1, .(y2, .(y3, y4)))))) Defined Rule Symbols: bubble_1, bsort_1, butlast_1, last_1 Defined Pair Symbols: BSORT_1, BUBBLE_1, LAST_1, BUTLAST_1 Compound Symbols: c2_3, c2_2, c5_1, c_1, c6_1, c9_1, c12_1, c4_1, c1_1, c3_1 ---------------------------------------- (181) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace BSORT(.(z0, .(z1, .(z2, z3)))) -> c2(LAST(.(if(<=(z0, z1), .(z1, if(<=(z0, z2), .(z2, bubble(.(z0, z3))), .(z0, bubble(.(z2, z3))))), .(z0, bubble(.(z1, .(z2, z3))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3)))))))), BSORT(butlast(bubble(.(z0, .(z1, .(z2, z3)))))), BUBBLE(.(z0, .(z1, .(z2, z3))))) by BSORT(.(z0, .(z1, .(z2, z3)))) -> c2(LAST(.(if(<=(z0, z1), .(z1, if(<=(z0, z2), .(z2, bubble(.(z0, z3))), .(z0, bubble(.(z2, z3))))), .(z0, bubble(.(z1, .(z2, z3))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3)))))))), BSORT(butlast(if(<=(z0, z1), .(z1, bubble(.(z0, .(z2, z3)))), .(z0, bubble(.(z1, .(z2, z3))))))), BUBBLE(.(z0, .(z1, .(z2, z3))))) ---------------------------------------- (182) Obligation: Complexity Dependency Tuples Problem Rules: bubble(.(z0, nil)) -> .(z0, nil) bubble(.(z0, .(z1, z2))) -> if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))) bsort(nil) -> nil bsort(.(z0, z1)) -> last(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1)))))) butlast(nil) -> nil butlast(.(z0, nil)) -> nil butlast(.(z0, .(z1, z2))) -> .(z0, butlast(.(z1, z2))) last(.(z0, nil)) -> z0 last(.(z0, .(z1, z2))) -> last(.(z1, z2)) Tuples: BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil))))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil))))) BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c5(BUBBLE(.(z0, .(y1, y2)))) BSORT(.(z0, nil)) -> c(LAST(.(.(z0, nil), bsort(nil)))) BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c6(BUBBLE(.(z1, .(y1, y2)))) BUBBLE(.(z0, .(z1, .(y1, .(y2, y3))))) -> c6(BUBBLE(.(z1, .(y1, .(y2, y3))))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, nil))), .(z0, .(z1, nil))), bsort(butlast(bubble(.(z0, .(z1, nil))))))), BSORT(butlast(bubble(.(z0, .(z1, nil)))))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, .(z0, nil)), .(z0, bubble(.(z1, nil)))), bsort(butlast(bubble(.(z0, .(z1, nil))))))), BSORT(butlast(bubble(.(z0, .(z1, nil)))))) LAST(.(z0, .(z1, .(y1, y2)))) -> c9(LAST(.(z1, .(y1, y2)))) BUTLAST(.(z0, .(z1, .(y1, y2)))) -> c12(BUTLAST(.(z1, .(y1, y2)))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c4(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, .(z2, z3)))), .(z0, if(<=(z1, z2), .(z2, bubble(.(z1, z3))), .(z1, bubble(.(z2, z3)))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3))))))))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c4(BUBBLE(.(z0, .(z1, .(z2, z3))))) BSORT(.(z0, .(y1, .(y2, y3)))) -> c(BUBBLE(.(z0, .(y1, .(y2, y3))))) BSORT(.(z0, .(y1, .(y2, .(y3, y4))))) -> c(BUBBLE(.(z0, .(y1, .(y2, .(y3, y4)))))) BSORT(.(z0, .(z1, .(y2, y3)))) -> c1(BUBBLE(.(z0, .(z1, .(y2, y3))))) BSORT(.(z0, .(z1, .(y2, .(y3, y4))))) -> c1(BUBBLE(.(z0, .(z1, .(y2, .(y3, y4)))))) BSORT(.(z0, .(z1, .(y2, y3)))) -> c3(BUBBLE(.(z0, .(z1, .(y2, y3))))) BSORT(.(z0, .(z1, .(y2, .(y3, y4))))) -> c3(BUBBLE(.(z0, .(z1, .(y2, .(y3, y4)))))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c2(LAST(.(if(<=(z0, z1), .(z1, if(<=(z0, z2), .(z2, bubble(.(z0, z3))), .(z0, bubble(.(z2, z3))))), .(z0, bubble(.(z1, .(z2, z3))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3)))))))), BSORT(butlast(if(<=(z0, z1), .(z1, bubble(.(z0, .(z2, z3)))), .(z0, bubble(.(z1, .(z2, z3))))))), BUBBLE(.(z0, .(z1, .(z2, z3))))) S tuples: BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil))))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil))))) LAST(.(z0, .(z1, .(y1, y2)))) -> c9(LAST(.(z1, .(y1, y2)))) K tuples: BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c5(BUBBLE(.(z0, .(y1, y2)))) BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c6(BUBBLE(.(z1, .(y1, y2)))) BUBBLE(.(z0, .(z1, .(y1, .(y2, y3))))) -> c6(BUBBLE(.(z1, .(y1, .(y2, y3))))) BUTLAST(.(z0, .(z1, .(y1, y2)))) -> c12(BUTLAST(.(z1, .(y1, y2)))) BSORT(.(z0, .(y1, .(y2, y3)))) -> c(BUBBLE(.(z0, .(y1, .(y2, y3))))) BSORT(.(z0, .(y1, .(y2, .(y3, y4))))) -> c(BUBBLE(.(z0, .(y1, .(y2, .(y3, y4)))))) Defined Rule Symbols: bubble_1, bsort_1, butlast_1, last_1 Defined Pair Symbols: BSORT_1, BUBBLE_1, LAST_1, BUTLAST_1 Compound Symbols: c2_2, c5_1, c_1, c6_1, c9_1, c12_1, c4_1, c1_1, c3_1, c2_3 ---------------------------------------- (183) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (184) Obligation: Complexity Dependency Tuples Problem Rules: bubble(.(z0, nil)) -> .(z0, nil) bubble(.(z0, .(z1, z2))) -> if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))) bsort(nil) -> nil bsort(.(z0, z1)) -> last(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1)))))) butlast(nil) -> nil butlast(.(z0, nil)) -> nil butlast(.(z0, .(z1, z2))) -> .(z0, butlast(.(z1, z2))) last(.(z0, nil)) -> z0 last(.(z0, .(z1, z2))) -> last(.(z1, z2)) Tuples: BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil))))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil))))) BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c5(BUBBLE(.(z0, .(y1, y2)))) BSORT(.(z0, nil)) -> c(LAST(.(.(z0, nil), bsort(nil)))) BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c6(BUBBLE(.(z1, .(y1, y2)))) BUBBLE(.(z0, .(z1, .(y1, .(y2, y3))))) -> c6(BUBBLE(.(z1, .(y1, .(y2, y3))))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, nil))), .(z0, .(z1, nil))), bsort(butlast(bubble(.(z0, .(z1, nil))))))), BSORT(butlast(bubble(.(z0, .(z1, nil)))))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, .(z0, nil)), .(z0, bubble(.(z1, nil)))), bsort(butlast(bubble(.(z0, .(z1, nil))))))), BSORT(butlast(bubble(.(z0, .(z1, nil)))))) LAST(.(z0, .(z1, .(y1, y2)))) -> c9(LAST(.(z1, .(y1, y2)))) BUTLAST(.(z0, .(z1, .(y1, y2)))) -> c12(BUTLAST(.(z1, .(y1, y2)))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c4(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, .(z2, z3)))), .(z0, if(<=(z1, z2), .(z2, bubble(.(z1, z3))), .(z1, bubble(.(z2, z3)))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3))))))))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c4(BUBBLE(.(z0, .(z1, .(z2, z3))))) BSORT(.(z0, .(y1, .(y2, y3)))) -> c(BUBBLE(.(z0, .(y1, .(y2, y3))))) BSORT(.(z0, .(y1, .(y2, .(y3, y4))))) -> c(BUBBLE(.(z0, .(y1, .(y2, .(y3, y4)))))) BSORT(.(z0, .(z1, .(y2, y3)))) -> c1(BUBBLE(.(z0, .(z1, .(y2, y3))))) BSORT(.(z0, .(z1, .(y2, .(y3, y4))))) -> c1(BUBBLE(.(z0, .(z1, .(y2, .(y3, y4)))))) BSORT(.(z0, .(z1, .(y2, y3)))) -> c3(BUBBLE(.(z0, .(z1, .(y2, y3))))) BSORT(.(z0, .(z1, .(y2, .(y3, y4))))) -> c3(BUBBLE(.(z0, .(z1, .(y2, .(y3, y4)))))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c2(LAST(.(if(<=(z0, z1), .(z1, if(<=(z0, z2), .(z2, bubble(.(z0, z3))), .(z0, bubble(.(z2, z3))))), .(z0, bubble(.(z1, .(z2, z3))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3)))))))), BUBBLE(.(z0, .(z1, .(z2, z3))))) S tuples: BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil))))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil))))) LAST(.(z0, .(z1, .(y1, y2)))) -> c9(LAST(.(z1, .(y1, y2)))) K tuples: BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c5(BUBBLE(.(z0, .(y1, y2)))) BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c6(BUBBLE(.(z1, .(y1, y2)))) BUBBLE(.(z0, .(z1, .(y1, .(y2, y3))))) -> c6(BUBBLE(.(z1, .(y1, .(y2, y3))))) BUTLAST(.(z0, .(z1, .(y1, y2)))) -> c12(BUTLAST(.(z1, .(y1, y2)))) BSORT(.(z0, .(y1, .(y2, y3)))) -> c(BUBBLE(.(z0, .(y1, .(y2, y3))))) BSORT(.(z0, .(y1, .(y2, .(y3, y4))))) -> c(BUBBLE(.(z0, .(y1, .(y2, .(y3, y4)))))) Defined Rule Symbols: bubble_1, bsort_1, butlast_1, last_1 Defined Pair Symbols: BSORT_1, BUBBLE_1, LAST_1, BUTLAST_1 Compound Symbols: c2_2, c5_1, c_1, c6_1, c9_1, c12_1, c4_1, c1_1, c3_1 ---------------------------------------- (185) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (186) Obligation: Complexity Dependency Tuples Problem Rules: bubble(.(z0, nil)) -> .(z0, nil) bubble(.(z0, .(z1, z2))) -> if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))) bsort(nil) -> nil bsort(.(z0, z1)) -> last(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1)))))) butlast(nil) -> nil butlast(.(z0, nil)) -> nil butlast(.(z0, .(z1, z2))) -> .(z0, butlast(.(z1, z2))) last(.(z0, nil)) -> z0 last(.(z0, .(z1, z2))) -> last(.(z1, z2)) Tuples: BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil))))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil))))) BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c5(BUBBLE(.(z0, .(y1, y2)))) BSORT(.(z0, nil)) -> c(LAST(.(.(z0, nil), bsort(nil)))) BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c6(BUBBLE(.(z1, .(y1, y2)))) BUBBLE(.(z0, .(z1, .(y1, .(y2, y3))))) -> c6(BUBBLE(.(z1, .(y1, .(y2, y3))))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, nil))), .(z0, .(z1, nil))), bsort(butlast(bubble(.(z0, .(z1, nil))))))), BSORT(butlast(bubble(.(z0, .(z1, nil)))))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, .(z0, nil)), .(z0, bubble(.(z1, nil)))), bsort(butlast(bubble(.(z0, .(z1, nil))))))), BSORT(butlast(bubble(.(z0, .(z1, nil)))))) LAST(.(z0, .(z1, .(y1, y2)))) -> c9(LAST(.(z1, .(y1, y2)))) BUTLAST(.(z0, .(z1, .(y1, y2)))) -> c12(BUTLAST(.(z1, .(y1, y2)))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c4(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, .(z2, z3)))), .(z0, if(<=(z1, z2), .(z2, bubble(.(z1, z3))), .(z1, bubble(.(z2, z3)))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3))))))))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c4(BUBBLE(.(z0, .(z1, .(z2, z3))))) BSORT(.(z0, .(y1, .(y2, y3)))) -> c(BUBBLE(.(z0, .(y1, .(y2, y3))))) BSORT(.(z0, .(y1, .(y2, .(y3, y4))))) -> c(BUBBLE(.(z0, .(y1, .(y2, .(y3, y4)))))) BSORT(.(z0, .(z1, .(y2, y3)))) -> c1(BUBBLE(.(z0, .(z1, .(y2, y3))))) BSORT(.(z0, .(z1, .(y2, .(y3, y4))))) -> c1(BUBBLE(.(z0, .(z1, .(y2, .(y3, y4)))))) BSORT(.(z0, .(z1, .(y2, y3)))) -> c3(BUBBLE(.(z0, .(z1, .(y2, y3))))) BSORT(.(z0, .(z1, .(y2, .(y3, y4))))) -> c3(BUBBLE(.(z0, .(z1, .(y2, .(y3, y4)))))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c7(LAST(.(if(<=(z0, z1), .(z1, if(<=(z0, z2), .(z2, bubble(.(z0, z3))), .(z0, bubble(.(z2, z3))))), .(z0, bubble(.(z1, .(z2, z3))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3))))))))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c7(BUBBLE(.(z0, .(z1, .(z2, z3))))) S tuples: BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil))))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil))))) LAST(.(z0, .(z1, .(y1, y2)))) -> c9(LAST(.(z1, .(y1, y2)))) K tuples: BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c5(BUBBLE(.(z0, .(y1, y2)))) BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c6(BUBBLE(.(z1, .(y1, y2)))) BUBBLE(.(z0, .(z1, .(y1, .(y2, y3))))) -> c6(BUBBLE(.(z1, .(y1, .(y2, y3))))) BUTLAST(.(z0, .(z1, .(y1, y2)))) -> c12(BUTLAST(.(z1, .(y1, y2)))) BSORT(.(z0, .(y1, .(y2, y3)))) -> c(BUBBLE(.(z0, .(y1, .(y2, y3))))) BSORT(.(z0, .(y1, .(y2, .(y3, y4))))) -> c(BUBBLE(.(z0, .(y1, .(y2, .(y3, y4)))))) Defined Rule Symbols: bubble_1, bsort_1, butlast_1, last_1 Defined Pair Symbols: BSORT_1, BUBBLE_1, LAST_1, BUTLAST_1 Compound Symbols: c2_2, c5_1, c_1, c6_1, c9_1, c12_1, c4_1, c1_1, c3_1, c7_1 ---------------------------------------- (187) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(bubble(.(z0, nil))))) by BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(.(z0, nil)))) ---------------------------------------- (188) Obligation: Complexity Dependency Tuples Problem Rules: bubble(.(z0, nil)) -> .(z0, nil) bubble(.(z0, .(z1, z2))) -> if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))) bsort(nil) -> nil bsort(.(z0, z1)) -> last(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1)))))) butlast(nil) -> nil butlast(.(z0, nil)) -> nil butlast(.(z0, .(z1, z2))) -> .(z0, butlast(.(z1, z2))) last(.(z0, nil)) -> z0 last(.(z0, .(z1, z2))) -> last(.(z1, z2)) Tuples: BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil))))) BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c5(BUBBLE(.(z0, .(y1, y2)))) BSORT(.(z0, nil)) -> c(LAST(.(.(z0, nil), bsort(nil)))) BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c6(BUBBLE(.(z1, .(y1, y2)))) BUBBLE(.(z0, .(z1, .(y1, .(y2, y3))))) -> c6(BUBBLE(.(z1, .(y1, .(y2, y3))))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, nil))), .(z0, .(z1, nil))), bsort(butlast(bubble(.(z0, .(z1, nil))))))), BSORT(butlast(bubble(.(z0, .(z1, nil)))))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, .(z0, nil)), .(z0, bubble(.(z1, nil)))), bsort(butlast(bubble(.(z0, .(z1, nil))))))), BSORT(butlast(bubble(.(z0, .(z1, nil)))))) LAST(.(z0, .(z1, .(y1, y2)))) -> c9(LAST(.(z1, .(y1, y2)))) BUTLAST(.(z0, .(z1, .(y1, y2)))) -> c12(BUTLAST(.(z1, .(y1, y2)))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c4(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, .(z2, z3)))), .(z0, if(<=(z1, z2), .(z2, bubble(.(z1, z3))), .(z1, bubble(.(z2, z3)))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3))))))))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c4(BUBBLE(.(z0, .(z1, .(z2, z3))))) BSORT(.(z0, .(y1, .(y2, y3)))) -> c(BUBBLE(.(z0, .(y1, .(y2, y3))))) BSORT(.(z0, .(y1, .(y2, .(y3, y4))))) -> c(BUBBLE(.(z0, .(y1, .(y2, .(y3, y4)))))) BSORT(.(z0, .(z1, .(y2, y3)))) -> c1(BUBBLE(.(z0, .(z1, .(y2, y3))))) BSORT(.(z0, .(z1, .(y2, .(y3, y4))))) -> c1(BUBBLE(.(z0, .(z1, .(y2, .(y3, y4)))))) BSORT(.(z0, .(z1, .(y2, y3)))) -> c3(BUBBLE(.(z0, .(z1, .(y2, y3))))) BSORT(.(z0, .(z1, .(y2, .(y3, y4))))) -> c3(BUBBLE(.(z0, .(z1, .(y2, .(y3, y4)))))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c7(LAST(.(if(<=(z0, z1), .(z1, if(<=(z0, z2), .(z2, bubble(.(z0, z3))), .(z0, bubble(.(z2, z3))))), .(z0, bubble(.(z1, .(z2, z3))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3))))))))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c7(BUBBLE(.(z0, .(z1, .(z2, z3))))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(.(z0, nil)))) S tuples: BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil))))) LAST(.(z0, .(z1, .(y1, y2)))) -> c9(LAST(.(z1, .(y1, y2)))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(.(z0, nil)))) K tuples: BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c5(BUBBLE(.(z0, .(y1, y2)))) BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c6(BUBBLE(.(z1, .(y1, y2)))) BUBBLE(.(z0, .(z1, .(y1, .(y2, y3))))) -> c6(BUBBLE(.(z1, .(y1, .(y2, y3))))) BUTLAST(.(z0, .(z1, .(y1, y2)))) -> c12(BUTLAST(.(z1, .(y1, y2)))) BSORT(.(z0, .(y1, .(y2, y3)))) -> c(BUBBLE(.(z0, .(y1, .(y2, y3))))) BSORT(.(z0, .(y1, .(y2, .(y3, y4))))) -> c(BUBBLE(.(z0, .(y1, .(y2, .(y3, y4)))))) Defined Rule Symbols: bubble_1, bsort_1, butlast_1, last_1 Defined Pair Symbols: BSORT_1, BUBBLE_1, LAST_1, BUTLAST_1 Compound Symbols: c2_2, c5_1, c_1, c6_1, c9_1, c12_1, c4_1, c1_1, c3_1, c7_1 ---------------------------------------- (189) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil))))) by BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(.(z0, nil)))) ---------------------------------------- (190) Obligation: Complexity Dependency Tuples Problem Rules: bubble(.(z0, nil)) -> .(z0, nil) bubble(.(z0, .(z1, z2))) -> if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))) bsort(nil) -> nil bsort(.(z0, z1)) -> last(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1)))))) butlast(nil) -> nil butlast(.(z0, nil)) -> nil butlast(.(z0, .(z1, z2))) -> .(z0, butlast(.(z1, z2))) last(.(z0, nil)) -> z0 last(.(z0, .(z1, z2))) -> last(.(z1, z2)) Tuples: BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil))))) BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c5(BUBBLE(.(z0, .(y1, y2)))) BSORT(.(z0, nil)) -> c(LAST(.(.(z0, nil), bsort(nil)))) BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c6(BUBBLE(.(z1, .(y1, y2)))) BUBBLE(.(z0, .(z1, .(y1, .(y2, y3))))) -> c6(BUBBLE(.(z1, .(y1, .(y2, y3))))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, nil))), .(z0, .(z1, nil))), bsort(butlast(bubble(.(z0, .(z1, nil))))))), BSORT(butlast(bubble(.(z0, .(z1, nil)))))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, .(z0, nil)), .(z0, bubble(.(z1, nil)))), bsort(butlast(bubble(.(z0, .(z1, nil))))))), BSORT(butlast(bubble(.(z0, .(z1, nil)))))) LAST(.(z0, .(z1, .(y1, y2)))) -> c9(LAST(.(z1, .(y1, y2)))) BUTLAST(.(z0, .(z1, .(y1, y2)))) -> c12(BUTLAST(.(z1, .(y1, y2)))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c4(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, .(z2, z3)))), .(z0, if(<=(z1, z2), .(z2, bubble(.(z1, z3))), .(z1, bubble(.(z2, z3)))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3))))))))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c4(BUBBLE(.(z0, .(z1, .(z2, z3))))) BSORT(.(z0, .(y1, .(y2, y3)))) -> c(BUBBLE(.(z0, .(y1, .(y2, y3))))) BSORT(.(z0, .(y1, .(y2, .(y3, y4))))) -> c(BUBBLE(.(z0, .(y1, .(y2, .(y3, y4)))))) BSORT(.(z0, .(z1, .(y2, y3)))) -> c1(BUBBLE(.(z0, .(z1, .(y2, y3))))) BSORT(.(z0, .(z1, .(y2, .(y3, y4))))) -> c1(BUBBLE(.(z0, .(z1, .(y2, .(y3, y4)))))) BSORT(.(z0, .(z1, .(y2, y3)))) -> c3(BUBBLE(.(z0, .(z1, .(y2, y3))))) BSORT(.(z0, .(z1, .(y2, .(y3, y4))))) -> c3(BUBBLE(.(z0, .(z1, .(y2, .(y3, y4)))))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c7(LAST(.(if(<=(z0, z1), .(z1, if(<=(z0, z2), .(z2, bubble(.(z0, z3))), .(z0, bubble(.(z2, z3))))), .(z0, bubble(.(z1, .(z2, z3))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3))))))))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c7(BUBBLE(.(z0, .(z1, .(z2, z3))))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(.(z0, nil)))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(.(z0, nil)))) S tuples: LAST(.(z0, .(z1, .(y1, y2)))) -> c9(LAST(.(z1, .(y1, y2)))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(.(z0, nil)))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(.(z0, nil)))) K tuples: BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c5(BUBBLE(.(z0, .(y1, y2)))) BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c6(BUBBLE(.(z1, .(y1, y2)))) BUBBLE(.(z0, .(z1, .(y1, .(y2, y3))))) -> c6(BUBBLE(.(z1, .(y1, .(y2, y3))))) BUTLAST(.(z0, .(z1, .(y1, y2)))) -> c12(BUTLAST(.(z1, .(y1, y2)))) BSORT(.(z0, .(y1, .(y2, y3)))) -> c(BUBBLE(.(z0, .(y1, .(y2, y3))))) BSORT(.(z0, .(y1, .(y2, .(y3, y4))))) -> c(BUBBLE(.(z0, .(y1, .(y2, .(y3, y4)))))) Defined Rule Symbols: bubble_1, bsort_1, butlast_1, last_1 Defined Pair Symbols: BSORT_1, BUBBLE_1, LAST_1, BUTLAST_1 Compound Symbols: c2_2, c5_1, c_1, c6_1, c9_1, c12_1, c4_1, c1_1, c3_1, c7_1 ---------------------------------------- (191) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c5(BUBBLE(.(z0, .(y1, y2)))) by BUBBLE(.(z0, .(z1, .(z2, .(y2, y3))))) -> c5(BUBBLE(.(z0, .(z2, .(y2, y3))))) BUBBLE(.(z0, .(z1, .(z2, .(y2, .(y3, y4)))))) -> c5(BUBBLE(.(z0, .(z2, .(y2, .(y3, y4)))))) ---------------------------------------- (192) Obligation: Complexity Dependency Tuples Problem Rules: bubble(.(z0, nil)) -> .(z0, nil) bubble(.(z0, .(z1, z2))) -> if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))) bsort(nil) -> nil bsort(.(z0, z1)) -> last(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1)))))) butlast(nil) -> nil butlast(.(z0, nil)) -> nil butlast(.(z0, .(z1, z2))) -> .(z0, butlast(.(z1, z2))) last(.(z0, nil)) -> z0 last(.(z0, .(z1, z2))) -> last(.(z1, z2)) Tuples: BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil))))) BSORT(.(z0, nil)) -> c(LAST(.(.(z0, nil), bsort(nil)))) BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c6(BUBBLE(.(z1, .(y1, y2)))) BUBBLE(.(z0, .(z1, .(y1, .(y2, y3))))) -> c6(BUBBLE(.(z1, .(y1, .(y2, y3))))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, nil))), .(z0, .(z1, nil))), bsort(butlast(bubble(.(z0, .(z1, nil))))))), BSORT(butlast(bubble(.(z0, .(z1, nil)))))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, .(z0, nil)), .(z0, bubble(.(z1, nil)))), bsort(butlast(bubble(.(z0, .(z1, nil))))))), BSORT(butlast(bubble(.(z0, .(z1, nil)))))) LAST(.(z0, .(z1, .(y1, y2)))) -> c9(LAST(.(z1, .(y1, y2)))) BUTLAST(.(z0, .(z1, .(y1, y2)))) -> c12(BUTLAST(.(z1, .(y1, y2)))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c4(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, .(z2, z3)))), .(z0, if(<=(z1, z2), .(z2, bubble(.(z1, z3))), .(z1, bubble(.(z2, z3)))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3))))))))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c4(BUBBLE(.(z0, .(z1, .(z2, z3))))) BSORT(.(z0, .(y1, .(y2, y3)))) -> c(BUBBLE(.(z0, .(y1, .(y2, y3))))) BSORT(.(z0, .(y1, .(y2, .(y3, y4))))) -> c(BUBBLE(.(z0, .(y1, .(y2, .(y3, y4)))))) BSORT(.(z0, .(z1, .(y2, y3)))) -> c1(BUBBLE(.(z0, .(z1, .(y2, y3))))) BSORT(.(z0, .(z1, .(y2, .(y3, y4))))) -> c1(BUBBLE(.(z0, .(z1, .(y2, .(y3, y4)))))) BSORT(.(z0, .(z1, .(y2, y3)))) -> c3(BUBBLE(.(z0, .(z1, .(y2, y3))))) BSORT(.(z0, .(z1, .(y2, .(y3, y4))))) -> c3(BUBBLE(.(z0, .(z1, .(y2, .(y3, y4)))))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c7(LAST(.(if(<=(z0, z1), .(z1, if(<=(z0, z2), .(z2, bubble(.(z0, z3))), .(z0, bubble(.(z2, z3))))), .(z0, bubble(.(z1, .(z2, z3))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3))))))))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c7(BUBBLE(.(z0, .(z1, .(z2, z3))))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(.(z0, nil)))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(.(z0, nil)))) BUBBLE(.(z0, .(z1, .(z2, .(y2, y3))))) -> c5(BUBBLE(.(z0, .(z2, .(y2, y3))))) BUBBLE(.(z0, .(z1, .(z2, .(y2, .(y3, y4)))))) -> c5(BUBBLE(.(z0, .(z2, .(y2, .(y3, y4)))))) S tuples: LAST(.(z0, .(z1, .(y1, y2)))) -> c9(LAST(.(z1, .(y1, y2)))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(.(z0, nil)))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(.(z0, nil)))) K tuples: BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c6(BUBBLE(.(z1, .(y1, y2)))) BUBBLE(.(z0, .(z1, .(y1, .(y2, y3))))) -> c6(BUBBLE(.(z1, .(y1, .(y2, y3))))) BUTLAST(.(z0, .(z1, .(y1, y2)))) -> c12(BUTLAST(.(z1, .(y1, y2)))) BSORT(.(z0, .(y1, .(y2, y3)))) -> c(BUBBLE(.(z0, .(y1, .(y2, y3))))) BSORT(.(z0, .(y1, .(y2, .(y3, y4))))) -> c(BUBBLE(.(z0, .(y1, .(y2, .(y3, y4)))))) BUBBLE(.(z0, .(z1, .(z2, .(y2, y3))))) -> c5(BUBBLE(.(z0, .(z2, .(y2, y3))))) BUBBLE(.(z0, .(z1, .(z2, .(y2, .(y3, y4)))))) -> c5(BUBBLE(.(z0, .(z2, .(y2, .(y3, y4)))))) Defined Rule Symbols: bubble_1, bsort_1, butlast_1, last_1 Defined Pair Symbols: BSORT_1, BUBBLE_1, LAST_1, BUTLAST_1 Compound Symbols: c2_2, c_1, c6_1, c9_1, c12_1, c4_1, c1_1, c3_1, c7_1, c5_1 ---------------------------------------- (193) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(bubble(.(z0, nil))))) by BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(.(z0, nil)))) ---------------------------------------- (194) Obligation: Complexity Dependency Tuples Problem Rules: bubble(.(z0, nil)) -> .(z0, nil) bubble(.(z0, .(z1, z2))) -> if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))) bsort(nil) -> nil bsort(.(z0, z1)) -> last(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1)))))) butlast(nil) -> nil butlast(.(z0, nil)) -> nil butlast(.(z0, .(z1, z2))) -> .(z0, butlast(.(z1, z2))) last(.(z0, nil)) -> z0 last(.(z0, .(z1, z2))) -> last(.(z1, z2)) Tuples: BSORT(.(z0, nil)) -> c(LAST(.(.(z0, nil), bsort(nil)))) BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c6(BUBBLE(.(z1, .(y1, y2)))) BUBBLE(.(z0, .(z1, .(y1, .(y2, y3))))) -> c6(BUBBLE(.(z1, .(y1, .(y2, y3))))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, nil))), .(z0, .(z1, nil))), bsort(butlast(bubble(.(z0, .(z1, nil))))))), BSORT(butlast(bubble(.(z0, .(z1, nil)))))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, .(z0, nil)), .(z0, bubble(.(z1, nil)))), bsort(butlast(bubble(.(z0, .(z1, nil))))))), BSORT(butlast(bubble(.(z0, .(z1, nil)))))) LAST(.(z0, .(z1, .(y1, y2)))) -> c9(LAST(.(z1, .(y1, y2)))) BUTLAST(.(z0, .(z1, .(y1, y2)))) -> c12(BUTLAST(.(z1, .(y1, y2)))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c4(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, .(z2, z3)))), .(z0, if(<=(z1, z2), .(z2, bubble(.(z1, z3))), .(z1, bubble(.(z2, z3)))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3))))))))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c4(BUBBLE(.(z0, .(z1, .(z2, z3))))) BSORT(.(z0, .(y1, .(y2, y3)))) -> c(BUBBLE(.(z0, .(y1, .(y2, y3))))) BSORT(.(z0, .(y1, .(y2, .(y3, y4))))) -> c(BUBBLE(.(z0, .(y1, .(y2, .(y3, y4)))))) BSORT(.(z0, .(z1, .(y2, y3)))) -> c1(BUBBLE(.(z0, .(z1, .(y2, y3))))) BSORT(.(z0, .(z1, .(y2, .(y3, y4))))) -> c1(BUBBLE(.(z0, .(z1, .(y2, .(y3, y4)))))) BSORT(.(z0, .(z1, .(y2, y3)))) -> c3(BUBBLE(.(z0, .(z1, .(y2, y3))))) BSORT(.(z0, .(z1, .(y2, .(y3, y4))))) -> c3(BUBBLE(.(z0, .(z1, .(y2, .(y3, y4)))))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c7(LAST(.(if(<=(z0, z1), .(z1, if(<=(z0, z2), .(z2, bubble(.(z0, z3))), .(z0, bubble(.(z2, z3))))), .(z0, bubble(.(z1, .(z2, z3))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3))))))))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c7(BUBBLE(.(z0, .(z1, .(z2, z3))))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(.(z0, nil)))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(.(z0, nil)))) BUBBLE(.(z0, .(z1, .(z2, .(y2, y3))))) -> c5(BUBBLE(.(z0, .(z2, .(y2, y3))))) BUBBLE(.(z0, .(z1, .(z2, .(y2, .(y3, y4)))))) -> c5(BUBBLE(.(z0, .(z2, .(y2, .(y3, y4)))))) S tuples: LAST(.(z0, .(z1, .(y1, y2)))) -> c9(LAST(.(z1, .(y1, y2)))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(.(z0, nil)))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(.(z0, nil)))) K tuples: BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c6(BUBBLE(.(z1, .(y1, y2)))) BUBBLE(.(z0, .(z1, .(y1, .(y2, y3))))) -> c6(BUBBLE(.(z1, .(y1, .(y2, y3))))) BUTLAST(.(z0, .(z1, .(y1, y2)))) -> c12(BUTLAST(.(z1, .(y1, y2)))) BSORT(.(z0, .(y1, .(y2, y3)))) -> c(BUBBLE(.(z0, .(y1, .(y2, y3))))) BSORT(.(z0, .(y1, .(y2, .(y3, y4))))) -> c(BUBBLE(.(z0, .(y1, .(y2, .(y3, y4)))))) BUBBLE(.(z0, .(z1, .(z2, .(y2, y3))))) -> c5(BUBBLE(.(z0, .(z2, .(y2, y3))))) BUBBLE(.(z0, .(z1, .(z2, .(y2, .(y3, y4)))))) -> c5(BUBBLE(.(z0, .(z2, .(y2, .(y3, y4)))))) Defined Rule Symbols: bubble_1, bsort_1, butlast_1, last_1 Defined Pair Symbols: BSORT_1, BUBBLE_1, LAST_1, BUTLAST_1 Compound Symbols: c_1, c6_1, c2_2, c9_1, c12_1, c4_1, c1_1, c3_1, c7_1, c5_1 ---------------------------------------- (195) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace BSORT(.(z0, nil)) -> c(LAST(.(.(z0, nil), bsort(nil)))) by BSORT(.(z0, nil)) -> c(LAST(.(.(z0, nil), nil))) ---------------------------------------- (196) Obligation: Complexity Dependency Tuples Problem Rules: bubble(.(z0, nil)) -> .(z0, nil) bubble(.(z0, .(z1, z2))) -> if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))) bsort(nil) -> nil bsort(.(z0, z1)) -> last(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1)))))) butlast(nil) -> nil butlast(.(z0, nil)) -> nil butlast(.(z0, .(z1, z2))) -> .(z0, butlast(.(z1, z2))) last(.(z0, nil)) -> z0 last(.(z0, .(z1, z2))) -> last(.(z1, z2)) Tuples: BSORT(.(z0, nil)) -> c(LAST(.(.(z0, nil), bsort(nil)))) BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c6(BUBBLE(.(z1, .(y1, y2)))) BUBBLE(.(z0, .(z1, .(y1, .(y2, y3))))) -> c6(BUBBLE(.(z1, .(y1, .(y2, y3))))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, nil))), .(z0, .(z1, nil))), bsort(butlast(bubble(.(z0, .(z1, nil))))))), BSORT(butlast(bubble(.(z0, .(z1, nil)))))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, .(z0, nil)), .(z0, bubble(.(z1, nil)))), bsort(butlast(bubble(.(z0, .(z1, nil))))))), BSORT(butlast(bubble(.(z0, .(z1, nil)))))) LAST(.(z0, .(z1, .(y1, y2)))) -> c9(LAST(.(z1, .(y1, y2)))) BUTLAST(.(z0, .(z1, .(y1, y2)))) -> c12(BUTLAST(.(z1, .(y1, y2)))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c4(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, .(z2, z3)))), .(z0, if(<=(z1, z2), .(z2, bubble(.(z1, z3))), .(z1, bubble(.(z2, z3)))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3))))))))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c4(BUBBLE(.(z0, .(z1, .(z2, z3))))) BSORT(.(z0, .(y1, .(y2, y3)))) -> c(BUBBLE(.(z0, .(y1, .(y2, y3))))) BSORT(.(z0, .(y1, .(y2, .(y3, y4))))) -> c(BUBBLE(.(z0, .(y1, .(y2, .(y3, y4)))))) BSORT(.(z0, .(z1, .(y2, y3)))) -> c1(BUBBLE(.(z0, .(z1, .(y2, y3))))) BSORT(.(z0, .(z1, .(y2, .(y3, y4))))) -> c1(BUBBLE(.(z0, .(z1, .(y2, .(y3, y4)))))) BSORT(.(z0, .(z1, .(y2, y3)))) -> c3(BUBBLE(.(z0, .(z1, .(y2, y3))))) BSORT(.(z0, .(z1, .(y2, .(y3, y4))))) -> c3(BUBBLE(.(z0, .(z1, .(y2, .(y3, y4)))))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c7(LAST(.(if(<=(z0, z1), .(z1, if(<=(z0, z2), .(z2, bubble(.(z0, z3))), .(z0, bubble(.(z2, z3))))), .(z0, bubble(.(z1, .(z2, z3))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3))))))))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c7(BUBBLE(.(z0, .(z1, .(z2, z3))))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(.(z0, nil)))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(.(z0, nil)))) BUBBLE(.(z0, .(z1, .(z2, .(y2, y3))))) -> c5(BUBBLE(.(z0, .(z2, .(y2, y3))))) BUBBLE(.(z0, .(z1, .(z2, .(y2, .(y3, y4)))))) -> c5(BUBBLE(.(z0, .(z2, .(y2, .(y3, y4)))))) BSORT(.(z0, nil)) -> c(LAST(.(.(z0, nil), nil))) S tuples: LAST(.(z0, .(z1, .(y1, y2)))) -> c9(LAST(.(z1, .(y1, y2)))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(.(z0, nil)))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(.(z0, nil)))) K tuples: BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c6(BUBBLE(.(z1, .(y1, y2)))) BUBBLE(.(z0, .(z1, .(y1, .(y2, y3))))) -> c6(BUBBLE(.(z1, .(y1, .(y2, y3))))) BUTLAST(.(z0, .(z1, .(y1, y2)))) -> c12(BUTLAST(.(z1, .(y1, y2)))) BSORT(.(z0, .(y1, .(y2, y3)))) -> c(BUBBLE(.(z0, .(y1, .(y2, y3))))) BSORT(.(z0, .(y1, .(y2, .(y3, y4))))) -> c(BUBBLE(.(z0, .(y1, .(y2, .(y3, y4)))))) BUBBLE(.(z0, .(z1, .(z2, .(y2, y3))))) -> c5(BUBBLE(.(z0, .(z2, .(y2, y3))))) BUBBLE(.(z0, .(z1, .(z2, .(y2, .(y3, y4)))))) -> c5(BUBBLE(.(z0, .(z2, .(y2, .(y3, y4)))))) Defined Rule Symbols: bubble_1, bsort_1, butlast_1, last_1 Defined Pair Symbols: BSORT_1, BUBBLE_1, LAST_1, BUTLAST_1 Compound Symbols: c_1, c6_1, c2_2, c9_1, c12_1, c4_1, c1_1, c3_1, c7_1, c5_1 ---------------------------------------- (197) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: BSORT(.(z0, nil)) -> c(LAST(.(.(z0, nil), nil))) ---------------------------------------- (198) Obligation: Complexity Dependency Tuples Problem Rules: bubble(.(z0, nil)) -> .(z0, nil) bubble(.(z0, .(z1, z2))) -> if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))) bsort(nil) -> nil bsort(.(z0, z1)) -> last(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1)))))) butlast(nil) -> nil butlast(.(z0, nil)) -> nil butlast(.(z0, .(z1, z2))) -> .(z0, butlast(.(z1, z2))) last(.(z0, nil)) -> z0 last(.(z0, .(z1, z2))) -> last(.(z1, z2)) Tuples: BSORT(.(z0, nil)) -> c(LAST(.(.(z0, nil), bsort(nil)))) BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c6(BUBBLE(.(z1, .(y1, y2)))) BUBBLE(.(z0, .(z1, .(y1, .(y2, y3))))) -> c6(BUBBLE(.(z1, .(y1, .(y2, y3))))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, nil))), .(z0, .(z1, nil))), bsort(butlast(bubble(.(z0, .(z1, nil))))))), BSORT(butlast(bubble(.(z0, .(z1, nil)))))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, .(z0, nil)), .(z0, bubble(.(z1, nil)))), bsort(butlast(bubble(.(z0, .(z1, nil))))))), BSORT(butlast(bubble(.(z0, .(z1, nil)))))) LAST(.(z0, .(z1, .(y1, y2)))) -> c9(LAST(.(z1, .(y1, y2)))) BUTLAST(.(z0, .(z1, .(y1, y2)))) -> c12(BUTLAST(.(z1, .(y1, y2)))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c4(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, .(z2, z3)))), .(z0, if(<=(z1, z2), .(z2, bubble(.(z1, z3))), .(z1, bubble(.(z2, z3)))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3))))))))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c4(BUBBLE(.(z0, .(z1, .(z2, z3))))) BSORT(.(z0, .(y1, .(y2, y3)))) -> c(BUBBLE(.(z0, .(y1, .(y2, y3))))) BSORT(.(z0, .(y1, .(y2, .(y3, y4))))) -> c(BUBBLE(.(z0, .(y1, .(y2, .(y3, y4)))))) BSORT(.(z0, .(z1, .(y2, y3)))) -> c1(BUBBLE(.(z0, .(z1, .(y2, y3))))) BSORT(.(z0, .(z1, .(y2, .(y3, y4))))) -> c1(BUBBLE(.(z0, .(z1, .(y2, .(y3, y4)))))) BSORT(.(z0, .(z1, .(y2, y3)))) -> c3(BUBBLE(.(z0, .(z1, .(y2, y3))))) BSORT(.(z0, .(z1, .(y2, .(y3, y4))))) -> c3(BUBBLE(.(z0, .(z1, .(y2, .(y3, y4)))))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c7(LAST(.(if(<=(z0, z1), .(z1, if(<=(z0, z2), .(z2, bubble(.(z0, z3))), .(z0, bubble(.(z2, z3))))), .(z0, bubble(.(z1, .(z2, z3))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3))))))))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c7(BUBBLE(.(z0, .(z1, .(z2, z3))))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(.(z0, nil)))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(.(z0, nil)))) BUBBLE(.(z0, .(z1, .(z2, .(y2, y3))))) -> c5(BUBBLE(.(z0, .(z2, .(y2, y3))))) BUBBLE(.(z0, .(z1, .(z2, .(y2, .(y3, y4)))))) -> c5(BUBBLE(.(z0, .(z2, .(y2, .(y3, y4)))))) S tuples: LAST(.(z0, .(z1, .(y1, y2)))) -> c9(LAST(.(z1, .(y1, y2)))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(.(z0, nil)))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(.(z0, nil)))) K tuples: BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c6(BUBBLE(.(z1, .(y1, y2)))) BUBBLE(.(z0, .(z1, .(y1, .(y2, y3))))) -> c6(BUBBLE(.(z1, .(y1, .(y2, y3))))) BUTLAST(.(z0, .(z1, .(y1, y2)))) -> c12(BUTLAST(.(z1, .(y1, y2)))) BSORT(.(z0, .(y1, .(y2, y3)))) -> c(BUBBLE(.(z0, .(y1, .(y2, y3))))) BSORT(.(z0, .(y1, .(y2, .(y3, y4))))) -> c(BUBBLE(.(z0, .(y1, .(y2, .(y3, y4)))))) BUBBLE(.(z0, .(z1, .(z2, .(y2, y3))))) -> c5(BUBBLE(.(z0, .(z2, .(y2, y3))))) BUBBLE(.(z0, .(z1, .(z2, .(y2, .(y3, y4)))))) -> c5(BUBBLE(.(z0, .(z2, .(y2, .(y3, y4)))))) Defined Rule Symbols: bubble_1, bsort_1, butlast_1, last_1 Defined Pair Symbols: BSORT_1, BUBBLE_1, LAST_1, BUTLAST_1 Compound Symbols: c_1, c6_1, c2_2, c9_1, c12_1, c4_1, c1_1, c3_1, c7_1, c5_1 ---------------------------------------- (199) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace BSORT(.(z0, nil)) -> c(LAST(.(.(z0, nil), bsort(nil)))) by none ---------------------------------------- (200) Obligation: Complexity Dependency Tuples Problem Rules: bubble(.(z0, nil)) -> .(z0, nil) bubble(.(z0, .(z1, z2))) -> if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))) bsort(nil) -> nil bsort(.(z0, z1)) -> last(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1)))))) butlast(nil) -> nil butlast(.(z0, nil)) -> nil butlast(.(z0, .(z1, z2))) -> .(z0, butlast(.(z1, z2))) last(.(z0, nil)) -> z0 last(.(z0, .(z1, z2))) -> last(.(z1, z2)) Tuples: BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c6(BUBBLE(.(z1, .(y1, y2)))) BUBBLE(.(z0, .(z1, .(y1, .(y2, y3))))) -> c6(BUBBLE(.(z1, .(y1, .(y2, y3))))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, nil))), .(z0, .(z1, nil))), bsort(butlast(bubble(.(z0, .(z1, nil))))))), BSORT(butlast(bubble(.(z0, .(z1, nil)))))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, .(z0, nil)), .(z0, bubble(.(z1, nil)))), bsort(butlast(bubble(.(z0, .(z1, nil))))))), BSORT(butlast(bubble(.(z0, .(z1, nil)))))) LAST(.(z0, .(z1, .(y1, y2)))) -> c9(LAST(.(z1, .(y1, y2)))) BUTLAST(.(z0, .(z1, .(y1, y2)))) -> c12(BUTLAST(.(z1, .(y1, y2)))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c4(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, .(z2, z3)))), .(z0, if(<=(z1, z2), .(z2, bubble(.(z1, z3))), .(z1, bubble(.(z2, z3)))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3))))))))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c4(BUBBLE(.(z0, .(z1, .(z2, z3))))) BSORT(.(z0, .(y1, .(y2, y3)))) -> c(BUBBLE(.(z0, .(y1, .(y2, y3))))) BSORT(.(z0, .(y1, .(y2, .(y3, y4))))) -> c(BUBBLE(.(z0, .(y1, .(y2, .(y3, y4)))))) BSORT(.(z0, .(z1, .(y2, y3)))) -> c1(BUBBLE(.(z0, .(z1, .(y2, y3))))) BSORT(.(z0, .(z1, .(y2, .(y3, y4))))) -> c1(BUBBLE(.(z0, .(z1, .(y2, .(y3, y4)))))) BSORT(.(z0, .(z1, .(y2, y3)))) -> c3(BUBBLE(.(z0, .(z1, .(y2, y3))))) BSORT(.(z0, .(z1, .(y2, .(y3, y4))))) -> c3(BUBBLE(.(z0, .(z1, .(y2, .(y3, y4)))))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c7(LAST(.(if(<=(z0, z1), .(z1, if(<=(z0, z2), .(z2, bubble(.(z0, z3))), .(z0, bubble(.(z2, z3))))), .(z0, bubble(.(z1, .(z2, z3))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3))))))))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c7(BUBBLE(.(z0, .(z1, .(z2, z3))))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(.(z0, nil)))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(.(z0, nil)))) BUBBLE(.(z0, .(z1, .(z2, .(y2, y3))))) -> c5(BUBBLE(.(z0, .(z2, .(y2, y3))))) BUBBLE(.(z0, .(z1, .(z2, .(y2, .(y3, y4)))))) -> c5(BUBBLE(.(z0, .(z2, .(y2, .(y3, y4)))))) S tuples: LAST(.(z0, .(z1, .(y1, y2)))) -> c9(LAST(.(z1, .(y1, y2)))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(.(z0, nil)))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(.(z0, nil)))) K tuples: BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c6(BUBBLE(.(z1, .(y1, y2)))) BUBBLE(.(z0, .(z1, .(y1, .(y2, y3))))) -> c6(BUBBLE(.(z1, .(y1, .(y2, y3))))) BUTLAST(.(z0, .(z1, .(y1, y2)))) -> c12(BUTLAST(.(z1, .(y1, y2)))) BSORT(.(z0, .(y1, .(y2, y3)))) -> c(BUBBLE(.(z0, .(y1, .(y2, y3))))) BSORT(.(z0, .(y1, .(y2, .(y3, y4))))) -> c(BUBBLE(.(z0, .(y1, .(y2, .(y3, y4)))))) BUBBLE(.(z0, .(z1, .(z2, .(y2, y3))))) -> c5(BUBBLE(.(z0, .(z2, .(y2, y3))))) BUBBLE(.(z0, .(z1, .(z2, .(y2, .(y3, y4)))))) -> c5(BUBBLE(.(z0, .(z2, .(y2, .(y3, y4)))))) Defined Rule Symbols: bubble_1, bsort_1, butlast_1, last_1 Defined Pair Symbols: BUBBLE_1, BSORT_1, LAST_1, BUTLAST_1 Compound Symbols: c6_1, c2_2, c9_1, c12_1, c4_1, c_1, c1_1, c3_1, c7_1, c5_1 ---------------------------------------- (201) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, nil))), .(z0, .(z1, nil))), bsort(butlast(bubble(.(z0, .(z1, nil))))))), BSORT(butlast(bubble(.(z0, .(z1, nil)))))) by BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, nil))), .(z0, .(z1, nil))), bsort(butlast(bubble(.(z0, .(z1, nil))))))), BSORT(butlast(if(<=(z0, z1), .(z1, bubble(.(z0, nil))), .(z0, bubble(.(z1, nil))))))) ---------------------------------------- (202) Obligation: Complexity Dependency Tuples Problem Rules: bubble(.(z0, nil)) -> .(z0, nil) bubble(.(z0, .(z1, z2))) -> if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))) bsort(nil) -> nil bsort(.(z0, z1)) -> last(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1)))))) butlast(nil) -> nil butlast(.(z0, nil)) -> nil butlast(.(z0, .(z1, z2))) -> .(z0, butlast(.(z1, z2))) last(.(z0, nil)) -> z0 last(.(z0, .(z1, z2))) -> last(.(z1, z2)) Tuples: BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c6(BUBBLE(.(z1, .(y1, y2)))) BUBBLE(.(z0, .(z1, .(y1, .(y2, y3))))) -> c6(BUBBLE(.(z1, .(y1, .(y2, y3))))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, .(z0, nil)), .(z0, bubble(.(z1, nil)))), bsort(butlast(bubble(.(z0, .(z1, nil))))))), BSORT(butlast(bubble(.(z0, .(z1, nil)))))) LAST(.(z0, .(z1, .(y1, y2)))) -> c9(LAST(.(z1, .(y1, y2)))) BUTLAST(.(z0, .(z1, .(y1, y2)))) -> c12(BUTLAST(.(z1, .(y1, y2)))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c4(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, .(z2, z3)))), .(z0, if(<=(z1, z2), .(z2, bubble(.(z1, z3))), .(z1, bubble(.(z2, z3)))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3))))))))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c4(BUBBLE(.(z0, .(z1, .(z2, z3))))) BSORT(.(z0, .(y1, .(y2, y3)))) -> c(BUBBLE(.(z0, .(y1, .(y2, y3))))) BSORT(.(z0, .(y1, .(y2, .(y3, y4))))) -> c(BUBBLE(.(z0, .(y1, .(y2, .(y3, y4)))))) BSORT(.(z0, .(z1, .(y2, y3)))) -> c1(BUBBLE(.(z0, .(z1, .(y2, y3))))) BSORT(.(z0, .(z1, .(y2, .(y3, y4))))) -> c1(BUBBLE(.(z0, .(z1, .(y2, .(y3, y4)))))) BSORT(.(z0, .(z1, .(y2, y3)))) -> c3(BUBBLE(.(z0, .(z1, .(y2, y3))))) BSORT(.(z0, .(z1, .(y2, .(y3, y4))))) -> c3(BUBBLE(.(z0, .(z1, .(y2, .(y3, y4)))))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c7(LAST(.(if(<=(z0, z1), .(z1, if(<=(z0, z2), .(z2, bubble(.(z0, z3))), .(z0, bubble(.(z2, z3))))), .(z0, bubble(.(z1, .(z2, z3))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3))))))))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c7(BUBBLE(.(z0, .(z1, .(z2, z3))))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(.(z0, nil)))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(.(z0, nil)))) BUBBLE(.(z0, .(z1, .(z2, .(y2, y3))))) -> c5(BUBBLE(.(z0, .(z2, .(y2, y3))))) BUBBLE(.(z0, .(z1, .(z2, .(y2, .(y3, y4)))))) -> c5(BUBBLE(.(z0, .(z2, .(y2, .(y3, y4)))))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, nil))), .(z0, .(z1, nil))), bsort(butlast(bubble(.(z0, .(z1, nil))))))), BSORT(butlast(if(<=(z0, z1), .(z1, bubble(.(z0, nil))), .(z0, bubble(.(z1, nil))))))) S tuples: LAST(.(z0, .(z1, .(y1, y2)))) -> c9(LAST(.(z1, .(y1, y2)))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(.(z0, nil)))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(.(z0, nil)))) K tuples: BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c6(BUBBLE(.(z1, .(y1, y2)))) BUBBLE(.(z0, .(z1, .(y1, .(y2, y3))))) -> c6(BUBBLE(.(z1, .(y1, .(y2, y3))))) BUTLAST(.(z0, .(z1, .(y1, y2)))) -> c12(BUTLAST(.(z1, .(y1, y2)))) BSORT(.(z0, .(y1, .(y2, y3)))) -> c(BUBBLE(.(z0, .(y1, .(y2, y3))))) BSORT(.(z0, .(y1, .(y2, .(y3, y4))))) -> c(BUBBLE(.(z0, .(y1, .(y2, .(y3, y4)))))) BUBBLE(.(z0, .(z1, .(z2, .(y2, y3))))) -> c5(BUBBLE(.(z0, .(z2, .(y2, y3))))) BUBBLE(.(z0, .(z1, .(z2, .(y2, .(y3, y4)))))) -> c5(BUBBLE(.(z0, .(z2, .(y2, .(y3, y4)))))) Defined Rule Symbols: bubble_1, bsort_1, butlast_1, last_1 Defined Pair Symbols: BUBBLE_1, BSORT_1, LAST_1, BUTLAST_1 Compound Symbols: c6_1, c2_2, c9_1, c12_1, c4_1, c_1, c1_1, c3_1, c7_1, c5_1 ---------------------------------------- (203) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (204) Obligation: Complexity Dependency Tuples Problem Rules: bubble(.(z0, nil)) -> .(z0, nil) bubble(.(z0, .(z1, z2))) -> if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))) bsort(nil) -> nil bsort(.(z0, z1)) -> last(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1)))))) butlast(nil) -> nil butlast(.(z0, nil)) -> nil butlast(.(z0, .(z1, z2))) -> .(z0, butlast(.(z1, z2))) last(.(z0, nil)) -> z0 last(.(z0, .(z1, z2))) -> last(.(z1, z2)) Tuples: BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c6(BUBBLE(.(z1, .(y1, y2)))) BUBBLE(.(z0, .(z1, .(y1, .(y2, y3))))) -> c6(BUBBLE(.(z1, .(y1, .(y2, y3))))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, .(z0, nil)), .(z0, bubble(.(z1, nil)))), bsort(butlast(bubble(.(z0, .(z1, nil))))))), BSORT(butlast(bubble(.(z0, .(z1, nil)))))) LAST(.(z0, .(z1, .(y1, y2)))) -> c9(LAST(.(z1, .(y1, y2)))) BUTLAST(.(z0, .(z1, .(y1, y2)))) -> c12(BUTLAST(.(z1, .(y1, y2)))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c4(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, .(z2, z3)))), .(z0, if(<=(z1, z2), .(z2, bubble(.(z1, z3))), .(z1, bubble(.(z2, z3)))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3))))))))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c4(BUBBLE(.(z0, .(z1, .(z2, z3))))) BSORT(.(z0, .(y1, .(y2, y3)))) -> c(BUBBLE(.(z0, .(y1, .(y2, y3))))) BSORT(.(z0, .(y1, .(y2, .(y3, y4))))) -> c(BUBBLE(.(z0, .(y1, .(y2, .(y3, y4)))))) BSORT(.(z0, .(z1, .(y2, y3)))) -> c1(BUBBLE(.(z0, .(z1, .(y2, y3))))) BSORT(.(z0, .(z1, .(y2, .(y3, y4))))) -> c1(BUBBLE(.(z0, .(z1, .(y2, .(y3, y4)))))) BSORT(.(z0, .(z1, .(y2, y3)))) -> c3(BUBBLE(.(z0, .(z1, .(y2, y3))))) BSORT(.(z0, .(z1, .(y2, .(y3, y4))))) -> c3(BUBBLE(.(z0, .(z1, .(y2, .(y3, y4)))))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c7(LAST(.(if(<=(z0, z1), .(z1, if(<=(z0, z2), .(z2, bubble(.(z0, z3))), .(z0, bubble(.(z2, z3))))), .(z0, bubble(.(z1, .(z2, z3))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3))))))))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c7(BUBBLE(.(z0, .(z1, .(z2, z3))))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(.(z0, nil)))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(.(z0, nil)))) BUBBLE(.(z0, .(z1, .(z2, .(y2, y3))))) -> c5(BUBBLE(.(z0, .(z2, .(y2, y3))))) BUBBLE(.(z0, .(z1, .(z2, .(y2, .(y3, y4)))))) -> c5(BUBBLE(.(z0, .(z2, .(y2, .(y3, y4)))))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, nil))), .(z0, .(z1, nil))), bsort(butlast(bubble(.(z0, .(z1, nil)))))))) S tuples: LAST(.(z0, .(z1, .(y1, y2)))) -> c9(LAST(.(z1, .(y1, y2)))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(.(z0, nil)))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(.(z0, nil)))) K tuples: BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c6(BUBBLE(.(z1, .(y1, y2)))) BUBBLE(.(z0, .(z1, .(y1, .(y2, y3))))) -> c6(BUBBLE(.(z1, .(y1, .(y2, y3))))) BUTLAST(.(z0, .(z1, .(y1, y2)))) -> c12(BUTLAST(.(z1, .(y1, y2)))) BSORT(.(z0, .(y1, .(y2, y3)))) -> c(BUBBLE(.(z0, .(y1, .(y2, y3))))) BSORT(.(z0, .(y1, .(y2, .(y3, y4))))) -> c(BUBBLE(.(z0, .(y1, .(y2, .(y3, y4)))))) BUBBLE(.(z0, .(z1, .(z2, .(y2, y3))))) -> c5(BUBBLE(.(z0, .(z2, .(y2, y3))))) BUBBLE(.(z0, .(z1, .(z2, .(y2, .(y3, y4)))))) -> c5(BUBBLE(.(z0, .(z2, .(y2, .(y3, y4)))))) Defined Rule Symbols: bubble_1, bsort_1, butlast_1, last_1 Defined Pair Symbols: BUBBLE_1, BSORT_1, LAST_1, BUTLAST_1 Compound Symbols: c6_1, c2_2, c9_1, c12_1, c4_1, c_1, c1_1, c3_1, c7_1, c5_1, c2_1 ---------------------------------------- (205) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, .(z0, nil)), .(z0, bubble(.(z1, nil)))), bsort(butlast(bubble(.(z0, .(z1, nil))))))), BSORT(butlast(bubble(.(z0, .(z1, nil)))))) by BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, .(z0, nil)), .(z0, bubble(.(z1, nil)))), bsort(butlast(bubble(.(z0, .(z1, nil))))))), BSORT(butlast(if(<=(z0, z1), .(z1, bubble(.(z0, nil))), .(z0, bubble(.(z1, nil))))))) ---------------------------------------- (206) Obligation: Complexity Dependency Tuples Problem Rules: bubble(.(z0, nil)) -> .(z0, nil) bubble(.(z0, .(z1, z2))) -> if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))) bsort(nil) -> nil bsort(.(z0, z1)) -> last(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1)))))) butlast(nil) -> nil butlast(.(z0, nil)) -> nil butlast(.(z0, .(z1, z2))) -> .(z0, butlast(.(z1, z2))) last(.(z0, nil)) -> z0 last(.(z0, .(z1, z2))) -> last(.(z1, z2)) Tuples: BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c6(BUBBLE(.(z1, .(y1, y2)))) BUBBLE(.(z0, .(z1, .(y1, .(y2, y3))))) -> c6(BUBBLE(.(z1, .(y1, .(y2, y3))))) LAST(.(z0, .(z1, .(y1, y2)))) -> c9(LAST(.(z1, .(y1, y2)))) BUTLAST(.(z0, .(z1, .(y1, y2)))) -> c12(BUTLAST(.(z1, .(y1, y2)))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c4(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, .(z2, z3)))), .(z0, if(<=(z1, z2), .(z2, bubble(.(z1, z3))), .(z1, bubble(.(z2, z3)))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3))))))))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c4(BUBBLE(.(z0, .(z1, .(z2, z3))))) BSORT(.(z0, .(y1, .(y2, y3)))) -> c(BUBBLE(.(z0, .(y1, .(y2, y3))))) BSORT(.(z0, .(y1, .(y2, .(y3, y4))))) -> c(BUBBLE(.(z0, .(y1, .(y2, .(y3, y4)))))) BSORT(.(z0, .(z1, .(y2, y3)))) -> c1(BUBBLE(.(z0, .(z1, .(y2, y3))))) BSORT(.(z0, .(z1, .(y2, .(y3, y4))))) -> c1(BUBBLE(.(z0, .(z1, .(y2, .(y3, y4)))))) BSORT(.(z0, .(z1, .(y2, y3)))) -> c3(BUBBLE(.(z0, .(z1, .(y2, y3))))) BSORT(.(z0, .(z1, .(y2, .(y3, y4))))) -> c3(BUBBLE(.(z0, .(z1, .(y2, .(y3, y4)))))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c7(LAST(.(if(<=(z0, z1), .(z1, if(<=(z0, z2), .(z2, bubble(.(z0, z3))), .(z0, bubble(.(z2, z3))))), .(z0, bubble(.(z1, .(z2, z3))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3))))))))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c7(BUBBLE(.(z0, .(z1, .(z2, z3))))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(.(z0, nil)))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(.(z0, nil)))) BUBBLE(.(z0, .(z1, .(z2, .(y2, y3))))) -> c5(BUBBLE(.(z0, .(z2, .(y2, y3))))) BUBBLE(.(z0, .(z1, .(z2, .(y2, .(y3, y4)))))) -> c5(BUBBLE(.(z0, .(z2, .(y2, .(y3, y4)))))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, nil))), .(z0, .(z1, nil))), bsort(butlast(bubble(.(z0, .(z1, nil)))))))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, .(z0, nil)), .(z0, bubble(.(z1, nil)))), bsort(butlast(bubble(.(z0, .(z1, nil))))))), BSORT(butlast(if(<=(z0, z1), .(z1, bubble(.(z0, nil))), .(z0, bubble(.(z1, nil))))))) S tuples: LAST(.(z0, .(z1, .(y1, y2)))) -> c9(LAST(.(z1, .(y1, y2)))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(.(z0, nil)))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(.(z0, nil)))) K tuples: BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c6(BUBBLE(.(z1, .(y1, y2)))) BUBBLE(.(z0, .(z1, .(y1, .(y2, y3))))) -> c6(BUBBLE(.(z1, .(y1, .(y2, y3))))) BUTLAST(.(z0, .(z1, .(y1, y2)))) -> c12(BUTLAST(.(z1, .(y1, y2)))) BSORT(.(z0, .(y1, .(y2, y3)))) -> c(BUBBLE(.(z0, .(y1, .(y2, y3))))) BSORT(.(z0, .(y1, .(y2, .(y3, y4))))) -> c(BUBBLE(.(z0, .(y1, .(y2, .(y3, y4)))))) BUBBLE(.(z0, .(z1, .(z2, .(y2, y3))))) -> c5(BUBBLE(.(z0, .(z2, .(y2, y3))))) BUBBLE(.(z0, .(z1, .(z2, .(y2, .(y3, y4)))))) -> c5(BUBBLE(.(z0, .(z2, .(y2, .(y3, y4)))))) Defined Rule Symbols: bubble_1, bsort_1, butlast_1, last_1 Defined Pair Symbols: BUBBLE_1, LAST_1, BUTLAST_1, BSORT_1 Compound Symbols: c6_1, c9_1, c12_1, c4_1, c_1, c1_1, c3_1, c7_1, c2_2, c5_1, c2_1 ---------------------------------------- (207) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (208) Obligation: Complexity Dependency Tuples Problem Rules: bubble(.(z0, nil)) -> .(z0, nil) bubble(.(z0, .(z1, z2))) -> if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))) bsort(nil) -> nil bsort(.(z0, z1)) -> last(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1)))))) butlast(nil) -> nil butlast(.(z0, nil)) -> nil butlast(.(z0, .(z1, z2))) -> .(z0, butlast(.(z1, z2))) last(.(z0, nil)) -> z0 last(.(z0, .(z1, z2))) -> last(.(z1, z2)) Tuples: BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c6(BUBBLE(.(z1, .(y1, y2)))) BUBBLE(.(z0, .(z1, .(y1, .(y2, y3))))) -> c6(BUBBLE(.(z1, .(y1, .(y2, y3))))) LAST(.(z0, .(z1, .(y1, y2)))) -> c9(LAST(.(z1, .(y1, y2)))) BUTLAST(.(z0, .(z1, .(y1, y2)))) -> c12(BUTLAST(.(z1, .(y1, y2)))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c4(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, .(z2, z3)))), .(z0, if(<=(z1, z2), .(z2, bubble(.(z1, z3))), .(z1, bubble(.(z2, z3)))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3))))))))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c4(BUBBLE(.(z0, .(z1, .(z2, z3))))) BSORT(.(z0, .(y1, .(y2, y3)))) -> c(BUBBLE(.(z0, .(y1, .(y2, y3))))) BSORT(.(z0, .(y1, .(y2, .(y3, y4))))) -> c(BUBBLE(.(z0, .(y1, .(y2, .(y3, y4)))))) BSORT(.(z0, .(z1, .(y2, y3)))) -> c1(BUBBLE(.(z0, .(z1, .(y2, y3))))) BSORT(.(z0, .(z1, .(y2, .(y3, y4))))) -> c1(BUBBLE(.(z0, .(z1, .(y2, .(y3, y4)))))) BSORT(.(z0, .(z1, .(y2, y3)))) -> c3(BUBBLE(.(z0, .(z1, .(y2, y3))))) BSORT(.(z0, .(z1, .(y2, .(y3, y4))))) -> c3(BUBBLE(.(z0, .(z1, .(y2, .(y3, y4)))))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c7(LAST(.(if(<=(z0, z1), .(z1, if(<=(z0, z2), .(z2, bubble(.(z0, z3))), .(z0, bubble(.(z2, z3))))), .(z0, bubble(.(z1, .(z2, z3))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3))))))))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c7(BUBBLE(.(z0, .(z1, .(z2, z3))))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(.(z0, nil)))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(.(z0, nil)))) BUBBLE(.(z0, .(z1, .(z2, .(y2, y3))))) -> c5(BUBBLE(.(z0, .(z2, .(y2, y3))))) BUBBLE(.(z0, .(z1, .(z2, .(y2, .(y3, y4)))))) -> c5(BUBBLE(.(z0, .(z2, .(y2, .(y3, y4)))))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, nil))), .(z0, .(z1, nil))), bsort(butlast(bubble(.(z0, .(z1, nil)))))))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, .(z0, nil)), .(z0, bubble(.(z1, nil)))), bsort(butlast(bubble(.(z0, .(z1, nil)))))))) S tuples: LAST(.(z0, .(z1, .(y1, y2)))) -> c9(LAST(.(z1, .(y1, y2)))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(.(z0, nil)))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(.(z0, nil)))) K tuples: BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c6(BUBBLE(.(z1, .(y1, y2)))) BUBBLE(.(z0, .(z1, .(y1, .(y2, y3))))) -> c6(BUBBLE(.(z1, .(y1, .(y2, y3))))) BUTLAST(.(z0, .(z1, .(y1, y2)))) -> c12(BUTLAST(.(z1, .(y1, y2)))) BSORT(.(z0, .(y1, .(y2, y3)))) -> c(BUBBLE(.(z0, .(y1, .(y2, y3))))) BSORT(.(z0, .(y1, .(y2, .(y3, y4))))) -> c(BUBBLE(.(z0, .(y1, .(y2, .(y3, y4)))))) BUBBLE(.(z0, .(z1, .(z2, .(y2, y3))))) -> c5(BUBBLE(.(z0, .(z2, .(y2, y3))))) BUBBLE(.(z0, .(z1, .(z2, .(y2, .(y3, y4)))))) -> c5(BUBBLE(.(z0, .(z2, .(y2, .(y3, y4)))))) Defined Rule Symbols: bubble_1, bsort_1, butlast_1, last_1 Defined Pair Symbols: BUBBLE_1, LAST_1, BUTLAST_1, BSORT_1 Compound Symbols: c6_1, c9_1, c12_1, c4_1, c_1, c1_1, c3_1, c7_1, c2_2, c5_1, c2_1 ---------------------------------------- (209) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(.(z0, nil)))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(.(z0, nil)))) We considered the (Usable) Rules: butlast(.(z0, nil)) -> nil And the Tuples: BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c6(BUBBLE(.(z1, .(y1, y2)))) BUBBLE(.(z0, .(z1, .(y1, .(y2, y3))))) -> c6(BUBBLE(.(z1, .(y1, .(y2, y3))))) LAST(.(z0, .(z1, .(y1, y2)))) -> c9(LAST(.(z1, .(y1, y2)))) BUTLAST(.(z0, .(z1, .(y1, y2)))) -> c12(BUTLAST(.(z1, .(y1, y2)))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c4(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, .(z2, z3)))), .(z0, if(<=(z1, z2), .(z2, bubble(.(z1, z3))), .(z1, bubble(.(z2, z3)))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3))))))))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c4(BUBBLE(.(z0, .(z1, .(z2, z3))))) BSORT(.(z0, .(y1, .(y2, y3)))) -> c(BUBBLE(.(z0, .(y1, .(y2, y3))))) BSORT(.(z0, .(y1, .(y2, .(y3, y4))))) -> c(BUBBLE(.(z0, .(y1, .(y2, .(y3, y4)))))) BSORT(.(z0, .(z1, .(y2, y3)))) -> c1(BUBBLE(.(z0, .(z1, .(y2, y3))))) BSORT(.(z0, .(z1, .(y2, .(y3, y4))))) -> c1(BUBBLE(.(z0, .(z1, .(y2, .(y3, y4)))))) BSORT(.(z0, .(z1, .(y2, y3)))) -> c3(BUBBLE(.(z0, .(z1, .(y2, y3))))) BSORT(.(z0, .(z1, .(y2, .(y3, y4))))) -> c3(BUBBLE(.(z0, .(z1, .(y2, .(y3, y4)))))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c7(LAST(.(if(<=(z0, z1), .(z1, if(<=(z0, z2), .(z2, bubble(.(z0, z3))), .(z0, bubble(.(z2, z3))))), .(z0, bubble(.(z1, .(z2, z3))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3))))))))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c7(BUBBLE(.(z0, .(z1, .(z2, z3))))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(.(z0, nil)))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(.(z0, nil)))) BUBBLE(.(z0, .(z1, .(z2, .(y2, y3))))) -> c5(BUBBLE(.(z0, .(z2, .(y2, y3))))) BUBBLE(.(z0, .(z1, .(z2, .(y2, .(y3, y4)))))) -> c5(BUBBLE(.(z0, .(z2, .(y2, .(y3, y4)))))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, nil))), .(z0, .(z1, nil))), bsort(butlast(bubble(.(z0, .(z1, nil)))))))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, .(z0, nil)), .(z0, bubble(.(z1, nil)))), bsort(butlast(bubble(.(z0, .(z1, nil)))))))) The order we found is given by the following interpretation: Polynomial interpretation : POL(.(x_1, x_2)) = [1] POL(<=(x_1, x_2)) = x_2 POL(BSORT(x_1)) = x_1 POL(BUBBLE(x_1)) = 0 POL(BUTLAST(x_1)) = x_1 POL(LAST(x_1)) = 0 POL(bsort(x_1)) = [1] POL(bubble(x_1)) = 0 POL(butlast(x_1)) = 0 POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c2(x_1)) = x_1 POL(c2(x_1, x_2)) = x_1 + x_2 POL(c3(x_1)) = x_1 POL(c4(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c9(x_1)) = x_1 POL(if(x_1, x_2, x_3)) = [1] + x_1 POL(last(x_1)) = [1] POL(nil) = 0 ---------------------------------------- (210) Obligation: Complexity Dependency Tuples Problem Rules: bubble(.(z0, nil)) -> .(z0, nil) bubble(.(z0, .(z1, z2))) -> if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))) bsort(nil) -> nil bsort(.(z0, z1)) -> last(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1)))))) butlast(nil) -> nil butlast(.(z0, nil)) -> nil butlast(.(z0, .(z1, z2))) -> .(z0, butlast(.(z1, z2))) last(.(z0, nil)) -> z0 last(.(z0, .(z1, z2))) -> last(.(z1, z2)) Tuples: BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c6(BUBBLE(.(z1, .(y1, y2)))) BUBBLE(.(z0, .(z1, .(y1, .(y2, y3))))) -> c6(BUBBLE(.(z1, .(y1, .(y2, y3))))) LAST(.(z0, .(z1, .(y1, y2)))) -> c9(LAST(.(z1, .(y1, y2)))) BUTLAST(.(z0, .(z1, .(y1, y2)))) -> c12(BUTLAST(.(z1, .(y1, y2)))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c4(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, .(z2, z3)))), .(z0, if(<=(z1, z2), .(z2, bubble(.(z1, z3))), .(z1, bubble(.(z2, z3)))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3))))))))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c4(BUBBLE(.(z0, .(z1, .(z2, z3))))) BSORT(.(z0, .(y1, .(y2, y3)))) -> c(BUBBLE(.(z0, .(y1, .(y2, y3))))) BSORT(.(z0, .(y1, .(y2, .(y3, y4))))) -> c(BUBBLE(.(z0, .(y1, .(y2, .(y3, y4)))))) BSORT(.(z0, .(z1, .(y2, y3)))) -> c1(BUBBLE(.(z0, .(z1, .(y2, y3))))) BSORT(.(z0, .(z1, .(y2, .(y3, y4))))) -> c1(BUBBLE(.(z0, .(z1, .(y2, .(y3, y4)))))) BSORT(.(z0, .(z1, .(y2, y3)))) -> c3(BUBBLE(.(z0, .(z1, .(y2, y3))))) BSORT(.(z0, .(z1, .(y2, .(y3, y4))))) -> c3(BUBBLE(.(z0, .(z1, .(y2, .(y3, y4)))))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c7(LAST(.(if(<=(z0, z1), .(z1, if(<=(z0, z2), .(z2, bubble(.(z0, z3))), .(z0, bubble(.(z2, z3))))), .(z0, bubble(.(z1, .(z2, z3))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3))))))))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c7(BUBBLE(.(z0, .(z1, .(z2, z3))))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(.(z0, nil)))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(.(z0, nil)))) BUBBLE(.(z0, .(z1, .(z2, .(y2, y3))))) -> c5(BUBBLE(.(z0, .(z2, .(y2, y3))))) BUBBLE(.(z0, .(z1, .(z2, .(y2, .(y3, y4)))))) -> c5(BUBBLE(.(z0, .(z2, .(y2, .(y3, y4)))))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, nil))), .(z0, .(z1, nil))), bsort(butlast(bubble(.(z0, .(z1, nil)))))))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, .(z0, nil)), .(z0, bubble(.(z1, nil)))), bsort(butlast(bubble(.(z0, .(z1, nil)))))))) S tuples: LAST(.(z0, .(z1, .(y1, y2)))) -> c9(LAST(.(z1, .(y1, y2)))) K tuples: BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c6(BUBBLE(.(z1, .(y1, y2)))) BUBBLE(.(z0, .(z1, .(y1, .(y2, y3))))) -> c6(BUBBLE(.(z1, .(y1, .(y2, y3))))) BUTLAST(.(z0, .(z1, .(y1, y2)))) -> c12(BUTLAST(.(z1, .(y1, y2)))) BSORT(.(z0, .(y1, .(y2, y3)))) -> c(BUBBLE(.(z0, .(y1, .(y2, y3))))) BSORT(.(z0, .(y1, .(y2, .(y3, y4))))) -> c(BUBBLE(.(z0, .(y1, .(y2, .(y3, y4)))))) BUBBLE(.(z0, .(z1, .(z2, .(y2, y3))))) -> c5(BUBBLE(.(z0, .(z2, .(y2, y3))))) BUBBLE(.(z0, .(z1, .(z2, .(y2, .(y3, y4)))))) -> c5(BUBBLE(.(z0, .(z2, .(y2, .(y3, y4)))))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(.(z0, nil)))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(.(z0, nil)))) Defined Rule Symbols: bubble_1, bsort_1, butlast_1, last_1 Defined Pair Symbols: BUBBLE_1, LAST_1, BUTLAST_1, BSORT_1 Compound Symbols: c6_1, c9_1, c12_1, c4_1, c_1, c1_1, c3_1, c7_1, c2_2, c5_1, c2_1 ---------------------------------------- (211) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace BUBBLE(.(z0, .(z1, .(y1, y2)))) -> c6(BUBBLE(.(z1, .(y1, y2)))) by BUBBLE(.(z0, .(z1, .(z2, .(y2, y3))))) -> c6(BUBBLE(.(z1, .(z2, .(y2, y3))))) BUBBLE(.(z0, .(z1, .(z2, .(y2, .(y3, y4)))))) -> c6(BUBBLE(.(z1, .(z2, .(y2, .(y3, y4)))))) BUBBLE(.(z0, .(z1, .(z2, .(y2, .(y3, .(y4, y5))))))) -> c6(BUBBLE(.(z1, .(z2, .(y2, .(y3, .(y4, y5))))))) ---------------------------------------- (212) Obligation: Complexity Dependency Tuples Problem Rules: bubble(.(z0, nil)) -> .(z0, nil) bubble(.(z0, .(z1, z2))) -> if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))) bsort(nil) -> nil bsort(.(z0, z1)) -> last(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1)))))) butlast(nil) -> nil butlast(.(z0, nil)) -> nil butlast(.(z0, .(z1, z2))) -> .(z0, butlast(.(z1, z2))) last(.(z0, nil)) -> z0 last(.(z0, .(z1, z2))) -> last(.(z1, z2)) Tuples: BUBBLE(.(z0, .(z1, .(y1, .(y2, y3))))) -> c6(BUBBLE(.(z1, .(y1, .(y2, y3))))) LAST(.(z0, .(z1, .(y1, y2)))) -> c9(LAST(.(z1, .(y1, y2)))) BUTLAST(.(z0, .(z1, .(y1, y2)))) -> c12(BUTLAST(.(z1, .(y1, y2)))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c4(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, .(z2, z3)))), .(z0, if(<=(z1, z2), .(z2, bubble(.(z1, z3))), .(z1, bubble(.(z2, z3)))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3))))))))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c4(BUBBLE(.(z0, .(z1, .(z2, z3))))) BSORT(.(z0, .(y1, .(y2, y3)))) -> c(BUBBLE(.(z0, .(y1, .(y2, y3))))) BSORT(.(z0, .(y1, .(y2, .(y3, y4))))) -> c(BUBBLE(.(z0, .(y1, .(y2, .(y3, y4)))))) BSORT(.(z0, .(z1, .(y2, y3)))) -> c1(BUBBLE(.(z0, .(z1, .(y2, y3))))) BSORT(.(z0, .(z1, .(y2, .(y3, y4))))) -> c1(BUBBLE(.(z0, .(z1, .(y2, .(y3, y4)))))) BSORT(.(z0, .(z1, .(y2, y3)))) -> c3(BUBBLE(.(z0, .(z1, .(y2, y3))))) BSORT(.(z0, .(z1, .(y2, .(y3, y4))))) -> c3(BUBBLE(.(z0, .(z1, .(y2, .(y3, y4)))))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c7(LAST(.(if(<=(z0, z1), .(z1, if(<=(z0, z2), .(z2, bubble(.(z0, z3))), .(z0, bubble(.(z2, z3))))), .(z0, bubble(.(z1, .(z2, z3))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3))))))))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c7(BUBBLE(.(z0, .(z1, .(z2, z3))))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(.(z0, nil)))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(.(z0, nil)))) BUBBLE(.(z0, .(z1, .(z2, .(y2, y3))))) -> c5(BUBBLE(.(z0, .(z2, .(y2, y3))))) BUBBLE(.(z0, .(z1, .(z2, .(y2, .(y3, y4)))))) -> c5(BUBBLE(.(z0, .(z2, .(y2, .(y3, y4)))))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, nil))), .(z0, .(z1, nil))), bsort(butlast(bubble(.(z0, .(z1, nil)))))))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, .(z0, nil)), .(z0, bubble(.(z1, nil)))), bsort(butlast(bubble(.(z0, .(z1, nil)))))))) BUBBLE(.(z0, .(z1, .(z2, .(y2, .(y3, y4)))))) -> c6(BUBBLE(.(z1, .(z2, .(y2, .(y3, y4)))))) BUBBLE(.(z0, .(z1, .(z2, .(y2, .(y3, .(y4, y5))))))) -> c6(BUBBLE(.(z1, .(z2, .(y2, .(y3, .(y4, y5))))))) S tuples: LAST(.(z0, .(z1, .(y1, y2)))) -> c9(LAST(.(z1, .(y1, y2)))) K tuples: BUBBLE(.(z0, .(z1, .(y1, .(y2, y3))))) -> c6(BUBBLE(.(z1, .(y1, .(y2, y3))))) BUTLAST(.(z0, .(z1, .(y1, y2)))) -> c12(BUTLAST(.(z1, .(y1, y2)))) BSORT(.(z0, .(y1, .(y2, y3)))) -> c(BUBBLE(.(z0, .(y1, .(y2, y3))))) BSORT(.(z0, .(y1, .(y2, .(y3, y4))))) -> c(BUBBLE(.(z0, .(y1, .(y2, .(y3, y4)))))) BUBBLE(.(z0, .(z1, .(z2, .(y2, y3))))) -> c5(BUBBLE(.(z0, .(z2, .(y2, y3))))) BUBBLE(.(z0, .(z1, .(z2, .(y2, .(y3, y4)))))) -> c5(BUBBLE(.(z0, .(z2, .(y2, .(y3, y4)))))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(.(z0, nil)))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(.(z0, nil)))) BUBBLE(.(z0, .(z1, .(z2, .(y2, .(y3, y4)))))) -> c6(BUBBLE(.(z1, .(z2, .(y2, .(y3, y4)))))) BUBBLE(.(z0, .(z1, .(z2, .(y2, .(y3, .(y4, y5))))))) -> c6(BUBBLE(.(z1, .(z2, .(y2, .(y3, .(y4, y5))))))) Defined Rule Symbols: bubble_1, bsort_1, butlast_1, last_1 Defined Pair Symbols: BUBBLE_1, LAST_1, BUTLAST_1, BSORT_1 Compound Symbols: c6_1, c9_1, c12_1, c4_1, c_1, c1_1, c3_1, c7_1, c2_2, c5_1, c2_1 ---------------------------------------- (213) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace BUBBLE(.(z0, .(z1, .(y1, .(y2, y3))))) -> c6(BUBBLE(.(z1, .(y1, .(y2, y3))))) by BUBBLE(.(z0, .(z1, .(z2, .(z3, .(y3, y4)))))) -> c6(BUBBLE(.(z1, .(z2, .(z3, .(y3, y4)))))) BUBBLE(.(z0, .(z1, .(z2, .(z3, .(y3, .(y4, y5))))))) -> c6(BUBBLE(.(z1, .(z2, .(z3, .(y3, .(y4, y5))))))) BUBBLE(.(z0, .(z1, .(z2, .(z3, .(y3, .(y4, .(y5, y6)))))))) -> c6(BUBBLE(.(z1, .(z2, .(z3, .(y3, .(y4, .(y5, y6)))))))) ---------------------------------------- (214) Obligation: Complexity Dependency Tuples Problem Rules: bubble(.(z0, nil)) -> .(z0, nil) bubble(.(z0, .(z1, z2))) -> if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))) bsort(nil) -> nil bsort(.(z0, z1)) -> last(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1)))))) butlast(nil) -> nil butlast(.(z0, nil)) -> nil butlast(.(z0, .(z1, z2))) -> .(z0, butlast(.(z1, z2))) last(.(z0, nil)) -> z0 last(.(z0, .(z1, z2))) -> last(.(z1, z2)) Tuples: BUBBLE(.(z0, .(z1, .(y1, .(y2, y3))))) -> c6(BUBBLE(.(z1, .(y1, .(y2, y3))))) LAST(.(z0, .(z1, .(y1, y2)))) -> c9(LAST(.(z1, .(y1, y2)))) BUTLAST(.(z0, .(z1, .(y1, y2)))) -> c12(BUTLAST(.(z1, .(y1, y2)))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c4(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, .(z2, z3)))), .(z0, if(<=(z1, z2), .(z2, bubble(.(z1, z3))), .(z1, bubble(.(z2, z3)))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3))))))))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c4(BUBBLE(.(z0, .(z1, .(z2, z3))))) BSORT(.(z0, .(y1, .(y2, y3)))) -> c(BUBBLE(.(z0, .(y1, .(y2, y3))))) BSORT(.(z0, .(y1, .(y2, .(y3, y4))))) -> c(BUBBLE(.(z0, .(y1, .(y2, .(y3, y4)))))) BSORT(.(z0, .(z1, .(y2, y3)))) -> c1(BUBBLE(.(z0, .(z1, .(y2, y3))))) BSORT(.(z0, .(z1, .(y2, .(y3, y4))))) -> c1(BUBBLE(.(z0, .(z1, .(y2, .(y3, y4)))))) BSORT(.(z0, .(z1, .(y2, y3)))) -> c3(BUBBLE(.(z0, .(z1, .(y2, y3))))) BSORT(.(z0, .(z1, .(y2, .(y3, y4))))) -> c3(BUBBLE(.(z0, .(z1, .(y2, .(y3, y4)))))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c7(LAST(.(if(<=(z0, z1), .(z1, if(<=(z0, z2), .(z2, bubble(.(z0, z3))), .(z0, bubble(.(z2, z3))))), .(z0, bubble(.(z1, .(z2, z3))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3))))))))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c7(BUBBLE(.(z0, .(z1, .(z2, z3))))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(.(z0, nil)))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(.(z0, nil)))) BUBBLE(.(z0, .(z1, .(z2, .(y2, y3))))) -> c5(BUBBLE(.(z0, .(z2, .(y2, y3))))) BUBBLE(.(z0, .(z1, .(z2, .(y2, .(y3, y4)))))) -> c5(BUBBLE(.(z0, .(z2, .(y2, .(y3, y4)))))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, nil))), .(z0, .(z1, nil))), bsort(butlast(bubble(.(z0, .(z1, nil)))))))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, .(z0, nil)), .(z0, bubble(.(z1, nil)))), bsort(butlast(bubble(.(z0, .(z1, nil)))))))) BUBBLE(.(z0, .(z1, .(z2, .(y2, .(y3, y4)))))) -> c6(BUBBLE(.(z1, .(z2, .(y2, .(y3, y4)))))) BUBBLE(.(z0, .(z1, .(z2, .(y2, .(y3, .(y4, y5))))))) -> c6(BUBBLE(.(z1, .(z2, .(y2, .(y3, .(y4, y5))))))) BUBBLE(.(z0, .(z1, .(z2, .(z3, .(y3, .(y4, .(y5, y6)))))))) -> c6(BUBBLE(.(z1, .(z2, .(z3, .(y3, .(y4, .(y5, y6)))))))) S tuples: LAST(.(z0, .(z1, .(y1, y2)))) -> c9(LAST(.(z1, .(y1, y2)))) K tuples: BUTLAST(.(z0, .(z1, .(y1, y2)))) -> c12(BUTLAST(.(z1, .(y1, y2)))) BSORT(.(z0, .(y1, .(y2, y3)))) -> c(BUBBLE(.(z0, .(y1, .(y2, y3))))) BSORT(.(z0, .(y1, .(y2, .(y3, y4))))) -> c(BUBBLE(.(z0, .(y1, .(y2, .(y3, y4)))))) BUBBLE(.(z0, .(z1, .(z2, .(y2, y3))))) -> c5(BUBBLE(.(z0, .(z2, .(y2, y3))))) BUBBLE(.(z0, .(z1, .(z2, .(y2, .(y3, y4)))))) -> c5(BUBBLE(.(z0, .(z2, .(y2, .(y3, y4)))))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(.(z0, nil)))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(.(z0, nil)))) BUBBLE(.(z0, .(z1, .(z2, .(y2, .(y3, y4)))))) -> c6(BUBBLE(.(z1, .(z2, .(y2, .(y3, y4)))))) BUBBLE(.(z0, .(z1, .(z2, .(y2, .(y3, .(y4, y5))))))) -> c6(BUBBLE(.(z1, .(z2, .(y2, .(y3, .(y4, y5))))))) BUBBLE(.(z0, .(z1, .(z2, .(z3, .(y3, .(y4, .(y5, y6)))))))) -> c6(BUBBLE(.(z1, .(z2, .(z3, .(y3, .(y4, .(y5, y6)))))))) Defined Rule Symbols: bubble_1, bsort_1, butlast_1, last_1 Defined Pair Symbols: BUBBLE_1, LAST_1, BUTLAST_1, BSORT_1 Compound Symbols: c6_1, c9_1, c12_1, c4_1, c_1, c1_1, c3_1, c7_1, c2_2, c5_1, c2_1 ---------------------------------------- (215) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace LAST(.(z0, .(z1, .(y1, y2)))) -> c9(LAST(.(z1, .(y1, y2)))) by LAST(.(z0, .(z1, .(z2, .(y2, y3))))) -> c9(LAST(.(z1, .(z2, .(y2, y3))))) ---------------------------------------- (216) Obligation: Complexity Dependency Tuples Problem Rules: bubble(.(z0, nil)) -> .(z0, nil) bubble(.(z0, .(z1, z2))) -> if(<=(z0, z1), .(z1, bubble(.(z0, z2))), .(z0, bubble(.(z1, z2)))) bsort(nil) -> nil bsort(.(z0, z1)) -> last(.(bubble(.(z0, z1)), bsort(butlast(bubble(.(z0, z1)))))) butlast(nil) -> nil butlast(.(z0, nil)) -> nil butlast(.(z0, .(z1, z2))) -> .(z0, butlast(.(z1, z2))) last(.(z0, nil)) -> z0 last(.(z0, .(z1, z2))) -> last(.(z1, z2)) Tuples: BUBBLE(.(z0, .(z1, .(y1, .(y2, y3))))) -> c6(BUBBLE(.(z1, .(y1, .(y2, y3))))) BUTLAST(.(z0, .(z1, .(y1, y2)))) -> c12(BUTLAST(.(z1, .(y1, y2)))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c4(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, .(z2, z3)))), .(z0, if(<=(z1, z2), .(z2, bubble(.(z1, z3))), .(z1, bubble(.(z2, z3)))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3))))))))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c4(BUBBLE(.(z0, .(z1, .(z2, z3))))) BSORT(.(z0, .(y1, .(y2, y3)))) -> c(BUBBLE(.(z0, .(y1, .(y2, y3))))) BSORT(.(z0, .(y1, .(y2, .(y3, y4))))) -> c(BUBBLE(.(z0, .(y1, .(y2, .(y3, y4)))))) BSORT(.(z0, .(z1, .(y2, y3)))) -> c1(BUBBLE(.(z0, .(z1, .(y2, y3))))) BSORT(.(z0, .(z1, .(y2, .(y3, y4))))) -> c1(BUBBLE(.(z0, .(z1, .(y2, .(y3, y4)))))) BSORT(.(z0, .(z1, .(y2, y3)))) -> c3(BUBBLE(.(z0, .(z1, .(y2, y3))))) BSORT(.(z0, .(z1, .(y2, .(y3, y4))))) -> c3(BUBBLE(.(z0, .(z1, .(y2, .(y3, y4)))))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c7(LAST(.(if(<=(z0, z1), .(z1, if(<=(z0, z2), .(z2, bubble(.(z0, z3))), .(z0, bubble(.(z2, z3))))), .(z0, bubble(.(z1, .(z2, z3))))), bsort(butlast(bubble(.(z0, .(z1, .(z2, z3))))))))) BSORT(.(z0, .(z1, .(z2, z3)))) -> c7(BUBBLE(.(z0, .(z1, .(z2, z3))))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(.(z0, nil)))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(.(z0, nil)))) BUBBLE(.(z0, .(z1, .(z2, .(y2, y3))))) -> c5(BUBBLE(.(z0, .(z2, .(y2, y3))))) BUBBLE(.(z0, .(z1, .(z2, .(y2, .(y3, y4)))))) -> c5(BUBBLE(.(z0, .(z2, .(y2, .(y3, y4)))))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, bubble(.(z0, nil))), .(z0, .(z1, nil))), bsort(butlast(bubble(.(z0, .(z1, nil)))))))) BSORT(.(z0, .(z1, nil))) -> c2(LAST(.(if(<=(z0, z1), .(z1, .(z0, nil)), .(z0, bubble(.(z1, nil)))), bsort(butlast(bubble(.(z0, .(z1, nil)))))))) BUBBLE(.(z0, .(z1, .(z2, .(y2, .(y3, y4)))))) -> c6(BUBBLE(.(z1, .(z2, .(y2, .(y3, y4)))))) BUBBLE(.(z0, .(z1, .(z2, .(y2, .(y3, .(y4, y5))))))) -> c6(BUBBLE(.(z1, .(z2, .(y2, .(y3, .(y4, y5))))))) BUBBLE(.(z0, .(z1, .(z2, .(z3, .(y3, .(y4, .(y5, y6)))))))) -> c6(BUBBLE(.(z1, .(z2, .(z3, .(y3, .(y4, .(y5, y6)))))))) LAST(.(z0, .(z1, .(z2, .(y2, y3))))) -> c9(LAST(.(z1, .(z2, .(y2, y3))))) S tuples: LAST(.(z0, .(z1, .(z2, .(y2, y3))))) -> c9(LAST(.(z1, .(z2, .(y2, y3))))) K tuples: BUTLAST(.(z0, .(z1, .(y1, y2)))) -> c12(BUTLAST(.(z1, .(y1, y2)))) BSORT(.(z0, .(y1, .(y2, y3)))) -> c(BUBBLE(.(z0, .(y1, .(y2, y3))))) BSORT(.(z0, .(y1, .(y2, .(y3, y4))))) -> c(BUBBLE(.(z0, .(y1, .(y2, .(y3, y4)))))) BUBBLE(.(z0, .(z1, .(z2, .(y2, y3))))) -> c5(BUBBLE(.(z0, .(z2, .(y2, y3))))) BUBBLE(.(z0, .(z1, .(z2, .(y2, .(y3, y4)))))) -> c5(BUBBLE(.(z0, .(z2, .(y2, .(y3, y4)))))) BSORT(.(z0, nil)) -> c2(LAST(.(bubble(.(z0, nil)), bsort(nil))), BSORT(butlast(.(z0, nil)))) BSORT(.(z0, nil)) -> c2(LAST(.(.(z0, nil), bsort(butlast(.(z0, nil))))), BSORT(butlast(.(z0, nil)))) BUBBLE(.(z0, .(z1, .(z2, .(y2, .(y3, y4)))))) -> c6(BUBBLE(.(z1, .(z2, .(y2, .(y3, y4)))))) BUBBLE(.(z0, .(z1, .(z2, .(y2, .(y3, .(y4, y5))))))) -> c6(BUBBLE(.(z1, .(z2, .(y2, .(y3, .(y4, y5))))))) BUBBLE(.(z0, .(z1, .(z2, .(z3, .(y3, .(y4, .(y5, y6)))))))) -> c6(BUBBLE(.(z1, .(z2, .(z3, .(y3, .(y4, .(y5, y6)))))))) Defined Rule Symbols: bubble_1, bsort_1, butlast_1, last_1 Defined Pair Symbols: BUBBLE_1, BUTLAST_1, BSORT_1, LAST_1 Compound Symbols: c6_1, c12_1, c4_1, c_1, c1_1, c3_1, c7_1, c2_2, c5_1, c2_1, c9_1