KILLED proof of input_3DwaxjF5RO.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). (0) CpxTRS (1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxTRS (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (4) CpxTRS (5) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxWeightedTrs (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (8) CpxTypedWeightedTrs (9) CompletionProof [UPPER BOUND(ID), 0 ms] (10) CpxTypedWeightedCompleteTrs (11) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (12) CpxTypedWeightedCompleteTrs (13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (14) CpxRNTS (15) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (16) CpxRNTS (17) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms] (18) CpxRNTS (19) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (20) CpxRNTS (21) IntTrsBoundProof [UPPER BOUND(ID), 118 ms] (22) CpxRNTS (23) IntTrsBoundProof [UPPER BOUND(ID), 26 ms] (24) CpxRNTS (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (26) CpxRNTS (27) IntTrsBoundProof [UPPER BOUND(ID), 315 ms] (28) CpxRNTS (29) IntTrsBoundProof [UPPER BOUND(ID), 139 ms] (30) CpxRNTS (31) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (32) CpxRNTS (33) IntTrsBoundProof [UPPER BOUND(ID), 254 ms] (34) CpxRNTS (35) IntTrsBoundProof [UPPER BOUND(ID), 117 ms] (36) CpxRNTS (37) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (38) CpxRNTS (39) IntTrsBoundProof [UPPER BOUND(ID), 1248 ms] (40) CpxRNTS (41) IntTrsBoundProof [UPPER BOUND(ID), 508 ms] (42) CpxRNTS (43) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (44) CpxRNTS (45) IntTrsBoundProof [UPPER BOUND(ID), 964 ms] (46) CpxRNTS (47) IntTrsBoundProof [UPPER BOUND(ID), 360 ms] (48) CpxRNTS (49) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (50) CpxRNTS (51) IntTrsBoundProof [UPPER BOUND(ID), 1523 ms] (52) CpxRNTS (53) IntTrsBoundProof [UPPER BOUND(ID), 536 ms] (54) CpxRNTS (55) CompletionProof [UPPER BOUND(ID), 0 ms] (56) CpxTypedWeightedCompleteTrs (57) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (58) CpxRNTS (59) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (60) CdtProblem (61) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (62) CdtProblem (63) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (64) CdtProblem (65) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (66) CdtProblem (67) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (68) CdtProblem (69) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (70) CdtProblem (71) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (72) CdtProblem (73) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (74) CdtProblem (75) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (76) CdtProblem (77) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (78) CdtProblem (79) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (80) CdtProblem (81) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (82) CdtProblem (83) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (84) CdtProblem (85) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms] (86) CdtProblem (87) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (88) CdtProblem (89) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (90) CdtProblem (91) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 2 ms] (92) CdtProblem (93) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (94) CdtProblem (95) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (96) CdtProblem (97) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (98) CdtProblem (99) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (100) CdtProblem (101) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms] (102) CdtProblem (103) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (104) CdtProblem (105) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (106) CdtProblem (107) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (108) CdtProblem (109) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (110) CdtProblem (111) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 67 ms] (112) CdtProblem (113) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (114) CdtProblem (115) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (116) CdtProblem (117) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (118) CdtProblem (119) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 1 ms] (120) CdtProblem (121) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (122) CdtProblem (123) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (124) CdtProblem (125) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (126) CdtProblem (127) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (128) CdtProblem (129) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (130) CdtProblem (131) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (132) CdtProblem (133) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (134) CdtProblem (135) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms] (136) CdtProblem (137) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (138) CdtProblem (139) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms] (140) CdtProblem (141) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (142) CdtProblem (143) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (144) CdtProblem (145) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (146) CdtProblem (147) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (148) CdtProblem (149) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (150) CdtProblem (151) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (152) CdtProblem (153) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (154) CdtProblem (155) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (156) CdtProblem (157) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 272 ms] (158) CdtProblem (159) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (160) CdtProblem (161) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (162) CdtProblem (163) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 253 ms] (164) CdtProblem (165) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 3 ms] (166) CdtProblem (167) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (168) CdtProblem (169) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 345 ms] (170) CdtProblem (171) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (172) CdtProblem (173) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (174) CdtProblem (175) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (176) CdtProblem (177) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 343 ms] (178) CdtProblem (179) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 1 ms] (180) CdtProblem (181) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (182) CdtProblem (183) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (184) CdtProblem (185) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (186) CdtProblem (187) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (188) CdtProblem (189) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (190) CdtProblem (191) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 395 ms] (192) CdtProblem (193) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 322 ms] (194) CdtProblem (195) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (196) CdtProblem (197) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (198) CdtProblem (199) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (200) CdtProblem (201) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (202) CdtProblem (203) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (204) CdtProblem (205) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (206) CdtProblem (207) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (208) CdtProblem (209) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (210) CdtProblem (211) CdtRewritingProof [BOTH BOUNDS(ID, ID), 1 ms] (212) CdtProblem ---------------------------------------- (0) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: max(nil) -> 0 max(cons(x, nil)) -> x max(cons(x, cons(y, xs))) -> if1(ge(x, y), x, y, xs) if1(true, x, y, xs) -> max(cons(x, xs)) if1(false, x, y, xs) -> max(cons(y, xs)) del(x, nil) -> nil del(x, cons(y, xs)) -> if2(eq(x, y), x, y, xs) if2(true, x, y, xs) -> xs if2(false, x, y, xs) -> cons(y, del(x, xs)) eq(0, 0) -> true eq(0, s(y)) -> false eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) sort(xs) -> if3(empty(xs), xs) if3(true, xs) -> nil if3(false, xs) -> sort(del(max(xs), xs)) empty(nil) -> true empty(cons(x, xs)) -> false ge(x, 0) -> true ge(0, s(x)) -> false ge(s(x), s(y)) -> ge(x, y) S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (1) RenamingProof (BOTH BOUNDS(ID, ID)) Renamed function symbols to avoid clashes with predefined symbol. ---------------------------------------- (2) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: max(nil) -> 0' max(cons(x, nil)) -> x max(cons(x, cons(y, xs))) -> if1(ge(x, y), x, y, xs) if1(true, x, y, xs) -> max(cons(x, xs)) if1(false, x, y, xs) -> max(cons(y, xs)) del(x, nil) -> nil del(x, cons(y, xs)) -> if2(eq(x, y), x, y, xs) if2(true, x, y, xs) -> xs if2(false, x, y, xs) -> cons(y, del(x, xs)) eq(0', 0') -> true eq(0', s(y)) -> false eq(s(x), 0') -> false eq(s(x), s(y)) -> eq(x, y) sort(xs) -> if3(empty(xs), xs) if3(true, xs) -> nil if3(false, xs) -> sort(del(max(xs), xs)) empty(nil) -> true empty(cons(x, xs)) -> false ge(x, 0') -> true ge(0', s(x)) -> false ge(s(x), s(y)) -> ge(x, y) S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (3) RelTrsToTrsProof (UPPER BOUND(ID)) transformed relative TRS to TRS ---------------------------------------- (4) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: max(nil) -> 0 max(cons(x, nil)) -> x max(cons(x, cons(y, xs))) -> if1(ge(x, y), x, y, xs) if1(true, x, y, xs) -> max(cons(x, xs)) if1(false, x, y, xs) -> max(cons(y, xs)) del(x, nil) -> nil del(x, cons(y, xs)) -> if2(eq(x, y), x, y, xs) if2(true, x, y, xs) -> xs if2(false, x, y, xs) -> cons(y, del(x, xs)) eq(0, 0) -> true eq(0, s(y)) -> false eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) sort(xs) -> if3(empty(xs), xs) if3(true, xs) -> nil if3(false, xs) -> sort(del(max(xs), xs)) empty(nil) -> true empty(cons(x, xs)) -> false ge(x, 0) -> true ge(0, s(x)) -> false ge(s(x), s(y)) -> ge(x, y) S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (5) RelTrsToWeightedTrsProof (UPPER BOUND(ID)) Transformed relative TRS to weighted TRS ---------------------------------------- (6) Obligation: The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: max(nil) -> 0 [1] max(cons(x, nil)) -> x [1] max(cons(x, cons(y, xs))) -> if1(ge(x, y), x, y, xs) [1] if1(true, x, y, xs) -> max(cons(x, xs)) [1] if1(false, x, y, xs) -> max(cons(y, xs)) [1] del(x, nil) -> nil [1] del(x, cons(y, xs)) -> if2(eq(x, y), x, y, xs) [1] if2(true, x, y, xs) -> xs [1] if2(false, x, y, xs) -> cons(y, del(x, xs)) [1] eq(0, 0) -> true [1] eq(0, s(y)) -> false [1] eq(s(x), 0) -> false [1] eq(s(x), s(y)) -> eq(x, y) [1] sort(xs) -> if3(empty(xs), xs) [1] if3(true, xs) -> nil [1] if3(false, xs) -> sort(del(max(xs), xs)) [1] empty(nil) -> true [1] empty(cons(x, xs)) -> false [1] ge(x, 0) -> true [1] ge(0, s(x)) -> false [1] ge(s(x), s(y)) -> ge(x, y) [1] Rewrite Strategy: INNERMOST ---------------------------------------- (7) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (8) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: max(nil) -> 0 [1] max(cons(x, nil)) -> x [1] max(cons(x, cons(y, xs))) -> if1(ge(x, y), x, y, xs) [1] if1(true, x, y, xs) -> max(cons(x, xs)) [1] if1(false, x, y, xs) -> max(cons(y, xs)) [1] del(x, nil) -> nil [1] del(x, cons(y, xs)) -> if2(eq(x, y), x, y, xs) [1] if2(true, x, y, xs) -> xs [1] if2(false, x, y, xs) -> cons(y, del(x, xs)) [1] eq(0, 0) -> true [1] eq(0, s(y)) -> false [1] eq(s(x), 0) -> false [1] eq(s(x), s(y)) -> eq(x, y) [1] sort(xs) -> if3(empty(xs), xs) [1] if3(true, xs) -> nil [1] if3(false, xs) -> sort(del(max(xs), xs)) [1] empty(nil) -> true [1] empty(cons(x, xs)) -> false [1] ge(x, 0) -> true [1] ge(0, s(x)) -> false [1] ge(s(x), s(y)) -> ge(x, y) [1] The TRS has the following type information: max :: nil:cons -> 0:s nil :: nil:cons 0 :: 0:s cons :: 0:s -> nil:cons -> nil:cons if1 :: true:false -> 0:s -> 0:s -> nil:cons -> 0:s ge :: 0:s -> 0:s -> true:false true :: true:false false :: true:false del :: 0:s -> nil:cons -> nil:cons if2 :: true:false -> 0:s -> 0:s -> nil:cons -> nil:cons eq :: 0:s -> 0:s -> true:false s :: 0:s -> 0:s sort :: nil:cons -> nil:cons if3 :: true:false -> nil:cons -> nil:cons empty :: nil:cons -> true:false Rewrite Strategy: INNERMOST ---------------------------------------- (9) CompletionProof (UPPER BOUND(ID)) The transformation into a RNTS is sound, since: (a) The obligation is a constructor system where every type has a constant constructor, (b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols: sort_1 if3_2 (c) The following functions are completely defined: del_2 max_1 eq_2 empty_1 ge_2 if2_4 if1_4 Due to the following rules being added: none And the following fresh constants: none ---------------------------------------- (10) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: max(nil) -> 0 [1] max(cons(x, nil)) -> x [1] max(cons(x, cons(y, xs))) -> if1(ge(x, y), x, y, xs) [1] if1(true, x, y, xs) -> max(cons(x, xs)) [1] if1(false, x, y, xs) -> max(cons(y, xs)) [1] del(x, nil) -> nil [1] del(x, cons(y, xs)) -> if2(eq(x, y), x, y, xs) [1] if2(true, x, y, xs) -> xs [1] if2(false, x, y, xs) -> cons(y, del(x, xs)) [1] eq(0, 0) -> true [1] eq(0, s(y)) -> false [1] eq(s(x), 0) -> false [1] eq(s(x), s(y)) -> eq(x, y) [1] sort(xs) -> if3(empty(xs), xs) [1] if3(true, xs) -> nil [1] if3(false, xs) -> sort(del(max(xs), xs)) [1] empty(nil) -> true [1] empty(cons(x, xs)) -> false [1] ge(x, 0) -> true [1] ge(0, s(x)) -> false [1] ge(s(x), s(y)) -> ge(x, y) [1] The TRS has the following type information: max :: nil:cons -> 0:s nil :: nil:cons 0 :: 0:s cons :: 0:s -> nil:cons -> nil:cons if1 :: true:false -> 0:s -> 0:s -> nil:cons -> 0:s ge :: 0:s -> 0:s -> true:false true :: true:false false :: true:false del :: 0:s -> nil:cons -> nil:cons if2 :: true:false -> 0:s -> 0:s -> nil:cons -> nil:cons eq :: 0:s -> 0:s -> true:false s :: 0:s -> 0:s sort :: nil:cons -> nil:cons if3 :: true:false -> nil:cons -> nil:cons empty :: nil:cons -> true:false Rewrite Strategy: INNERMOST ---------------------------------------- (11) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (12) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: max(nil) -> 0 [1] max(cons(x, nil)) -> x [1] max(cons(x, cons(0, xs))) -> if1(true, x, 0, xs) [2] max(cons(0, cons(s(x'), xs))) -> if1(false, 0, s(x'), xs) [2] max(cons(s(x''), cons(s(y'), xs))) -> if1(ge(x'', y'), s(x''), s(y'), xs) [2] if1(true, x, y, xs) -> max(cons(x, xs)) [1] if1(false, x, y, xs) -> max(cons(y, xs)) [1] del(x, nil) -> nil [1] del(0, cons(0, xs)) -> if2(true, 0, 0, xs) [2] del(0, cons(s(y''), xs)) -> if2(false, 0, s(y''), xs) [2] del(s(x1), cons(0, xs)) -> if2(false, s(x1), 0, xs) [2] del(s(x2), cons(s(y1), xs)) -> if2(eq(x2, y1), s(x2), s(y1), xs) [2] if2(true, x, y, xs) -> xs [1] if2(false, x, y, xs) -> cons(y, del(x, xs)) [1] eq(0, 0) -> true [1] eq(0, s(y)) -> false [1] eq(s(x), 0) -> false [1] eq(s(x), s(y)) -> eq(x, y) [1] sort(nil) -> if3(true, nil) [2] sort(cons(x3, xs')) -> if3(false, cons(x3, xs')) [2] if3(true, xs) -> nil [1] if3(false, nil) -> sort(del(0, nil)) [2] if3(false, cons(x4, nil)) -> sort(del(x4, cons(x4, nil))) [2] if3(false, cons(x5, cons(y2, xs''))) -> sort(del(if1(ge(x5, y2), x5, y2, xs''), cons(x5, cons(y2, xs'')))) [2] empty(nil) -> true [1] empty(cons(x, xs)) -> false [1] ge(x, 0) -> true [1] ge(0, s(x)) -> false [1] ge(s(x), s(y)) -> ge(x, y) [1] The TRS has the following type information: max :: nil:cons -> 0:s nil :: nil:cons 0 :: 0:s cons :: 0:s -> nil:cons -> nil:cons if1 :: true:false -> 0:s -> 0:s -> nil:cons -> 0:s ge :: 0:s -> 0:s -> true:false true :: true:false false :: true:false del :: 0:s -> nil:cons -> nil:cons if2 :: true:false -> 0:s -> 0:s -> nil:cons -> nil:cons eq :: 0:s -> 0:s -> true:false s :: 0:s -> 0:s sort :: nil:cons -> nil:cons if3 :: true:false -> nil:cons -> nil:cons empty :: nil:cons -> true:false Rewrite Strategy: INNERMOST ---------------------------------------- (13) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: nil => 0 0 => 0 true => 1 false => 0 ---------------------------------------- (14) Obligation: Complexity RNTS consisting of the following rules: del(z, z') -{ 2 }-> if2(eq(x2, y1), 1 + x2, 1 + y1, xs) :|: y1 >= 0, xs >= 0, z' = 1 + (1 + y1) + xs, z = 1 + x2, x2 >= 0 del(z, z') -{ 2 }-> if2(1, 0, 0, xs) :|: xs >= 0, z' = 1 + 0 + xs, z = 0 del(z, z') -{ 2 }-> if2(0, 0, 1 + y'', xs) :|: xs >= 0, z' = 1 + (1 + y'') + xs, y'' >= 0, z = 0 del(z, z') -{ 2 }-> if2(0, 1 + x1, 0, xs) :|: xs >= 0, x1 >= 0, z' = 1 + 0 + xs, z = 1 + x1 del(z, z') -{ 1 }-> 0 :|: x >= 0, z = x, z' = 0 empty(z) -{ 1 }-> 1 :|: z = 0 empty(z) -{ 1 }-> 0 :|: z = 1 + x + xs, xs >= 0, x >= 0 eq(z, z') -{ 1 }-> eq(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 eq(z, z') -{ 1 }-> 0 :|: z' = 1 + y, y >= 0, z = 0 eq(z, z') -{ 1 }-> 0 :|: x >= 0, z = 1 + x, z' = 0 ge(z, z') -{ 1 }-> ge(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x ge(z, z') -{ 1 }-> 1 :|: x >= 0, z = x, z' = 0 ge(z, z') -{ 1 }-> 0 :|: z' = 1 + x, x >= 0, z = 0 if1(z, z', z'', z1) -{ 1 }-> max(1 + x + xs) :|: xs >= 0, z' = x, z'' = y, z = 1, x >= 0, y >= 0, z1 = xs if1(z, z', z'', z1) -{ 1 }-> max(1 + y + xs) :|: xs >= 0, z' = x, z'' = y, x >= 0, y >= 0, z = 0, z1 = xs if2(z, z', z'', z1) -{ 1 }-> xs :|: xs >= 0, z' = x, z'' = y, z = 1, x >= 0, y >= 0, z1 = xs if2(z, z', z'', z1) -{ 1 }-> 1 + y + del(x, xs) :|: xs >= 0, z' = x, z'' = y, x >= 0, y >= 0, z = 0, z1 = xs if3(z, z') -{ 2 }-> sort(del(x4, 1 + x4 + 0)) :|: x4 >= 0, z = 0, z' = 1 + x4 + 0 if3(z, z') -{ 2 }-> sort(del(if1(ge(x5, y2), x5, y2, xs''), 1 + x5 + (1 + y2 + xs''))) :|: z' = 1 + x5 + (1 + y2 + xs''), x5 >= 0, xs'' >= 0, y2 >= 0, z = 0 if3(z, z') -{ 2 }-> sort(del(0, 0)) :|: z = 0, z' = 0 if3(z, z') -{ 1 }-> 0 :|: xs >= 0, z = 1, z' = xs max(z) -{ 1 }-> x :|: x >= 0, z = 1 + x + 0 max(z) -{ 2 }-> if1(ge(x'', y'), 1 + x'', 1 + y', xs) :|: xs >= 0, z = 1 + (1 + x'') + (1 + (1 + y') + xs), y' >= 0, x'' >= 0 max(z) -{ 2 }-> if1(1, x, 0, xs) :|: xs >= 0, x >= 0, z = 1 + x + (1 + 0 + xs) max(z) -{ 2 }-> if1(0, 0, 1 + x', xs) :|: z = 1 + 0 + (1 + (1 + x') + xs), xs >= 0, x' >= 0 max(z) -{ 1 }-> 0 :|: z = 0 sort(z) -{ 2 }-> if3(1, 0) :|: z = 0 sort(z) -{ 2 }-> if3(0, 1 + x3 + xs') :|: xs' >= 0, z = 1 + x3 + xs', x3 >= 0 ---------------------------------------- (15) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (16) Obligation: Complexity RNTS consisting of the following rules: del(z, z') -{ 2 }-> if2(eq(z - 1, y1), 1 + (z - 1), 1 + y1, xs) :|: y1 >= 0, xs >= 0, z' = 1 + (1 + y1) + xs, z - 1 >= 0 del(z, z') -{ 2 }-> if2(1, 0, 0, z' - 1) :|: z' - 1 >= 0, z = 0 del(z, z') -{ 2 }-> if2(0, 0, 1 + y'', xs) :|: xs >= 0, z' = 1 + (1 + y'') + xs, y'' >= 0, z = 0 del(z, z') -{ 2 }-> if2(0, 1 + (z - 1), 0, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 del(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 empty(z) -{ 1 }-> 1 :|: z = 0 empty(z) -{ 1 }-> 0 :|: z = 1 + x + xs, xs >= 0, x >= 0 eq(z, z') -{ 1 }-> eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 eq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 eq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 ge(z, z') -{ 1 }-> ge(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 ge(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 ge(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 if1(z, z', z'', z1) -{ 1 }-> max(1 + z' + z1) :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0 if1(z, z', z'', z1) -{ 1 }-> max(1 + z'' + z1) :|: z1 >= 0, z' >= 0, z'' >= 0, z = 0 if2(z, z', z'', z1) -{ 1 }-> z1 :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0 if2(z, z', z'', z1) -{ 1 }-> 1 + z'' + del(z', z1) :|: z1 >= 0, z' >= 0, z'' >= 0, z = 0 if3(z, z') -{ 2 }-> sort(del(if1(ge(x5, y2), x5, y2, xs''), 1 + x5 + (1 + y2 + xs''))) :|: z' = 1 + x5 + (1 + y2 + xs''), x5 >= 0, xs'' >= 0, y2 >= 0, z = 0 if3(z, z') -{ 2 }-> sort(del(0, 0)) :|: z = 0, z' = 0 if3(z, z') -{ 2 }-> sort(del(z' - 1, 1 + (z' - 1) + 0)) :|: z' - 1 >= 0, z = 0 if3(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 1 max(z) -{ 2 }-> if1(ge(x'', y'), 1 + x'', 1 + y', xs) :|: xs >= 0, z = 1 + (1 + x'') + (1 + (1 + y') + xs), y' >= 0, x'' >= 0 max(z) -{ 2 }-> if1(1, x, 0, xs) :|: xs >= 0, x >= 0, z = 1 + x + (1 + 0 + xs) max(z) -{ 2 }-> if1(0, 0, 1 + x', xs) :|: z = 1 + 0 + (1 + (1 + x') + xs), xs >= 0, x' >= 0 max(z) -{ 1 }-> 0 :|: z = 0 max(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 sort(z) -{ 2 }-> if3(1, 0) :|: z = 0 sort(z) -{ 2 }-> if3(0, 1 + x3 + xs') :|: xs' >= 0, z = 1 + x3 + xs', x3 >= 0 ---------------------------------------- (17) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { empty } { eq } { ge } { del, if2 } { max, if1 } { sort, if3 } ---------------------------------------- (18) Obligation: Complexity RNTS consisting of the following rules: del(z, z') -{ 2 }-> if2(eq(z - 1, y1), 1 + (z - 1), 1 + y1, xs) :|: y1 >= 0, xs >= 0, z' = 1 + (1 + y1) + xs, z - 1 >= 0 del(z, z') -{ 2 }-> if2(1, 0, 0, z' - 1) :|: z' - 1 >= 0, z = 0 del(z, z') -{ 2 }-> if2(0, 0, 1 + y'', xs) :|: xs >= 0, z' = 1 + (1 + y'') + xs, y'' >= 0, z = 0 del(z, z') -{ 2 }-> if2(0, 1 + (z - 1), 0, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 del(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 empty(z) -{ 1 }-> 1 :|: z = 0 empty(z) -{ 1 }-> 0 :|: z = 1 + x + xs, xs >= 0, x >= 0 eq(z, z') -{ 1 }-> eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 eq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 eq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 ge(z, z') -{ 1 }-> ge(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 ge(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 ge(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 if1(z, z', z'', z1) -{ 1 }-> max(1 + z' + z1) :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0 if1(z, z', z'', z1) -{ 1 }-> max(1 + z'' + z1) :|: z1 >= 0, z' >= 0, z'' >= 0, z = 0 if2(z, z', z'', z1) -{ 1 }-> z1 :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0 if2(z, z', z'', z1) -{ 1 }-> 1 + z'' + del(z', z1) :|: z1 >= 0, z' >= 0, z'' >= 0, z = 0 if3(z, z') -{ 2 }-> sort(del(if1(ge(x5, y2), x5, y2, xs''), 1 + x5 + (1 + y2 + xs''))) :|: z' = 1 + x5 + (1 + y2 + xs''), x5 >= 0, xs'' >= 0, y2 >= 0, z = 0 if3(z, z') -{ 2 }-> sort(del(0, 0)) :|: z = 0, z' = 0 if3(z, z') -{ 2 }-> sort(del(z' - 1, 1 + (z' - 1) + 0)) :|: z' - 1 >= 0, z = 0 if3(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 1 max(z) -{ 2 }-> if1(ge(x'', y'), 1 + x'', 1 + y', xs) :|: xs >= 0, z = 1 + (1 + x'') + (1 + (1 + y') + xs), y' >= 0, x'' >= 0 max(z) -{ 2 }-> if1(1, x, 0, xs) :|: xs >= 0, x >= 0, z = 1 + x + (1 + 0 + xs) max(z) -{ 2 }-> if1(0, 0, 1 + x', xs) :|: z = 1 + 0 + (1 + (1 + x') + xs), xs >= 0, x' >= 0 max(z) -{ 1 }-> 0 :|: z = 0 max(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 sort(z) -{ 2 }-> if3(1, 0) :|: z = 0 sort(z) -{ 2 }-> if3(0, 1 + x3 + xs') :|: xs' >= 0, z = 1 + x3 + xs', x3 >= 0 Function symbols to be analyzed: {empty}, {eq}, {ge}, {del,if2}, {max,if1}, {sort,if3} ---------------------------------------- (19) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: del(z, z') -{ 2 }-> if2(eq(z - 1, y1), 1 + (z - 1), 1 + y1, xs) :|: y1 >= 0, xs >= 0, z' = 1 + (1 + y1) + xs, z - 1 >= 0 del(z, z') -{ 2 }-> if2(1, 0, 0, z' - 1) :|: z' - 1 >= 0, z = 0 del(z, z') -{ 2 }-> if2(0, 0, 1 + y'', xs) :|: xs >= 0, z' = 1 + (1 + y'') + xs, y'' >= 0, z = 0 del(z, z') -{ 2 }-> if2(0, 1 + (z - 1), 0, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 del(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 empty(z) -{ 1 }-> 1 :|: z = 0 empty(z) -{ 1 }-> 0 :|: z = 1 + x + xs, xs >= 0, x >= 0 eq(z, z') -{ 1 }-> eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 eq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 eq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 ge(z, z') -{ 1 }-> ge(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 ge(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 ge(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 if1(z, z', z'', z1) -{ 1 }-> max(1 + z' + z1) :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0 if1(z, z', z'', z1) -{ 1 }-> max(1 + z'' + z1) :|: z1 >= 0, z' >= 0, z'' >= 0, z = 0 if2(z, z', z'', z1) -{ 1 }-> z1 :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0 if2(z, z', z'', z1) -{ 1 }-> 1 + z'' + del(z', z1) :|: z1 >= 0, z' >= 0, z'' >= 0, z = 0 if3(z, z') -{ 2 }-> sort(del(if1(ge(x5, y2), x5, y2, xs''), 1 + x5 + (1 + y2 + xs''))) :|: z' = 1 + x5 + (1 + y2 + xs''), x5 >= 0, xs'' >= 0, y2 >= 0, z = 0 if3(z, z') -{ 2 }-> sort(del(0, 0)) :|: z = 0, z' = 0 if3(z, z') -{ 2 }-> sort(del(z' - 1, 1 + (z' - 1) + 0)) :|: z' - 1 >= 0, z = 0 if3(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 1 max(z) -{ 2 }-> if1(ge(x'', y'), 1 + x'', 1 + y', xs) :|: xs >= 0, z = 1 + (1 + x'') + (1 + (1 + y') + xs), y' >= 0, x'' >= 0 max(z) -{ 2 }-> if1(1, x, 0, xs) :|: xs >= 0, x >= 0, z = 1 + x + (1 + 0 + xs) max(z) -{ 2 }-> if1(0, 0, 1 + x', xs) :|: z = 1 + 0 + (1 + (1 + x') + xs), xs >= 0, x' >= 0 max(z) -{ 1 }-> 0 :|: z = 0 max(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 sort(z) -{ 2 }-> if3(1, 0) :|: z = 0 sort(z) -{ 2 }-> if3(0, 1 + x3 + xs') :|: xs' >= 0, z = 1 + x3 + xs', x3 >= 0 Function symbols to be analyzed: {empty}, {eq}, {ge}, {del,if2}, {max,if1}, {sort,if3} ---------------------------------------- (21) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: empty after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (22) Obligation: Complexity RNTS consisting of the following rules: del(z, z') -{ 2 }-> if2(eq(z - 1, y1), 1 + (z - 1), 1 + y1, xs) :|: y1 >= 0, xs >= 0, z' = 1 + (1 + y1) + xs, z - 1 >= 0 del(z, z') -{ 2 }-> if2(1, 0, 0, z' - 1) :|: z' - 1 >= 0, z = 0 del(z, z') -{ 2 }-> if2(0, 0, 1 + y'', xs) :|: xs >= 0, z' = 1 + (1 + y'') + xs, y'' >= 0, z = 0 del(z, z') -{ 2 }-> if2(0, 1 + (z - 1), 0, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 del(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 empty(z) -{ 1 }-> 1 :|: z = 0 empty(z) -{ 1 }-> 0 :|: z = 1 + x + xs, xs >= 0, x >= 0 eq(z, z') -{ 1 }-> eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 eq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 eq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 ge(z, z') -{ 1 }-> ge(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 ge(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 ge(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 if1(z, z', z'', z1) -{ 1 }-> max(1 + z' + z1) :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0 if1(z, z', z'', z1) -{ 1 }-> max(1 + z'' + z1) :|: z1 >= 0, z' >= 0, z'' >= 0, z = 0 if2(z, z', z'', z1) -{ 1 }-> z1 :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0 if2(z, z', z'', z1) -{ 1 }-> 1 + z'' + del(z', z1) :|: z1 >= 0, z' >= 0, z'' >= 0, z = 0 if3(z, z') -{ 2 }-> sort(del(if1(ge(x5, y2), x5, y2, xs''), 1 + x5 + (1 + y2 + xs''))) :|: z' = 1 + x5 + (1 + y2 + xs''), x5 >= 0, xs'' >= 0, y2 >= 0, z = 0 if3(z, z') -{ 2 }-> sort(del(0, 0)) :|: z = 0, z' = 0 if3(z, z') -{ 2 }-> sort(del(z' - 1, 1 + (z' - 1) + 0)) :|: z' - 1 >= 0, z = 0 if3(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 1 max(z) -{ 2 }-> if1(ge(x'', y'), 1 + x'', 1 + y', xs) :|: xs >= 0, z = 1 + (1 + x'') + (1 + (1 + y') + xs), y' >= 0, x'' >= 0 max(z) -{ 2 }-> if1(1, x, 0, xs) :|: xs >= 0, x >= 0, z = 1 + x + (1 + 0 + xs) max(z) -{ 2 }-> if1(0, 0, 1 + x', xs) :|: z = 1 + 0 + (1 + (1 + x') + xs), xs >= 0, x' >= 0 max(z) -{ 1 }-> 0 :|: z = 0 max(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 sort(z) -{ 2 }-> if3(1, 0) :|: z = 0 sort(z) -{ 2 }-> if3(0, 1 + x3 + xs') :|: xs' >= 0, z = 1 + x3 + xs', x3 >= 0 Function symbols to be analyzed: {empty}, {eq}, {ge}, {del,if2}, {max,if1}, {sort,if3} Previous analysis results are: empty: runtime: ?, size: O(1) [1] ---------------------------------------- (23) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: empty after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (24) Obligation: Complexity RNTS consisting of the following rules: del(z, z') -{ 2 }-> if2(eq(z - 1, y1), 1 + (z - 1), 1 + y1, xs) :|: y1 >= 0, xs >= 0, z' = 1 + (1 + y1) + xs, z - 1 >= 0 del(z, z') -{ 2 }-> if2(1, 0, 0, z' - 1) :|: z' - 1 >= 0, z = 0 del(z, z') -{ 2 }-> if2(0, 0, 1 + y'', xs) :|: xs >= 0, z' = 1 + (1 + y'') + xs, y'' >= 0, z = 0 del(z, z') -{ 2 }-> if2(0, 1 + (z - 1), 0, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 del(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 empty(z) -{ 1 }-> 1 :|: z = 0 empty(z) -{ 1 }-> 0 :|: z = 1 + x + xs, xs >= 0, x >= 0 eq(z, z') -{ 1 }-> eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 eq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 eq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 ge(z, z') -{ 1 }-> ge(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 ge(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 ge(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 if1(z, z', z'', z1) -{ 1 }-> max(1 + z' + z1) :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0 if1(z, z', z'', z1) -{ 1 }-> max(1 + z'' + z1) :|: z1 >= 0, z' >= 0, z'' >= 0, z = 0 if2(z, z', z'', z1) -{ 1 }-> z1 :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0 if2(z, z', z'', z1) -{ 1 }-> 1 + z'' + del(z', z1) :|: z1 >= 0, z' >= 0, z'' >= 0, z = 0 if3(z, z') -{ 2 }-> sort(del(if1(ge(x5, y2), x5, y2, xs''), 1 + x5 + (1 + y2 + xs''))) :|: z' = 1 + x5 + (1 + y2 + xs''), x5 >= 0, xs'' >= 0, y2 >= 0, z = 0 if3(z, z') -{ 2 }-> sort(del(0, 0)) :|: z = 0, z' = 0 if3(z, z') -{ 2 }-> sort(del(z' - 1, 1 + (z' - 1) + 0)) :|: z' - 1 >= 0, z = 0 if3(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 1 max(z) -{ 2 }-> if1(ge(x'', y'), 1 + x'', 1 + y', xs) :|: xs >= 0, z = 1 + (1 + x'') + (1 + (1 + y') + xs), y' >= 0, x'' >= 0 max(z) -{ 2 }-> if1(1, x, 0, xs) :|: xs >= 0, x >= 0, z = 1 + x + (1 + 0 + xs) max(z) -{ 2 }-> if1(0, 0, 1 + x', xs) :|: z = 1 + 0 + (1 + (1 + x') + xs), xs >= 0, x' >= 0 max(z) -{ 1 }-> 0 :|: z = 0 max(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 sort(z) -{ 2 }-> if3(1, 0) :|: z = 0 sort(z) -{ 2 }-> if3(0, 1 + x3 + xs') :|: xs' >= 0, z = 1 + x3 + xs', x3 >= 0 Function symbols to be analyzed: {eq}, {ge}, {del,if2}, {max,if1}, {sort,if3} Previous analysis results are: empty: runtime: O(1) [1], size: O(1) [1] ---------------------------------------- (25) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (26) Obligation: Complexity RNTS consisting of the following rules: del(z, z') -{ 2 }-> if2(eq(z - 1, y1), 1 + (z - 1), 1 + y1, xs) :|: y1 >= 0, xs >= 0, z' = 1 + (1 + y1) + xs, z - 1 >= 0 del(z, z') -{ 2 }-> if2(1, 0, 0, z' - 1) :|: z' - 1 >= 0, z = 0 del(z, z') -{ 2 }-> if2(0, 0, 1 + y'', xs) :|: xs >= 0, z' = 1 + (1 + y'') + xs, y'' >= 0, z = 0 del(z, z') -{ 2 }-> if2(0, 1 + (z - 1), 0, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 del(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 empty(z) -{ 1 }-> 1 :|: z = 0 empty(z) -{ 1 }-> 0 :|: z = 1 + x + xs, xs >= 0, x >= 0 eq(z, z') -{ 1 }-> eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 eq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 eq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 ge(z, z') -{ 1 }-> ge(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 ge(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 ge(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 if1(z, z', z'', z1) -{ 1 }-> max(1 + z' + z1) :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0 if1(z, z', z'', z1) -{ 1 }-> max(1 + z'' + z1) :|: z1 >= 0, z' >= 0, z'' >= 0, z = 0 if2(z, z', z'', z1) -{ 1 }-> z1 :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0 if2(z, z', z'', z1) -{ 1 }-> 1 + z'' + del(z', z1) :|: z1 >= 0, z' >= 0, z'' >= 0, z = 0 if3(z, z') -{ 2 }-> sort(del(if1(ge(x5, y2), x5, y2, xs''), 1 + x5 + (1 + y2 + xs''))) :|: z' = 1 + x5 + (1 + y2 + xs''), x5 >= 0, xs'' >= 0, y2 >= 0, z = 0 if3(z, z') -{ 2 }-> sort(del(0, 0)) :|: z = 0, z' = 0 if3(z, z') -{ 2 }-> sort(del(z' - 1, 1 + (z' - 1) + 0)) :|: z' - 1 >= 0, z = 0 if3(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 1 max(z) -{ 2 }-> if1(ge(x'', y'), 1 + x'', 1 + y', xs) :|: xs >= 0, z = 1 + (1 + x'') + (1 + (1 + y') + xs), y' >= 0, x'' >= 0 max(z) -{ 2 }-> if1(1, x, 0, xs) :|: xs >= 0, x >= 0, z = 1 + x + (1 + 0 + xs) max(z) -{ 2 }-> if1(0, 0, 1 + x', xs) :|: z = 1 + 0 + (1 + (1 + x') + xs), xs >= 0, x' >= 0 max(z) -{ 1 }-> 0 :|: z = 0 max(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 sort(z) -{ 2 }-> if3(1, 0) :|: z = 0 sort(z) -{ 2 }-> if3(0, 1 + x3 + xs') :|: xs' >= 0, z = 1 + x3 + xs', x3 >= 0 Function symbols to be analyzed: {eq}, {ge}, {del,if2}, {max,if1}, {sort,if3} Previous analysis results are: empty: runtime: O(1) [1], size: O(1) [1] ---------------------------------------- (27) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: eq after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: del(z, z') -{ 2 }-> if2(eq(z - 1, y1), 1 + (z - 1), 1 + y1, xs) :|: y1 >= 0, xs >= 0, z' = 1 + (1 + y1) + xs, z - 1 >= 0 del(z, z') -{ 2 }-> if2(1, 0, 0, z' - 1) :|: z' - 1 >= 0, z = 0 del(z, z') -{ 2 }-> if2(0, 0, 1 + y'', xs) :|: xs >= 0, z' = 1 + (1 + y'') + xs, y'' >= 0, z = 0 del(z, z') -{ 2 }-> if2(0, 1 + (z - 1), 0, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 del(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 empty(z) -{ 1 }-> 1 :|: z = 0 empty(z) -{ 1 }-> 0 :|: z = 1 + x + xs, xs >= 0, x >= 0 eq(z, z') -{ 1 }-> eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 eq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 eq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 ge(z, z') -{ 1 }-> ge(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 ge(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 ge(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 if1(z, z', z'', z1) -{ 1 }-> max(1 + z' + z1) :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0 if1(z, z', z'', z1) -{ 1 }-> max(1 + z'' + z1) :|: z1 >= 0, z' >= 0, z'' >= 0, z = 0 if2(z, z', z'', z1) -{ 1 }-> z1 :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0 if2(z, z', z'', z1) -{ 1 }-> 1 + z'' + del(z', z1) :|: z1 >= 0, z' >= 0, z'' >= 0, z = 0 if3(z, z') -{ 2 }-> sort(del(if1(ge(x5, y2), x5, y2, xs''), 1 + x5 + (1 + y2 + xs''))) :|: z' = 1 + x5 + (1 + y2 + xs''), x5 >= 0, xs'' >= 0, y2 >= 0, z = 0 if3(z, z') -{ 2 }-> sort(del(0, 0)) :|: z = 0, z' = 0 if3(z, z') -{ 2 }-> sort(del(z' - 1, 1 + (z' - 1) + 0)) :|: z' - 1 >= 0, z = 0 if3(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 1 max(z) -{ 2 }-> if1(ge(x'', y'), 1 + x'', 1 + y', xs) :|: xs >= 0, z = 1 + (1 + x'') + (1 + (1 + y') + xs), y' >= 0, x'' >= 0 max(z) -{ 2 }-> if1(1, x, 0, xs) :|: xs >= 0, x >= 0, z = 1 + x + (1 + 0 + xs) max(z) -{ 2 }-> if1(0, 0, 1 + x', xs) :|: z = 1 + 0 + (1 + (1 + x') + xs), xs >= 0, x' >= 0 max(z) -{ 1 }-> 0 :|: z = 0 max(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 sort(z) -{ 2 }-> if3(1, 0) :|: z = 0 sort(z) -{ 2 }-> if3(0, 1 + x3 + xs') :|: xs' >= 0, z = 1 + x3 + xs', x3 >= 0 Function symbols to be analyzed: {eq}, {ge}, {del,if2}, {max,if1}, {sort,if3} Previous analysis results are: empty: runtime: O(1) [1], size: O(1) [1] eq: runtime: ?, size: O(1) [1] ---------------------------------------- (29) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: eq after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 3 + z' ---------------------------------------- (30) Obligation: Complexity RNTS consisting of the following rules: del(z, z') -{ 2 }-> if2(eq(z - 1, y1), 1 + (z - 1), 1 + y1, xs) :|: y1 >= 0, xs >= 0, z' = 1 + (1 + y1) + xs, z - 1 >= 0 del(z, z') -{ 2 }-> if2(1, 0, 0, z' - 1) :|: z' - 1 >= 0, z = 0 del(z, z') -{ 2 }-> if2(0, 0, 1 + y'', xs) :|: xs >= 0, z' = 1 + (1 + y'') + xs, y'' >= 0, z = 0 del(z, z') -{ 2 }-> if2(0, 1 + (z - 1), 0, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 del(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 empty(z) -{ 1 }-> 1 :|: z = 0 empty(z) -{ 1 }-> 0 :|: z = 1 + x + xs, xs >= 0, x >= 0 eq(z, z') -{ 1 }-> eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 eq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 eq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 ge(z, z') -{ 1 }-> ge(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 ge(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 ge(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 if1(z, z', z'', z1) -{ 1 }-> max(1 + z' + z1) :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0 if1(z, z', z'', z1) -{ 1 }-> max(1 + z'' + z1) :|: z1 >= 0, z' >= 0, z'' >= 0, z = 0 if2(z, z', z'', z1) -{ 1 }-> z1 :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0 if2(z, z', z'', z1) -{ 1 }-> 1 + z'' + del(z', z1) :|: z1 >= 0, z' >= 0, z'' >= 0, z = 0 if3(z, z') -{ 2 }-> sort(del(if1(ge(x5, y2), x5, y2, xs''), 1 + x5 + (1 + y2 + xs''))) :|: z' = 1 + x5 + (1 + y2 + xs''), x5 >= 0, xs'' >= 0, y2 >= 0, z = 0 if3(z, z') -{ 2 }-> sort(del(0, 0)) :|: z = 0, z' = 0 if3(z, z') -{ 2 }-> sort(del(z' - 1, 1 + (z' - 1) + 0)) :|: z' - 1 >= 0, z = 0 if3(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 1 max(z) -{ 2 }-> if1(ge(x'', y'), 1 + x'', 1 + y', xs) :|: xs >= 0, z = 1 + (1 + x'') + (1 + (1 + y') + xs), y' >= 0, x'' >= 0 max(z) -{ 2 }-> if1(1, x, 0, xs) :|: xs >= 0, x >= 0, z = 1 + x + (1 + 0 + xs) max(z) -{ 2 }-> if1(0, 0, 1 + x', xs) :|: z = 1 + 0 + (1 + (1 + x') + xs), xs >= 0, x' >= 0 max(z) -{ 1 }-> 0 :|: z = 0 max(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 sort(z) -{ 2 }-> if3(1, 0) :|: z = 0 sort(z) -{ 2 }-> if3(0, 1 + x3 + xs') :|: xs' >= 0, z = 1 + x3 + xs', x3 >= 0 Function symbols to be analyzed: {ge}, {del,if2}, {max,if1}, {sort,if3} Previous analysis results are: empty: runtime: O(1) [1], size: O(1) [1] eq: runtime: O(n^1) [3 + z'], size: O(1) [1] ---------------------------------------- (31) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (32) Obligation: Complexity RNTS consisting of the following rules: del(z, z') -{ 5 + y1 }-> if2(s, 1 + (z - 1), 1 + y1, xs) :|: s >= 0, s <= 1, y1 >= 0, xs >= 0, z' = 1 + (1 + y1) + xs, z - 1 >= 0 del(z, z') -{ 2 }-> if2(1, 0, 0, z' - 1) :|: z' - 1 >= 0, z = 0 del(z, z') -{ 2 }-> if2(0, 0, 1 + y'', xs) :|: xs >= 0, z' = 1 + (1 + y'') + xs, y'' >= 0, z = 0 del(z, z') -{ 2 }-> if2(0, 1 + (z - 1), 0, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 del(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 empty(z) -{ 1 }-> 1 :|: z = 0 empty(z) -{ 1 }-> 0 :|: z = 1 + x + xs, xs >= 0, x >= 0 eq(z, z') -{ 3 + z' }-> s' :|: s' >= 0, s' <= 1, z - 1 >= 0, z' - 1 >= 0 eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 eq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 eq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 ge(z, z') -{ 1 }-> ge(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 ge(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 ge(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 if1(z, z', z'', z1) -{ 1 }-> max(1 + z' + z1) :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0 if1(z, z', z'', z1) -{ 1 }-> max(1 + z'' + z1) :|: z1 >= 0, z' >= 0, z'' >= 0, z = 0 if2(z, z', z'', z1) -{ 1 }-> z1 :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0 if2(z, z', z'', z1) -{ 1 }-> 1 + z'' + del(z', z1) :|: z1 >= 0, z' >= 0, z'' >= 0, z = 0 if3(z, z') -{ 2 }-> sort(del(if1(ge(x5, y2), x5, y2, xs''), 1 + x5 + (1 + y2 + xs''))) :|: z' = 1 + x5 + (1 + y2 + xs''), x5 >= 0, xs'' >= 0, y2 >= 0, z = 0 if3(z, z') -{ 2 }-> sort(del(0, 0)) :|: z = 0, z' = 0 if3(z, z') -{ 2 }-> sort(del(z' - 1, 1 + (z' - 1) + 0)) :|: z' - 1 >= 0, z = 0 if3(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 1 max(z) -{ 2 }-> if1(ge(x'', y'), 1 + x'', 1 + y', xs) :|: xs >= 0, z = 1 + (1 + x'') + (1 + (1 + y') + xs), y' >= 0, x'' >= 0 max(z) -{ 2 }-> if1(1, x, 0, xs) :|: xs >= 0, x >= 0, z = 1 + x + (1 + 0 + xs) max(z) -{ 2 }-> if1(0, 0, 1 + x', xs) :|: z = 1 + 0 + (1 + (1 + x') + xs), xs >= 0, x' >= 0 max(z) -{ 1 }-> 0 :|: z = 0 max(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 sort(z) -{ 2 }-> if3(1, 0) :|: z = 0 sort(z) -{ 2 }-> if3(0, 1 + x3 + xs') :|: xs' >= 0, z = 1 + x3 + xs', x3 >= 0 Function symbols to be analyzed: {ge}, {del,if2}, {max,if1}, {sort,if3} Previous analysis results are: empty: runtime: O(1) [1], size: O(1) [1] eq: runtime: O(n^1) [3 + z'], size: O(1) [1] ---------------------------------------- (33) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: ge after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (34) Obligation: Complexity RNTS consisting of the following rules: del(z, z') -{ 5 + y1 }-> if2(s, 1 + (z - 1), 1 + y1, xs) :|: s >= 0, s <= 1, y1 >= 0, xs >= 0, z' = 1 + (1 + y1) + xs, z - 1 >= 0 del(z, z') -{ 2 }-> if2(1, 0, 0, z' - 1) :|: z' - 1 >= 0, z = 0 del(z, z') -{ 2 }-> if2(0, 0, 1 + y'', xs) :|: xs >= 0, z' = 1 + (1 + y'') + xs, y'' >= 0, z = 0 del(z, z') -{ 2 }-> if2(0, 1 + (z - 1), 0, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 del(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 empty(z) -{ 1 }-> 1 :|: z = 0 empty(z) -{ 1 }-> 0 :|: z = 1 + x + xs, xs >= 0, x >= 0 eq(z, z') -{ 3 + z' }-> s' :|: s' >= 0, s' <= 1, z - 1 >= 0, z' - 1 >= 0 eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 eq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 eq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 ge(z, z') -{ 1 }-> ge(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 ge(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 ge(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 if1(z, z', z'', z1) -{ 1 }-> max(1 + z' + z1) :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0 if1(z, z', z'', z1) -{ 1 }-> max(1 + z'' + z1) :|: z1 >= 0, z' >= 0, z'' >= 0, z = 0 if2(z, z', z'', z1) -{ 1 }-> z1 :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0 if2(z, z', z'', z1) -{ 1 }-> 1 + z'' + del(z', z1) :|: z1 >= 0, z' >= 0, z'' >= 0, z = 0 if3(z, z') -{ 2 }-> sort(del(if1(ge(x5, y2), x5, y2, xs''), 1 + x5 + (1 + y2 + xs''))) :|: z' = 1 + x5 + (1 + y2 + xs''), x5 >= 0, xs'' >= 0, y2 >= 0, z = 0 if3(z, z') -{ 2 }-> sort(del(0, 0)) :|: z = 0, z' = 0 if3(z, z') -{ 2 }-> sort(del(z' - 1, 1 + (z' - 1) + 0)) :|: z' - 1 >= 0, z = 0 if3(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 1 max(z) -{ 2 }-> if1(ge(x'', y'), 1 + x'', 1 + y', xs) :|: xs >= 0, z = 1 + (1 + x'') + (1 + (1 + y') + xs), y' >= 0, x'' >= 0 max(z) -{ 2 }-> if1(1, x, 0, xs) :|: xs >= 0, x >= 0, z = 1 + x + (1 + 0 + xs) max(z) -{ 2 }-> if1(0, 0, 1 + x', xs) :|: z = 1 + 0 + (1 + (1 + x') + xs), xs >= 0, x' >= 0 max(z) -{ 1 }-> 0 :|: z = 0 max(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 sort(z) -{ 2 }-> if3(1, 0) :|: z = 0 sort(z) -{ 2 }-> if3(0, 1 + x3 + xs') :|: xs' >= 0, z = 1 + x3 + xs', x3 >= 0 Function symbols to be analyzed: {ge}, {del,if2}, {max,if1}, {sort,if3} Previous analysis results are: empty: runtime: O(1) [1], size: O(1) [1] eq: runtime: O(n^1) [3 + z'], size: O(1) [1] ge: runtime: ?, size: O(1) [1] ---------------------------------------- (35) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: ge after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2 + z' ---------------------------------------- (36) Obligation: Complexity RNTS consisting of the following rules: del(z, z') -{ 5 + y1 }-> if2(s, 1 + (z - 1), 1 + y1, xs) :|: s >= 0, s <= 1, y1 >= 0, xs >= 0, z' = 1 + (1 + y1) + xs, z - 1 >= 0 del(z, z') -{ 2 }-> if2(1, 0, 0, z' - 1) :|: z' - 1 >= 0, z = 0 del(z, z') -{ 2 }-> if2(0, 0, 1 + y'', xs) :|: xs >= 0, z' = 1 + (1 + y'') + xs, y'' >= 0, z = 0 del(z, z') -{ 2 }-> if2(0, 1 + (z - 1), 0, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 del(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 empty(z) -{ 1 }-> 1 :|: z = 0 empty(z) -{ 1 }-> 0 :|: z = 1 + x + xs, xs >= 0, x >= 0 eq(z, z') -{ 3 + z' }-> s' :|: s' >= 0, s' <= 1, z - 1 >= 0, z' - 1 >= 0 eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 eq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 eq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 ge(z, z') -{ 1 }-> ge(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 ge(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 ge(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 if1(z, z', z'', z1) -{ 1 }-> max(1 + z' + z1) :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0 if1(z, z', z'', z1) -{ 1 }-> max(1 + z'' + z1) :|: z1 >= 0, z' >= 0, z'' >= 0, z = 0 if2(z, z', z'', z1) -{ 1 }-> z1 :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0 if2(z, z', z'', z1) -{ 1 }-> 1 + z'' + del(z', z1) :|: z1 >= 0, z' >= 0, z'' >= 0, z = 0 if3(z, z') -{ 2 }-> sort(del(if1(ge(x5, y2), x5, y2, xs''), 1 + x5 + (1 + y2 + xs''))) :|: z' = 1 + x5 + (1 + y2 + xs''), x5 >= 0, xs'' >= 0, y2 >= 0, z = 0 if3(z, z') -{ 2 }-> sort(del(0, 0)) :|: z = 0, z' = 0 if3(z, z') -{ 2 }-> sort(del(z' - 1, 1 + (z' - 1) + 0)) :|: z' - 1 >= 0, z = 0 if3(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 1 max(z) -{ 2 }-> if1(ge(x'', y'), 1 + x'', 1 + y', xs) :|: xs >= 0, z = 1 + (1 + x'') + (1 + (1 + y') + xs), y' >= 0, x'' >= 0 max(z) -{ 2 }-> if1(1, x, 0, xs) :|: xs >= 0, x >= 0, z = 1 + x + (1 + 0 + xs) max(z) -{ 2 }-> if1(0, 0, 1 + x', xs) :|: z = 1 + 0 + (1 + (1 + x') + xs), xs >= 0, x' >= 0 max(z) -{ 1 }-> 0 :|: z = 0 max(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 sort(z) -{ 2 }-> if3(1, 0) :|: z = 0 sort(z) -{ 2 }-> if3(0, 1 + x3 + xs') :|: xs' >= 0, z = 1 + x3 + xs', x3 >= 0 Function symbols to be analyzed: {del,if2}, {max,if1}, {sort,if3} Previous analysis results are: empty: runtime: O(1) [1], size: O(1) [1] eq: runtime: O(n^1) [3 + z'], size: O(1) [1] ge: runtime: O(n^1) [2 + z'], size: O(1) [1] ---------------------------------------- (37) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (38) Obligation: Complexity RNTS consisting of the following rules: del(z, z') -{ 5 + y1 }-> if2(s, 1 + (z - 1), 1 + y1, xs) :|: s >= 0, s <= 1, y1 >= 0, xs >= 0, z' = 1 + (1 + y1) + xs, z - 1 >= 0 del(z, z') -{ 2 }-> if2(1, 0, 0, z' - 1) :|: z' - 1 >= 0, z = 0 del(z, z') -{ 2 }-> if2(0, 0, 1 + y'', xs) :|: xs >= 0, z' = 1 + (1 + y'') + xs, y'' >= 0, z = 0 del(z, z') -{ 2 }-> if2(0, 1 + (z - 1), 0, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 del(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 empty(z) -{ 1 }-> 1 :|: z = 0 empty(z) -{ 1 }-> 0 :|: z = 1 + x + xs, xs >= 0, x >= 0 eq(z, z') -{ 3 + z' }-> s' :|: s' >= 0, s' <= 1, z - 1 >= 0, z' - 1 >= 0 eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 eq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 eq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 ge(z, z') -{ 2 + z' }-> s2 :|: s2 >= 0, s2 <= 1, z - 1 >= 0, z' - 1 >= 0 ge(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 ge(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 if1(z, z', z'', z1) -{ 1 }-> max(1 + z' + z1) :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0 if1(z, z', z'', z1) -{ 1 }-> max(1 + z'' + z1) :|: z1 >= 0, z' >= 0, z'' >= 0, z = 0 if2(z, z', z'', z1) -{ 1 }-> z1 :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0 if2(z, z', z'', z1) -{ 1 }-> 1 + z'' + del(z', z1) :|: z1 >= 0, z' >= 0, z'' >= 0, z = 0 if3(z, z') -{ 4 + y2 }-> sort(del(if1(s1, x5, y2, xs''), 1 + x5 + (1 + y2 + xs''))) :|: s1 >= 0, s1 <= 1, z' = 1 + x5 + (1 + y2 + xs''), x5 >= 0, xs'' >= 0, y2 >= 0, z = 0 if3(z, z') -{ 2 }-> sort(del(0, 0)) :|: z = 0, z' = 0 if3(z, z') -{ 2 }-> sort(del(z' - 1, 1 + (z' - 1) + 0)) :|: z' - 1 >= 0, z = 0 if3(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 1 max(z) -{ 4 + y' }-> if1(s'', 1 + x'', 1 + y', xs) :|: s'' >= 0, s'' <= 1, xs >= 0, z = 1 + (1 + x'') + (1 + (1 + y') + xs), y' >= 0, x'' >= 0 max(z) -{ 2 }-> if1(1, x, 0, xs) :|: xs >= 0, x >= 0, z = 1 + x + (1 + 0 + xs) max(z) -{ 2 }-> if1(0, 0, 1 + x', xs) :|: z = 1 + 0 + (1 + (1 + x') + xs), xs >= 0, x' >= 0 max(z) -{ 1 }-> 0 :|: z = 0 max(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 sort(z) -{ 2 }-> if3(1, 0) :|: z = 0 sort(z) -{ 2 }-> if3(0, 1 + x3 + xs') :|: xs' >= 0, z = 1 + x3 + xs', x3 >= 0 Function symbols to be analyzed: {del,if2}, {max,if1}, {sort,if3} Previous analysis results are: empty: runtime: O(1) [1], size: O(1) [1] eq: runtime: O(n^1) [3 + z'], size: O(1) [1] ge: runtime: O(n^1) [2 + z'], size: O(1) [1] ---------------------------------------- (39) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: del after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z' Computed SIZE bound using CoFloCo for: if2 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z'' + z1 ---------------------------------------- (40) Obligation: Complexity RNTS consisting of the following rules: del(z, z') -{ 5 + y1 }-> if2(s, 1 + (z - 1), 1 + y1, xs) :|: s >= 0, s <= 1, y1 >= 0, xs >= 0, z' = 1 + (1 + y1) + xs, z - 1 >= 0 del(z, z') -{ 2 }-> if2(1, 0, 0, z' - 1) :|: z' - 1 >= 0, z = 0 del(z, z') -{ 2 }-> if2(0, 0, 1 + y'', xs) :|: xs >= 0, z' = 1 + (1 + y'') + xs, y'' >= 0, z = 0 del(z, z') -{ 2 }-> if2(0, 1 + (z - 1), 0, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 del(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 empty(z) -{ 1 }-> 1 :|: z = 0 empty(z) -{ 1 }-> 0 :|: z = 1 + x + xs, xs >= 0, x >= 0 eq(z, z') -{ 3 + z' }-> s' :|: s' >= 0, s' <= 1, z - 1 >= 0, z' - 1 >= 0 eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 eq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 eq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 ge(z, z') -{ 2 + z' }-> s2 :|: s2 >= 0, s2 <= 1, z - 1 >= 0, z' - 1 >= 0 ge(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 ge(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 if1(z, z', z'', z1) -{ 1 }-> max(1 + z' + z1) :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0 if1(z, z', z'', z1) -{ 1 }-> max(1 + z'' + z1) :|: z1 >= 0, z' >= 0, z'' >= 0, z = 0 if2(z, z', z'', z1) -{ 1 }-> z1 :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0 if2(z, z', z'', z1) -{ 1 }-> 1 + z'' + del(z', z1) :|: z1 >= 0, z' >= 0, z'' >= 0, z = 0 if3(z, z') -{ 4 + y2 }-> sort(del(if1(s1, x5, y2, xs''), 1 + x5 + (1 + y2 + xs''))) :|: s1 >= 0, s1 <= 1, z' = 1 + x5 + (1 + y2 + xs''), x5 >= 0, xs'' >= 0, y2 >= 0, z = 0 if3(z, z') -{ 2 }-> sort(del(0, 0)) :|: z = 0, z' = 0 if3(z, z') -{ 2 }-> sort(del(z' - 1, 1 + (z' - 1) + 0)) :|: z' - 1 >= 0, z = 0 if3(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 1 max(z) -{ 4 + y' }-> if1(s'', 1 + x'', 1 + y', xs) :|: s'' >= 0, s'' <= 1, xs >= 0, z = 1 + (1 + x'') + (1 + (1 + y') + xs), y' >= 0, x'' >= 0 max(z) -{ 2 }-> if1(1, x, 0, xs) :|: xs >= 0, x >= 0, z = 1 + x + (1 + 0 + xs) max(z) -{ 2 }-> if1(0, 0, 1 + x', xs) :|: z = 1 + 0 + (1 + (1 + x') + xs), xs >= 0, x' >= 0 max(z) -{ 1 }-> 0 :|: z = 0 max(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 sort(z) -{ 2 }-> if3(1, 0) :|: z = 0 sort(z) -{ 2 }-> if3(0, 1 + x3 + xs') :|: xs' >= 0, z = 1 + x3 + xs', x3 >= 0 Function symbols to be analyzed: {del,if2}, {max,if1}, {sort,if3} Previous analysis results are: empty: runtime: O(1) [1], size: O(1) [1] eq: runtime: O(n^1) [3 + z'], size: O(1) [1] ge: runtime: O(n^1) [2 + z'], size: O(1) [1] del: runtime: ?, size: O(n^1) [z'] if2: runtime: ?, size: O(n^1) [1 + z'' + z1] ---------------------------------------- (41) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: del after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 49 + 17*z' Computed RUNTIME bound using CoFloCo for: if2 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 50 + 17*z1 ---------------------------------------- (42) Obligation: Complexity RNTS consisting of the following rules: del(z, z') -{ 5 + y1 }-> if2(s, 1 + (z - 1), 1 + y1, xs) :|: s >= 0, s <= 1, y1 >= 0, xs >= 0, z' = 1 + (1 + y1) + xs, z - 1 >= 0 del(z, z') -{ 2 }-> if2(1, 0, 0, z' - 1) :|: z' - 1 >= 0, z = 0 del(z, z') -{ 2 }-> if2(0, 0, 1 + y'', xs) :|: xs >= 0, z' = 1 + (1 + y'') + xs, y'' >= 0, z = 0 del(z, z') -{ 2 }-> if2(0, 1 + (z - 1), 0, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 del(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 empty(z) -{ 1 }-> 1 :|: z = 0 empty(z) -{ 1 }-> 0 :|: z = 1 + x + xs, xs >= 0, x >= 0 eq(z, z') -{ 3 + z' }-> s' :|: s' >= 0, s' <= 1, z - 1 >= 0, z' - 1 >= 0 eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 eq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 eq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 ge(z, z') -{ 2 + z' }-> s2 :|: s2 >= 0, s2 <= 1, z - 1 >= 0, z' - 1 >= 0 ge(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 ge(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 if1(z, z', z'', z1) -{ 1 }-> max(1 + z' + z1) :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0 if1(z, z', z'', z1) -{ 1 }-> max(1 + z'' + z1) :|: z1 >= 0, z' >= 0, z'' >= 0, z = 0 if2(z, z', z'', z1) -{ 1 }-> z1 :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0 if2(z, z', z'', z1) -{ 1 }-> 1 + z'' + del(z', z1) :|: z1 >= 0, z' >= 0, z'' >= 0, z = 0 if3(z, z') -{ 4 + y2 }-> sort(del(if1(s1, x5, y2, xs''), 1 + x5 + (1 + y2 + xs''))) :|: s1 >= 0, s1 <= 1, z' = 1 + x5 + (1 + y2 + xs''), x5 >= 0, xs'' >= 0, y2 >= 0, z = 0 if3(z, z') -{ 2 }-> sort(del(0, 0)) :|: z = 0, z' = 0 if3(z, z') -{ 2 }-> sort(del(z' - 1, 1 + (z' - 1) + 0)) :|: z' - 1 >= 0, z = 0 if3(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 1 max(z) -{ 4 + y' }-> if1(s'', 1 + x'', 1 + y', xs) :|: s'' >= 0, s'' <= 1, xs >= 0, z = 1 + (1 + x'') + (1 + (1 + y') + xs), y' >= 0, x'' >= 0 max(z) -{ 2 }-> if1(1, x, 0, xs) :|: xs >= 0, x >= 0, z = 1 + x + (1 + 0 + xs) max(z) -{ 2 }-> if1(0, 0, 1 + x', xs) :|: z = 1 + 0 + (1 + (1 + x') + xs), xs >= 0, x' >= 0 max(z) -{ 1 }-> 0 :|: z = 0 max(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 sort(z) -{ 2 }-> if3(1, 0) :|: z = 0 sort(z) -{ 2 }-> if3(0, 1 + x3 + xs') :|: xs' >= 0, z = 1 + x3 + xs', x3 >= 0 Function symbols to be analyzed: {max,if1}, {sort,if3} Previous analysis results are: empty: runtime: O(1) [1], size: O(1) [1] eq: runtime: O(n^1) [3 + z'], size: O(1) [1] ge: runtime: O(n^1) [2 + z'], size: O(1) [1] del: runtime: O(n^1) [49 + 17*z'], size: O(n^1) [z'] if2: runtime: O(n^1) [50 + 17*z1], size: O(n^1) [1 + z'' + z1] ---------------------------------------- (43) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (44) Obligation: Complexity RNTS consisting of the following rules: del(z, z') -{ 35 + 17*z' }-> s3 :|: s3 >= 0, s3 <= 0 + (z' - 1) + 1, z' - 1 >= 0, z = 0 del(z, z') -{ 52 + 17*xs }-> s4 :|: s4 >= 0, s4 <= 1 + y'' + xs + 1, xs >= 0, z' = 1 + (1 + y'') + xs, y'' >= 0, z = 0 del(z, z') -{ 35 + 17*z' }-> s5 :|: s5 >= 0, s5 <= 0 + (z' - 1) + 1, z' - 1 >= 0, z - 1 >= 0 del(z, z') -{ 55 + 17*xs + y1 }-> s6 :|: s6 >= 0, s6 <= 1 + y1 + xs + 1, s >= 0, s <= 1, y1 >= 0, xs >= 0, z' = 1 + (1 + y1) + xs, z - 1 >= 0 del(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 empty(z) -{ 1 }-> 1 :|: z = 0 empty(z) -{ 1 }-> 0 :|: z = 1 + x + xs, xs >= 0, x >= 0 eq(z, z') -{ 3 + z' }-> s' :|: s' >= 0, s' <= 1, z - 1 >= 0, z' - 1 >= 0 eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 eq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 eq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 ge(z, z') -{ 2 + z' }-> s2 :|: s2 >= 0, s2 <= 1, z - 1 >= 0, z' - 1 >= 0 ge(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 ge(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 if1(z, z', z'', z1) -{ 1 }-> max(1 + z' + z1) :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0 if1(z, z', z'', z1) -{ 1 }-> max(1 + z'' + z1) :|: z1 >= 0, z' >= 0, z'' >= 0, z = 0 if2(z, z', z'', z1) -{ 1 }-> z1 :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0 if2(z, z', z'', z1) -{ 50 + 17*z1 }-> 1 + z'' + s7 :|: s7 >= 0, s7 <= z1, z1 >= 0, z' >= 0, z'' >= 0, z = 0 if3(z, z') -{ 51 }-> sort(s8) :|: s8 >= 0, s8 <= 0, z = 0, z' = 0 if3(z, z') -{ 51 + 17*z' }-> sort(s9) :|: s9 >= 0, s9 <= 1 + (z' - 1) + 0, z' - 1 >= 0, z = 0 if3(z, z') -{ 4 + y2 }-> sort(del(if1(s1, x5, y2, xs''), 1 + x5 + (1 + y2 + xs''))) :|: s1 >= 0, s1 <= 1, z' = 1 + x5 + (1 + y2 + xs''), x5 >= 0, xs'' >= 0, y2 >= 0, z = 0 if3(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 1 max(z) -{ 4 + y' }-> if1(s'', 1 + x'', 1 + y', xs) :|: s'' >= 0, s'' <= 1, xs >= 0, z = 1 + (1 + x'') + (1 + (1 + y') + xs), y' >= 0, x'' >= 0 max(z) -{ 2 }-> if1(1, x, 0, xs) :|: xs >= 0, x >= 0, z = 1 + x + (1 + 0 + xs) max(z) -{ 2 }-> if1(0, 0, 1 + x', xs) :|: z = 1 + 0 + (1 + (1 + x') + xs), xs >= 0, x' >= 0 max(z) -{ 1 }-> 0 :|: z = 0 max(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 sort(z) -{ 2 }-> if3(1, 0) :|: z = 0 sort(z) -{ 2 }-> if3(0, 1 + x3 + xs') :|: xs' >= 0, z = 1 + x3 + xs', x3 >= 0 Function symbols to be analyzed: {max,if1}, {sort,if3} Previous analysis results are: empty: runtime: O(1) [1], size: O(1) [1] eq: runtime: O(n^1) [3 + z'], size: O(1) [1] ge: runtime: O(n^1) [2 + z'], size: O(1) [1] del: runtime: O(n^1) [49 + 17*z'], size: O(n^1) [z'] if2: runtime: O(n^1) [50 + 17*z1], size: O(n^1) [1 + z'' + z1] ---------------------------------------- (45) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: max after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z Computed SIZE bound using CoFloCo for: if1 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z' + z'' + z1 ---------------------------------------- (46) Obligation: Complexity RNTS consisting of the following rules: del(z, z') -{ 35 + 17*z' }-> s3 :|: s3 >= 0, s3 <= 0 + (z' - 1) + 1, z' - 1 >= 0, z = 0 del(z, z') -{ 52 + 17*xs }-> s4 :|: s4 >= 0, s4 <= 1 + y'' + xs + 1, xs >= 0, z' = 1 + (1 + y'') + xs, y'' >= 0, z = 0 del(z, z') -{ 35 + 17*z' }-> s5 :|: s5 >= 0, s5 <= 0 + (z' - 1) + 1, z' - 1 >= 0, z - 1 >= 0 del(z, z') -{ 55 + 17*xs + y1 }-> s6 :|: s6 >= 0, s6 <= 1 + y1 + xs + 1, s >= 0, s <= 1, y1 >= 0, xs >= 0, z' = 1 + (1 + y1) + xs, z - 1 >= 0 del(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 empty(z) -{ 1 }-> 1 :|: z = 0 empty(z) -{ 1 }-> 0 :|: z = 1 + x + xs, xs >= 0, x >= 0 eq(z, z') -{ 3 + z' }-> s' :|: s' >= 0, s' <= 1, z - 1 >= 0, z' - 1 >= 0 eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 eq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 eq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 ge(z, z') -{ 2 + z' }-> s2 :|: s2 >= 0, s2 <= 1, z - 1 >= 0, z' - 1 >= 0 ge(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 ge(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 if1(z, z', z'', z1) -{ 1 }-> max(1 + z' + z1) :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0 if1(z, z', z'', z1) -{ 1 }-> max(1 + z'' + z1) :|: z1 >= 0, z' >= 0, z'' >= 0, z = 0 if2(z, z', z'', z1) -{ 1 }-> z1 :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0 if2(z, z', z'', z1) -{ 50 + 17*z1 }-> 1 + z'' + s7 :|: s7 >= 0, s7 <= z1, z1 >= 0, z' >= 0, z'' >= 0, z = 0 if3(z, z') -{ 51 }-> sort(s8) :|: s8 >= 0, s8 <= 0, z = 0, z' = 0 if3(z, z') -{ 51 + 17*z' }-> sort(s9) :|: s9 >= 0, s9 <= 1 + (z' - 1) + 0, z' - 1 >= 0, z = 0 if3(z, z') -{ 4 + y2 }-> sort(del(if1(s1, x5, y2, xs''), 1 + x5 + (1 + y2 + xs''))) :|: s1 >= 0, s1 <= 1, z' = 1 + x5 + (1 + y2 + xs''), x5 >= 0, xs'' >= 0, y2 >= 0, z = 0 if3(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 1 max(z) -{ 4 + y' }-> if1(s'', 1 + x'', 1 + y', xs) :|: s'' >= 0, s'' <= 1, xs >= 0, z = 1 + (1 + x'') + (1 + (1 + y') + xs), y' >= 0, x'' >= 0 max(z) -{ 2 }-> if1(1, x, 0, xs) :|: xs >= 0, x >= 0, z = 1 + x + (1 + 0 + xs) max(z) -{ 2 }-> if1(0, 0, 1 + x', xs) :|: z = 1 + 0 + (1 + (1 + x') + xs), xs >= 0, x' >= 0 max(z) -{ 1 }-> 0 :|: z = 0 max(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 sort(z) -{ 2 }-> if3(1, 0) :|: z = 0 sort(z) -{ 2 }-> if3(0, 1 + x3 + xs') :|: xs' >= 0, z = 1 + x3 + xs', x3 >= 0 Function symbols to be analyzed: {max,if1}, {sort,if3} Previous analysis results are: empty: runtime: O(1) [1], size: O(1) [1] eq: runtime: O(n^1) [3 + z'], size: O(1) [1] ge: runtime: O(n^1) [2 + z'], size: O(1) [1] del: runtime: O(n^1) [49 + 17*z'], size: O(n^1) [z'] if2: runtime: O(n^1) [50 + 17*z1], size: O(n^1) [1 + z'' + z1] max: runtime: ?, size: O(n^1) [z] if1: runtime: ?, size: O(n^1) [1 + z' + z'' + z1] ---------------------------------------- (47) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: max after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 16 + 7*z + 2*z^2 Computed RUNTIME bound using KoAT for: if1 after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 52 + 11*z' + 4*z'*z1 + 2*z'^2 + 11*z'' + 4*z''*z1 + 2*z''^2 + 22*z1 + 4*z1^2 ---------------------------------------- (48) Obligation: Complexity RNTS consisting of the following rules: del(z, z') -{ 35 + 17*z' }-> s3 :|: s3 >= 0, s3 <= 0 + (z' - 1) + 1, z' - 1 >= 0, z = 0 del(z, z') -{ 52 + 17*xs }-> s4 :|: s4 >= 0, s4 <= 1 + y'' + xs + 1, xs >= 0, z' = 1 + (1 + y'') + xs, y'' >= 0, z = 0 del(z, z') -{ 35 + 17*z' }-> s5 :|: s5 >= 0, s5 <= 0 + (z' - 1) + 1, z' - 1 >= 0, z - 1 >= 0 del(z, z') -{ 55 + 17*xs + y1 }-> s6 :|: s6 >= 0, s6 <= 1 + y1 + xs + 1, s >= 0, s <= 1, y1 >= 0, xs >= 0, z' = 1 + (1 + y1) + xs, z - 1 >= 0 del(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 empty(z) -{ 1 }-> 1 :|: z = 0 empty(z) -{ 1 }-> 0 :|: z = 1 + x + xs, xs >= 0, x >= 0 eq(z, z') -{ 3 + z' }-> s' :|: s' >= 0, s' <= 1, z - 1 >= 0, z' - 1 >= 0 eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 eq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 eq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 ge(z, z') -{ 2 + z' }-> s2 :|: s2 >= 0, s2 <= 1, z - 1 >= 0, z' - 1 >= 0 ge(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 ge(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 if1(z, z', z'', z1) -{ 1 }-> max(1 + z' + z1) :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0 if1(z, z', z'', z1) -{ 1 }-> max(1 + z'' + z1) :|: z1 >= 0, z' >= 0, z'' >= 0, z = 0 if2(z, z', z'', z1) -{ 1 }-> z1 :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0 if2(z, z', z'', z1) -{ 50 + 17*z1 }-> 1 + z'' + s7 :|: s7 >= 0, s7 <= z1, z1 >= 0, z' >= 0, z'' >= 0, z = 0 if3(z, z') -{ 51 }-> sort(s8) :|: s8 >= 0, s8 <= 0, z = 0, z' = 0 if3(z, z') -{ 51 + 17*z' }-> sort(s9) :|: s9 >= 0, s9 <= 1 + (z' - 1) + 0, z' - 1 >= 0, z = 0 if3(z, z') -{ 4 + y2 }-> sort(del(if1(s1, x5, y2, xs''), 1 + x5 + (1 + y2 + xs''))) :|: s1 >= 0, s1 <= 1, z' = 1 + x5 + (1 + y2 + xs''), x5 >= 0, xs'' >= 0, y2 >= 0, z = 0 if3(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 1 max(z) -{ 4 + y' }-> if1(s'', 1 + x'', 1 + y', xs) :|: s'' >= 0, s'' <= 1, xs >= 0, z = 1 + (1 + x'') + (1 + (1 + y') + xs), y' >= 0, x'' >= 0 max(z) -{ 2 }-> if1(1, x, 0, xs) :|: xs >= 0, x >= 0, z = 1 + x + (1 + 0 + xs) max(z) -{ 2 }-> if1(0, 0, 1 + x', xs) :|: z = 1 + 0 + (1 + (1 + x') + xs), xs >= 0, x' >= 0 max(z) -{ 1 }-> 0 :|: z = 0 max(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 sort(z) -{ 2 }-> if3(1, 0) :|: z = 0 sort(z) -{ 2 }-> if3(0, 1 + x3 + xs') :|: xs' >= 0, z = 1 + x3 + xs', x3 >= 0 Function symbols to be analyzed: {sort,if3} Previous analysis results are: empty: runtime: O(1) [1], size: O(1) [1] eq: runtime: O(n^1) [3 + z'], size: O(1) [1] ge: runtime: O(n^1) [2 + z'], size: O(1) [1] del: runtime: O(n^1) [49 + 17*z'], size: O(n^1) [z'] if2: runtime: O(n^1) [50 + 17*z1], size: O(n^1) [1 + z'' + z1] max: runtime: O(n^2) [16 + 7*z + 2*z^2], size: O(n^1) [z] if1: runtime: O(n^2) [52 + 11*z' + 4*z'*z1 + 2*z'^2 + 11*z'' + 4*z''*z1 + 2*z''^2 + 22*z1 + 4*z1^2], size: O(n^1) [1 + z' + z'' + z1] ---------------------------------------- (49) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (50) Obligation: Complexity RNTS consisting of the following rules: del(z, z') -{ 35 + 17*z' }-> s3 :|: s3 >= 0, s3 <= 0 + (z' - 1) + 1, z' - 1 >= 0, z = 0 del(z, z') -{ 52 + 17*xs }-> s4 :|: s4 >= 0, s4 <= 1 + y'' + xs + 1, xs >= 0, z' = 1 + (1 + y'') + xs, y'' >= 0, z = 0 del(z, z') -{ 35 + 17*z' }-> s5 :|: s5 >= 0, s5 <= 0 + (z' - 1) + 1, z' - 1 >= 0, z - 1 >= 0 del(z, z') -{ 55 + 17*xs + y1 }-> s6 :|: s6 >= 0, s6 <= 1 + y1 + xs + 1, s >= 0, s <= 1, y1 >= 0, xs >= 0, z' = 1 + (1 + y1) + xs, z - 1 >= 0 del(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 empty(z) -{ 1 }-> 1 :|: z = 0 empty(z) -{ 1 }-> 0 :|: z = 1 + x + xs, xs >= 0, x >= 0 eq(z, z') -{ 3 + z' }-> s' :|: s' >= 0, s' <= 1, z - 1 >= 0, z' - 1 >= 0 eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 eq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 eq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 ge(z, z') -{ 2 + z' }-> s2 :|: s2 >= 0, s2 <= 1, z - 1 >= 0, z' - 1 >= 0 ge(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 ge(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 if1(z, z', z'', z1) -{ 26 + 11*z' + 4*z'*z1 + 2*z'^2 + 11*z1 + 2*z1^2 }-> s13 :|: s13 >= 0, s13 <= 1 + z' + z1, z1 >= 0, z = 1, z' >= 0, z'' >= 0 if1(z, z', z'', z1) -{ 26 + 11*z'' + 4*z''*z1 + 2*z''^2 + 11*z1 + 2*z1^2 }-> s14 :|: s14 >= 0, s14 <= 1 + z'' + z1, z1 >= 0, z' >= 0, z'' >= 0, z = 0 if2(z, z', z'', z1) -{ 1 }-> z1 :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0 if2(z, z', z'', z1) -{ 50 + 17*z1 }-> 1 + z'' + s7 :|: s7 >= 0, s7 <= z1, z1 >= 0, z' >= 0, z'' >= 0, z = 0 if3(z, z') -{ 139 + 28*x5 + 4*x5*xs'' + 2*x5^2 + 39*xs'' + 4*xs''*y2 + 4*xs''^2 + 29*y2 + 2*y2^2 }-> sort(s16) :|: s15 >= 0, s15 <= y2 + xs'' + 1 + x5, s16 >= 0, s16 <= 1 + x5 + (1 + y2 + xs''), s1 >= 0, s1 <= 1, z' = 1 + x5 + (1 + y2 + xs''), x5 >= 0, xs'' >= 0, y2 >= 0, z = 0 if3(z, z') -{ 51 }-> sort(s8) :|: s8 >= 0, s8 <= 0, z = 0, z' = 0 if3(z, z') -{ 51 + 17*z' }-> sort(s9) :|: s9 >= 0, s9 <= 1 + (z' - 1) + 0, z' - 1 >= 0, z = 0 if3(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 1 max(z) -{ 54 + 11*x + 4*x*xs + 2*x^2 + 22*xs + 4*xs^2 }-> s10 :|: s10 >= 0, s10 <= 0 + xs + 1 + x, xs >= 0, x >= 0, z = 1 + x + (1 + 0 + xs) max(z) -{ 67 + 15*x' + 4*x'*xs + 2*x'^2 + 26*xs + 4*xs^2 }-> s11 :|: s11 >= 0, s11 <= 1 + x' + xs + 1 + 0, z = 1 + 0 + (1 + (1 + x') + xs), xs >= 0, x' >= 0 max(z) -{ 82 + 15*x'' + 4*x''*xs + 2*x''^2 + 30*xs + 4*xs*y' + 4*xs^2 + 16*y' + 2*y'^2 }-> s12 :|: s12 >= 0, s12 <= 1 + y' + xs + 1 + (1 + x''), s'' >= 0, s'' <= 1, xs >= 0, z = 1 + (1 + x'') + (1 + (1 + y') + xs), y' >= 0, x'' >= 0 max(z) -{ 1 }-> 0 :|: z = 0 max(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 sort(z) -{ 2 }-> if3(1, 0) :|: z = 0 sort(z) -{ 2 }-> if3(0, 1 + x3 + xs') :|: xs' >= 0, z = 1 + x3 + xs', x3 >= 0 Function symbols to be analyzed: {sort,if3} Previous analysis results are: empty: runtime: O(1) [1], size: O(1) [1] eq: runtime: O(n^1) [3 + z'], size: O(1) [1] ge: runtime: O(n^1) [2 + z'], size: O(1) [1] del: runtime: O(n^1) [49 + 17*z'], size: O(n^1) [z'] if2: runtime: O(n^1) [50 + 17*z1], size: O(n^1) [1 + z'' + z1] max: runtime: O(n^2) [16 + 7*z + 2*z^2], size: O(n^1) [z] if1: runtime: O(n^2) [52 + 11*z' + 4*z'*z1 + 2*z'^2 + 11*z'' + 4*z''*z1 + 2*z''^2 + 22*z1 + 4*z1^2], size: O(n^1) [1 + z' + z'' + z1] ---------------------------------------- (51) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: sort after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 Computed SIZE bound using CoFloCo for: if3 after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (52) Obligation: Complexity RNTS consisting of the following rules: del(z, z') -{ 35 + 17*z' }-> s3 :|: s3 >= 0, s3 <= 0 + (z' - 1) + 1, z' - 1 >= 0, z = 0 del(z, z') -{ 52 + 17*xs }-> s4 :|: s4 >= 0, s4 <= 1 + y'' + xs + 1, xs >= 0, z' = 1 + (1 + y'') + xs, y'' >= 0, z = 0 del(z, z') -{ 35 + 17*z' }-> s5 :|: s5 >= 0, s5 <= 0 + (z' - 1) + 1, z' - 1 >= 0, z - 1 >= 0 del(z, z') -{ 55 + 17*xs + y1 }-> s6 :|: s6 >= 0, s6 <= 1 + y1 + xs + 1, s >= 0, s <= 1, y1 >= 0, xs >= 0, z' = 1 + (1 + y1) + xs, z - 1 >= 0 del(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 empty(z) -{ 1 }-> 1 :|: z = 0 empty(z) -{ 1 }-> 0 :|: z = 1 + x + xs, xs >= 0, x >= 0 eq(z, z') -{ 3 + z' }-> s' :|: s' >= 0, s' <= 1, z - 1 >= 0, z' - 1 >= 0 eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 eq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 eq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 ge(z, z') -{ 2 + z' }-> s2 :|: s2 >= 0, s2 <= 1, z - 1 >= 0, z' - 1 >= 0 ge(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 ge(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 if1(z, z', z'', z1) -{ 26 + 11*z' + 4*z'*z1 + 2*z'^2 + 11*z1 + 2*z1^2 }-> s13 :|: s13 >= 0, s13 <= 1 + z' + z1, z1 >= 0, z = 1, z' >= 0, z'' >= 0 if1(z, z', z'', z1) -{ 26 + 11*z'' + 4*z''*z1 + 2*z''^2 + 11*z1 + 2*z1^2 }-> s14 :|: s14 >= 0, s14 <= 1 + z'' + z1, z1 >= 0, z' >= 0, z'' >= 0, z = 0 if2(z, z', z'', z1) -{ 1 }-> z1 :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0 if2(z, z', z'', z1) -{ 50 + 17*z1 }-> 1 + z'' + s7 :|: s7 >= 0, s7 <= z1, z1 >= 0, z' >= 0, z'' >= 0, z = 0 if3(z, z') -{ 139 + 28*x5 + 4*x5*xs'' + 2*x5^2 + 39*xs'' + 4*xs''*y2 + 4*xs''^2 + 29*y2 + 2*y2^2 }-> sort(s16) :|: s15 >= 0, s15 <= y2 + xs'' + 1 + x5, s16 >= 0, s16 <= 1 + x5 + (1 + y2 + xs''), s1 >= 0, s1 <= 1, z' = 1 + x5 + (1 + y2 + xs''), x5 >= 0, xs'' >= 0, y2 >= 0, z = 0 if3(z, z') -{ 51 }-> sort(s8) :|: s8 >= 0, s8 <= 0, z = 0, z' = 0 if3(z, z') -{ 51 + 17*z' }-> sort(s9) :|: s9 >= 0, s9 <= 1 + (z' - 1) + 0, z' - 1 >= 0, z = 0 if3(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 1 max(z) -{ 54 + 11*x + 4*x*xs + 2*x^2 + 22*xs + 4*xs^2 }-> s10 :|: s10 >= 0, s10 <= 0 + xs + 1 + x, xs >= 0, x >= 0, z = 1 + x + (1 + 0 + xs) max(z) -{ 67 + 15*x' + 4*x'*xs + 2*x'^2 + 26*xs + 4*xs^2 }-> s11 :|: s11 >= 0, s11 <= 1 + x' + xs + 1 + 0, z = 1 + 0 + (1 + (1 + x') + xs), xs >= 0, x' >= 0 max(z) -{ 82 + 15*x'' + 4*x''*xs + 2*x''^2 + 30*xs + 4*xs*y' + 4*xs^2 + 16*y' + 2*y'^2 }-> s12 :|: s12 >= 0, s12 <= 1 + y' + xs + 1 + (1 + x''), s'' >= 0, s'' <= 1, xs >= 0, z = 1 + (1 + x'') + (1 + (1 + y') + xs), y' >= 0, x'' >= 0 max(z) -{ 1 }-> 0 :|: z = 0 max(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 sort(z) -{ 2 }-> if3(1, 0) :|: z = 0 sort(z) -{ 2 }-> if3(0, 1 + x3 + xs') :|: xs' >= 0, z = 1 + x3 + xs', x3 >= 0 Function symbols to be analyzed: {sort,if3} Previous analysis results are: empty: runtime: O(1) [1], size: O(1) [1] eq: runtime: O(n^1) [3 + z'], size: O(1) [1] ge: runtime: O(n^1) [2 + z'], size: O(1) [1] del: runtime: O(n^1) [49 + 17*z'], size: O(n^1) [z'] if2: runtime: O(n^1) [50 + 17*z1], size: O(n^1) [1 + z'' + z1] max: runtime: O(n^2) [16 + 7*z + 2*z^2], size: O(n^1) [z] if1: runtime: O(n^2) [52 + 11*z' + 4*z'*z1 + 2*z'^2 + 11*z'' + 4*z''*z1 + 2*z''^2 + 22*z1 + 4*z1^2], size: O(n^1) [1 + z' + z'' + z1] sort: runtime: ?, size: O(1) [0] if3: runtime: ?, size: O(1) [0] ---------------------------------------- (53) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: sort after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (54) Obligation: Complexity RNTS consisting of the following rules: del(z, z') -{ 35 + 17*z' }-> s3 :|: s3 >= 0, s3 <= 0 + (z' - 1) + 1, z' - 1 >= 0, z = 0 del(z, z') -{ 52 + 17*xs }-> s4 :|: s4 >= 0, s4 <= 1 + y'' + xs + 1, xs >= 0, z' = 1 + (1 + y'') + xs, y'' >= 0, z = 0 del(z, z') -{ 35 + 17*z' }-> s5 :|: s5 >= 0, s5 <= 0 + (z' - 1) + 1, z' - 1 >= 0, z - 1 >= 0 del(z, z') -{ 55 + 17*xs + y1 }-> s6 :|: s6 >= 0, s6 <= 1 + y1 + xs + 1, s >= 0, s <= 1, y1 >= 0, xs >= 0, z' = 1 + (1 + y1) + xs, z - 1 >= 0 del(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 empty(z) -{ 1 }-> 1 :|: z = 0 empty(z) -{ 1 }-> 0 :|: z = 1 + x + xs, xs >= 0, x >= 0 eq(z, z') -{ 3 + z' }-> s' :|: s' >= 0, s' <= 1, z - 1 >= 0, z' - 1 >= 0 eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 eq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 eq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 ge(z, z') -{ 2 + z' }-> s2 :|: s2 >= 0, s2 <= 1, z - 1 >= 0, z' - 1 >= 0 ge(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 ge(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 if1(z, z', z'', z1) -{ 26 + 11*z' + 4*z'*z1 + 2*z'^2 + 11*z1 + 2*z1^2 }-> s13 :|: s13 >= 0, s13 <= 1 + z' + z1, z1 >= 0, z = 1, z' >= 0, z'' >= 0 if1(z, z', z'', z1) -{ 26 + 11*z'' + 4*z''*z1 + 2*z''^2 + 11*z1 + 2*z1^2 }-> s14 :|: s14 >= 0, s14 <= 1 + z'' + z1, z1 >= 0, z' >= 0, z'' >= 0, z = 0 if2(z, z', z'', z1) -{ 1 }-> z1 :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0 if2(z, z', z'', z1) -{ 50 + 17*z1 }-> 1 + z'' + s7 :|: s7 >= 0, s7 <= z1, z1 >= 0, z' >= 0, z'' >= 0, z = 0 if3(z, z') -{ 139 + 28*x5 + 4*x5*xs'' + 2*x5^2 + 39*xs'' + 4*xs''*y2 + 4*xs''^2 + 29*y2 + 2*y2^2 }-> sort(s16) :|: s15 >= 0, s15 <= y2 + xs'' + 1 + x5, s16 >= 0, s16 <= 1 + x5 + (1 + y2 + xs''), s1 >= 0, s1 <= 1, z' = 1 + x5 + (1 + y2 + xs''), x5 >= 0, xs'' >= 0, y2 >= 0, z = 0 if3(z, z') -{ 51 }-> sort(s8) :|: s8 >= 0, s8 <= 0, z = 0, z' = 0 if3(z, z') -{ 51 + 17*z' }-> sort(s9) :|: s9 >= 0, s9 <= 1 + (z' - 1) + 0, z' - 1 >= 0, z = 0 if3(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 1 max(z) -{ 54 + 11*x + 4*x*xs + 2*x^2 + 22*xs + 4*xs^2 }-> s10 :|: s10 >= 0, s10 <= 0 + xs + 1 + x, xs >= 0, x >= 0, z = 1 + x + (1 + 0 + xs) max(z) -{ 67 + 15*x' + 4*x'*xs + 2*x'^2 + 26*xs + 4*xs^2 }-> s11 :|: s11 >= 0, s11 <= 1 + x' + xs + 1 + 0, z = 1 + 0 + (1 + (1 + x') + xs), xs >= 0, x' >= 0 max(z) -{ 82 + 15*x'' + 4*x''*xs + 2*x''^2 + 30*xs + 4*xs*y' + 4*xs^2 + 16*y' + 2*y'^2 }-> s12 :|: s12 >= 0, s12 <= 1 + y' + xs + 1 + (1 + x''), s'' >= 0, s'' <= 1, xs >= 0, z = 1 + (1 + x'') + (1 + (1 + y') + xs), y' >= 0, x'' >= 0 max(z) -{ 1 }-> 0 :|: z = 0 max(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 sort(z) -{ 2 }-> if3(1, 0) :|: z = 0 sort(z) -{ 2 }-> if3(0, 1 + x3 + xs') :|: xs' >= 0, z = 1 + x3 + xs', x3 >= 0 Function symbols to be analyzed: {sort,if3} Previous analysis results are: empty: runtime: O(1) [1], size: O(1) [1] eq: runtime: O(n^1) [3 + z'], size: O(1) [1] ge: runtime: O(n^1) [2 + z'], size: O(1) [1] del: runtime: O(n^1) [49 + 17*z'], size: O(n^1) [z'] if2: runtime: O(n^1) [50 + 17*z1], size: O(n^1) [1 + z'' + z1] max: runtime: O(n^2) [16 + 7*z + 2*z^2], size: O(n^1) [z] if1: runtime: O(n^2) [52 + 11*z' + 4*z'*z1 + 2*z'^2 + 11*z'' + 4*z''*z1 + 2*z''^2 + 22*z1 + 4*z1^2], size: O(n^1) [1 + z' + z'' + z1] sort: runtime: INF, size: O(1) [0] if3: runtime: ?, size: O(1) [0] ---------------------------------------- (55) CompletionProof (UPPER BOUND(ID)) The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added: none And the following fresh constants: none ---------------------------------------- (56) Obligation: Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: max(nil) -> 0 [1] max(cons(x, nil)) -> x [1] max(cons(x, cons(y, xs))) -> if1(ge(x, y), x, y, xs) [1] if1(true, x, y, xs) -> max(cons(x, xs)) [1] if1(false, x, y, xs) -> max(cons(y, xs)) [1] del(x, nil) -> nil [1] del(x, cons(y, xs)) -> if2(eq(x, y), x, y, xs) [1] if2(true, x, y, xs) -> xs [1] if2(false, x, y, xs) -> cons(y, del(x, xs)) [1] eq(0, 0) -> true [1] eq(0, s(y)) -> false [1] eq(s(x), 0) -> false [1] eq(s(x), s(y)) -> eq(x, y) [1] sort(xs) -> if3(empty(xs), xs) [1] if3(true, xs) -> nil [1] if3(false, xs) -> sort(del(max(xs), xs)) [1] empty(nil) -> true [1] empty(cons(x, xs)) -> false [1] ge(x, 0) -> true [1] ge(0, s(x)) -> false [1] ge(s(x), s(y)) -> ge(x, y) [1] The TRS has the following type information: max :: nil:cons -> 0:s nil :: nil:cons 0 :: 0:s cons :: 0:s -> nil:cons -> nil:cons if1 :: true:false -> 0:s -> 0:s -> nil:cons -> 0:s ge :: 0:s -> 0:s -> true:false true :: true:false false :: true:false del :: 0:s -> nil:cons -> nil:cons if2 :: true:false -> 0:s -> 0:s -> nil:cons -> nil:cons eq :: 0:s -> 0:s -> true:false s :: 0:s -> 0:s sort :: nil:cons -> nil:cons if3 :: true:false -> nil:cons -> nil:cons empty :: nil:cons -> true:false Rewrite Strategy: INNERMOST ---------------------------------------- (57) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: nil => 0 0 => 0 true => 1 false => 0 ---------------------------------------- (58) Obligation: Complexity RNTS consisting of the following rules: del(z, z') -{ 1 }-> if2(eq(x, y), x, y, xs) :|: xs >= 0, z' = 1 + y + xs, x >= 0, y >= 0, z = x del(z, z') -{ 1 }-> 0 :|: x >= 0, z = x, z' = 0 empty(z) -{ 1 }-> 1 :|: z = 0 empty(z) -{ 1 }-> 0 :|: z = 1 + x + xs, xs >= 0, x >= 0 eq(z, z') -{ 1 }-> eq(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 eq(z, z') -{ 1 }-> 0 :|: z' = 1 + y, y >= 0, z = 0 eq(z, z') -{ 1 }-> 0 :|: x >= 0, z = 1 + x, z' = 0 ge(z, z') -{ 1 }-> ge(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x ge(z, z') -{ 1 }-> 1 :|: x >= 0, z = x, z' = 0 ge(z, z') -{ 1 }-> 0 :|: z' = 1 + x, x >= 0, z = 0 if1(z, z', z'', z1) -{ 1 }-> max(1 + x + xs) :|: xs >= 0, z' = x, z'' = y, z = 1, x >= 0, y >= 0, z1 = xs if1(z, z', z'', z1) -{ 1 }-> max(1 + y + xs) :|: xs >= 0, z' = x, z'' = y, x >= 0, y >= 0, z = 0, z1 = xs if2(z, z', z'', z1) -{ 1 }-> xs :|: xs >= 0, z' = x, z'' = y, z = 1, x >= 0, y >= 0, z1 = xs if2(z, z', z'', z1) -{ 1 }-> 1 + y + del(x, xs) :|: xs >= 0, z' = x, z'' = y, x >= 0, y >= 0, z = 0, z1 = xs if3(z, z') -{ 1 }-> sort(del(max(xs), xs)) :|: xs >= 0, z' = xs, z = 0 if3(z, z') -{ 1 }-> 0 :|: xs >= 0, z = 1, z' = xs max(z) -{ 1 }-> x :|: x >= 0, z = 1 + x + 0 max(z) -{ 1 }-> if1(ge(x, y), x, y, xs) :|: xs >= 0, x >= 0, y >= 0, z = 1 + x + (1 + y + xs) max(z) -{ 1 }-> 0 :|: z = 0 sort(z) -{ 1 }-> if3(empty(xs), xs) :|: xs >= 0, z = xs Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (59) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (60) Obligation: Complexity Dependency Tuples Problem Rules: max(nil) -> 0 max(cons(z0, nil)) -> z0 max(cons(z0, cons(z1, z2))) -> if1(ge(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> max(cons(z0, z2)) if1(false, z0, z1, z2) -> max(cons(z1, z2)) del(z0, nil) -> nil del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) if2(true, z0, z1, z2) -> z2 if2(false, z0, z1, z2) -> cons(z1, del(z0, z2)) eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) sort(z0) -> if3(empty(z0), z0) if3(true, z0) -> nil if3(false, z0) -> sort(del(max(z0), z0)) empty(nil) -> true empty(cons(z0, z1)) -> false ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) Tuples: MAX(nil) -> c MAX(cons(z0, nil)) -> c1 MAX(cons(z0, cons(z1, z2))) -> c2(IF1(ge(z0, z1), z0, z1, z2), GE(z0, z1)) IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) DEL(z0, nil) -> c5 DEL(z0, cons(z1, z2)) -> c6(IF2(eq(z0, z1), z0, z1, z2), EQ(z0, z1)) IF2(true, z0, z1, z2) -> c7 IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(0, 0) -> c9 EQ(0, s(z0)) -> c10 EQ(s(z0), 0) -> c11 EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) SORT(z0) -> c13(IF3(empty(z0), z0), EMPTY(z0)) IF3(true, z0) -> c14 IF3(false, z0) -> c15(SORT(del(max(z0), z0)), DEL(max(z0), z0), MAX(z0)) EMPTY(nil) -> c16 EMPTY(cons(z0, z1)) -> c17 GE(z0, 0) -> c18 GE(0, s(z0)) -> c19 GE(s(z0), s(z1)) -> c20(GE(z0, z1)) S tuples: MAX(nil) -> c MAX(cons(z0, nil)) -> c1 MAX(cons(z0, cons(z1, z2))) -> c2(IF1(ge(z0, z1), z0, z1, z2), GE(z0, z1)) IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) DEL(z0, nil) -> c5 DEL(z0, cons(z1, z2)) -> c6(IF2(eq(z0, z1), z0, z1, z2), EQ(z0, z1)) IF2(true, z0, z1, z2) -> c7 IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(0, 0) -> c9 EQ(0, s(z0)) -> c10 EQ(s(z0), 0) -> c11 EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) SORT(z0) -> c13(IF3(empty(z0), z0), EMPTY(z0)) IF3(true, z0) -> c14 IF3(false, z0) -> c15(SORT(del(max(z0), z0)), DEL(max(z0), z0), MAX(z0)) EMPTY(nil) -> c16 EMPTY(cons(z0, z1)) -> c17 GE(z0, 0) -> c18 GE(0, s(z0)) -> c19 GE(s(z0), s(z1)) -> c20(GE(z0, z1)) K tuples:none Defined Rule Symbols: max_1, if1_4, del_2, if2_4, eq_2, sort_1, if3_2, empty_1, ge_2 Defined Pair Symbols: MAX_1, IF1_4, DEL_2, IF2_4, EQ_2, SORT_1, IF3_2, EMPTY_1, GE_2 Compound Symbols: c, c1, c2_2, c3_1, c4_1, c5, c6_2, c7, c8_1, c9, c10, c11, c12_1, c13_2, c14, c15_3, c16, c17, c18, c19, c20_1 ---------------------------------------- (61) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 12 trailing nodes: GE(0, s(z0)) -> c19 IF2(true, z0, z1, z2) -> c7 MAX(cons(z0, nil)) -> c1 EMPTY(nil) -> c16 IF3(true, z0) -> c14 EQ(s(z0), 0) -> c11 MAX(nil) -> c GE(z0, 0) -> c18 EQ(0, s(z0)) -> c10 EMPTY(cons(z0, z1)) -> c17 EQ(0, 0) -> c9 DEL(z0, nil) -> c5 ---------------------------------------- (62) Obligation: Complexity Dependency Tuples Problem Rules: max(nil) -> 0 max(cons(z0, nil)) -> z0 max(cons(z0, cons(z1, z2))) -> if1(ge(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> max(cons(z0, z2)) if1(false, z0, z1, z2) -> max(cons(z1, z2)) del(z0, nil) -> nil del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) if2(true, z0, z1, z2) -> z2 if2(false, z0, z1, z2) -> cons(z1, del(z0, z2)) eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) sort(z0) -> if3(empty(z0), z0) if3(true, z0) -> nil if3(false, z0) -> sort(del(max(z0), z0)) empty(nil) -> true empty(cons(z0, z1)) -> false ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) Tuples: MAX(cons(z0, cons(z1, z2))) -> c2(IF1(ge(z0, z1), z0, z1, z2), GE(z0, z1)) IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) DEL(z0, cons(z1, z2)) -> c6(IF2(eq(z0, z1), z0, z1, z2), EQ(z0, z1)) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) SORT(z0) -> c13(IF3(empty(z0), z0), EMPTY(z0)) IF3(false, z0) -> c15(SORT(del(max(z0), z0)), DEL(max(z0), z0), MAX(z0)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) S tuples: MAX(cons(z0, cons(z1, z2))) -> c2(IF1(ge(z0, z1), z0, z1, z2), GE(z0, z1)) IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) DEL(z0, cons(z1, z2)) -> c6(IF2(eq(z0, z1), z0, z1, z2), EQ(z0, z1)) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) SORT(z0) -> c13(IF3(empty(z0), z0), EMPTY(z0)) IF3(false, z0) -> c15(SORT(del(max(z0), z0)), DEL(max(z0), z0), MAX(z0)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) K tuples:none Defined Rule Symbols: max_1, if1_4, del_2, if2_4, eq_2, sort_1, if3_2, empty_1, ge_2 Defined Pair Symbols: MAX_1, IF1_4, DEL_2, IF2_4, EQ_2, SORT_1, IF3_2, GE_2 Compound Symbols: c2_2, c3_1, c4_1, c6_2, c8_1, c12_1, c13_2, c15_3, c20_1 ---------------------------------------- (63) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (64) Obligation: Complexity Dependency Tuples Problem Rules: max(nil) -> 0 max(cons(z0, nil)) -> z0 max(cons(z0, cons(z1, z2))) -> if1(ge(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> max(cons(z0, z2)) if1(false, z0, z1, z2) -> max(cons(z1, z2)) del(z0, nil) -> nil del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) if2(true, z0, z1, z2) -> z2 if2(false, z0, z1, z2) -> cons(z1, del(z0, z2)) eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) sort(z0) -> if3(empty(z0), z0) if3(true, z0) -> nil if3(false, z0) -> sort(del(max(z0), z0)) empty(nil) -> true empty(cons(z0, z1)) -> false ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) Tuples: MAX(cons(z0, cons(z1, z2))) -> c2(IF1(ge(z0, z1), z0, z1, z2), GE(z0, z1)) IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) DEL(z0, cons(z1, z2)) -> c6(IF2(eq(z0, z1), z0, z1, z2), EQ(z0, z1)) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) IF3(false, z0) -> c15(SORT(del(max(z0), z0)), DEL(max(z0), z0), MAX(z0)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) SORT(z0) -> c13(IF3(empty(z0), z0)) S tuples: MAX(cons(z0, cons(z1, z2))) -> c2(IF1(ge(z0, z1), z0, z1, z2), GE(z0, z1)) IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) DEL(z0, cons(z1, z2)) -> c6(IF2(eq(z0, z1), z0, z1, z2), EQ(z0, z1)) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) IF3(false, z0) -> c15(SORT(del(max(z0), z0)), DEL(max(z0), z0), MAX(z0)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) SORT(z0) -> c13(IF3(empty(z0), z0)) K tuples:none Defined Rule Symbols: max_1, if1_4, del_2, if2_4, eq_2, sort_1, if3_2, empty_1, ge_2 Defined Pair Symbols: MAX_1, IF1_4, DEL_2, IF2_4, EQ_2, IF3_2, GE_2, SORT_1 Compound Symbols: c2_2, c3_1, c4_1, c6_2, c8_1, c12_1, c15_3, c20_1, c13_1 ---------------------------------------- (65) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: sort(z0) -> if3(empty(z0), z0) if3(true, z0) -> nil if3(false, z0) -> sort(del(max(z0), z0)) ---------------------------------------- (66) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) del(z0, nil) -> nil del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) max(nil) -> 0 max(cons(z0, nil)) -> z0 max(cons(z0, cons(z1, z2))) -> if1(ge(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> max(cons(z0, z2)) if1(false, z0, z1, z2) -> max(cons(z1, z2)) if2(true, z0, z1, z2) -> z2 if2(false, z0, z1, z2) -> cons(z1, del(z0, z2)) empty(nil) -> true empty(cons(z0, z1)) -> false Tuples: MAX(cons(z0, cons(z1, z2))) -> c2(IF1(ge(z0, z1), z0, z1, z2), GE(z0, z1)) IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) DEL(z0, cons(z1, z2)) -> c6(IF2(eq(z0, z1), z0, z1, z2), EQ(z0, z1)) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) IF3(false, z0) -> c15(SORT(del(max(z0), z0)), DEL(max(z0), z0), MAX(z0)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) SORT(z0) -> c13(IF3(empty(z0), z0)) S tuples: MAX(cons(z0, cons(z1, z2))) -> c2(IF1(ge(z0, z1), z0, z1, z2), GE(z0, z1)) IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) DEL(z0, cons(z1, z2)) -> c6(IF2(eq(z0, z1), z0, z1, z2), EQ(z0, z1)) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) IF3(false, z0) -> c15(SORT(del(max(z0), z0)), DEL(max(z0), z0), MAX(z0)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) SORT(z0) -> c13(IF3(empty(z0), z0)) K tuples:none Defined Rule Symbols: ge_2, eq_2, del_2, max_1, if1_4, if2_4, empty_1 Defined Pair Symbols: MAX_1, IF1_4, DEL_2, IF2_4, EQ_2, IF3_2, GE_2, SORT_1 Compound Symbols: c2_2, c3_1, c4_1, c6_2, c8_1, c12_1, c15_3, c20_1, c13_1 ---------------------------------------- (67) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MAX(cons(z0, cons(z1, z2))) -> c2(IF1(ge(z0, z1), z0, z1, z2), GE(z0, z1)) by MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2), GE(z0, 0)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2), GE(0, s(z0))) MAX(cons(s(z0), cons(s(z1), x2))) -> c2(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1))) ---------------------------------------- (68) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) del(z0, nil) -> nil del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) max(nil) -> 0 max(cons(z0, nil)) -> z0 max(cons(z0, cons(z1, z2))) -> if1(ge(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> max(cons(z0, z2)) if1(false, z0, z1, z2) -> max(cons(z1, z2)) if2(true, z0, z1, z2) -> z2 if2(false, z0, z1, z2) -> cons(z1, del(z0, z2)) empty(nil) -> true empty(cons(z0, z1)) -> false Tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) DEL(z0, cons(z1, z2)) -> c6(IF2(eq(z0, z1), z0, z1, z2), EQ(z0, z1)) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) IF3(false, z0) -> c15(SORT(del(max(z0), z0)), DEL(max(z0), z0), MAX(z0)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) SORT(z0) -> c13(IF3(empty(z0), z0)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2), GE(z0, 0)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2), GE(0, s(z0))) MAX(cons(s(z0), cons(s(z1), x2))) -> c2(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1))) S tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) DEL(z0, cons(z1, z2)) -> c6(IF2(eq(z0, z1), z0, z1, z2), EQ(z0, z1)) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) IF3(false, z0) -> c15(SORT(del(max(z0), z0)), DEL(max(z0), z0), MAX(z0)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) SORT(z0) -> c13(IF3(empty(z0), z0)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2), GE(z0, 0)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2), GE(0, s(z0))) MAX(cons(s(z0), cons(s(z1), x2))) -> c2(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1))) K tuples:none Defined Rule Symbols: ge_2, eq_2, del_2, max_1, if1_4, if2_4, empty_1 Defined Pair Symbols: IF1_4, DEL_2, IF2_4, EQ_2, IF3_2, GE_2, SORT_1, MAX_1 Compound Symbols: c3_1, c4_1, c6_2, c8_1, c12_1, c15_3, c20_1, c13_1, c2_2 ---------------------------------------- (69) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (70) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) del(z0, nil) -> nil del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) max(nil) -> 0 max(cons(z0, nil)) -> z0 max(cons(z0, cons(z1, z2))) -> if1(ge(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> max(cons(z0, z2)) if1(false, z0, z1, z2) -> max(cons(z1, z2)) if2(true, z0, z1, z2) -> z2 if2(false, z0, z1, z2) -> cons(z1, del(z0, z2)) empty(nil) -> true empty(cons(z0, z1)) -> false Tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) DEL(z0, cons(z1, z2)) -> c6(IF2(eq(z0, z1), z0, z1, z2), EQ(z0, z1)) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) IF3(false, z0) -> c15(SORT(del(max(z0), z0)), DEL(max(z0), z0), MAX(z0)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) SORT(z0) -> c13(IF3(empty(z0), z0)) MAX(cons(s(z0), cons(s(z1), x2))) -> c2(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1))) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) S tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) DEL(z0, cons(z1, z2)) -> c6(IF2(eq(z0, z1), z0, z1, z2), EQ(z0, z1)) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) IF3(false, z0) -> c15(SORT(del(max(z0), z0)), DEL(max(z0), z0), MAX(z0)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) SORT(z0) -> c13(IF3(empty(z0), z0)) MAX(cons(s(z0), cons(s(z1), x2))) -> c2(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1))) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) K tuples:none Defined Rule Symbols: ge_2, eq_2, del_2, max_1, if1_4, if2_4, empty_1 Defined Pair Symbols: IF1_4, DEL_2, IF2_4, EQ_2, IF3_2, GE_2, SORT_1, MAX_1 Compound Symbols: c3_1, c4_1, c6_2, c8_1, c12_1, c15_3, c20_1, c13_1, c2_2, c2_1 ---------------------------------------- (71) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace DEL(z0, cons(z1, z2)) -> c6(IF2(eq(z0, z1), z0, z1, z2), EQ(z0, z1)) by DEL(0, cons(0, x2)) -> c6(IF2(true, 0, 0, x2), EQ(0, 0)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2), EQ(0, s(z0))) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2), EQ(s(z0), 0)) DEL(s(z0), cons(s(z1), x2)) -> c6(IF2(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1))) ---------------------------------------- (72) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) del(z0, nil) -> nil del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) max(nil) -> 0 max(cons(z0, nil)) -> z0 max(cons(z0, cons(z1, z2))) -> if1(ge(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> max(cons(z0, z2)) if1(false, z0, z1, z2) -> max(cons(z1, z2)) if2(true, z0, z1, z2) -> z2 if2(false, z0, z1, z2) -> cons(z1, del(z0, z2)) empty(nil) -> true empty(cons(z0, z1)) -> false Tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) IF3(false, z0) -> c15(SORT(del(max(z0), z0)), DEL(max(z0), z0), MAX(z0)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) SORT(z0) -> c13(IF3(empty(z0), z0)) MAX(cons(s(z0), cons(s(z1), x2))) -> c2(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1))) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(0, x2)) -> c6(IF2(true, 0, 0, x2), EQ(0, 0)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2), EQ(0, s(z0))) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2), EQ(s(z0), 0)) DEL(s(z0), cons(s(z1), x2)) -> c6(IF2(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1))) S tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) IF3(false, z0) -> c15(SORT(del(max(z0), z0)), DEL(max(z0), z0), MAX(z0)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) SORT(z0) -> c13(IF3(empty(z0), z0)) MAX(cons(s(z0), cons(s(z1), x2))) -> c2(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1))) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(0, x2)) -> c6(IF2(true, 0, 0, x2), EQ(0, 0)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2), EQ(0, s(z0))) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2), EQ(s(z0), 0)) DEL(s(z0), cons(s(z1), x2)) -> c6(IF2(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1))) K tuples:none Defined Rule Symbols: ge_2, eq_2, del_2, max_1, if1_4, if2_4, empty_1 Defined Pair Symbols: IF1_4, IF2_4, EQ_2, IF3_2, GE_2, SORT_1, MAX_1, DEL_2 Compound Symbols: c3_1, c4_1, c8_1, c12_1, c15_3, c20_1, c13_1, c2_2, c2_1, c6_2 ---------------------------------------- (73) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: DEL(0, cons(0, x2)) -> c6(IF2(true, 0, 0, x2), EQ(0, 0)) ---------------------------------------- (74) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) del(z0, nil) -> nil del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) max(nil) -> 0 max(cons(z0, nil)) -> z0 max(cons(z0, cons(z1, z2))) -> if1(ge(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> max(cons(z0, z2)) if1(false, z0, z1, z2) -> max(cons(z1, z2)) if2(true, z0, z1, z2) -> z2 if2(false, z0, z1, z2) -> cons(z1, del(z0, z2)) empty(nil) -> true empty(cons(z0, z1)) -> false Tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) IF3(false, z0) -> c15(SORT(del(max(z0), z0)), DEL(max(z0), z0), MAX(z0)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) SORT(z0) -> c13(IF3(empty(z0), z0)) MAX(cons(s(z0), cons(s(z1), x2))) -> c2(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1))) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2), EQ(0, s(z0))) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2), EQ(s(z0), 0)) DEL(s(z0), cons(s(z1), x2)) -> c6(IF2(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1))) S tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) IF3(false, z0) -> c15(SORT(del(max(z0), z0)), DEL(max(z0), z0), MAX(z0)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) SORT(z0) -> c13(IF3(empty(z0), z0)) MAX(cons(s(z0), cons(s(z1), x2))) -> c2(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1))) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2), EQ(0, s(z0))) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2), EQ(s(z0), 0)) DEL(s(z0), cons(s(z1), x2)) -> c6(IF2(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1))) K tuples:none Defined Rule Symbols: ge_2, eq_2, del_2, max_1, if1_4, if2_4, empty_1 Defined Pair Symbols: IF1_4, IF2_4, EQ_2, IF3_2, GE_2, SORT_1, MAX_1, DEL_2 Compound Symbols: c3_1, c4_1, c8_1, c12_1, c15_3, c20_1, c13_1, c2_2, c2_1, c6_2 ---------------------------------------- (75) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (76) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) del(z0, nil) -> nil del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) max(nil) -> 0 max(cons(z0, nil)) -> z0 max(cons(z0, cons(z1, z2))) -> if1(ge(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> max(cons(z0, z2)) if1(false, z0, z1, z2) -> max(cons(z1, z2)) if2(true, z0, z1, z2) -> z2 if2(false, z0, z1, z2) -> cons(z1, del(z0, z2)) empty(nil) -> true empty(cons(z0, z1)) -> false Tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) IF3(false, z0) -> c15(SORT(del(max(z0), z0)), DEL(max(z0), z0), MAX(z0)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) SORT(z0) -> c13(IF3(empty(z0), z0)) MAX(cons(s(z0), cons(s(z1), x2))) -> c2(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1))) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(s(z0), cons(s(z1), x2)) -> c6(IF2(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1))) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) S tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) IF3(false, z0) -> c15(SORT(del(max(z0), z0)), DEL(max(z0), z0), MAX(z0)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) SORT(z0) -> c13(IF3(empty(z0), z0)) MAX(cons(s(z0), cons(s(z1), x2))) -> c2(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1))) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(s(z0), cons(s(z1), x2)) -> c6(IF2(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1))) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) K tuples:none Defined Rule Symbols: ge_2, eq_2, del_2, max_1, if1_4, if2_4, empty_1 Defined Pair Symbols: IF1_4, IF2_4, EQ_2, IF3_2, GE_2, SORT_1, MAX_1, DEL_2 Compound Symbols: c3_1, c4_1, c8_1, c12_1, c15_3, c20_1, c13_1, c2_2, c2_1, c6_2, c6_1 ---------------------------------------- (77) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF3(false, z0) -> c15(SORT(del(max(z0), z0)), DEL(max(z0), z0), MAX(z0)) by IF3(false, nil) -> c15(SORT(del(max(nil), nil)), DEL(0, nil), MAX(nil)) IF3(false, cons(z0, nil)) -> c15(SORT(del(max(cons(z0, nil)), cons(z0, nil))), DEL(z0, cons(z0, nil)), MAX(cons(z0, nil))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(del(max(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) IF3(false, x0) -> c15(SORT(del(max(x0), x0))) ---------------------------------------- (78) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) del(z0, nil) -> nil del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) max(nil) -> 0 max(cons(z0, nil)) -> z0 max(cons(z0, cons(z1, z2))) -> if1(ge(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> max(cons(z0, z2)) if1(false, z0, z1, z2) -> max(cons(z1, z2)) if2(true, z0, z1, z2) -> z2 if2(false, z0, z1, z2) -> cons(z1, del(z0, z2)) empty(nil) -> true empty(cons(z0, z1)) -> false Tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) SORT(z0) -> c13(IF3(empty(z0), z0)) MAX(cons(s(z0), cons(s(z1), x2))) -> c2(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1))) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(s(z0), cons(s(z1), x2)) -> c6(IF2(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1))) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) IF3(false, nil) -> c15(SORT(del(max(nil), nil)), DEL(0, nil), MAX(nil)) IF3(false, cons(z0, nil)) -> c15(SORT(del(max(cons(z0, nil)), cons(z0, nil))), DEL(z0, cons(z0, nil)), MAX(cons(z0, nil))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(del(max(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) IF3(false, x0) -> c15(SORT(del(max(x0), x0))) S tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) SORT(z0) -> c13(IF3(empty(z0), z0)) MAX(cons(s(z0), cons(s(z1), x2))) -> c2(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1))) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(s(z0), cons(s(z1), x2)) -> c6(IF2(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1))) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) IF3(false, nil) -> c15(SORT(del(max(nil), nil)), DEL(0, nil), MAX(nil)) IF3(false, cons(z0, nil)) -> c15(SORT(del(max(cons(z0, nil)), cons(z0, nil))), DEL(z0, cons(z0, nil)), MAX(cons(z0, nil))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(del(max(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) IF3(false, x0) -> c15(SORT(del(max(x0), x0))) K tuples:none Defined Rule Symbols: ge_2, eq_2, del_2, max_1, if1_4, if2_4, empty_1 Defined Pair Symbols: IF1_4, IF2_4, EQ_2, GE_2, SORT_1, MAX_1, DEL_2, IF3_2 Compound Symbols: c3_1, c4_1, c8_1, c12_1, c20_1, c13_1, c2_2, c2_1, c6_2, c6_1, c15_3, c15_1 ---------------------------------------- (79) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (80) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) del(z0, nil) -> nil del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) max(nil) -> 0 max(cons(z0, nil)) -> z0 max(cons(z0, cons(z1, z2))) -> if1(ge(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> max(cons(z0, z2)) if1(false, z0, z1, z2) -> max(cons(z1, z2)) if2(true, z0, z1, z2) -> z2 if2(false, z0, z1, z2) -> cons(z1, del(z0, z2)) empty(nil) -> true empty(cons(z0, z1)) -> false Tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) SORT(z0) -> c13(IF3(empty(z0), z0)) MAX(cons(s(z0), cons(s(z1), x2))) -> c2(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1))) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(s(z0), cons(s(z1), x2)) -> c6(IF2(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1))) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(del(max(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) IF3(false, x0) -> c15(SORT(del(max(x0), x0))) IF3(false, nil) -> c15(SORT(del(max(nil), nil))) IF3(false, cons(z0, nil)) -> c15(SORT(del(max(cons(z0, nil)), cons(z0, nil))), DEL(z0, cons(z0, nil))) S tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) SORT(z0) -> c13(IF3(empty(z0), z0)) MAX(cons(s(z0), cons(s(z1), x2))) -> c2(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1))) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(s(z0), cons(s(z1), x2)) -> c6(IF2(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1))) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(del(max(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) IF3(false, x0) -> c15(SORT(del(max(x0), x0))) IF3(false, nil) -> c15(SORT(del(max(nil), nil))) IF3(false, cons(z0, nil)) -> c15(SORT(del(max(cons(z0, nil)), cons(z0, nil))), DEL(z0, cons(z0, nil))) K tuples:none Defined Rule Symbols: ge_2, eq_2, del_2, max_1, if1_4, if2_4, empty_1 Defined Pair Symbols: IF1_4, IF2_4, EQ_2, GE_2, SORT_1, MAX_1, DEL_2, IF3_2 Compound Symbols: c3_1, c4_1, c8_1, c12_1, c20_1, c13_1, c2_2, c2_1, c6_2, c6_1, c15_3, c15_1, c15_2 ---------------------------------------- (81) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace SORT(z0) -> c13(IF3(empty(z0), z0)) by SORT(nil) -> c13(IF3(true, nil)) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) ---------------------------------------- (82) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) del(z0, nil) -> nil del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) max(nil) -> 0 max(cons(z0, nil)) -> z0 max(cons(z0, cons(z1, z2))) -> if1(ge(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> max(cons(z0, z2)) if1(false, z0, z1, z2) -> max(cons(z1, z2)) if2(true, z0, z1, z2) -> z2 if2(false, z0, z1, z2) -> cons(z1, del(z0, z2)) empty(nil) -> true empty(cons(z0, z1)) -> false Tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(s(z0), cons(s(z1), x2))) -> c2(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1))) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(s(z0), cons(s(z1), x2)) -> c6(IF2(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1))) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(del(max(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) IF3(false, x0) -> c15(SORT(del(max(x0), x0))) IF3(false, nil) -> c15(SORT(del(max(nil), nil))) IF3(false, cons(z0, nil)) -> c15(SORT(del(max(cons(z0, nil)), cons(z0, nil))), DEL(z0, cons(z0, nil))) SORT(nil) -> c13(IF3(true, nil)) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) S tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(s(z0), cons(s(z1), x2))) -> c2(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1))) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(s(z0), cons(s(z1), x2)) -> c6(IF2(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1))) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(del(max(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) IF3(false, x0) -> c15(SORT(del(max(x0), x0))) IF3(false, nil) -> c15(SORT(del(max(nil), nil))) IF3(false, cons(z0, nil)) -> c15(SORT(del(max(cons(z0, nil)), cons(z0, nil))), DEL(z0, cons(z0, nil))) SORT(nil) -> c13(IF3(true, nil)) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) K tuples:none Defined Rule Symbols: ge_2, eq_2, del_2, max_1, if1_4, if2_4, empty_1 Defined Pair Symbols: IF1_4, IF2_4, EQ_2, GE_2, MAX_1, DEL_2, IF3_2, SORT_1 Compound Symbols: c3_1, c4_1, c8_1, c12_1, c20_1, c2_2, c2_1, c6_2, c6_1, c15_3, c15_1, c15_2, c13_1 ---------------------------------------- (83) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: SORT(nil) -> c13(IF3(true, nil)) ---------------------------------------- (84) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) del(z0, nil) -> nil del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) max(nil) -> 0 max(cons(z0, nil)) -> z0 max(cons(z0, cons(z1, z2))) -> if1(ge(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> max(cons(z0, z2)) if1(false, z0, z1, z2) -> max(cons(z1, z2)) if2(true, z0, z1, z2) -> z2 if2(false, z0, z1, z2) -> cons(z1, del(z0, z2)) empty(nil) -> true empty(cons(z0, z1)) -> false Tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(s(z0), cons(s(z1), x2))) -> c2(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1))) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(s(z0), cons(s(z1), x2)) -> c6(IF2(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1))) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(del(max(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) IF3(false, x0) -> c15(SORT(del(max(x0), x0))) IF3(false, nil) -> c15(SORT(del(max(nil), nil))) IF3(false, cons(z0, nil)) -> c15(SORT(del(max(cons(z0, nil)), cons(z0, nil))), DEL(z0, cons(z0, nil))) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) S tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(s(z0), cons(s(z1), x2))) -> c2(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1))) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(s(z0), cons(s(z1), x2)) -> c6(IF2(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1))) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(del(max(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) IF3(false, x0) -> c15(SORT(del(max(x0), x0))) IF3(false, nil) -> c15(SORT(del(max(nil), nil))) IF3(false, cons(z0, nil)) -> c15(SORT(del(max(cons(z0, nil)), cons(z0, nil))), DEL(z0, cons(z0, nil))) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) K tuples:none Defined Rule Symbols: ge_2, eq_2, del_2, max_1, if1_4, if2_4, empty_1 Defined Pair Symbols: IF1_4, IF2_4, EQ_2, GE_2, MAX_1, DEL_2, IF3_2, SORT_1 Compound Symbols: c3_1, c4_1, c8_1, c12_1, c20_1, c2_2, c2_1, c6_2, c6_1, c15_3, c15_1, c15_2, c13_1 ---------------------------------------- (85) CdtKnowledgeProof (BOTH BOUNDS(ID, ID)) The following tuples could be moved from S to K by knowledge propagation: IF3(false, nil) -> c15(SORT(del(max(nil), nil))) ---------------------------------------- (86) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) del(z0, nil) -> nil del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) max(nil) -> 0 max(cons(z0, nil)) -> z0 max(cons(z0, cons(z1, z2))) -> if1(ge(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> max(cons(z0, z2)) if1(false, z0, z1, z2) -> max(cons(z1, z2)) if2(true, z0, z1, z2) -> z2 if2(false, z0, z1, z2) -> cons(z1, del(z0, z2)) empty(nil) -> true empty(cons(z0, z1)) -> false Tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(s(z0), cons(s(z1), x2))) -> c2(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1))) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(s(z0), cons(s(z1), x2)) -> c6(IF2(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1))) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(del(max(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) IF3(false, x0) -> c15(SORT(del(max(x0), x0))) IF3(false, nil) -> c15(SORT(del(max(nil), nil))) IF3(false, cons(z0, nil)) -> c15(SORT(del(max(cons(z0, nil)), cons(z0, nil))), DEL(z0, cons(z0, nil))) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) S tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(s(z0), cons(s(z1), x2))) -> c2(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1))) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(s(z0), cons(s(z1), x2)) -> c6(IF2(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1))) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(del(max(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) IF3(false, x0) -> c15(SORT(del(max(x0), x0))) IF3(false, cons(z0, nil)) -> c15(SORT(del(max(cons(z0, nil)), cons(z0, nil))), DEL(z0, cons(z0, nil))) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) K tuples: IF3(false, nil) -> c15(SORT(del(max(nil), nil))) Defined Rule Symbols: ge_2, eq_2, del_2, max_1, if1_4, if2_4, empty_1 Defined Pair Symbols: IF1_4, IF2_4, EQ_2, GE_2, MAX_1, DEL_2, IF3_2, SORT_1 Compound Symbols: c3_1, c4_1, c8_1, c12_1, c20_1, c2_2, c2_1, c6_2, c6_1, c15_3, c15_1, c15_2, c13_1 ---------------------------------------- (87) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: empty(nil) -> true empty(cons(z0, z1)) -> false ---------------------------------------- (88) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil max(cons(z0, cons(z1, z2))) -> if1(ge(z0, z1), z0, z1, z2) max(cons(z0, nil)) -> z0 max(nil) -> 0 if1(true, z0, z1, z2) -> max(cons(z0, z2)) if1(false, z0, z1, z2) -> max(cons(z1, z2)) if2(true, z0, z1, z2) -> z2 if2(false, z0, z1, z2) -> cons(z1, del(z0, z2)) Tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(s(z0), cons(s(z1), x2))) -> c2(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1))) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(s(z0), cons(s(z1), x2)) -> c6(IF2(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1))) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(del(max(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) IF3(false, x0) -> c15(SORT(del(max(x0), x0))) IF3(false, nil) -> c15(SORT(del(max(nil), nil))) IF3(false, cons(z0, nil)) -> c15(SORT(del(max(cons(z0, nil)), cons(z0, nil))), DEL(z0, cons(z0, nil))) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) S tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(s(z0), cons(s(z1), x2))) -> c2(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1))) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(s(z0), cons(s(z1), x2)) -> c6(IF2(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1))) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(del(max(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) IF3(false, x0) -> c15(SORT(del(max(x0), x0))) IF3(false, cons(z0, nil)) -> c15(SORT(del(max(cons(z0, nil)), cons(z0, nil))), DEL(z0, cons(z0, nil))) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) K tuples: IF3(false, nil) -> c15(SORT(del(max(nil), nil))) Defined Rule Symbols: ge_2, eq_2, del_2, max_1, if1_4, if2_4 Defined Pair Symbols: IF1_4, IF2_4, EQ_2, GE_2, MAX_1, DEL_2, IF3_2, SORT_1 Compound Symbols: c3_1, c4_1, c8_1, c12_1, c20_1, c2_2, c2_1, c6_2, c6_1, c15_3, c15_1, c15_2, c13_1 ---------------------------------------- (89) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MAX(cons(s(z0), cons(s(z1), x2))) -> c2(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1))) by MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c2(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) ---------------------------------------- (90) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil max(cons(z0, cons(z1, z2))) -> if1(ge(z0, z1), z0, z1, z2) max(cons(z0, nil)) -> z0 max(nil) -> 0 if1(true, z0, z1, z2) -> max(cons(z0, z2)) if1(false, z0, z1, z2) -> max(cons(z1, z2)) if2(true, z0, z1, z2) -> z2 if2(false, z0, z1, z2) -> cons(z1, del(z0, z2)) Tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(s(z0), cons(s(z1), x2)) -> c6(IF2(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1))) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(del(max(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) IF3(false, x0) -> c15(SORT(del(max(x0), x0))) IF3(false, nil) -> c15(SORT(del(max(nil), nil))) IF3(false, cons(z0, nil)) -> c15(SORT(del(max(cons(z0, nil)), cons(z0, nil))), DEL(z0, cons(z0, nil))) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c2(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) S tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(s(z0), cons(s(z1), x2)) -> c6(IF2(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1))) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(del(max(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) IF3(false, x0) -> c15(SORT(del(max(x0), x0))) IF3(false, cons(z0, nil)) -> c15(SORT(del(max(cons(z0, nil)), cons(z0, nil))), DEL(z0, cons(z0, nil))) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c2(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) K tuples: IF3(false, nil) -> c15(SORT(del(max(nil), nil))) Defined Rule Symbols: ge_2, eq_2, del_2, max_1, if1_4, if2_4 Defined Pair Symbols: IF1_4, IF2_4, EQ_2, GE_2, MAX_1, DEL_2, IF3_2, SORT_1 Compound Symbols: c3_1, c4_1, c8_1, c12_1, c20_1, c2_1, c6_2, c6_1, c15_3, c15_1, c15_2, c13_1, c2_2 ---------------------------------------- (91) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace DEL(s(z0), cons(s(z1), x2)) -> c6(IF2(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1))) by DEL(s(0), cons(s(0), x2)) -> c6(IF2(true, s(0), s(0), x2), EQ(s(0), s(0))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c6(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) ---------------------------------------- (92) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil max(cons(z0, cons(z1, z2))) -> if1(ge(z0, z1), z0, z1, z2) max(cons(z0, nil)) -> z0 max(nil) -> 0 if1(true, z0, z1, z2) -> max(cons(z0, z2)) if1(false, z0, z1, z2) -> max(cons(z1, z2)) if2(true, z0, z1, z2) -> z2 if2(false, z0, z1, z2) -> cons(z1, del(z0, z2)) Tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(del(max(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) IF3(false, x0) -> c15(SORT(del(max(x0), x0))) IF3(false, nil) -> c15(SORT(del(max(nil), nil))) IF3(false, cons(z0, nil)) -> c15(SORT(del(max(cons(z0, nil)), cons(z0, nil))), DEL(z0, cons(z0, nil))) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c2(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(IF2(true, s(0), s(0), x2), EQ(s(0), s(0))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c6(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) S tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(del(max(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) IF3(false, x0) -> c15(SORT(del(max(x0), x0))) IF3(false, cons(z0, nil)) -> c15(SORT(del(max(cons(z0, nil)), cons(z0, nil))), DEL(z0, cons(z0, nil))) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c2(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(IF2(true, s(0), s(0), x2), EQ(s(0), s(0))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c6(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) K tuples: IF3(false, nil) -> c15(SORT(del(max(nil), nil))) Defined Rule Symbols: ge_2, eq_2, del_2, max_1, if1_4, if2_4 Defined Pair Symbols: IF1_4, IF2_4, EQ_2, GE_2, MAX_1, DEL_2, IF3_2, SORT_1 Compound Symbols: c3_1, c4_1, c8_1, c12_1, c20_1, c2_1, c6_1, c15_3, c15_1, c15_2, c13_1, c2_2, c6_2 ---------------------------------------- (93) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (94) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil max(cons(z0, cons(z1, z2))) -> if1(ge(z0, z1), z0, z1, z2) max(cons(z0, nil)) -> z0 max(nil) -> 0 if1(true, z0, z1, z2) -> max(cons(z0, z2)) if1(false, z0, z1, z2) -> max(cons(z1, z2)) if2(true, z0, z1, z2) -> z2 if2(false, z0, z1, z2) -> cons(z1, del(z0, z2)) Tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(del(max(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) IF3(false, x0) -> c15(SORT(del(max(x0), x0))) IF3(false, nil) -> c15(SORT(del(max(nil), nil))) IF3(false, cons(z0, nil)) -> c15(SORT(del(max(cons(z0, nil)), cons(z0, nil))), DEL(z0, cons(z0, nil))) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c2(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c6(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) S tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(del(max(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) IF3(false, x0) -> c15(SORT(del(max(x0), x0))) IF3(false, cons(z0, nil)) -> c15(SORT(del(max(cons(z0, nil)), cons(z0, nil))), DEL(z0, cons(z0, nil))) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c2(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c6(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) K tuples: IF3(false, nil) -> c15(SORT(del(max(nil), nil))) Defined Rule Symbols: ge_2, eq_2, del_2, max_1, if1_4, if2_4 Defined Pair Symbols: IF1_4, IF2_4, EQ_2, GE_2, MAX_1, DEL_2, IF3_2, SORT_1 Compound Symbols: c3_1, c4_1, c8_1, c12_1, c20_1, c2_1, c6_1, c15_3, c15_1, c15_2, c13_1, c2_2, c6_2 ---------------------------------------- (95) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(del(max(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) by IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(max(cons(z1, cons(x1, x2))), z1), max(cons(z1, cons(x1, x2))), z1, cons(x1, x2))), DEL(if1(ge(z1, x1), z1, x1, x2), cons(z1, cons(x1, x2))), MAX(cons(z1, cons(x1, x2)))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(del(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) ---------------------------------------- (96) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil max(cons(z0, cons(z1, z2))) -> if1(ge(z0, z1), z0, z1, z2) max(cons(z0, nil)) -> z0 max(nil) -> 0 if1(true, z0, z1, z2) -> max(cons(z0, z2)) if1(false, z0, z1, z2) -> max(cons(z1, z2)) if2(true, z0, z1, z2) -> z2 if2(false, z0, z1, z2) -> cons(z1, del(z0, z2)) Tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) IF3(false, x0) -> c15(SORT(del(max(x0), x0))) IF3(false, nil) -> c15(SORT(del(max(nil), nil))) IF3(false, cons(z0, nil)) -> c15(SORT(del(max(cons(z0, nil)), cons(z0, nil))), DEL(z0, cons(z0, nil))) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c2(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c6(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(max(cons(z1, cons(x1, x2))), z1), max(cons(z1, cons(x1, x2))), z1, cons(x1, x2))), DEL(if1(ge(z1, x1), z1, x1, x2), cons(z1, cons(x1, x2))), MAX(cons(z1, cons(x1, x2)))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(del(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) S tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) IF3(false, x0) -> c15(SORT(del(max(x0), x0))) IF3(false, cons(z0, nil)) -> c15(SORT(del(max(cons(z0, nil)), cons(z0, nil))), DEL(z0, cons(z0, nil))) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c2(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c6(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(max(cons(z1, cons(x1, x2))), z1), max(cons(z1, cons(x1, x2))), z1, cons(x1, x2))), DEL(if1(ge(z1, x1), z1, x1, x2), cons(z1, cons(x1, x2))), MAX(cons(z1, cons(x1, x2)))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(del(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) K tuples: IF3(false, nil) -> c15(SORT(del(max(nil), nil))) Defined Rule Symbols: ge_2, eq_2, del_2, max_1, if1_4, if2_4 Defined Pair Symbols: IF1_4, IF2_4, EQ_2, GE_2, MAX_1, DEL_2, IF3_2, SORT_1 Compound Symbols: c3_1, c4_1, c8_1, c12_1, c20_1, c2_1, c6_1, c15_1, c15_2, c13_1, c2_2, c6_2, c15_3 ---------------------------------------- (97) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF3(false, x0) -> c15(SORT(del(max(x0), x0))) by IF3(false, cons(z1, z2)) -> c15(SORT(if2(eq(max(cons(z1, z2)), z1), max(cons(z1, z2)), z1, z2))) IF3(false, nil) -> c15(SORT(nil)) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(del(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))))) IF3(false, cons(z0, nil)) -> c15(SORT(del(z0, cons(z0, nil)))) IF3(false, nil) -> c15(SORT(del(0, nil))) ---------------------------------------- (98) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil max(cons(z0, cons(z1, z2))) -> if1(ge(z0, z1), z0, z1, z2) max(cons(z0, nil)) -> z0 max(nil) -> 0 if1(true, z0, z1, z2) -> max(cons(z0, z2)) if1(false, z0, z1, z2) -> max(cons(z1, z2)) if2(true, z0, z1, z2) -> z2 if2(false, z0, z1, z2) -> cons(z1, del(z0, z2)) Tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) IF3(false, nil) -> c15(SORT(del(max(nil), nil))) IF3(false, cons(z0, nil)) -> c15(SORT(del(max(cons(z0, nil)), cons(z0, nil))), DEL(z0, cons(z0, nil))) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c2(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c6(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(max(cons(z1, cons(x1, x2))), z1), max(cons(z1, cons(x1, x2))), z1, cons(x1, x2))), DEL(if1(ge(z1, x1), z1, x1, x2), cons(z1, cons(x1, x2))), MAX(cons(z1, cons(x1, x2)))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(del(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) IF3(false, cons(z1, z2)) -> c15(SORT(if2(eq(max(cons(z1, z2)), z1), max(cons(z1, z2)), z1, z2))) IF3(false, nil) -> c15(SORT(nil)) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(del(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))))) IF3(false, cons(z0, nil)) -> c15(SORT(del(z0, cons(z0, nil)))) IF3(false, nil) -> c15(SORT(del(0, nil))) S tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) IF3(false, cons(z0, nil)) -> c15(SORT(del(max(cons(z0, nil)), cons(z0, nil))), DEL(z0, cons(z0, nil))) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c2(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c6(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(max(cons(z1, cons(x1, x2))), z1), max(cons(z1, cons(x1, x2))), z1, cons(x1, x2))), DEL(if1(ge(z1, x1), z1, x1, x2), cons(z1, cons(x1, x2))), MAX(cons(z1, cons(x1, x2)))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(del(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) IF3(false, cons(z1, z2)) -> c15(SORT(if2(eq(max(cons(z1, z2)), z1), max(cons(z1, z2)), z1, z2))) IF3(false, nil) -> c15(SORT(nil)) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(del(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))))) IF3(false, cons(z0, nil)) -> c15(SORT(del(z0, cons(z0, nil)))) IF3(false, nil) -> c15(SORT(del(0, nil))) K tuples: IF3(false, nil) -> c15(SORT(del(max(nil), nil))) Defined Rule Symbols: ge_2, eq_2, del_2, max_1, if1_4, if2_4 Defined Pair Symbols: IF1_4, IF2_4, EQ_2, GE_2, MAX_1, DEL_2, IF3_2, SORT_1 Compound Symbols: c3_1, c4_1, c8_1, c12_1, c20_1, c2_1, c6_1, c15_1, c15_2, c13_1, c2_2, c6_2, c15_3 ---------------------------------------- (99) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: IF3(false, nil) -> c15(SORT(nil)) ---------------------------------------- (100) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil max(cons(z0, cons(z1, z2))) -> if1(ge(z0, z1), z0, z1, z2) max(cons(z0, nil)) -> z0 max(nil) -> 0 if1(true, z0, z1, z2) -> max(cons(z0, z2)) if1(false, z0, z1, z2) -> max(cons(z1, z2)) if2(true, z0, z1, z2) -> z2 if2(false, z0, z1, z2) -> cons(z1, del(z0, z2)) Tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) IF3(false, nil) -> c15(SORT(del(max(nil), nil))) IF3(false, cons(z0, nil)) -> c15(SORT(del(max(cons(z0, nil)), cons(z0, nil))), DEL(z0, cons(z0, nil))) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c2(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c6(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(max(cons(z1, cons(x1, x2))), z1), max(cons(z1, cons(x1, x2))), z1, cons(x1, x2))), DEL(if1(ge(z1, x1), z1, x1, x2), cons(z1, cons(x1, x2))), MAX(cons(z1, cons(x1, x2)))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(del(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) IF3(false, cons(z1, z2)) -> c15(SORT(if2(eq(max(cons(z1, z2)), z1), max(cons(z1, z2)), z1, z2))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(del(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))))) IF3(false, cons(z0, nil)) -> c15(SORT(del(z0, cons(z0, nil)))) IF3(false, nil) -> c15(SORT(del(0, nil))) S tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) IF3(false, cons(z0, nil)) -> c15(SORT(del(max(cons(z0, nil)), cons(z0, nil))), DEL(z0, cons(z0, nil))) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c2(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c6(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(max(cons(z1, cons(x1, x2))), z1), max(cons(z1, cons(x1, x2))), z1, cons(x1, x2))), DEL(if1(ge(z1, x1), z1, x1, x2), cons(z1, cons(x1, x2))), MAX(cons(z1, cons(x1, x2)))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(del(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) IF3(false, cons(z1, z2)) -> c15(SORT(if2(eq(max(cons(z1, z2)), z1), max(cons(z1, z2)), z1, z2))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(del(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))))) IF3(false, cons(z0, nil)) -> c15(SORT(del(z0, cons(z0, nil)))) IF3(false, nil) -> c15(SORT(del(0, nil))) K tuples: IF3(false, nil) -> c15(SORT(del(max(nil), nil))) Defined Rule Symbols: ge_2, eq_2, del_2, max_1, if1_4, if2_4 Defined Pair Symbols: IF1_4, IF2_4, EQ_2, GE_2, MAX_1, DEL_2, IF3_2, SORT_1 Compound Symbols: c3_1, c4_1, c8_1, c12_1, c20_1, c2_1, c6_1, c15_1, c15_2, c13_1, c2_2, c6_2, c15_3 ---------------------------------------- (101) CdtKnowledgeProof (BOTH BOUNDS(ID, ID)) The following tuples could be moved from S to K by knowledge propagation: IF3(false, nil) -> c15(SORT(del(0, nil))) ---------------------------------------- (102) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil max(cons(z0, cons(z1, z2))) -> if1(ge(z0, z1), z0, z1, z2) max(cons(z0, nil)) -> z0 max(nil) -> 0 if1(true, z0, z1, z2) -> max(cons(z0, z2)) if1(false, z0, z1, z2) -> max(cons(z1, z2)) if2(true, z0, z1, z2) -> z2 if2(false, z0, z1, z2) -> cons(z1, del(z0, z2)) Tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) IF3(false, nil) -> c15(SORT(del(max(nil), nil))) IF3(false, cons(z0, nil)) -> c15(SORT(del(max(cons(z0, nil)), cons(z0, nil))), DEL(z0, cons(z0, nil))) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c2(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c6(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(max(cons(z1, cons(x1, x2))), z1), max(cons(z1, cons(x1, x2))), z1, cons(x1, x2))), DEL(if1(ge(z1, x1), z1, x1, x2), cons(z1, cons(x1, x2))), MAX(cons(z1, cons(x1, x2)))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(del(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) IF3(false, cons(z1, z2)) -> c15(SORT(if2(eq(max(cons(z1, z2)), z1), max(cons(z1, z2)), z1, z2))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(del(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))))) IF3(false, cons(z0, nil)) -> c15(SORT(del(z0, cons(z0, nil)))) IF3(false, nil) -> c15(SORT(del(0, nil))) S tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) IF3(false, cons(z0, nil)) -> c15(SORT(del(max(cons(z0, nil)), cons(z0, nil))), DEL(z0, cons(z0, nil))) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c2(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c6(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(max(cons(z1, cons(x1, x2))), z1), max(cons(z1, cons(x1, x2))), z1, cons(x1, x2))), DEL(if1(ge(z1, x1), z1, x1, x2), cons(z1, cons(x1, x2))), MAX(cons(z1, cons(x1, x2)))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(del(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) IF3(false, cons(z1, z2)) -> c15(SORT(if2(eq(max(cons(z1, z2)), z1), max(cons(z1, z2)), z1, z2))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(del(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))))) IF3(false, cons(z0, nil)) -> c15(SORT(del(z0, cons(z0, nil)))) K tuples: IF3(false, nil) -> c15(SORT(del(max(nil), nil))) IF3(false, nil) -> c15(SORT(del(0, nil))) Defined Rule Symbols: ge_2, eq_2, del_2, max_1, if1_4, if2_4 Defined Pair Symbols: IF1_4, IF2_4, EQ_2, GE_2, MAX_1, DEL_2, IF3_2, SORT_1 Compound Symbols: c3_1, c4_1, c8_1, c12_1, c20_1, c2_1, c6_1, c15_1, c15_2, c13_1, c2_2, c6_2, c15_3 ---------------------------------------- (103) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF3(false, nil) -> c15(SORT(del(max(nil), nil))) by IF3(false, nil) -> c15(SORT(nil)) IF3(false, nil) -> c15(SORT(del(0, nil))) ---------------------------------------- (104) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil max(cons(z0, cons(z1, z2))) -> if1(ge(z0, z1), z0, z1, z2) max(cons(z0, nil)) -> z0 max(nil) -> 0 if1(true, z0, z1, z2) -> max(cons(z0, z2)) if1(false, z0, z1, z2) -> max(cons(z1, z2)) if2(true, z0, z1, z2) -> z2 if2(false, z0, z1, z2) -> cons(z1, del(z0, z2)) Tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) IF3(false, cons(z0, nil)) -> c15(SORT(del(max(cons(z0, nil)), cons(z0, nil))), DEL(z0, cons(z0, nil))) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c2(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c6(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(max(cons(z1, cons(x1, x2))), z1), max(cons(z1, cons(x1, x2))), z1, cons(x1, x2))), DEL(if1(ge(z1, x1), z1, x1, x2), cons(z1, cons(x1, x2))), MAX(cons(z1, cons(x1, x2)))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(del(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) IF3(false, cons(z1, z2)) -> c15(SORT(if2(eq(max(cons(z1, z2)), z1), max(cons(z1, z2)), z1, z2))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(del(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))))) IF3(false, cons(z0, nil)) -> c15(SORT(del(z0, cons(z0, nil)))) IF3(false, nil) -> c15(SORT(del(0, nil))) IF3(false, nil) -> c15(SORT(nil)) S tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) IF3(false, cons(z0, nil)) -> c15(SORT(del(max(cons(z0, nil)), cons(z0, nil))), DEL(z0, cons(z0, nil))) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c2(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c6(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(max(cons(z1, cons(x1, x2))), z1), max(cons(z1, cons(x1, x2))), z1, cons(x1, x2))), DEL(if1(ge(z1, x1), z1, x1, x2), cons(z1, cons(x1, x2))), MAX(cons(z1, cons(x1, x2)))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(del(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) IF3(false, cons(z1, z2)) -> c15(SORT(if2(eq(max(cons(z1, z2)), z1), max(cons(z1, z2)), z1, z2))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(del(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))))) IF3(false, cons(z0, nil)) -> c15(SORT(del(z0, cons(z0, nil)))) K tuples: IF3(false, nil) -> c15(SORT(del(max(nil), nil))) IF3(false, nil) -> c15(SORT(del(0, nil))) Defined Rule Symbols: ge_2, eq_2, del_2, max_1, if1_4, if2_4 Defined Pair Symbols: IF1_4, IF2_4, EQ_2, GE_2, MAX_1, DEL_2, IF3_2, SORT_1 Compound Symbols: c3_1, c4_1, c8_1, c12_1, c20_1, c2_1, c6_1, c15_2, c13_1, c2_2, c6_2, c15_3, c15_1 ---------------------------------------- (105) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: IF3(false, nil) -> c15(SORT(nil)) ---------------------------------------- (106) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil max(cons(z0, cons(z1, z2))) -> if1(ge(z0, z1), z0, z1, z2) max(cons(z0, nil)) -> z0 max(nil) -> 0 if1(true, z0, z1, z2) -> max(cons(z0, z2)) if1(false, z0, z1, z2) -> max(cons(z1, z2)) if2(true, z0, z1, z2) -> z2 if2(false, z0, z1, z2) -> cons(z1, del(z0, z2)) Tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) IF3(false, cons(z0, nil)) -> c15(SORT(del(max(cons(z0, nil)), cons(z0, nil))), DEL(z0, cons(z0, nil))) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c2(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c6(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(max(cons(z1, cons(x1, x2))), z1), max(cons(z1, cons(x1, x2))), z1, cons(x1, x2))), DEL(if1(ge(z1, x1), z1, x1, x2), cons(z1, cons(x1, x2))), MAX(cons(z1, cons(x1, x2)))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(del(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) IF3(false, cons(z1, z2)) -> c15(SORT(if2(eq(max(cons(z1, z2)), z1), max(cons(z1, z2)), z1, z2))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(del(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))))) IF3(false, cons(z0, nil)) -> c15(SORT(del(z0, cons(z0, nil)))) IF3(false, nil) -> c15(SORT(del(0, nil))) S tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) IF3(false, cons(z0, nil)) -> c15(SORT(del(max(cons(z0, nil)), cons(z0, nil))), DEL(z0, cons(z0, nil))) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c2(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c6(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(max(cons(z1, cons(x1, x2))), z1), max(cons(z1, cons(x1, x2))), z1, cons(x1, x2))), DEL(if1(ge(z1, x1), z1, x1, x2), cons(z1, cons(x1, x2))), MAX(cons(z1, cons(x1, x2)))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(del(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) IF3(false, cons(z1, z2)) -> c15(SORT(if2(eq(max(cons(z1, z2)), z1), max(cons(z1, z2)), z1, z2))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(del(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))))) IF3(false, cons(z0, nil)) -> c15(SORT(del(z0, cons(z0, nil)))) K tuples: IF3(false, nil) -> c15(SORT(del(0, nil))) Defined Rule Symbols: ge_2, eq_2, del_2, max_1, if1_4, if2_4 Defined Pair Symbols: IF1_4, IF2_4, EQ_2, GE_2, MAX_1, DEL_2, IF3_2, SORT_1 Compound Symbols: c3_1, c4_1, c8_1, c12_1, c20_1, c2_1, c6_1, c15_2, c13_1, c2_2, c6_2, c15_3, c15_1 ---------------------------------------- (107) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: max(nil) -> 0 ---------------------------------------- (108) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil max(cons(z0, nil)) -> z0 max(cons(z0, cons(z1, z2))) -> if1(ge(z0, z1), z0, z1, z2) if2(true, z0, z1, z2) -> z2 if2(false, z0, z1, z2) -> cons(z1, del(z0, z2)) eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) if1(true, z0, z1, z2) -> max(cons(z0, z2)) if1(false, z0, z1, z2) -> max(cons(z1, z2)) Tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) IF3(false, cons(z0, nil)) -> c15(SORT(del(max(cons(z0, nil)), cons(z0, nil))), DEL(z0, cons(z0, nil))) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c2(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c6(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(max(cons(z1, cons(x1, x2))), z1), max(cons(z1, cons(x1, x2))), z1, cons(x1, x2))), DEL(if1(ge(z1, x1), z1, x1, x2), cons(z1, cons(x1, x2))), MAX(cons(z1, cons(x1, x2)))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(del(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) IF3(false, cons(z1, z2)) -> c15(SORT(if2(eq(max(cons(z1, z2)), z1), max(cons(z1, z2)), z1, z2))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(del(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))))) IF3(false, cons(z0, nil)) -> c15(SORT(del(z0, cons(z0, nil)))) IF3(false, nil) -> c15(SORT(del(0, nil))) S tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) IF3(false, cons(z0, nil)) -> c15(SORT(del(max(cons(z0, nil)), cons(z0, nil))), DEL(z0, cons(z0, nil))) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c2(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c6(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(max(cons(z1, cons(x1, x2))), z1), max(cons(z1, cons(x1, x2))), z1, cons(x1, x2))), DEL(if1(ge(z1, x1), z1, x1, x2), cons(z1, cons(x1, x2))), MAX(cons(z1, cons(x1, x2)))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(del(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) IF3(false, cons(z1, z2)) -> c15(SORT(if2(eq(max(cons(z1, z2)), z1), max(cons(z1, z2)), z1, z2))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(del(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))))) IF3(false, cons(z0, nil)) -> c15(SORT(del(z0, cons(z0, nil)))) K tuples: IF3(false, nil) -> c15(SORT(del(0, nil))) Defined Rule Symbols: del_2, max_1, if2_4, eq_2, ge_2, if1_4 Defined Pair Symbols: IF1_4, IF2_4, EQ_2, GE_2, MAX_1, DEL_2, IF3_2, SORT_1 Compound Symbols: c3_1, c4_1, c8_1, c12_1, c20_1, c2_1, c6_1, c15_2, c13_1, c2_2, c6_2, c15_3, c15_1 ---------------------------------------- (109) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF3(false, cons(z0, nil)) -> c15(SORT(del(max(cons(z0, nil)), cons(z0, nil))), DEL(z0, cons(z0, nil))) by IF3(false, cons(z1, nil)) -> c15(SORT(if2(eq(max(cons(z1, nil)), z1), max(cons(z1, nil)), z1, nil)), DEL(z1, cons(z1, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(del(z0, cons(z0, nil))), DEL(z0, cons(z0, nil))) IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) ---------------------------------------- (110) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil max(cons(z0, nil)) -> z0 max(cons(z0, cons(z1, z2))) -> if1(ge(z0, z1), z0, z1, z2) if2(true, z0, z1, z2) -> z2 if2(false, z0, z1, z2) -> cons(z1, del(z0, z2)) eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) if1(true, z0, z1, z2) -> max(cons(z0, z2)) if1(false, z0, z1, z2) -> max(cons(z1, z2)) Tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c2(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c6(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(max(cons(z1, cons(x1, x2))), z1), max(cons(z1, cons(x1, x2))), z1, cons(x1, x2))), DEL(if1(ge(z1, x1), z1, x1, x2), cons(z1, cons(x1, x2))), MAX(cons(z1, cons(x1, x2)))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(del(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) IF3(false, cons(z1, z2)) -> c15(SORT(if2(eq(max(cons(z1, z2)), z1), max(cons(z1, z2)), z1, z2))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(del(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))))) IF3(false, cons(z0, nil)) -> c15(SORT(del(z0, cons(z0, nil)))) IF3(false, nil) -> c15(SORT(del(0, nil))) IF3(false, cons(z1, nil)) -> c15(SORT(if2(eq(max(cons(z1, nil)), z1), max(cons(z1, nil)), z1, nil)), DEL(z1, cons(z1, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(del(z0, cons(z0, nil))), DEL(z0, cons(z0, nil))) IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) S tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c2(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c6(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(max(cons(z1, cons(x1, x2))), z1), max(cons(z1, cons(x1, x2))), z1, cons(x1, x2))), DEL(if1(ge(z1, x1), z1, x1, x2), cons(z1, cons(x1, x2))), MAX(cons(z1, cons(x1, x2)))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(del(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) IF3(false, cons(z1, z2)) -> c15(SORT(if2(eq(max(cons(z1, z2)), z1), max(cons(z1, z2)), z1, z2))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(del(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))))) IF3(false, cons(z0, nil)) -> c15(SORT(del(z0, cons(z0, nil)))) IF3(false, cons(z1, nil)) -> c15(SORT(if2(eq(max(cons(z1, nil)), z1), max(cons(z1, nil)), z1, nil)), DEL(z1, cons(z1, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(del(z0, cons(z0, nil))), DEL(z0, cons(z0, nil))) IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) K tuples: IF3(false, nil) -> c15(SORT(del(0, nil))) Defined Rule Symbols: del_2, max_1, if2_4, eq_2, ge_2, if1_4 Defined Pair Symbols: IF1_4, IF2_4, EQ_2, GE_2, MAX_1, DEL_2, SORT_1, IF3_2 Compound Symbols: c3_1, c4_1, c8_1, c12_1, c20_1, c2_1, c6_1, c13_1, c2_2, c6_2, c15_3, c15_1, c15_2 ---------------------------------------- (111) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) We considered the (Usable) Rules: if2(false, z0, z1, z2) -> cons(z1, del(z0, z2)) del(z0, nil) -> nil if2(true, z0, z1, z2) -> z2 del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) And the Tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c2(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c6(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(max(cons(z1, cons(x1, x2))), z1), max(cons(z1, cons(x1, x2))), z1, cons(x1, x2))), DEL(if1(ge(z1, x1), z1, x1, x2), cons(z1, cons(x1, x2))), MAX(cons(z1, cons(x1, x2)))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(del(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) IF3(false, cons(z1, z2)) -> c15(SORT(if2(eq(max(cons(z1, z2)), z1), max(cons(z1, z2)), z1, z2))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(del(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))))) IF3(false, cons(z0, nil)) -> c15(SORT(del(z0, cons(z0, nil)))) IF3(false, nil) -> c15(SORT(del(0, nil))) IF3(false, cons(z1, nil)) -> c15(SORT(if2(eq(max(cons(z1, nil)), z1), max(cons(z1, nil)), z1, nil)), DEL(z1, cons(z1, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(del(z0, cons(z0, nil))), DEL(z0, cons(z0, nil))) IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(DEL(x_1, x_2)) = 0 POL(EQ(x_1, x_2)) = 0 POL(GE(x_1, x_2)) = 0 POL(IF1(x_1, x_2, x_3, x_4)) = 0 POL(IF2(x_1, x_2, x_3, x_4)) = x_3 POL(IF3(x_1, x_2)) = x_1 + x_2 POL(MAX(x_1)) = 0 POL(SORT(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c13(x_1)) = x_1 POL(c15(x_1)) = x_1 POL(c15(x_1, x_2)) = x_1 + x_2 POL(c15(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c2(x_1)) = x_1 POL(c2(x_1, x_2)) = x_1 + x_2 POL(c20(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c4(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c6(x_1, x_2)) = x_1 + x_2 POL(c8(x_1)) = x_1 POL(cons(x_1, x_2)) = x_2 POL(del(x_1, x_2)) = x_2 POL(eq(x_1, x_2)) = 0 POL(false) = 0 POL(ge(x_1, x_2)) = 0 POL(if1(x_1, x_2, x_3, x_4)) = [1] + x_2 + x_3 POL(if2(x_1, x_2, x_3, x_4)) = x_4 POL(max(x_1)) = [1] POL(nil) = [1] POL(s(x_1)) = 0 POL(true) = 0 ---------------------------------------- (112) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil max(cons(z0, nil)) -> z0 max(cons(z0, cons(z1, z2))) -> if1(ge(z0, z1), z0, z1, z2) if2(true, z0, z1, z2) -> z2 if2(false, z0, z1, z2) -> cons(z1, del(z0, z2)) eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) if1(true, z0, z1, z2) -> max(cons(z0, z2)) if1(false, z0, z1, z2) -> max(cons(z1, z2)) Tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c2(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c6(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(max(cons(z1, cons(x1, x2))), z1), max(cons(z1, cons(x1, x2))), z1, cons(x1, x2))), DEL(if1(ge(z1, x1), z1, x1, x2), cons(z1, cons(x1, x2))), MAX(cons(z1, cons(x1, x2)))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(del(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) IF3(false, cons(z1, z2)) -> c15(SORT(if2(eq(max(cons(z1, z2)), z1), max(cons(z1, z2)), z1, z2))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(del(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))))) IF3(false, cons(z0, nil)) -> c15(SORT(del(z0, cons(z0, nil)))) IF3(false, nil) -> c15(SORT(del(0, nil))) IF3(false, cons(z1, nil)) -> c15(SORT(if2(eq(max(cons(z1, nil)), z1), max(cons(z1, nil)), z1, nil)), DEL(z1, cons(z1, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(del(z0, cons(z0, nil))), DEL(z0, cons(z0, nil))) IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) S tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c2(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c6(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(max(cons(z1, cons(x1, x2))), z1), max(cons(z1, cons(x1, x2))), z1, cons(x1, x2))), DEL(if1(ge(z1, x1), z1, x1, x2), cons(z1, cons(x1, x2))), MAX(cons(z1, cons(x1, x2)))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(del(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) IF3(false, cons(z1, z2)) -> c15(SORT(if2(eq(max(cons(z1, z2)), z1), max(cons(z1, z2)), z1, z2))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(del(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))))) IF3(false, cons(z0, nil)) -> c15(SORT(del(z0, cons(z0, nil)))) IF3(false, cons(z1, nil)) -> c15(SORT(if2(eq(max(cons(z1, nil)), z1), max(cons(z1, nil)), z1, nil)), DEL(z1, cons(z1, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(del(z0, cons(z0, nil))), DEL(z0, cons(z0, nil))) K tuples: IF3(false, nil) -> c15(SORT(del(0, nil))) IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) Defined Rule Symbols: del_2, max_1, if2_4, eq_2, ge_2, if1_4 Defined Pair Symbols: IF1_4, IF2_4, EQ_2, GE_2, MAX_1, DEL_2, SORT_1, IF3_2 Compound Symbols: c3_1, c4_1, c8_1, c12_1, c20_1, c2_1, c6_1, c13_1, c2_2, c6_2, c15_3, c15_1, c15_2 ---------------------------------------- (113) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MAX(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c2(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1)))) by MAX(cons(s(s(z0)), cons(s(s(0)), x2))) -> c2(IF1(true, s(s(z0)), s(s(0)), x2), GE(s(s(z0)), s(s(0)))) MAX(cons(s(s(0)), cons(s(s(s(z0))), x2))) -> c2(IF1(false, s(s(0)), s(s(s(z0))), x2), GE(s(s(0)), s(s(s(z0))))) MAX(cons(s(s(s(z0))), cons(s(s(s(z1))), x2))) -> c2(IF1(ge(z0, z1), s(s(s(z0))), s(s(s(z1))), x2), GE(s(s(s(z0))), s(s(s(z1))))) MAX(cons(s(s(x0)), cons(s(s(x1)), x2))) -> c2(GE(s(s(x0)), s(s(x1)))) ---------------------------------------- (114) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil max(cons(z0, nil)) -> z0 max(cons(z0, cons(z1, z2))) -> if1(ge(z0, z1), z0, z1, z2) if2(true, z0, z1, z2) -> z2 if2(false, z0, z1, z2) -> cons(z1, del(z0, z2)) eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) if1(true, z0, z1, z2) -> max(cons(z0, z2)) if1(false, z0, z1, z2) -> max(cons(z1, z2)) Tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c6(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(max(cons(z1, cons(x1, x2))), z1), max(cons(z1, cons(x1, x2))), z1, cons(x1, x2))), DEL(if1(ge(z1, x1), z1, x1, x2), cons(z1, cons(x1, x2))), MAX(cons(z1, cons(x1, x2)))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(del(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) IF3(false, cons(z1, z2)) -> c15(SORT(if2(eq(max(cons(z1, z2)), z1), max(cons(z1, z2)), z1, z2))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(del(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))))) IF3(false, cons(z0, nil)) -> c15(SORT(del(z0, cons(z0, nil)))) IF3(false, nil) -> c15(SORT(del(0, nil))) IF3(false, cons(z1, nil)) -> c15(SORT(if2(eq(max(cons(z1, nil)), z1), max(cons(z1, nil)), z1, nil)), DEL(z1, cons(z1, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(del(z0, cons(z0, nil))), DEL(z0, cons(z0, nil))) IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) MAX(cons(s(s(z0)), cons(s(s(0)), x2))) -> c2(IF1(true, s(s(z0)), s(s(0)), x2), GE(s(s(z0)), s(s(0)))) MAX(cons(s(s(0)), cons(s(s(s(z0))), x2))) -> c2(IF1(false, s(s(0)), s(s(s(z0))), x2), GE(s(s(0)), s(s(s(z0))))) MAX(cons(s(s(s(z0))), cons(s(s(s(z1))), x2))) -> c2(IF1(ge(z0, z1), s(s(s(z0))), s(s(s(z1))), x2), GE(s(s(s(z0))), s(s(s(z1))))) MAX(cons(s(s(x0)), cons(s(s(x1)), x2))) -> c2(GE(s(s(x0)), s(s(x1)))) S tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c6(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(max(cons(z1, cons(x1, x2))), z1), max(cons(z1, cons(x1, x2))), z1, cons(x1, x2))), DEL(if1(ge(z1, x1), z1, x1, x2), cons(z1, cons(x1, x2))), MAX(cons(z1, cons(x1, x2)))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(del(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) IF3(false, cons(z1, z2)) -> c15(SORT(if2(eq(max(cons(z1, z2)), z1), max(cons(z1, z2)), z1, z2))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(del(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))))) IF3(false, cons(z0, nil)) -> c15(SORT(del(z0, cons(z0, nil)))) IF3(false, cons(z1, nil)) -> c15(SORT(if2(eq(max(cons(z1, nil)), z1), max(cons(z1, nil)), z1, nil)), DEL(z1, cons(z1, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(del(z0, cons(z0, nil))), DEL(z0, cons(z0, nil))) MAX(cons(s(s(z0)), cons(s(s(0)), x2))) -> c2(IF1(true, s(s(z0)), s(s(0)), x2), GE(s(s(z0)), s(s(0)))) MAX(cons(s(s(0)), cons(s(s(s(z0))), x2))) -> c2(IF1(false, s(s(0)), s(s(s(z0))), x2), GE(s(s(0)), s(s(s(z0))))) MAX(cons(s(s(s(z0))), cons(s(s(s(z1))), x2))) -> c2(IF1(ge(z0, z1), s(s(s(z0))), s(s(s(z1))), x2), GE(s(s(s(z0))), s(s(s(z1))))) MAX(cons(s(s(x0)), cons(s(s(x1)), x2))) -> c2(GE(s(s(x0)), s(s(x1)))) K tuples: IF3(false, nil) -> c15(SORT(del(0, nil))) IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) Defined Rule Symbols: del_2, max_1, if2_4, eq_2, ge_2, if1_4 Defined Pair Symbols: IF1_4, IF2_4, EQ_2, GE_2, MAX_1, DEL_2, SORT_1, IF3_2 Compound Symbols: c3_1, c4_1, c8_1, c12_1, c20_1, c2_1, c6_1, c13_1, c2_2, c6_2, c15_3, c15_1, c15_2 ---------------------------------------- (115) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c6(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) by DEL(s(s(0)), cons(s(s(0)), x2)) -> c6(IF2(true, s(s(0)), s(s(0)), x2), EQ(s(s(0)), s(s(0)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c6(IF2(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c6(IF2(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(s(z0))), cons(s(s(s(z1))), x2)) -> c6(IF2(eq(z0, z1), s(s(s(z0))), s(s(s(z1))), x2), EQ(s(s(s(z0))), s(s(s(z1))))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c6(EQ(s(s(x0)), s(s(x1)))) ---------------------------------------- (116) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil max(cons(z0, nil)) -> z0 max(cons(z0, cons(z1, z2))) -> if1(ge(z0, z1), z0, z1, z2) if2(true, z0, z1, z2) -> z2 if2(false, z0, z1, z2) -> cons(z1, del(z0, z2)) eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) if1(true, z0, z1, z2) -> max(cons(z0, z2)) if1(false, z0, z1, z2) -> max(cons(z1, z2)) Tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(max(cons(z1, cons(x1, x2))), z1), max(cons(z1, cons(x1, x2))), z1, cons(x1, x2))), DEL(if1(ge(z1, x1), z1, x1, x2), cons(z1, cons(x1, x2))), MAX(cons(z1, cons(x1, x2)))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(del(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) IF3(false, cons(z1, z2)) -> c15(SORT(if2(eq(max(cons(z1, z2)), z1), max(cons(z1, z2)), z1, z2))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(del(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))))) IF3(false, cons(z0, nil)) -> c15(SORT(del(z0, cons(z0, nil)))) IF3(false, nil) -> c15(SORT(del(0, nil))) IF3(false, cons(z1, nil)) -> c15(SORT(if2(eq(max(cons(z1, nil)), z1), max(cons(z1, nil)), z1, nil)), DEL(z1, cons(z1, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(del(z0, cons(z0, nil))), DEL(z0, cons(z0, nil))) IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) MAX(cons(s(s(z0)), cons(s(s(0)), x2))) -> c2(IF1(true, s(s(z0)), s(s(0)), x2), GE(s(s(z0)), s(s(0)))) MAX(cons(s(s(0)), cons(s(s(s(z0))), x2))) -> c2(IF1(false, s(s(0)), s(s(s(z0))), x2), GE(s(s(0)), s(s(s(z0))))) MAX(cons(s(s(s(z0))), cons(s(s(s(z1))), x2))) -> c2(IF1(ge(z0, z1), s(s(s(z0))), s(s(s(z1))), x2), GE(s(s(s(z0))), s(s(s(z1))))) MAX(cons(s(s(x0)), cons(s(s(x1)), x2))) -> c2(GE(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c6(IF2(true, s(s(0)), s(s(0)), x2), EQ(s(s(0)), s(s(0)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c6(IF2(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c6(IF2(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(s(z0))), cons(s(s(s(z1))), x2)) -> c6(IF2(eq(z0, z1), s(s(s(z0))), s(s(s(z1))), x2), EQ(s(s(s(z0))), s(s(s(z1))))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c6(EQ(s(s(x0)), s(s(x1)))) S tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(max(cons(z1, cons(x1, x2))), z1), max(cons(z1, cons(x1, x2))), z1, cons(x1, x2))), DEL(if1(ge(z1, x1), z1, x1, x2), cons(z1, cons(x1, x2))), MAX(cons(z1, cons(x1, x2)))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(del(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) IF3(false, cons(z1, z2)) -> c15(SORT(if2(eq(max(cons(z1, z2)), z1), max(cons(z1, z2)), z1, z2))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(del(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))))) IF3(false, cons(z0, nil)) -> c15(SORT(del(z0, cons(z0, nil)))) IF3(false, cons(z1, nil)) -> c15(SORT(if2(eq(max(cons(z1, nil)), z1), max(cons(z1, nil)), z1, nil)), DEL(z1, cons(z1, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(del(z0, cons(z0, nil))), DEL(z0, cons(z0, nil))) MAX(cons(s(s(z0)), cons(s(s(0)), x2))) -> c2(IF1(true, s(s(z0)), s(s(0)), x2), GE(s(s(z0)), s(s(0)))) MAX(cons(s(s(0)), cons(s(s(s(z0))), x2))) -> c2(IF1(false, s(s(0)), s(s(s(z0))), x2), GE(s(s(0)), s(s(s(z0))))) MAX(cons(s(s(s(z0))), cons(s(s(s(z1))), x2))) -> c2(IF1(ge(z0, z1), s(s(s(z0))), s(s(s(z1))), x2), GE(s(s(s(z0))), s(s(s(z1))))) MAX(cons(s(s(x0)), cons(s(s(x1)), x2))) -> c2(GE(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c6(IF2(true, s(s(0)), s(s(0)), x2), EQ(s(s(0)), s(s(0)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c6(IF2(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c6(IF2(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(s(z0))), cons(s(s(s(z1))), x2)) -> c6(IF2(eq(z0, z1), s(s(s(z0))), s(s(s(z1))), x2), EQ(s(s(s(z0))), s(s(s(z1))))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c6(EQ(s(s(x0)), s(s(x1)))) K tuples: IF3(false, nil) -> c15(SORT(del(0, nil))) IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) Defined Rule Symbols: del_2, max_1, if2_4, eq_2, ge_2, if1_4 Defined Pair Symbols: IF1_4, IF2_4, EQ_2, GE_2, MAX_1, DEL_2, SORT_1, IF3_2 Compound Symbols: c3_1, c4_1, c8_1, c12_1, c20_1, c2_1, c6_1, c13_1, c2_2, c6_2, c15_3, c15_1, c15_2 ---------------------------------------- (117) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (118) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil max(cons(z0, nil)) -> z0 max(cons(z0, cons(z1, z2))) -> if1(ge(z0, z1), z0, z1, z2) if2(true, z0, z1, z2) -> z2 if2(false, z0, z1, z2) -> cons(z1, del(z0, z2)) eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) if1(true, z0, z1, z2) -> max(cons(z0, z2)) if1(false, z0, z1, z2) -> max(cons(z1, z2)) Tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(max(cons(z1, cons(x1, x2))), z1), max(cons(z1, cons(x1, x2))), z1, cons(x1, x2))), DEL(if1(ge(z1, x1), z1, x1, x2), cons(z1, cons(x1, x2))), MAX(cons(z1, cons(x1, x2)))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(del(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) IF3(false, cons(z1, z2)) -> c15(SORT(if2(eq(max(cons(z1, z2)), z1), max(cons(z1, z2)), z1, z2))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(del(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))))) IF3(false, cons(z0, nil)) -> c15(SORT(del(z0, cons(z0, nil)))) IF3(false, nil) -> c15(SORT(del(0, nil))) IF3(false, cons(z1, nil)) -> c15(SORT(if2(eq(max(cons(z1, nil)), z1), max(cons(z1, nil)), z1, nil)), DEL(z1, cons(z1, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(del(z0, cons(z0, nil))), DEL(z0, cons(z0, nil))) IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) MAX(cons(s(s(z0)), cons(s(s(0)), x2))) -> c2(IF1(true, s(s(z0)), s(s(0)), x2), GE(s(s(z0)), s(s(0)))) MAX(cons(s(s(0)), cons(s(s(s(z0))), x2))) -> c2(IF1(false, s(s(0)), s(s(s(z0))), x2), GE(s(s(0)), s(s(s(z0))))) MAX(cons(s(s(s(z0))), cons(s(s(s(z1))), x2))) -> c2(IF1(ge(z0, z1), s(s(s(z0))), s(s(s(z1))), x2), GE(s(s(s(z0))), s(s(s(z1))))) MAX(cons(s(s(x0)), cons(s(s(x1)), x2))) -> c2(GE(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c6(IF2(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c6(IF2(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(s(z0))), cons(s(s(s(z1))), x2)) -> c6(IF2(eq(z0, z1), s(s(s(z0))), s(s(s(z1))), x2), EQ(s(s(s(z0))), s(s(s(z1))))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c6(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c6(EQ(s(s(0)), s(s(0)))) S tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(max(cons(z1, cons(x1, x2))), z1), max(cons(z1, cons(x1, x2))), z1, cons(x1, x2))), DEL(if1(ge(z1, x1), z1, x1, x2), cons(z1, cons(x1, x2))), MAX(cons(z1, cons(x1, x2)))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(del(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) IF3(false, cons(z1, z2)) -> c15(SORT(if2(eq(max(cons(z1, z2)), z1), max(cons(z1, z2)), z1, z2))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(del(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))))) IF3(false, cons(z0, nil)) -> c15(SORT(del(z0, cons(z0, nil)))) IF3(false, cons(z1, nil)) -> c15(SORT(if2(eq(max(cons(z1, nil)), z1), max(cons(z1, nil)), z1, nil)), DEL(z1, cons(z1, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(del(z0, cons(z0, nil))), DEL(z0, cons(z0, nil))) MAX(cons(s(s(z0)), cons(s(s(0)), x2))) -> c2(IF1(true, s(s(z0)), s(s(0)), x2), GE(s(s(z0)), s(s(0)))) MAX(cons(s(s(0)), cons(s(s(s(z0))), x2))) -> c2(IF1(false, s(s(0)), s(s(s(z0))), x2), GE(s(s(0)), s(s(s(z0))))) MAX(cons(s(s(s(z0))), cons(s(s(s(z1))), x2))) -> c2(IF1(ge(z0, z1), s(s(s(z0))), s(s(s(z1))), x2), GE(s(s(s(z0))), s(s(s(z1))))) MAX(cons(s(s(x0)), cons(s(s(x1)), x2))) -> c2(GE(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c6(IF2(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c6(IF2(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(s(z0))), cons(s(s(s(z1))), x2)) -> c6(IF2(eq(z0, z1), s(s(s(z0))), s(s(s(z1))), x2), EQ(s(s(s(z0))), s(s(s(z1))))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c6(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c6(EQ(s(s(0)), s(s(0)))) K tuples: IF3(false, nil) -> c15(SORT(del(0, nil))) IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) Defined Rule Symbols: del_2, max_1, if2_4, eq_2, ge_2, if1_4 Defined Pair Symbols: IF1_4, IF2_4, EQ_2, GE_2, MAX_1, DEL_2, SORT_1, IF3_2 Compound Symbols: c3_1, c4_1, c8_1, c12_1, c20_1, c2_1, c6_1, c13_1, c2_2, c6_2, c15_3, c15_1, c15_2 ---------------------------------------- (119) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(max(cons(z1, cons(x1, x2))), z1), max(cons(z1, cons(x1, x2))), z1, cons(x1, x2))), DEL(if1(ge(z1, x1), z1, x1, x2), cons(z1, cons(x1, x2))), MAX(cons(z1, cons(x1, x2)))) by IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(max(cons(z0, cons(z1, z2))), z0), if1(ge(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), z0, z1, z2), z0), max(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) ---------------------------------------- (120) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil max(cons(z0, nil)) -> z0 max(cons(z0, cons(z1, z2))) -> if1(ge(z0, z1), z0, z1, z2) if2(true, z0, z1, z2) -> z2 if2(false, z0, z1, z2) -> cons(z1, del(z0, z2)) eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) if1(true, z0, z1, z2) -> max(cons(z0, z2)) if1(false, z0, z1, z2) -> max(cons(z1, z2)) Tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(del(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) IF3(false, cons(z1, z2)) -> c15(SORT(if2(eq(max(cons(z1, z2)), z1), max(cons(z1, z2)), z1, z2))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(del(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))))) IF3(false, cons(z0, nil)) -> c15(SORT(del(z0, cons(z0, nil)))) IF3(false, nil) -> c15(SORT(del(0, nil))) IF3(false, cons(z1, nil)) -> c15(SORT(if2(eq(max(cons(z1, nil)), z1), max(cons(z1, nil)), z1, nil)), DEL(z1, cons(z1, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(del(z0, cons(z0, nil))), DEL(z0, cons(z0, nil))) IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) MAX(cons(s(s(z0)), cons(s(s(0)), x2))) -> c2(IF1(true, s(s(z0)), s(s(0)), x2), GE(s(s(z0)), s(s(0)))) MAX(cons(s(s(0)), cons(s(s(s(z0))), x2))) -> c2(IF1(false, s(s(0)), s(s(s(z0))), x2), GE(s(s(0)), s(s(s(z0))))) MAX(cons(s(s(s(z0))), cons(s(s(s(z1))), x2))) -> c2(IF1(ge(z0, z1), s(s(s(z0))), s(s(s(z1))), x2), GE(s(s(s(z0))), s(s(s(z1))))) MAX(cons(s(s(x0)), cons(s(s(x1)), x2))) -> c2(GE(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c6(IF2(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c6(IF2(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(s(z0))), cons(s(s(s(z1))), x2)) -> c6(IF2(eq(z0, z1), s(s(s(z0))), s(s(s(z1))), x2), EQ(s(s(s(z0))), s(s(s(z1))))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c6(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c6(EQ(s(s(0)), s(s(0)))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(max(cons(z0, cons(z1, z2))), z0), if1(ge(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), z0, z1, z2), z0), max(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) S tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(del(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) IF3(false, cons(z1, z2)) -> c15(SORT(if2(eq(max(cons(z1, z2)), z1), max(cons(z1, z2)), z1, z2))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(del(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))))) IF3(false, cons(z0, nil)) -> c15(SORT(del(z0, cons(z0, nil)))) IF3(false, cons(z1, nil)) -> c15(SORT(if2(eq(max(cons(z1, nil)), z1), max(cons(z1, nil)), z1, nil)), DEL(z1, cons(z1, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(del(z0, cons(z0, nil))), DEL(z0, cons(z0, nil))) MAX(cons(s(s(z0)), cons(s(s(0)), x2))) -> c2(IF1(true, s(s(z0)), s(s(0)), x2), GE(s(s(z0)), s(s(0)))) MAX(cons(s(s(0)), cons(s(s(s(z0))), x2))) -> c2(IF1(false, s(s(0)), s(s(s(z0))), x2), GE(s(s(0)), s(s(s(z0))))) MAX(cons(s(s(s(z0))), cons(s(s(s(z1))), x2))) -> c2(IF1(ge(z0, z1), s(s(s(z0))), s(s(s(z1))), x2), GE(s(s(s(z0))), s(s(s(z1))))) MAX(cons(s(s(x0)), cons(s(s(x1)), x2))) -> c2(GE(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c6(IF2(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c6(IF2(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(s(z0))), cons(s(s(s(z1))), x2)) -> c6(IF2(eq(z0, z1), s(s(s(z0))), s(s(s(z1))), x2), EQ(s(s(s(z0))), s(s(s(z1))))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c6(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c6(EQ(s(s(0)), s(s(0)))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(max(cons(z0, cons(z1, z2))), z0), if1(ge(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), z0, z1, z2), z0), max(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) K tuples: IF3(false, nil) -> c15(SORT(del(0, nil))) IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) Defined Rule Symbols: del_2, max_1, if2_4, eq_2, ge_2, if1_4 Defined Pair Symbols: IF1_4, IF2_4, EQ_2, GE_2, MAX_1, DEL_2, SORT_1, IF3_2 Compound Symbols: c3_1, c4_1, c8_1, c12_1, c20_1, c2_1, c6_1, c13_1, c2_2, c6_2, c15_3, c15_1, c15_2 ---------------------------------------- (121) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(del(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) by IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(if1(ge(z1, x1), z1, x1, x2), cons(z1, cons(x1, x2))), MAX(cons(z1, cons(x1, x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2)))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2)))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) ---------------------------------------- (122) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil max(cons(z0, nil)) -> z0 max(cons(z0, cons(z1, z2))) -> if1(ge(z0, z1), z0, z1, z2) if2(true, z0, z1, z2) -> z2 if2(false, z0, z1, z2) -> cons(z1, del(z0, z2)) eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) if1(true, z0, z1, z2) -> max(cons(z0, z2)) if1(false, z0, z1, z2) -> max(cons(z1, z2)) Tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) IF3(false, cons(z1, z2)) -> c15(SORT(if2(eq(max(cons(z1, z2)), z1), max(cons(z1, z2)), z1, z2))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(del(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))))) IF3(false, cons(z0, nil)) -> c15(SORT(del(z0, cons(z0, nil)))) IF3(false, nil) -> c15(SORT(del(0, nil))) IF3(false, cons(z1, nil)) -> c15(SORT(if2(eq(max(cons(z1, nil)), z1), max(cons(z1, nil)), z1, nil)), DEL(z1, cons(z1, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(del(z0, cons(z0, nil))), DEL(z0, cons(z0, nil))) IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) MAX(cons(s(s(z0)), cons(s(s(0)), x2))) -> c2(IF1(true, s(s(z0)), s(s(0)), x2), GE(s(s(z0)), s(s(0)))) MAX(cons(s(s(0)), cons(s(s(s(z0))), x2))) -> c2(IF1(false, s(s(0)), s(s(s(z0))), x2), GE(s(s(0)), s(s(s(z0))))) MAX(cons(s(s(s(z0))), cons(s(s(s(z1))), x2))) -> c2(IF1(ge(z0, z1), s(s(s(z0))), s(s(s(z1))), x2), GE(s(s(s(z0))), s(s(s(z1))))) MAX(cons(s(s(x0)), cons(s(s(x1)), x2))) -> c2(GE(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c6(IF2(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c6(IF2(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(s(z0))), cons(s(s(s(z1))), x2)) -> c6(IF2(eq(z0, z1), s(s(s(z0))), s(s(s(z1))), x2), EQ(s(s(s(z0))), s(s(s(z1))))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c6(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c6(EQ(s(s(0)), s(s(0)))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(max(cons(z0, cons(z1, z2))), z0), if1(ge(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), z0, z1, z2), z0), max(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(if1(ge(z1, x1), z1, x1, x2), cons(z1, cons(x1, x2))), MAX(cons(z1, cons(x1, x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2)))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2)))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) S tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) IF3(false, cons(z1, z2)) -> c15(SORT(if2(eq(max(cons(z1, z2)), z1), max(cons(z1, z2)), z1, z2))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(del(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))))) IF3(false, cons(z0, nil)) -> c15(SORT(del(z0, cons(z0, nil)))) IF3(false, cons(z1, nil)) -> c15(SORT(if2(eq(max(cons(z1, nil)), z1), max(cons(z1, nil)), z1, nil)), DEL(z1, cons(z1, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(del(z0, cons(z0, nil))), DEL(z0, cons(z0, nil))) MAX(cons(s(s(z0)), cons(s(s(0)), x2))) -> c2(IF1(true, s(s(z0)), s(s(0)), x2), GE(s(s(z0)), s(s(0)))) MAX(cons(s(s(0)), cons(s(s(s(z0))), x2))) -> c2(IF1(false, s(s(0)), s(s(s(z0))), x2), GE(s(s(0)), s(s(s(z0))))) MAX(cons(s(s(s(z0))), cons(s(s(s(z1))), x2))) -> c2(IF1(ge(z0, z1), s(s(s(z0))), s(s(s(z1))), x2), GE(s(s(s(z0))), s(s(s(z1))))) MAX(cons(s(s(x0)), cons(s(s(x1)), x2))) -> c2(GE(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c6(IF2(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c6(IF2(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(s(z0))), cons(s(s(s(z1))), x2)) -> c6(IF2(eq(z0, z1), s(s(s(z0))), s(s(s(z1))), x2), EQ(s(s(s(z0))), s(s(s(z1))))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c6(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c6(EQ(s(s(0)), s(s(0)))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(max(cons(z0, cons(z1, z2))), z0), if1(ge(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), z0, z1, z2), z0), max(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(if1(ge(z1, x1), z1, x1, x2), cons(z1, cons(x1, x2))), MAX(cons(z1, cons(x1, x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2)))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2)))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) K tuples: IF3(false, nil) -> c15(SORT(del(0, nil))) IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) Defined Rule Symbols: del_2, max_1, if2_4, eq_2, ge_2, if1_4 Defined Pair Symbols: IF1_4, IF2_4, EQ_2, GE_2, MAX_1, DEL_2, SORT_1, IF3_2 Compound Symbols: c3_1, c4_1, c8_1, c12_1, c20_1, c2_1, c6_1, c13_1, c2_2, c6_2, c15_1, c15_2, c15_3 ---------------------------------------- (123) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF3(false, cons(z1, z2)) -> c15(SORT(if2(eq(max(cons(z1, z2)), z1), max(cons(z1, z2)), z1, z2))) by IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(max(cons(z0, cons(z1, z2))), z0), if1(ge(z0, z1), z0, z1, z2), z0, cons(z1, z2)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), z0, z1, z2), z0), max(cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) ---------------------------------------- (124) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil max(cons(z0, nil)) -> z0 max(cons(z0, cons(z1, z2))) -> if1(ge(z0, z1), z0, z1, z2) if2(true, z0, z1, z2) -> z2 if2(false, z0, z1, z2) -> cons(z1, del(z0, z2)) eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) if1(true, z0, z1, z2) -> max(cons(z0, z2)) if1(false, z0, z1, z2) -> max(cons(z1, z2)) Tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(del(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))))) IF3(false, cons(z0, nil)) -> c15(SORT(del(z0, cons(z0, nil)))) IF3(false, nil) -> c15(SORT(del(0, nil))) IF3(false, cons(z1, nil)) -> c15(SORT(if2(eq(max(cons(z1, nil)), z1), max(cons(z1, nil)), z1, nil)), DEL(z1, cons(z1, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(del(z0, cons(z0, nil))), DEL(z0, cons(z0, nil))) IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) MAX(cons(s(s(z0)), cons(s(s(0)), x2))) -> c2(IF1(true, s(s(z0)), s(s(0)), x2), GE(s(s(z0)), s(s(0)))) MAX(cons(s(s(0)), cons(s(s(s(z0))), x2))) -> c2(IF1(false, s(s(0)), s(s(s(z0))), x2), GE(s(s(0)), s(s(s(z0))))) MAX(cons(s(s(s(z0))), cons(s(s(s(z1))), x2))) -> c2(IF1(ge(z0, z1), s(s(s(z0))), s(s(s(z1))), x2), GE(s(s(s(z0))), s(s(s(z1))))) MAX(cons(s(s(x0)), cons(s(s(x1)), x2))) -> c2(GE(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c6(IF2(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c6(IF2(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(s(z0))), cons(s(s(s(z1))), x2)) -> c6(IF2(eq(z0, z1), s(s(s(z0))), s(s(s(z1))), x2), EQ(s(s(s(z0))), s(s(s(z1))))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c6(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c6(EQ(s(s(0)), s(s(0)))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(max(cons(z0, cons(z1, z2))), z0), if1(ge(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), z0, z1, z2), z0), max(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(if1(ge(z1, x1), z1, x1, x2), cons(z1, cons(x1, x2))), MAX(cons(z1, cons(x1, x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2)))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2)))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(max(cons(z0, cons(z1, z2))), z0), if1(ge(z0, z1), z0, z1, z2), z0, cons(z1, z2)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), z0, z1, z2), z0), max(cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) S tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(del(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))))) IF3(false, cons(z0, nil)) -> c15(SORT(del(z0, cons(z0, nil)))) IF3(false, cons(z1, nil)) -> c15(SORT(if2(eq(max(cons(z1, nil)), z1), max(cons(z1, nil)), z1, nil)), DEL(z1, cons(z1, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(del(z0, cons(z0, nil))), DEL(z0, cons(z0, nil))) MAX(cons(s(s(z0)), cons(s(s(0)), x2))) -> c2(IF1(true, s(s(z0)), s(s(0)), x2), GE(s(s(z0)), s(s(0)))) MAX(cons(s(s(0)), cons(s(s(s(z0))), x2))) -> c2(IF1(false, s(s(0)), s(s(s(z0))), x2), GE(s(s(0)), s(s(s(z0))))) MAX(cons(s(s(s(z0))), cons(s(s(s(z1))), x2))) -> c2(IF1(ge(z0, z1), s(s(s(z0))), s(s(s(z1))), x2), GE(s(s(s(z0))), s(s(s(z1))))) MAX(cons(s(s(x0)), cons(s(s(x1)), x2))) -> c2(GE(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c6(IF2(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c6(IF2(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(s(z0))), cons(s(s(s(z1))), x2)) -> c6(IF2(eq(z0, z1), s(s(s(z0))), s(s(s(z1))), x2), EQ(s(s(s(z0))), s(s(s(z1))))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c6(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c6(EQ(s(s(0)), s(s(0)))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(max(cons(z0, cons(z1, z2))), z0), if1(ge(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), z0, z1, z2), z0), max(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(if1(ge(z1, x1), z1, x1, x2), cons(z1, cons(x1, x2))), MAX(cons(z1, cons(x1, x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2)))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2)))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(max(cons(z0, cons(z1, z2))), z0), if1(ge(z0, z1), z0, z1, z2), z0, cons(z1, z2)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), z0, z1, z2), z0), max(cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) K tuples: IF3(false, nil) -> c15(SORT(del(0, nil))) IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) Defined Rule Symbols: del_2, max_1, if2_4, eq_2, ge_2, if1_4 Defined Pair Symbols: IF1_4, IF2_4, EQ_2, GE_2, MAX_1, DEL_2, SORT_1, IF3_2 Compound Symbols: c3_1, c4_1, c8_1, c12_1, c20_1, c2_1, c6_1, c13_1, c2_2, c6_2, c15_1, c15_2, c15_3 ---------------------------------------- (125) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(del(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))))) by IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2))))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2))))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))))) ---------------------------------------- (126) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil max(cons(z0, nil)) -> z0 max(cons(z0, cons(z1, z2))) -> if1(ge(z0, z1), z0, z1, z2) if2(true, z0, z1, z2) -> z2 if2(false, z0, z1, z2) -> cons(z1, del(z0, z2)) eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) if1(true, z0, z1, z2) -> max(cons(z0, z2)) if1(false, z0, z1, z2) -> max(cons(z1, z2)) Tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) IF3(false, cons(z0, nil)) -> c15(SORT(del(z0, cons(z0, nil)))) IF3(false, nil) -> c15(SORT(del(0, nil))) IF3(false, cons(z1, nil)) -> c15(SORT(if2(eq(max(cons(z1, nil)), z1), max(cons(z1, nil)), z1, nil)), DEL(z1, cons(z1, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(del(z0, cons(z0, nil))), DEL(z0, cons(z0, nil))) IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) MAX(cons(s(s(z0)), cons(s(s(0)), x2))) -> c2(IF1(true, s(s(z0)), s(s(0)), x2), GE(s(s(z0)), s(s(0)))) MAX(cons(s(s(0)), cons(s(s(s(z0))), x2))) -> c2(IF1(false, s(s(0)), s(s(s(z0))), x2), GE(s(s(0)), s(s(s(z0))))) MAX(cons(s(s(s(z0))), cons(s(s(s(z1))), x2))) -> c2(IF1(ge(z0, z1), s(s(s(z0))), s(s(s(z1))), x2), GE(s(s(s(z0))), s(s(s(z1))))) MAX(cons(s(s(x0)), cons(s(s(x1)), x2))) -> c2(GE(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c6(IF2(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c6(IF2(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(s(z0))), cons(s(s(s(z1))), x2)) -> c6(IF2(eq(z0, z1), s(s(s(z0))), s(s(s(z1))), x2), EQ(s(s(s(z0))), s(s(s(z1))))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c6(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c6(EQ(s(s(0)), s(s(0)))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(max(cons(z0, cons(z1, z2))), z0), if1(ge(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), z0, z1, z2), z0), max(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(if1(ge(z1, x1), z1, x1, x2), cons(z1, cons(x1, x2))), MAX(cons(z1, cons(x1, x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2)))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2)))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(max(cons(z0, cons(z1, z2))), z0), if1(ge(z0, z1), z0, z1, z2), z0, cons(z1, z2)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), z0, z1, z2), z0), max(cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2))))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2))))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))))) S tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) IF3(false, cons(z0, nil)) -> c15(SORT(del(z0, cons(z0, nil)))) IF3(false, cons(z1, nil)) -> c15(SORT(if2(eq(max(cons(z1, nil)), z1), max(cons(z1, nil)), z1, nil)), DEL(z1, cons(z1, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(del(z0, cons(z0, nil))), DEL(z0, cons(z0, nil))) MAX(cons(s(s(z0)), cons(s(s(0)), x2))) -> c2(IF1(true, s(s(z0)), s(s(0)), x2), GE(s(s(z0)), s(s(0)))) MAX(cons(s(s(0)), cons(s(s(s(z0))), x2))) -> c2(IF1(false, s(s(0)), s(s(s(z0))), x2), GE(s(s(0)), s(s(s(z0))))) MAX(cons(s(s(s(z0))), cons(s(s(s(z1))), x2))) -> c2(IF1(ge(z0, z1), s(s(s(z0))), s(s(s(z1))), x2), GE(s(s(s(z0))), s(s(s(z1))))) MAX(cons(s(s(x0)), cons(s(s(x1)), x2))) -> c2(GE(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c6(IF2(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c6(IF2(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(s(z0))), cons(s(s(s(z1))), x2)) -> c6(IF2(eq(z0, z1), s(s(s(z0))), s(s(s(z1))), x2), EQ(s(s(s(z0))), s(s(s(z1))))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c6(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c6(EQ(s(s(0)), s(s(0)))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(max(cons(z0, cons(z1, z2))), z0), if1(ge(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), z0, z1, z2), z0), max(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(if1(ge(z1, x1), z1, x1, x2), cons(z1, cons(x1, x2))), MAX(cons(z1, cons(x1, x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2)))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2)))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(max(cons(z0, cons(z1, z2))), z0), if1(ge(z0, z1), z0, z1, z2), z0, cons(z1, z2)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), z0, z1, z2), z0), max(cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2))))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2))))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))))) K tuples: IF3(false, nil) -> c15(SORT(del(0, nil))) IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) Defined Rule Symbols: del_2, max_1, if2_4, eq_2, ge_2, if1_4 Defined Pair Symbols: IF1_4, IF2_4, EQ_2, GE_2, MAX_1, DEL_2, SORT_1, IF3_2 Compound Symbols: c3_1, c4_1, c8_1, c12_1, c20_1, c2_1, c6_1, c13_1, c2_2, c6_2, c15_1, c15_2, c15_3 ---------------------------------------- (127) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF3(false, cons(z0, nil)) -> c15(SORT(del(z0, cons(z0, nil)))) by IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil))) ---------------------------------------- (128) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil max(cons(z0, nil)) -> z0 max(cons(z0, cons(z1, z2))) -> if1(ge(z0, z1), z0, z1, z2) if2(true, z0, z1, z2) -> z2 if2(false, z0, z1, z2) -> cons(z1, del(z0, z2)) eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) if1(true, z0, z1, z2) -> max(cons(z0, z2)) if1(false, z0, z1, z2) -> max(cons(z1, z2)) Tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) IF3(false, nil) -> c15(SORT(del(0, nil))) IF3(false, cons(z1, nil)) -> c15(SORT(if2(eq(max(cons(z1, nil)), z1), max(cons(z1, nil)), z1, nil)), DEL(z1, cons(z1, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(del(z0, cons(z0, nil))), DEL(z0, cons(z0, nil))) IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) MAX(cons(s(s(z0)), cons(s(s(0)), x2))) -> c2(IF1(true, s(s(z0)), s(s(0)), x2), GE(s(s(z0)), s(s(0)))) MAX(cons(s(s(0)), cons(s(s(s(z0))), x2))) -> c2(IF1(false, s(s(0)), s(s(s(z0))), x2), GE(s(s(0)), s(s(s(z0))))) MAX(cons(s(s(s(z0))), cons(s(s(s(z1))), x2))) -> c2(IF1(ge(z0, z1), s(s(s(z0))), s(s(s(z1))), x2), GE(s(s(s(z0))), s(s(s(z1))))) MAX(cons(s(s(x0)), cons(s(s(x1)), x2))) -> c2(GE(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c6(IF2(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c6(IF2(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(s(z0))), cons(s(s(s(z1))), x2)) -> c6(IF2(eq(z0, z1), s(s(s(z0))), s(s(s(z1))), x2), EQ(s(s(s(z0))), s(s(s(z1))))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c6(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c6(EQ(s(s(0)), s(s(0)))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(max(cons(z0, cons(z1, z2))), z0), if1(ge(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), z0, z1, z2), z0), max(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(if1(ge(z1, x1), z1, x1, x2), cons(z1, cons(x1, x2))), MAX(cons(z1, cons(x1, x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2)))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2)))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(max(cons(z0, cons(z1, z2))), z0), if1(ge(z0, z1), z0, z1, z2), z0, cons(z1, z2)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), z0, z1, z2), z0), max(cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2))))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2))))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil))) S tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) IF3(false, cons(z1, nil)) -> c15(SORT(if2(eq(max(cons(z1, nil)), z1), max(cons(z1, nil)), z1, nil)), DEL(z1, cons(z1, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(del(z0, cons(z0, nil))), DEL(z0, cons(z0, nil))) MAX(cons(s(s(z0)), cons(s(s(0)), x2))) -> c2(IF1(true, s(s(z0)), s(s(0)), x2), GE(s(s(z0)), s(s(0)))) MAX(cons(s(s(0)), cons(s(s(s(z0))), x2))) -> c2(IF1(false, s(s(0)), s(s(s(z0))), x2), GE(s(s(0)), s(s(s(z0))))) MAX(cons(s(s(s(z0))), cons(s(s(s(z1))), x2))) -> c2(IF1(ge(z0, z1), s(s(s(z0))), s(s(s(z1))), x2), GE(s(s(s(z0))), s(s(s(z1))))) MAX(cons(s(s(x0)), cons(s(s(x1)), x2))) -> c2(GE(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c6(IF2(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c6(IF2(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(s(z0))), cons(s(s(s(z1))), x2)) -> c6(IF2(eq(z0, z1), s(s(s(z0))), s(s(s(z1))), x2), EQ(s(s(s(z0))), s(s(s(z1))))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c6(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c6(EQ(s(s(0)), s(s(0)))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(max(cons(z0, cons(z1, z2))), z0), if1(ge(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), z0, z1, z2), z0), max(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(if1(ge(z1, x1), z1, x1, x2), cons(z1, cons(x1, x2))), MAX(cons(z1, cons(x1, x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2)))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2)))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(max(cons(z0, cons(z1, z2))), z0), if1(ge(z0, z1), z0, z1, z2), z0, cons(z1, z2)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), z0, z1, z2), z0), max(cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2))))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2))))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil))) K tuples: IF3(false, nil) -> c15(SORT(del(0, nil))) IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) Defined Rule Symbols: del_2, max_1, if2_4, eq_2, ge_2, if1_4 Defined Pair Symbols: IF1_4, IF2_4, EQ_2, GE_2, MAX_1, DEL_2, SORT_1, IF3_2 Compound Symbols: c3_1, c4_1, c8_1, c12_1, c20_1, c2_1, c6_1, c13_1, c2_2, c6_2, c15_1, c15_2, c15_3 ---------------------------------------- (129) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF3(false, nil) -> c15(SORT(del(0, nil))) by IF3(false, nil) -> c15(SORT(nil)) ---------------------------------------- (130) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil max(cons(z0, nil)) -> z0 max(cons(z0, cons(z1, z2))) -> if1(ge(z0, z1), z0, z1, z2) if2(true, z0, z1, z2) -> z2 if2(false, z0, z1, z2) -> cons(z1, del(z0, z2)) eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) if1(true, z0, z1, z2) -> max(cons(z0, z2)) if1(false, z0, z1, z2) -> max(cons(z1, z2)) Tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) IF3(false, cons(z1, nil)) -> c15(SORT(if2(eq(max(cons(z1, nil)), z1), max(cons(z1, nil)), z1, nil)), DEL(z1, cons(z1, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(del(z0, cons(z0, nil))), DEL(z0, cons(z0, nil))) IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) MAX(cons(s(s(z0)), cons(s(s(0)), x2))) -> c2(IF1(true, s(s(z0)), s(s(0)), x2), GE(s(s(z0)), s(s(0)))) MAX(cons(s(s(0)), cons(s(s(s(z0))), x2))) -> c2(IF1(false, s(s(0)), s(s(s(z0))), x2), GE(s(s(0)), s(s(s(z0))))) MAX(cons(s(s(s(z0))), cons(s(s(s(z1))), x2))) -> c2(IF1(ge(z0, z1), s(s(s(z0))), s(s(s(z1))), x2), GE(s(s(s(z0))), s(s(s(z1))))) MAX(cons(s(s(x0)), cons(s(s(x1)), x2))) -> c2(GE(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c6(IF2(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c6(IF2(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(s(z0))), cons(s(s(s(z1))), x2)) -> c6(IF2(eq(z0, z1), s(s(s(z0))), s(s(s(z1))), x2), EQ(s(s(s(z0))), s(s(s(z1))))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c6(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c6(EQ(s(s(0)), s(s(0)))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(max(cons(z0, cons(z1, z2))), z0), if1(ge(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), z0, z1, z2), z0), max(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(if1(ge(z1, x1), z1, x1, x2), cons(z1, cons(x1, x2))), MAX(cons(z1, cons(x1, x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2)))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2)))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(max(cons(z0, cons(z1, z2))), z0), if1(ge(z0, z1), z0, z1, z2), z0, cons(z1, z2)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), z0, z1, z2), z0), max(cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2))))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2))))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil))) IF3(false, nil) -> c15(SORT(nil)) S tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) IF3(false, cons(z1, nil)) -> c15(SORT(if2(eq(max(cons(z1, nil)), z1), max(cons(z1, nil)), z1, nil)), DEL(z1, cons(z1, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(del(z0, cons(z0, nil))), DEL(z0, cons(z0, nil))) MAX(cons(s(s(z0)), cons(s(s(0)), x2))) -> c2(IF1(true, s(s(z0)), s(s(0)), x2), GE(s(s(z0)), s(s(0)))) MAX(cons(s(s(0)), cons(s(s(s(z0))), x2))) -> c2(IF1(false, s(s(0)), s(s(s(z0))), x2), GE(s(s(0)), s(s(s(z0))))) MAX(cons(s(s(s(z0))), cons(s(s(s(z1))), x2))) -> c2(IF1(ge(z0, z1), s(s(s(z0))), s(s(s(z1))), x2), GE(s(s(s(z0))), s(s(s(z1))))) MAX(cons(s(s(x0)), cons(s(s(x1)), x2))) -> c2(GE(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c6(IF2(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c6(IF2(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(s(z0))), cons(s(s(s(z1))), x2)) -> c6(IF2(eq(z0, z1), s(s(s(z0))), s(s(s(z1))), x2), EQ(s(s(s(z0))), s(s(s(z1))))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c6(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c6(EQ(s(s(0)), s(s(0)))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(max(cons(z0, cons(z1, z2))), z0), if1(ge(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), z0, z1, z2), z0), max(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(if1(ge(z1, x1), z1, x1, x2), cons(z1, cons(x1, x2))), MAX(cons(z1, cons(x1, x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2)))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2)))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(max(cons(z0, cons(z1, z2))), z0), if1(ge(z0, z1), z0, z1, z2), z0, cons(z1, z2)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), z0, z1, z2), z0), max(cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2))))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2))))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil))) K tuples: IF3(false, nil) -> c15(SORT(del(0, nil))) IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) Defined Rule Symbols: del_2, max_1, if2_4, eq_2, ge_2, if1_4 Defined Pair Symbols: IF1_4, IF2_4, EQ_2, GE_2, MAX_1, DEL_2, SORT_1, IF3_2 Compound Symbols: c3_1, c4_1, c8_1, c12_1, c20_1, c2_1, c6_1, c13_1, c2_2, c6_2, c15_2, c15_1, c15_3 ---------------------------------------- (131) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: IF3(false, nil) -> c15(SORT(nil)) ---------------------------------------- (132) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil max(cons(z0, nil)) -> z0 max(cons(z0, cons(z1, z2))) -> if1(ge(z0, z1), z0, z1, z2) if2(true, z0, z1, z2) -> z2 if2(false, z0, z1, z2) -> cons(z1, del(z0, z2)) eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) if1(true, z0, z1, z2) -> max(cons(z0, z2)) if1(false, z0, z1, z2) -> max(cons(z1, z2)) Tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) IF3(false, cons(z1, nil)) -> c15(SORT(if2(eq(max(cons(z1, nil)), z1), max(cons(z1, nil)), z1, nil)), DEL(z1, cons(z1, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(del(z0, cons(z0, nil))), DEL(z0, cons(z0, nil))) IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) MAX(cons(s(s(z0)), cons(s(s(0)), x2))) -> c2(IF1(true, s(s(z0)), s(s(0)), x2), GE(s(s(z0)), s(s(0)))) MAX(cons(s(s(0)), cons(s(s(s(z0))), x2))) -> c2(IF1(false, s(s(0)), s(s(s(z0))), x2), GE(s(s(0)), s(s(s(z0))))) MAX(cons(s(s(s(z0))), cons(s(s(s(z1))), x2))) -> c2(IF1(ge(z0, z1), s(s(s(z0))), s(s(s(z1))), x2), GE(s(s(s(z0))), s(s(s(z1))))) MAX(cons(s(s(x0)), cons(s(s(x1)), x2))) -> c2(GE(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c6(IF2(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c6(IF2(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(s(z0))), cons(s(s(s(z1))), x2)) -> c6(IF2(eq(z0, z1), s(s(s(z0))), s(s(s(z1))), x2), EQ(s(s(s(z0))), s(s(s(z1))))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c6(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c6(EQ(s(s(0)), s(s(0)))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(max(cons(z0, cons(z1, z2))), z0), if1(ge(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), z0, z1, z2), z0), max(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(if1(ge(z1, x1), z1, x1, x2), cons(z1, cons(x1, x2))), MAX(cons(z1, cons(x1, x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2)))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2)))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(max(cons(z0, cons(z1, z2))), z0), if1(ge(z0, z1), z0, z1, z2), z0, cons(z1, z2)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), z0, z1, z2), z0), max(cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2))))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2))))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil))) S tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) IF3(false, cons(z1, nil)) -> c15(SORT(if2(eq(max(cons(z1, nil)), z1), max(cons(z1, nil)), z1, nil)), DEL(z1, cons(z1, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(del(z0, cons(z0, nil))), DEL(z0, cons(z0, nil))) MAX(cons(s(s(z0)), cons(s(s(0)), x2))) -> c2(IF1(true, s(s(z0)), s(s(0)), x2), GE(s(s(z0)), s(s(0)))) MAX(cons(s(s(0)), cons(s(s(s(z0))), x2))) -> c2(IF1(false, s(s(0)), s(s(s(z0))), x2), GE(s(s(0)), s(s(s(z0))))) MAX(cons(s(s(s(z0))), cons(s(s(s(z1))), x2))) -> c2(IF1(ge(z0, z1), s(s(s(z0))), s(s(s(z1))), x2), GE(s(s(s(z0))), s(s(s(z1))))) MAX(cons(s(s(x0)), cons(s(s(x1)), x2))) -> c2(GE(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c6(IF2(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c6(IF2(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(s(z0))), cons(s(s(s(z1))), x2)) -> c6(IF2(eq(z0, z1), s(s(s(z0))), s(s(s(z1))), x2), EQ(s(s(s(z0))), s(s(s(z1))))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c6(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c6(EQ(s(s(0)), s(s(0)))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(max(cons(z0, cons(z1, z2))), z0), if1(ge(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), z0, z1, z2), z0), max(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(if1(ge(z1, x1), z1, x1, x2), cons(z1, cons(x1, x2))), MAX(cons(z1, cons(x1, x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2)))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2)))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(max(cons(z0, cons(z1, z2))), z0), if1(ge(z0, z1), z0, z1, z2), z0, cons(z1, z2)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), z0, z1, z2), z0), max(cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2))))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2))))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil))) K tuples: IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) Defined Rule Symbols: del_2, max_1, if2_4, eq_2, ge_2, if1_4 Defined Pair Symbols: IF1_4, IF2_4, EQ_2, GE_2, MAX_1, DEL_2, SORT_1, IF3_2 Compound Symbols: c3_1, c4_1, c8_1, c12_1, c20_1, c2_1, c6_1, c13_1, c2_2, c6_2, c15_2, c15_1, c15_3 ---------------------------------------- (133) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF3(false, cons(z1, nil)) -> c15(SORT(if2(eq(max(cons(z1, nil)), z1), max(cons(z1, nil)), z1, nil)), DEL(z1, cons(z1, nil))) by IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) ---------------------------------------- (134) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil max(cons(z0, nil)) -> z0 max(cons(z0, cons(z1, z2))) -> if1(ge(z0, z1), z0, z1, z2) if2(true, z0, z1, z2) -> z2 if2(false, z0, z1, z2) -> cons(z1, del(z0, z2)) eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) if1(true, z0, z1, z2) -> max(cons(z0, z2)) if1(false, z0, z1, z2) -> max(cons(z1, z2)) Tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) IF3(false, cons(z0, nil)) -> c15(SORT(del(z0, cons(z0, nil))), DEL(z0, cons(z0, nil))) IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) MAX(cons(s(s(z0)), cons(s(s(0)), x2))) -> c2(IF1(true, s(s(z0)), s(s(0)), x2), GE(s(s(z0)), s(s(0)))) MAX(cons(s(s(0)), cons(s(s(s(z0))), x2))) -> c2(IF1(false, s(s(0)), s(s(s(z0))), x2), GE(s(s(0)), s(s(s(z0))))) MAX(cons(s(s(s(z0))), cons(s(s(s(z1))), x2))) -> c2(IF1(ge(z0, z1), s(s(s(z0))), s(s(s(z1))), x2), GE(s(s(s(z0))), s(s(s(z1))))) MAX(cons(s(s(x0)), cons(s(s(x1)), x2))) -> c2(GE(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c6(IF2(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c6(IF2(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(s(z0))), cons(s(s(s(z1))), x2)) -> c6(IF2(eq(z0, z1), s(s(s(z0))), s(s(s(z1))), x2), EQ(s(s(s(z0))), s(s(s(z1))))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c6(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c6(EQ(s(s(0)), s(s(0)))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(max(cons(z0, cons(z1, z2))), z0), if1(ge(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), z0, z1, z2), z0), max(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(if1(ge(z1, x1), z1, x1, x2), cons(z1, cons(x1, x2))), MAX(cons(z1, cons(x1, x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2)))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2)))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(max(cons(z0, cons(z1, z2))), z0), if1(ge(z0, z1), z0, z1, z2), z0, cons(z1, z2)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), z0, z1, z2), z0), max(cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2))))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2))))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) S tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) IF3(false, cons(z0, nil)) -> c15(SORT(del(z0, cons(z0, nil))), DEL(z0, cons(z0, nil))) MAX(cons(s(s(z0)), cons(s(s(0)), x2))) -> c2(IF1(true, s(s(z0)), s(s(0)), x2), GE(s(s(z0)), s(s(0)))) MAX(cons(s(s(0)), cons(s(s(s(z0))), x2))) -> c2(IF1(false, s(s(0)), s(s(s(z0))), x2), GE(s(s(0)), s(s(s(z0))))) MAX(cons(s(s(s(z0))), cons(s(s(s(z1))), x2))) -> c2(IF1(ge(z0, z1), s(s(s(z0))), s(s(s(z1))), x2), GE(s(s(s(z0))), s(s(s(z1))))) MAX(cons(s(s(x0)), cons(s(s(x1)), x2))) -> c2(GE(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c6(IF2(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c6(IF2(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(s(z0))), cons(s(s(s(z1))), x2)) -> c6(IF2(eq(z0, z1), s(s(s(z0))), s(s(s(z1))), x2), EQ(s(s(s(z0))), s(s(s(z1))))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c6(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c6(EQ(s(s(0)), s(s(0)))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(max(cons(z0, cons(z1, z2))), z0), if1(ge(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), z0, z1, z2), z0), max(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(if1(ge(z1, x1), z1, x1, x2), cons(z1, cons(x1, x2))), MAX(cons(z1, cons(x1, x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2)))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2)))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(max(cons(z0, cons(z1, z2))), z0), if1(ge(z0, z1), z0, z1, z2), z0, cons(z1, z2)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), z0, z1, z2), z0), max(cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2))))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2))))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) K tuples: IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) Defined Rule Symbols: del_2, max_1, if2_4, eq_2, ge_2, if1_4 Defined Pair Symbols: IF1_4, IF2_4, EQ_2, GE_2, MAX_1, DEL_2, SORT_1, IF3_2 Compound Symbols: c3_1, c4_1, c8_1, c12_1, c20_1, c2_1, c6_1, c13_1, c2_2, c6_2, c15_2, c15_1, c15_3 ---------------------------------------- (135) CdtKnowledgeProof (BOTH BOUNDS(ID, ID)) The following tuples could be moved from S to K by knowledge propagation: IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) ---------------------------------------- (136) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil max(cons(z0, nil)) -> z0 max(cons(z0, cons(z1, z2))) -> if1(ge(z0, z1), z0, z1, z2) if2(true, z0, z1, z2) -> z2 if2(false, z0, z1, z2) -> cons(z1, del(z0, z2)) eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) if1(true, z0, z1, z2) -> max(cons(z0, z2)) if1(false, z0, z1, z2) -> max(cons(z1, z2)) Tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) IF3(false, cons(z0, nil)) -> c15(SORT(del(z0, cons(z0, nil))), DEL(z0, cons(z0, nil))) IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) MAX(cons(s(s(z0)), cons(s(s(0)), x2))) -> c2(IF1(true, s(s(z0)), s(s(0)), x2), GE(s(s(z0)), s(s(0)))) MAX(cons(s(s(0)), cons(s(s(s(z0))), x2))) -> c2(IF1(false, s(s(0)), s(s(s(z0))), x2), GE(s(s(0)), s(s(s(z0))))) MAX(cons(s(s(s(z0))), cons(s(s(s(z1))), x2))) -> c2(IF1(ge(z0, z1), s(s(s(z0))), s(s(s(z1))), x2), GE(s(s(s(z0))), s(s(s(z1))))) MAX(cons(s(s(x0)), cons(s(s(x1)), x2))) -> c2(GE(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c6(IF2(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c6(IF2(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(s(z0))), cons(s(s(s(z1))), x2)) -> c6(IF2(eq(z0, z1), s(s(s(z0))), s(s(s(z1))), x2), EQ(s(s(s(z0))), s(s(s(z1))))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c6(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c6(EQ(s(s(0)), s(s(0)))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(max(cons(z0, cons(z1, z2))), z0), if1(ge(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), z0, z1, z2), z0), max(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(if1(ge(z1, x1), z1, x1, x2), cons(z1, cons(x1, x2))), MAX(cons(z1, cons(x1, x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2)))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2)))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(max(cons(z0, cons(z1, z2))), z0), if1(ge(z0, z1), z0, z1, z2), z0, cons(z1, z2)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), z0, z1, z2), z0), max(cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2))))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2))))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) S tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) IF3(false, cons(z0, nil)) -> c15(SORT(del(z0, cons(z0, nil))), DEL(z0, cons(z0, nil))) MAX(cons(s(s(z0)), cons(s(s(0)), x2))) -> c2(IF1(true, s(s(z0)), s(s(0)), x2), GE(s(s(z0)), s(s(0)))) MAX(cons(s(s(0)), cons(s(s(s(z0))), x2))) -> c2(IF1(false, s(s(0)), s(s(s(z0))), x2), GE(s(s(0)), s(s(s(z0))))) MAX(cons(s(s(s(z0))), cons(s(s(s(z1))), x2))) -> c2(IF1(ge(z0, z1), s(s(s(z0))), s(s(s(z1))), x2), GE(s(s(s(z0))), s(s(s(z1))))) MAX(cons(s(s(x0)), cons(s(s(x1)), x2))) -> c2(GE(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c6(IF2(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c6(IF2(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(s(z0))), cons(s(s(s(z1))), x2)) -> c6(IF2(eq(z0, z1), s(s(s(z0))), s(s(s(z1))), x2), EQ(s(s(s(z0))), s(s(s(z1))))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c6(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c6(EQ(s(s(0)), s(s(0)))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(max(cons(z0, cons(z1, z2))), z0), if1(ge(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), z0, z1, z2), z0), max(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(if1(ge(z1, x1), z1, x1, x2), cons(z1, cons(x1, x2))), MAX(cons(z1, cons(x1, x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2)))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2)))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(max(cons(z0, cons(z1, z2))), z0), if1(ge(z0, z1), z0, z1, z2), z0, cons(z1, z2)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), z0, z1, z2), z0), max(cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2))))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2))))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) K tuples: IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) Defined Rule Symbols: del_2, max_1, if2_4, eq_2, ge_2, if1_4 Defined Pair Symbols: IF1_4, IF2_4, EQ_2, GE_2, MAX_1, DEL_2, SORT_1, IF3_2 Compound Symbols: c3_1, c4_1, c8_1, c12_1, c20_1, c2_1, c6_1, c13_1, c2_2, c6_2, c15_2, c15_1, c15_3 ---------------------------------------- (137) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF3(false, cons(z0, nil)) -> c15(SORT(del(z0, cons(z0, nil))), DEL(z0, cons(z0, nil))) by IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) ---------------------------------------- (138) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil max(cons(z0, nil)) -> z0 max(cons(z0, cons(z1, z2))) -> if1(ge(z0, z1), z0, z1, z2) if2(true, z0, z1, z2) -> z2 if2(false, z0, z1, z2) -> cons(z1, del(z0, z2)) eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) if1(true, z0, z1, z2) -> max(cons(z0, z2)) if1(false, z0, z1, z2) -> max(cons(z1, z2)) Tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) MAX(cons(s(s(z0)), cons(s(s(0)), x2))) -> c2(IF1(true, s(s(z0)), s(s(0)), x2), GE(s(s(z0)), s(s(0)))) MAX(cons(s(s(0)), cons(s(s(s(z0))), x2))) -> c2(IF1(false, s(s(0)), s(s(s(z0))), x2), GE(s(s(0)), s(s(s(z0))))) MAX(cons(s(s(s(z0))), cons(s(s(s(z1))), x2))) -> c2(IF1(ge(z0, z1), s(s(s(z0))), s(s(s(z1))), x2), GE(s(s(s(z0))), s(s(s(z1))))) MAX(cons(s(s(x0)), cons(s(s(x1)), x2))) -> c2(GE(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c6(IF2(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c6(IF2(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(s(z0))), cons(s(s(s(z1))), x2)) -> c6(IF2(eq(z0, z1), s(s(s(z0))), s(s(s(z1))), x2), EQ(s(s(s(z0))), s(s(s(z1))))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c6(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c6(EQ(s(s(0)), s(s(0)))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(max(cons(z0, cons(z1, z2))), z0), if1(ge(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), z0, z1, z2), z0), max(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(if1(ge(z1, x1), z1, x1, x2), cons(z1, cons(x1, x2))), MAX(cons(z1, cons(x1, x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2)))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2)))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(max(cons(z0, cons(z1, z2))), z0), if1(ge(z0, z1), z0, z1, z2), z0, cons(z1, z2)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), z0, z1, z2), z0), max(cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2))))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2))))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) S tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) MAX(cons(s(s(z0)), cons(s(s(0)), x2))) -> c2(IF1(true, s(s(z0)), s(s(0)), x2), GE(s(s(z0)), s(s(0)))) MAX(cons(s(s(0)), cons(s(s(s(z0))), x2))) -> c2(IF1(false, s(s(0)), s(s(s(z0))), x2), GE(s(s(0)), s(s(s(z0))))) MAX(cons(s(s(s(z0))), cons(s(s(s(z1))), x2))) -> c2(IF1(ge(z0, z1), s(s(s(z0))), s(s(s(z1))), x2), GE(s(s(s(z0))), s(s(s(z1))))) MAX(cons(s(s(x0)), cons(s(s(x1)), x2))) -> c2(GE(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c6(IF2(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c6(IF2(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(s(z0))), cons(s(s(s(z1))), x2)) -> c6(IF2(eq(z0, z1), s(s(s(z0))), s(s(s(z1))), x2), EQ(s(s(s(z0))), s(s(s(z1))))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c6(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c6(EQ(s(s(0)), s(s(0)))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(max(cons(z0, cons(z1, z2))), z0), if1(ge(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), z0, z1, z2), z0), max(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(if1(ge(z1, x1), z1, x1, x2), cons(z1, cons(x1, x2))), MAX(cons(z1, cons(x1, x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2)))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2)))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(max(cons(z0, cons(z1, z2))), z0), if1(ge(z0, z1), z0, z1, z2), z0, cons(z1, z2)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), z0, z1, z2), z0), max(cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2))))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2))))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) K tuples: IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) Defined Rule Symbols: del_2, max_1, if2_4, eq_2, ge_2, if1_4 Defined Pair Symbols: IF1_4, IF2_4, EQ_2, GE_2, MAX_1, DEL_2, SORT_1, IF3_2 Compound Symbols: c3_1, c4_1, c8_1, c12_1, c20_1, c2_1, c6_1, c13_1, c2_2, c6_2, c15_1, c15_3, c15_2 ---------------------------------------- (139) CdtKnowledgeProof (BOTH BOUNDS(ID, ID)) The following tuples could be moved from S to K by knowledge propagation: IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) ---------------------------------------- (140) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil max(cons(z0, nil)) -> z0 max(cons(z0, cons(z1, z2))) -> if1(ge(z0, z1), z0, z1, z2) if2(true, z0, z1, z2) -> z2 if2(false, z0, z1, z2) -> cons(z1, del(z0, z2)) eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) if1(true, z0, z1, z2) -> max(cons(z0, z2)) if1(false, z0, z1, z2) -> max(cons(z1, z2)) Tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) MAX(cons(s(s(z0)), cons(s(s(0)), x2))) -> c2(IF1(true, s(s(z0)), s(s(0)), x2), GE(s(s(z0)), s(s(0)))) MAX(cons(s(s(0)), cons(s(s(s(z0))), x2))) -> c2(IF1(false, s(s(0)), s(s(s(z0))), x2), GE(s(s(0)), s(s(s(z0))))) MAX(cons(s(s(s(z0))), cons(s(s(s(z1))), x2))) -> c2(IF1(ge(z0, z1), s(s(s(z0))), s(s(s(z1))), x2), GE(s(s(s(z0))), s(s(s(z1))))) MAX(cons(s(s(x0)), cons(s(s(x1)), x2))) -> c2(GE(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c6(IF2(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c6(IF2(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(s(z0))), cons(s(s(s(z1))), x2)) -> c6(IF2(eq(z0, z1), s(s(s(z0))), s(s(s(z1))), x2), EQ(s(s(s(z0))), s(s(s(z1))))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c6(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c6(EQ(s(s(0)), s(s(0)))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(max(cons(z0, cons(z1, z2))), z0), if1(ge(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), z0, z1, z2), z0), max(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(if1(ge(z1, x1), z1, x1, x2), cons(z1, cons(x1, x2))), MAX(cons(z1, cons(x1, x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2)))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2)))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(max(cons(z0, cons(z1, z2))), z0), if1(ge(z0, z1), z0, z1, z2), z0, cons(z1, z2)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), z0, z1, z2), z0), max(cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2))))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2))))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) S tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) MAX(cons(s(s(z0)), cons(s(s(0)), x2))) -> c2(IF1(true, s(s(z0)), s(s(0)), x2), GE(s(s(z0)), s(s(0)))) MAX(cons(s(s(0)), cons(s(s(s(z0))), x2))) -> c2(IF1(false, s(s(0)), s(s(s(z0))), x2), GE(s(s(0)), s(s(s(z0))))) MAX(cons(s(s(s(z0))), cons(s(s(s(z1))), x2))) -> c2(IF1(ge(z0, z1), s(s(s(z0))), s(s(s(z1))), x2), GE(s(s(s(z0))), s(s(s(z1))))) MAX(cons(s(s(x0)), cons(s(s(x1)), x2))) -> c2(GE(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c6(IF2(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c6(IF2(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(s(z0))), cons(s(s(s(z1))), x2)) -> c6(IF2(eq(z0, z1), s(s(s(z0))), s(s(s(z1))), x2), EQ(s(s(s(z0))), s(s(s(z1))))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c6(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c6(EQ(s(s(0)), s(s(0)))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(max(cons(z0, cons(z1, z2))), z0), if1(ge(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), z0, z1, z2), z0), max(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(if1(ge(z1, x1), z1, x1, x2), cons(z1, cons(x1, x2))), MAX(cons(z1, cons(x1, x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2)))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2)))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(max(cons(z0, cons(z1, z2))), z0), if1(ge(z0, z1), z0, z1, z2), z0, cons(z1, z2)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), z0, z1, z2), z0), max(cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2))))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2))))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) K tuples: IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) Defined Rule Symbols: del_2, max_1, if2_4, eq_2, ge_2, if1_4 Defined Pair Symbols: IF1_4, IF2_4, EQ_2, GE_2, MAX_1, DEL_2, SORT_1, IF3_2 Compound Symbols: c3_1, c4_1, c8_1, c12_1, c20_1, c2_1, c6_1, c13_1, c2_2, c6_2, c15_1, c15_3, c15_2 ---------------------------------------- (141) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MAX(cons(s(s(s(z0))), cons(s(s(s(z1))), x2))) -> c2(IF1(ge(z0, z1), s(s(s(z0))), s(s(s(z1))), x2), GE(s(s(s(z0))), s(s(s(z1))))) by MAX(cons(s(s(s(z0))), cons(s(s(s(0))), x2))) -> c2(IF1(true, s(s(s(z0))), s(s(s(0))), x2), GE(s(s(s(z0))), s(s(s(0))))) MAX(cons(s(s(s(0))), cons(s(s(s(s(z0)))), x2))) -> c2(IF1(false, s(s(s(0))), s(s(s(s(z0)))), x2), GE(s(s(s(0))), s(s(s(s(z0)))))) MAX(cons(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2))) -> c2(IF1(ge(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), GE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MAX(cons(s(s(s(x0))), cons(s(s(s(x1))), x2))) -> c2(GE(s(s(s(x0))), s(s(s(x1))))) ---------------------------------------- (142) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil max(cons(z0, nil)) -> z0 max(cons(z0, cons(z1, z2))) -> if1(ge(z0, z1), z0, z1, z2) if2(true, z0, z1, z2) -> z2 if2(false, z0, z1, z2) -> cons(z1, del(z0, z2)) eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) if1(true, z0, z1, z2) -> max(cons(z0, z2)) if1(false, z0, z1, z2) -> max(cons(z1, z2)) Tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) MAX(cons(s(s(z0)), cons(s(s(0)), x2))) -> c2(IF1(true, s(s(z0)), s(s(0)), x2), GE(s(s(z0)), s(s(0)))) MAX(cons(s(s(0)), cons(s(s(s(z0))), x2))) -> c2(IF1(false, s(s(0)), s(s(s(z0))), x2), GE(s(s(0)), s(s(s(z0))))) MAX(cons(s(s(x0)), cons(s(s(x1)), x2))) -> c2(GE(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c6(IF2(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c6(IF2(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(s(z0))), cons(s(s(s(z1))), x2)) -> c6(IF2(eq(z0, z1), s(s(s(z0))), s(s(s(z1))), x2), EQ(s(s(s(z0))), s(s(s(z1))))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c6(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c6(EQ(s(s(0)), s(s(0)))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(max(cons(z0, cons(z1, z2))), z0), if1(ge(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), z0, z1, z2), z0), max(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(if1(ge(z1, x1), z1, x1, x2), cons(z1, cons(x1, x2))), MAX(cons(z1, cons(x1, x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2)))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2)))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(max(cons(z0, cons(z1, z2))), z0), if1(ge(z0, z1), z0, z1, z2), z0, cons(z1, z2)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), z0, z1, z2), z0), max(cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2))))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2))))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) MAX(cons(s(s(s(z0))), cons(s(s(s(0))), x2))) -> c2(IF1(true, s(s(s(z0))), s(s(s(0))), x2), GE(s(s(s(z0))), s(s(s(0))))) MAX(cons(s(s(s(0))), cons(s(s(s(s(z0)))), x2))) -> c2(IF1(false, s(s(s(0))), s(s(s(s(z0)))), x2), GE(s(s(s(0))), s(s(s(s(z0)))))) MAX(cons(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2))) -> c2(IF1(ge(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), GE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MAX(cons(s(s(s(x0))), cons(s(s(s(x1))), x2))) -> c2(GE(s(s(s(x0))), s(s(s(x1))))) S tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) MAX(cons(s(s(z0)), cons(s(s(0)), x2))) -> c2(IF1(true, s(s(z0)), s(s(0)), x2), GE(s(s(z0)), s(s(0)))) MAX(cons(s(s(0)), cons(s(s(s(z0))), x2))) -> c2(IF1(false, s(s(0)), s(s(s(z0))), x2), GE(s(s(0)), s(s(s(z0))))) MAX(cons(s(s(x0)), cons(s(s(x1)), x2))) -> c2(GE(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c6(IF2(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c6(IF2(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(s(z0))), cons(s(s(s(z1))), x2)) -> c6(IF2(eq(z0, z1), s(s(s(z0))), s(s(s(z1))), x2), EQ(s(s(s(z0))), s(s(s(z1))))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c6(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c6(EQ(s(s(0)), s(s(0)))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(max(cons(z0, cons(z1, z2))), z0), if1(ge(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), z0, z1, z2), z0), max(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(if1(ge(z1, x1), z1, x1, x2), cons(z1, cons(x1, x2))), MAX(cons(z1, cons(x1, x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2)))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2)))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(max(cons(z0, cons(z1, z2))), z0), if1(ge(z0, z1), z0, z1, z2), z0, cons(z1, z2)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), z0, z1, z2), z0), max(cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2))))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2))))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) MAX(cons(s(s(s(z0))), cons(s(s(s(0))), x2))) -> c2(IF1(true, s(s(s(z0))), s(s(s(0))), x2), GE(s(s(s(z0))), s(s(s(0))))) MAX(cons(s(s(s(0))), cons(s(s(s(s(z0)))), x2))) -> c2(IF1(false, s(s(s(0))), s(s(s(s(z0)))), x2), GE(s(s(s(0))), s(s(s(s(z0)))))) MAX(cons(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2))) -> c2(IF1(ge(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), GE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MAX(cons(s(s(s(x0))), cons(s(s(s(x1))), x2))) -> c2(GE(s(s(s(x0))), s(s(s(x1))))) K tuples: IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) Defined Rule Symbols: del_2, max_1, if2_4, eq_2, ge_2, if1_4 Defined Pair Symbols: IF1_4, IF2_4, EQ_2, GE_2, MAX_1, DEL_2, SORT_1, IF3_2 Compound Symbols: c3_1, c4_1, c8_1, c12_1, c20_1, c2_1, c6_1, c13_1, c2_2, c6_2, c15_1, c15_3, c15_2 ---------------------------------------- (143) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace DEL(s(s(s(z0))), cons(s(s(s(z1))), x2)) -> c6(IF2(eq(z0, z1), s(s(s(z0))), s(s(s(z1))), x2), EQ(s(s(s(z0))), s(s(s(z1))))) by DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c6(IF2(true, s(s(s(0))), s(s(s(0))), x2), EQ(s(s(s(0))), s(s(s(0))))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c6(IF2(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c6(IF2(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c6(IF2(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c6(EQ(s(s(s(x0))), s(s(s(x1))))) ---------------------------------------- (144) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil max(cons(z0, nil)) -> z0 max(cons(z0, cons(z1, z2))) -> if1(ge(z0, z1), z0, z1, z2) if2(true, z0, z1, z2) -> z2 if2(false, z0, z1, z2) -> cons(z1, del(z0, z2)) eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) if1(true, z0, z1, z2) -> max(cons(z0, z2)) if1(false, z0, z1, z2) -> max(cons(z1, z2)) Tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) MAX(cons(s(s(z0)), cons(s(s(0)), x2))) -> c2(IF1(true, s(s(z0)), s(s(0)), x2), GE(s(s(z0)), s(s(0)))) MAX(cons(s(s(0)), cons(s(s(s(z0))), x2))) -> c2(IF1(false, s(s(0)), s(s(s(z0))), x2), GE(s(s(0)), s(s(s(z0))))) MAX(cons(s(s(x0)), cons(s(s(x1)), x2))) -> c2(GE(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c6(IF2(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c6(IF2(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c6(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c6(EQ(s(s(0)), s(s(0)))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(max(cons(z0, cons(z1, z2))), z0), if1(ge(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), z0, z1, z2), z0), max(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(if1(ge(z1, x1), z1, x1, x2), cons(z1, cons(x1, x2))), MAX(cons(z1, cons(x1, x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2)))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2)))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(max(cons(z0, cons(z1, z2))), z0), if1(ge(z0, z1), z0, z1, z2), z0, cons(z1, z2)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), z0, z1, z2), z0), max(cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2))))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2))))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) MAX(cons(s(s(s(z0))), cons(s(s(s(0))), x2))) -> c2(IF1(true, s(s(s(z0))), s(s(s(0))), x2), GE(s(s(s(z0))), s(s(s(0))))) MAX(cons(s(s(s(0))), cons(s(s(s(s(z0)))), x2))) -> c2(IF1(false, s(s(s(0))), s(s(s(s(z0)))), x2), GE(s(s(s(0))), s(s(s(s(z0)))))) MAX(cons(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2))) -> c2(IF1(ge(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), GE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MAX(cons(s(s(s(x0))), cons(s(s(s(x1))), x2))) -> c2(GE(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c6(IF2(true, s(s(s(0))), s(s(s(0))), x2), EQ(s(s(s(0))), s(s(s(0))))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c6(IF2(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c6(IF2(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c6(IF2(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c6(EQ(s(s(s(x0))), s(s(s(x1))))) S tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) MAX(cons(s(s(z0)), cons(s(s(0)), x2))) -> c2(IF1(true, s(s(z0)), s(s(0)), x2), GE(s(s(z0)), s(s(0)))) MAX(cons(s(s(0)), cons(s(s(s(z0))), x2))) -> c2(IF1(false, s(s(0)), s(s(s(z0))), x2), GE(s(s(0)), s(s(s(z0))))) MAX(cons(s(s(x0)), cons(s(s(x1)), x2))) -> c2(GE(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c6(IF2(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c6(IF2(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c6(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c6(EQ(s(s(0)), s(s(0)))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(max(cons(z0, cons(z1, z2))), z0), if1(ge(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), z0, z1, z2), z0), max(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(if1(ge(z1, x1), z1, x1, x2), cons(z1, cons(x1, x2))), MAX(cons(z1, cons(x1, x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2)))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2)))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(max(cons(z0, cons(z1, z2))), z0), if1(ge(z0, z1), z0, z1, z2), z0, cons(z1, z2)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), z0, z1, z2), z0), max(cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2))))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2))))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) MAX(cons(s(s(s(z0))), cons(s(s(s(0))), x2))) -> c2(IF1(true, s(s(s(z0))), s(s(s(0))), x2), GE(s(s(s(z0))), s(s(s(0))))) MAX(cons(s(s(s(0))), cons(s(s(s(s(z0)))), x2))) -> c2(IF1(false, s(s(s(0))), s(s(s(s(z0)))), x2), GE(s(s(s(0))), s(s(s(s(z0)))))) MAX(cons(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2))) -> c2(IF1(ge(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), GE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MAX(cons(s(s(s(x0))), cons(s(s(s(x1))), x2))) -> c2(GE(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c6(IF2(true, s(s(s(0))), s(s(s(0))), x2), EQ(s(s(s(0))), s(s(s(0))))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c6(IF2(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c6(IF2(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c6(IF2(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c6(EQ(s(s(s(x0))), s(s(s(x1))))) K tuples: IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) Defined Rule Symbols: del_2, max_1, if2_4, eq_2, ge_2, if1_4 Defined Pair Symbols: IF1_4, IF2_4, EQ_2, GE_2, MAX_1, DEL_2, SORT_1, IF3_2 Compound Symbols: c3_1, c4_1, c8_1, c12_1, c20_1, c2_1, c6_1, c13_1, c2_2, c6_2, c15_1, c15_3, c15_2 ---------------------------------------- (145) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (146) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil max(cons(z0, nil)) -> z0 max(cons(z0, cons(z1, z2))) -> if1(ge(z0, z1), z0, z1, z2) if2(true, z0, z1, z2) -> z2 if2(false, z0, z1, z2) -> cons(z1, del(z0, z2)) eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) if1(true, z0, z1, z2) -> max(cons(z0, z2)) if1(false, z0, z1, z2) -> max(cons(z1, z2)) Tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) MAX(cons(s(s(z0)), cons(s(s(0)), x2))) -> c2(IF1(true, s(s(z0)), s(s(0)), x2), GE(s(s(z0)), s(s(0)))) MAX(cons(s(s(0)), cons(s(s(s(z0))), x2))) -> c2(IF1(false, s(s(0)), s(s(s(z0))), x2), GE(s(s(0)), s(s(s(z0))))) MAX(cons(s(s(x0)), cons(s(s(x1)), x2))) -> c2(GE(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c6(IF2(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c6(IF2(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c6(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c6(EQ(s(s(0)), s(s(0)))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(max(cons(z0, cons(z1, z2))), z0), if1(ge(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), z0, z1, z2), z0), max(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(if1(ge(z1, x1), z1, x1, x2), cons(z1, cons(x1, x2))), MAX(cons(z1, cons(x1, x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2)))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2)))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(max(cons(z0, cons(z1, z2))), z0), if1(ge(z0, z1), z0, z1, z2), z0, cons(z1, z2)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), z0, z1, z2), z0), max(cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2))))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2))))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) MAX(cons(s(s(s(z0))), cons(s(s(s(0))), x2))) -> c2(IF1(true, s(s(s(z0))), s(s(s(0))), x2), GE(s(s(s(z0))), s(s(s(0))))) MAX(cons(s(s(s(0))), cons(s(s(s(s(z0)))), x2))) -> c2(IF1(false, s(s(s(0))), s(s(s(s(z0)))), x2), GE(s(s(s(0))), s(s(s(s(z0)))))) MAX(cons(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2))) -> c2(IF1(ge(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), GE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MAX(cons(s(s(s(x0))), cons(s(s(s(x1))), x2))) -> c2(GE(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c6(IF2(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c6(IF2(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c6(IF2(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c6(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c6(EQ(s(s(s(0))), s(s(s(0))))) S tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) MAX(cons(s(s(z0)), cons(s(s(0)), x2))) -> c2(IF1(true, s(s(z0)), s(s(0)), x2), GE(s(s(z0)), s(s(0)))) MAX(cons(s(s(0)), cons(s(s(s(z0))), x2))) -> c2(IF1(false, s(s(0)), s(s(s(z0))), x2), GE(s(s(0)), s(s(s(z0))))) MAX(cons(s(s(x0)), cons(s(s(x1)), x2))) -> c2(GE(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c6(IF2(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c6(IF2(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c6(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c6(EQ(s(s(0)), s(s(0)))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(max(cons(z0, cons(z1, z2))), z0), if1(ge(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), z0, z1, z2), z0), max(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(if1(ge(z1, x1), z1, x1, x2), cons(z1, cons(x1, x2))), MAX(cons(z1, cons(x1, x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2)))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2)))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(max(cons(z0, cons(z1, z2))), z0), if1(ge(z0, z1), z0, z1, z2), z0, cons(z1, z2)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), z0, z1, z2), z0), max(cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2))))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2))))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) MAX(cons(s(s(s(z0))), cons(s(s(s(0))), x2))) -> c2(IF1(true, s(s(s(z0))), s(s(s(0))), x2), GE(s(s(s(z0))), s(s(s(0))))) MAX(cons(s(s(s(0))), cons(s(s(s(s(z0)))), x2))) -> c2(IF1(false, s(s(s(0))), s(s(s(s(z0)))), x2), GE(s(s(s(0))), s(s(s(s(z0)))))) MAX(cons(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2))) -> c2(IF1(ge(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), GE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MAX(cons(s(s(s(x0))), cons(s(s(s(x1))), x2))) -> c2(GE(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c6(IF2(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c6(IF2(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c6(IF2(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c6(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c6(EQ(s(s(s(0))), s(s(s(0))))) K tuples: IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) Defined Rule Symbols: del_2, max_1, if2_4, eq_2, ge_2, if1_4 Defined Pair Symbols: IF1_4, IF2_4, EQ_2, GE_2, MAX_1, DEL_2, SORT_1, IF3_2 Compound Symbols: c3_1, c4_1, c8_1, c12_1, c20_1, c2_1, c6_1, c13_1, c2_2, c6_2, c15_1, c15_3, c15_2 ---------------------------------------- (147) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(max(cons(z0, cons(z1, z2))), z0), if1(ge(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) by IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(max(cons(z0, cons(0, x2))), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(max(cons(0, cons(s(z0), x2))), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(max(cons(s(z0), cons(s(z1), x2))), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), z0, z1, z2), z0), if1(ge(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) ---------------------------------------- (148) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil max(cons(z0, nil)) -> z0 max(cons(z0, cons(z1, z2))) -> if1(ge(z0, z1), z0, z1, z2) if2(true, z0, z1, z2) -> z2 if2(false, z0, z1, z2) -> cons(z1, del(z0, z2)) eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) if1(true, z0, z1, z2) -> max(cons(z0, z2)) if1(false, z0, z1, z2) -> max(cons(z1, z2)) Tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) MAX(cons(s(s(z0)), cons(s(s(0)), x2))) -> c2(IF1(true, s(s(z0)), s(s(0)), x2), GE(s(s(z0)), s(s(0)))) MAX(cons(s(s(0)), cons(s(s(s(z0))), x2))) -> c2(IF1(false, s(s(0)), s(s(s(z0))), x2), GE(s(s(0)), s(s(s(z0))))) MAX(cons(s(s(x0)), cons(s(s(x1)), x2))) -> c2(GE(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c6(IF2(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c6(IF2(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c6(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c6(EQ(s(s(0)), s(s(0)))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), z0, z1, z2), z0), max(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(if1(ge(z1, x1), z1, x1, x2), cons(z1, cons(x1, x2))), MAX(cons(z1, cons(x1, x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2)))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2)))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(max(cons(z0, cons(z1, z2))), z0), if1(ge(z0, z1), z0, z1, z2), z0, cons(z1, z2)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), z0, z1, z2), z0), max(cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2))))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2))))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) MAX(cons(s(s(s(z0))), cons(s(s(s(0))), x2))) -> c2(IF1(true, s(s(s(z0))), s(s(s(0))), x2), GE(s(s(s(z0))), s(s(s(0))))) MAX(cons(s(s(s(0))), cons(s(s(s(s(z0)))), x2))) -> c2(IF1(false, s(s(s(0))), s(s(s(s(z0)))), x2), GE(s(s(s(0))), s(s(s(s(z0)))))) MAX(cons(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2))) -> c2(IF1(ge(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), GE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MAX(cons(s(s(s(x0))), cons(s(s(s(x1))), x2))) -> c2(GE(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c6(IF2(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c6(IF2(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c6(IF2(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c6(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c6(EQ(s(s(s(0))), s(s(s(0))))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(max(cons(z0, cons(0, x2))), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(max(cons(0, cons(s(z0), x2))), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(max(cons(s(z0), cons(s(z1), x2))), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) S tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) MAX(cons(s(s(z0)), cons(s(s(0)), x2))) -> c2(IF1(true, s(s(z0)), s(s(0)), x2), GE(s(s(z0)), s(s(0)))) MAX(cons(s(s(0)), cons(s(s(s(z0))), x2))) -> c2(IF1(false, s(s(0)), s(s(s(z0))), x2), GE(s(s(0)), s(s(s(z0))))) MAX(cons(s(s(x0)), cons(s(s(x1)), x2))) -> c2(GE(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c6(IF2(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c6(IF2(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c6(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c6(EQ(s(s(0)), s(s(0)))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), z0, z1, z2), z0), max(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(if1(ge(z1, x1), z1, x1, x2), cons(z1, cons(x1, x2))), MAX(cons(z1, cons(x1, x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2)))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2)))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(max(cons(z0, cons(z1, z2))), z0), if1(ge(z0, z1), z0, z1, z2), z0, cons(z1, z2)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), z0, z1, z2), z0), max(cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2))))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2))))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) MAX(cons(s(s(s(z0))), cons(s(s(s(0))), x2))) -> c2(IF1(true, s(s(s(z0))), s(s(s(0))), x2), GE(s(s(s(z0))), s(s(s(0))))) MAX(cons(s(s(s(0))), cons(s(s(s(s(z0)))), x2))) -> c2(IF1(false, s(s(s(0))), s(s(s(s(z0)))), x2), GE(s(s(s(0))), s(s(s(s(z0)))))) MAX(cons(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2))) -> c2(IF1(ge(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), GE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MAX(cons(s(s(s(x0))), cons(s(s(s(x1))), x2))) -> c2(GE(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c6(IF2(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c6(IF2(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c6(IF2(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c6(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c6(EQ(s(s(s(0))), s(s(s(0))))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(max(cons(z0, cons(0, x2))), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(max(cons(0, cons(s(z0), x2))), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(max(cons(s(z0), cons(s(z1), x2))), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) K tuples: IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) Defined Rule Symbols: del_2, max_1, if2_4, eq_2, ge_2, if1_4 Defined Pair Symbols: IF1_4, IF2_4, EQ_2, GE_2, MAX_1, DEL_2, SORT_1, IF3_2 Compound Symbols: c3_1, c4_1, c8_1, c12_1, c20_1, c2_1, c6_1, c13_1, c2_2, c6_2, c15_1, c15_3, c15_2 ---------------------------------------- (149) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), z0, z1, z2), z0), max(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) by IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), z0, z1, z2), z0), if1(ge(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), max(cons(z0, cons(0, x2))), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), max(cons(0, cons(s(z0), x2))), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), max(cons(s(z0), cons(s(z1), x2))), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) ---------------------------------------- (150) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil max(cons(z0, nil)) -> z0 max(cons(z0, cons(z1, z2))) -> if1(ge(z0, z1), z0, z1, z2) if2(true, z0, z1, z2) -> z2 if2(false, z0, z1, z2) -> cons(z1, del(z0, z2)) eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) if1(true, z0, z1, z2) -> max(cons(z0, z2)) if1(false, z0, z1, z2) -> max(cons(z1, z2)) Tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) MAX(cons(s(s(z0)), cons(s(s(0)), x2))) -> c2(IF1(true, s(s(z0)), s(s(0)), x2), GE(s(s(z0)), s(s(0)))) MAX(cons(s(s(0)), cons(s(s(s(z0))), x2))) -> c2(IF1(false, s(s(0)), s(s(s(z0))), x2), GE(s(s(0)), s(s(s(z0))))) MAX(cons(s(s(x0)), cons(s(s(x1)), x2))) -> c2(GE(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c6(IF2(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c6(IF2(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c6(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c6(EQ(s(s(0)), s(s(0)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(if1(ge(z1, x1), z1, x1, x2), cons(z1, cons(x1, x2))), MAX(cons(z1, cons(x1, x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2)))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2)))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(max(cons(z0, cons(z1, z2))), z0), if1(ge(z0, z1), z0, z1, z2), z0, cons(z1, z2)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), z0, z1, z2), z0), max(cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2))))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2))))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) MAX(cons(s(s(s(z0))), cons(s(s(s(0))), x2))) -> c2(IF1(true, s(s(s(z0))), s(s(s(0))), x2), GE(s(s(s(z0))), s(s(s(0))))) MAX(cons(s(s(s(0))), cons(s(s(s(s(z0)))), x2))) -> c2(IF1(false, s(s(s(0))), s(s(s(s(z0)))), x2), GE(s(s(s(0))), s(s(s(s(z0)))))) MAX(cons(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2))) -> c2(IF1(ge(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), GE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MAX(cons(s(s(s(x0))), cons(s(s(s(x1))), x2))) -> c2(GE(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c6(IF2(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c6(IF2(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c6(IF2(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c6(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c6(EQ(s(s(s(0))), s(s(s(0))))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(max(cons(z0, cons(0, x2))), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(max(cons(0, cons(s(z0), x2))), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(max(cons(s(z0), cons(s(z1), x2))), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), max(cons(z0, cons(0, x2))), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), max(cons(0, cons(s(z0), x2))), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), max(cons(s(z0), cons(s(z1), x2))), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) S tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) MAX(cons(s(s(z0)), cons(s(s(0)), x2))) -> c2(IF1(true, s(s(z0)), s(s(0)), x2), GE(s(s(z0)), s(s(0)))) MAX(cons(s(s(0)), cons(s(s(s(z0))), x2))) -> c2(IF1(false, s(s(0)), s(s(s(z0))), x2), GE(s(s(0)), s(s(s(z0))))) MAX(cons(s(s(x0)), cons(s(s(x1)), x2))) -> c2(GE(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c6(IF2(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c6(IF2(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c6(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c6(EQ(s(s(0)), s(s(0)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(if1(ge(z1, x1), z1, x1, x2), cons(z1, cons(x1, x2))), MAX(cons(z1, cons(x1, x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2)))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2)))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(max(cons(z0, cons(z1, z2))), z0), if1(ge(z0, z1), z0, z1, z2), z0, cons(z1, z2)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), z0, z1, z2), z0), max(cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2))))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2))))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) MAX(cons(s(s(s(z0))), cons(s(s(s(0))), x2))) -> c2(IF1(true, s(s(s(z0))), s(s(s(0))), x2), GE(s(s(s(z0))), s(s(s(0))))) MAX(cons(s(s(s(0))), cons(s(s(s(s(z0)))), x2))) -> c2(IF1(false, s(s(s(0))), s(s(s(s(z0)))), x2), GE(s(s(s(0))), s(s(s(s(z0)))))) MAX(cons(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2))) -> c2(IF1(ge(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), GE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MAX(cons(s(s(s(x0))), cons(s(s(s(x1))), x2))) -> c2(GE(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c6(IF2(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c6(IF2(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c6(IF2(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c6(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c6(EQ(s(s(s(0))), s(s(s(0))))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(max(cons(z0, cons(0, x2))), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(max(cons(0, cons(s(z0), x2))), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(max(cons(s(z0), cons(s(z1), x2))), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), max(cons(z0, cons(0, x2))), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), max(cons(0, cons(s(z0), x2))), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), max(cons(s(z0), cons(s(z1), x2))), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) K tuples: IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) Defined Rule Symbols: del_2, max_1, if2_4, eq_2, ge_2, if1_4 Defined Pair Symbols: IF1_4, IF2_4, EQ_2, GE_2, MAX_1, DEL_2, SORT_1, IF3_2 Compound Symbols: c3_1, c4_1, c8_1, c12_1, c20_1, c2_1, c6_1, c13_1, c2_2, c6_2, c15_1, c15_3, c15_2 ---------------------------------------- (151) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(if1(ge(z1, x1), z1, x1, x2), cons(z1, cons(x1, x2))), MAX(cons(z1, cons(x1, x2)))) by IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(ge(z0, 0), z0, 0, x2), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(ge(0, s(z0)), 0, s(z0), x2), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), if1(ge(z0, 0), z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), if1(ge(0, s(z0)), 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) ---------------------------------------- (152) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil max(cons(z0, nil)) -> z0 max(cons(z0, cons(z1, z2))) -> if1(ge(z0, z1), z0, z1, z2) if2(true, z0, z1, z2) -> z2 if2(false, z0, z1, z2) -> cons(z1, del(z0, z2)) eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) if1(true, z0, z1, z2) -> max(cons(z0, z2)) if1(false, z0, z1, z2) -> max(cons(z1, z2)) Tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) MAX(cons(s(s(z0)), cons(s(s(0)), x2))) -> c2(IF1(true, s(s(z0)), s(s(0)), x2), GE(s(s(z0)), s(s(0)))) MAX(cons(s(s(0)), cons(s(s(s(z0))), x2))) -> c2(IF1(false, s(s(0)), s(s(s(z0))), x2), GE(s(s(0)), s(s(s(z0))))) MAX(cons(s(s(x0)), cons(s(s(x1)), x2))) -> c2(GE(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c6(IF2(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c6(IF2(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c6(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c6(EQ(s(s(0)), s(s(0)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(if1(ge(z1, x1), z1, x1, x2), cons(z1, cons(x1, x2))), MAX(cons(z1, cons(x1, x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2)))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2)))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(max(cons(z0, cons(z1, z2))), z0), if1(ge(z0, z1), z0, z1, z2), z0, cons(z1, z2)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), z0, z1, z2), z0), max(cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2))))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2))))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) MAX(cons(s(s(s(z0))), cons(s(s(s(0))), x2))) -> c2(IF1(true, s(s(s(z0))), s(s(s(0))), x2), GE(s(s(s(z0))), s(s(s(0))))) MAX(cons(s(s(s(0))), cons(s(s(s(s(z0)))), x2))) -> c2(IF1(false, s(s(s(0))), s(s(s(s(z0)))), x2), GE(s(s(s(0))), s(s(s(s(z0)))))) MAX(cons(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2))) -> c2(IF1(ge(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), GE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MAX(cons(s(s(s(x0))), cons(s(s(s(x1))), x2))) -> c2(GE(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c6(IF2(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c6(IF2(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c6(IF2(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c6(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c6(EQ(s(s(s(0))), s(s(s(0))))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(max(cons(z0, cons(0, x2))), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(max(cons(0, cons(s(z0), x2))), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(max(cons(s(z0), cons(s(z1), x2))), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), max(cons(z0, cons(0, x2))), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), max(cons(0, cons(s(z0), x2))), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), max(cons(s(z0), cons(s(z1), x2))), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(ge(z0, 0), z0, 0, x2), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(ge(0, s(z0)), 0, s(z0), x2), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), if1(ge(z0, 0), z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), if1(ge(0, s(z0)), 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) S tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) MAX(cons(s(s(z0)), cons(s(s(0)), x2))) -> c2(IF1(true, s(s(z0)), s(s(0)), x2), GE(s(s(z0)), s(s(0)))) MAX(cons(s(s(0)), cons(s(s(s(z0))), x2))) -> c2(IF1(false, s(s(0)), s(s(s(z0))), x2), GE(s(s(0)), s(s(s(z0))))) MAX(cons(s(s(x0)), cons(s(s(x1)), x2))) -> c2(GE(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c6(IF2(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c6(IF2(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c6(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c6(EQ(s(s(0)), s(s(0)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2)))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2)))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(max(cons(z0, cons(z1, z2))), z0), if1(ge(z0, z1), z0, z1, z2), z0, cons(z1, z2)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), z0, z1, z2), z0), max(cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2))))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2))))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) MAX(cons(s(s(s(z0))), cons(s(s(s(0))), x2))) -> c2(IF1(true, s(s(s(z0))), s(s(s(0))), x2), GE(s(s(s(z0))), s(s(s(0))))) MAX(cons(s(s(s(0))), cons(s(s(s(s(z0)))), x2))) -> c2(IF1(false, s(s(s(0))), s(s(s(s(z0)))), x2), GE(s(s(s(0))), s(s(s(s(z0)))))) MAX(cons(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2))) -> c2(IF1(ge(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), GE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MAX(cons(s(s(s(x0))), cons(s(s(s(x1))), x2))) -> c2(GE(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c6(IF2(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c6(IF2(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c6(IF2(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c6(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c6(EQ(s(s(s(0))), s(s(s(0))))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(max(cons(z0, cons(0, x2))), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(max(cons(0, cons(s(z0), x2))), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(max(cons(s(z0), cons(s(z1), x2))), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), max(cons(z0, cons(0, x2))), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), max(cons(0, cons(s(z0), x2))), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), max(cons(s(z0), cons(s(z1), x2))), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(ge(z0, 0), z0, 0, x2), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(ge(0, s(z0)), 0, s(z0), x2), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), if1(ge(z0, 0), z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), if1(ge(0, s(z0)), 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) K tuples: IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) Defined Rule Symbols: del_2, max_1, if2_4, eq_2, ge_2, if1_4 Defined Pair Symbols: IF1_4, IF2_4, EQ_2, GE_2, MAX_1, DEL_2, SORT_1, IF3_2 Compound Symbols: c3_1, c4_1, c8_1, c12_1, c20_1, c2_1, c6_1, c13_1, c2_2, c6_2, c15_1, c15_3, c15_2 ---------------------------------------- (153) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2)))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) by IF3(false, cons(z1, cons(0, x1))) -> c15(SORT(if2(eq(if1(true, z1, 0, x1), z1), if1(true, z1, 0, x1), z1, cons(0, x1))), DEL(if1(ge(z1, 0), z1, 0, x1), cons(z1, cons(0, x1))), MAX(cons(z1, cons(0, x1)))) IF3(false, cons(z0, cons(0, z2))) -> c15(SORT(del(max(cons(z0, z2)), cons(z0, cons(0, z2)))), DEL(if1(ge(z0, 0), z0, 0, z2), cons(z0, cons(0, z2))), MAX(cons(z0, cons(0, z2)))) IF3(false, cons(x0, cons(0, x1))) -> c15(DEL(if1(ge(x0, 0), x0, 0, x1), cons(x0, cons(0, x1))), MAX(cons(x0, cons(0, x1)))) ---------------------------------------- (154) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil max(cons(z0, nil)) -> z0 max(cons(z0, cons(z1, z2))) -> if1(ge(z0, z1), z0, z1, z2) if2(true, z0, z1, z2) -> z2 if2(false, z0, z1, z2) -> cons(z1, del(z0, z2)) eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) if1(true, z0, z1, z2) -> max(cons(z0, z2)) if1(false, z0, z1, z2) -> max(cons(z1, z2)) Tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) MAX(cons(s(s(z0)), cons(s(s(0)), x2))) -> c2(IF1(true, s(s(z0)), s(s(0)), x2), GE(s(s(z0)), s(s(0)))) MAX(cons(s(s(0)), cons(s(s(s(z0))), x2))) -> c2(IF1(false, s(s(0)), s(s(s(z0))), x2), GE(s(s(0)), s(s(s(z0))))) MAX(cons(s(s(x0)), cons(s(s(x1)), x2))) -> c2(GE(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c6(IF2(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c6(IF2(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c6(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c6(EQ(s(s(0)), s(s(0)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(if1(ge(z1, x1), z1, x1, x2), cons(z1, cons(x1, x2))), MAX(cons(z1, cons(x1, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2)))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(max(cons(z0, cons(z1, z2))), z0), if1(ge(z0, z1), z0, z1, z2), z0, cons(z1, z2)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), z0, z1, z2), z0), max(cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2))))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2))))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) MAX(cons(s(s(s(z0))), cons(s(s(s(0))), x2))) -> c2(IF1(true, s(s(s(z0))), s(s(s(0))), x2), GE(s(s(s(z0))), s(s(s(0))))) MAX(cons(s(s(s(0))), cons(s(s(s(s(z0)))), x2))) -> c2(IF1(false, s(s(s(0))), s(s(s(s(z0)))), x2), GE(s(s(s(0))), s(s(s(s(z0)))))) MAX(cons(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2))) -> c2(IF1(ge(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), GE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MAX(cons(s(s(s(x0))), cons(s(s(s(x1))), x2))) -> c2(GE(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c6(IF2(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c6(IF2(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c6(IF2(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c6(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c6(EQ(s(s(s(0))), s(s(s(0))))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(max(cons(z0, cons(0, x2))), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(max(cons(0, cons(s(z0), x2))), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(max(cons(s(z0), cons(s(z1), x2))), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), max(cons(z0, cons(0, x2))), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), max(cons(0, cons(s(z0), x2))), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), max(cons(s(z0), cons(s(z1), x2))), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(ge(z0, 0), z0, 0, x2), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(ge(0, s(z0)), 0, s(z0), x2), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), if1(ge(z0, 0), z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), if1(ge(0, s(z0)), 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z1, cons(0, x1))) -> c15(SORT(if2(eq(if1(true, z1, 0, x1), z1), if1(true, z1, 0, x1), z1, cons(0, x1))), DEL(if1(ge(z1, 0), z1, 0, x1), cons(z1, cons(0, x1))), MAX(cons(z1, cons(0, x1)))) IF3(false, cons(z0, cons(0, z2))) -> c15(SORT(del(max(cons(z0, z2)), cons(z0, cons(0, z2)))), DEL(if1(ge(z0, 0), z0, 0, z2), cons(z0, cons(0, z2))), MAX(cons(z0, cons(0, z2)))) IF3(false, cons(x0, cons(0, x1))) -> c15(DEL(if1(ge(x0, 0), x0, 0, x1), cons(x0, cons(0, x1))), MAX(cons(x0, cons(0, x1)))) S tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) MAX(cons(s(s(z0)), cons(s(s(0)), x2))) -> c2(IF1(true, s(s(z0)), s(s(0)), x2), GE(s(s(z0)), s(s(0)))) MAX(cons(s(s(0)), cons(s(s(s(z0))), x2))) -> c2(IF1(false, s(s(0)), s(s(s(z0))), x2), GE(s(s(0)), s(s(s(z0))))) MAX(cons(s(s(x0)), cons(s(s(x1)), x2))) -> c2(GE(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c6(IF2(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c6(IF2(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c6(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c6(EQ(s(s(0)), s(s(0)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2)))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(max(cons(z0, cons(z1, z2))), z0), if1(ge(z0, z1), z0, z1, z2), z0, cons(z1, z2)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), z0, z1, z2), z0), max(cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2))))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2))))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) MAX(cons(s(s(s(z0))), cons(s(s(s(0))), x2))) -> c2(IF1(true, s(s(s(z0))), s(s(s(0))), x2), GE(s(s(s(z0))), s(s(s(0))))) MAX(cons(s(s(s(0))), cons(s(s(s(s(z0)))), x2))) -> c2(IF1(false, s(s(s(0))), s(s(s(s(z0)))), x2), GE(s(s(s(0))), s(s(s(s(z0)))))) MAX(cons(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2))) -> c2(IF1(ge(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), GE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MAX(cons(s(s(s(x0))), cons(s(s(s(x1))), x2))) -> c2(GE(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c6(IF2(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c6(IF2(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c6(IF2(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c6(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c6(EQ(s(s(s(0))), s(s(s(0))))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(max(cons(z0, cons(0, x2))), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(max(cons(0, cons(s(z0), x2))), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(max(cons(s(z0), cons(s(z1), x2))), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), max(cons(z0, cons(0, x2))), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), max(cons(0, cons(s(z0), x2))), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), max(cons(s(z0), cons(s(z1), x2))), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(ge(z0, 0), z0, 0, x2), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(ge(0, s(z0)), 0, s(z0), x2), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), if1(ge(z0, 0), z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), if1(ge(0, s(z0)), 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z1, cons(0, x1))) -> c15(SORT(if2(eq(if1(true, z1, 0, x1), z1), if1(true, z1, 0, x1), z1, cons(0, x1))), DEL(if1(ge(z1, 0), z1, 0, x1), cons(z1, cons(0, x1))), MAX(cons(z1, cons(0, x1)))) IF3(false, cons(z0, cons(0, z2))) -> c15(SORT(del(max(cons(z0, z2)), cons(z0, cons(0, z2)))), DEL(if1(ge(z0, 0), z0, 0, z2), cons(z0, cons(0, z2))), MAX(cons(z0, cons(0, z2)))) IF3(false, cons(x0, cons(0, x1))) -> c15(DEL(if1(ge(x0, 0), x0, 0, x1), cons(x0, cons(0, x1))), MAX(cons(x0, cons(0, x1)))) K tuples: IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) Defined Rule Symbols: del_2, max_1, if2_4, eq_2, ge_2, if1_4 Defined Pair Symbols: IF1_4, IF2_4, EQ_2, GE_2, MAX_1, DEL_2, SORT_1, IF3_2 Compound Symbols: c3_1, c4_1, c8_1, c12_1, c20_1, c2_1, c6_1, c13_1, c2_2, c6_2, c15_1, c15_3, c15_2 ---------------------------------------- (155) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (156) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil max(cons(z0, nil)) -> z0 max(cons(z0, cons(z1, z2))) -> if1(ge(z0, z1), z0, z1, z2) if2(true, z0, z1, z2) -> z2 if2(false, z0, z1, z2) -> cons(z1, del(z0, z2)) eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) if1(true, z0, z1, z2) -> max(cons(z0, z2)) if1(false, z0, z1, z2) -> max(cons(z1, z2)) Tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) MAX(cons(s(s(z0)), cons(s(s(0)), x2))) -> c2(IF1(true, s(s(z0)), s(s(0)), x2), GE(s(s(z0)), s(s(0)))) MAX(cons(s(s(0)), cons(s(s(s(z0))), x2))) -> c2(IF1(false, s(s(0)), s(s(s(z0))), x2), GE(s(s(0)), s(s(s(z0))))) MAX(cons(s(s(x0)), cons(s(s(x1)), x2))) -> c2(GE(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c6(IF2(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c6(IF2(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c6(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c6(EQ(s(s(0)), s(s(0)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(if1(ge(z1, x1), z1, x1, x2), cons(z1, cons(x1, x2))), MAX(cons(z1, cons(x1, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2)))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(max(cons(z0, cons(z1, z2))), z0), if1(ge(z0, z1), z0, z1, z2), z0, cons(z1, z2)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), z0, z1, z2), z0), max(cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2))))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2))))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) MAX(cons(s(s(s(z0))), cons(s(s(s(0))), x2))) -> c2(IF1(true, s(s(s(z0))), s(s(s(0))), x2), GE(s(s(s(z0))), s(s(s(0))))) MAX(cons(s(s(s(0))), cons(s(s(s(s(z0)))), x2))) -> c2(IF1(false, s(s(s(0))), s(s(s(s(z0)))), x2), GE(s(s(s(0))), s(s(s(s(z0)))))) MAX(cons(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2))) -> c2(IF1(ge(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), GE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MAX(cons(s(s(s(x0))), cons(s(s(s(x1))), x2))) -> c2(GE(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c6(IF2(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c6(IF2(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c6(IF2(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c6(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c6(EQ(s(s(s(0))), s(s(s(0))))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(max(cons(z0, cons(0, x2))), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(max(cons(0, cons(s(z0), x2))), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(max(cons(s(z0), cons(s(z1), x2))), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), max(cons(z0, cons(0, x2))), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), max(cons(0, cons(s(z0), x2))), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), max(cons(s(z0), cons(s(z1), x2))), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(ge(z0, 0), z0, 0, x2), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(ge(0, s(z0)), 0, s(z0), x2), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), if1(ge(z0, 0), z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), if1(ge(0, s(z0)), 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z1, cons(0, x1))) -> c15(SORT(if2(eq(if1(true, z1, 0, x1), z1), if1(true, z1, 0, x1), z1, cons(0, x1))), DEL(if1(ge(z1, 0), z1, 0, x1), cons(z1, cons(0, x1))), MAX(cons(z1, cons(0, x1)))) IF3(false, cons(z0, cons(0, z2))) -> c15(SORT(del(max(cons(z0, z2)), cons(z0, cons(0, z2)))), DEL(if1(ge(z0, 0), z0, 0, z2), cons(z0, cons(0, z2))), MAX(cons(z0, cons(0, z2)))) IF3(false, cons(x0, cons(0, x1))) -> c(DEL(if1(ge(x0, 0), x0, 0, x1), cons(x0, cons(0, x1)))) IF3(false, cons(x0, cons(0, x1))) -> c(MAX(cons(x0, cons(0, x1)))) S tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) MAX(cons(s(s(z0)), cons(s(s(0)), x2))) -> c2(IF1(true, s(s(z0)), s(s(0)), x2), GE(s(s(z0)), s(s(0)))) MAX(cons(s(s(0)), cons(s(s(s(z0))), x2))) -> c2(IF1(false, s(s(0)), s(s(s(z0))), x2), GE(s(s(0)), s(s(s(z0))))) MAX(cons(s(s(x0)), cons(s(s(x1)), x2))) -> c2(GE(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c6(IF2(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c6(IF2(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c6(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c6(EQ(s(s(0)), s(s(0)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2)))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(max(cons(z0, cons(z1, z2))), z0), if1(ge(z0, z1), z0, z1, z2), z0, cons(z1, z2)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), z0, z1, z2), z0), max(cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2))))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2))))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) MAX(cons(s(s(s(z0))), cons(s(s(s(0))), x2))) -> c2(IF1(true, s(s(s(z0))), s(s(s(0))), x2), GE(s(s(s(z0))), s(s(s(0))))) MAX(cons(s(s(s(0))), cons(s(s(s(s(z0)))), x2))) -> c2(IF1(false, s(s(s(0))), s(s(s(s(z0)))), x2), GE(s(s(s(0))), s(s(s(s(z0)))))) MAX(cons(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2))) -> c2(IF1(ge(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), GE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MAX(cons(s(s(s(x0))), cons(s(s(s(x1))), x2))) -> c2(GE(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c6(IF2(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c6(IF2(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c6(IF2(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c6(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c6(EQ(s(s(s(0))), s(s(s(0))))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(max(cons(z0, cons(0, x2))), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(max(cons(0, cons(s(z0), x2))), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(max(cons(s(z0), cons(s(z1), x2))), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), max(cons(z0, cons(0, x2))), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), max(cons(0, cons(s(z0), x2))), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), max(cons(s(z0), cons(s(z1), x2))), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(ge(z0, 0), z0, 0, x2), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(ge(0, s(z0)), 0, s(z0), x2), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), if1(ge(z0, 0), z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), if1(ge(0, s(z0)), 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z1, cons(0, x1))) -> c15(SORT(if2(eq(if1(true, z1, 0, x1), z1), if1(true, z1, 0, x1), z1, cons(0, x1))), DEL(if1(ge(z1, 0), z1, 0, x1), cons(z1, cons(0, x1))), MAX(cons(z1, cons(0, x1)))) IF3(false, cons(z0, cons(0, z2))) -> c15(SORT(del(max(cons(z0, z2)), cons(z0, cons(0, z2)))), DEL(if1(ge(z0, 0), z0, 0, z2), cons(z0, cons(0, z2))), MAX(cons(z0, cons(0, z2)))) IF3(false, cons(x0, cons(0, x1))) -> c(DEL(if1(ge(x0, 0), x0, 0, x1), cons(x0, cons(0, x1)))) IF3(false, cons(x0, cons(0, x1))) -> c(MAX(cons(x0, cons(0, x1)))) K tuples: IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) Defined Rule Symbols: del_2, max_1, if2_4, eq_2, ge_2, if1_4 Defined Pair Symbols: IF1_4, IF2_4, EQ_2, GE_2, MAX_1, DEL_2, SORT_1, IF3_2 Compound Symbols: c3_1, c4_1, c8_1, c12_1, c20_1, c2_1, c6_1, c13_1, c2_2, c6_2, c15_1, c15_3, c15_2, c_1 ---------------------------------------- (157) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. IF3(false, cons(x0, cons(0, x1))) -> c(DEL(if1(ge(x0, 0), x0, 0, x1), cons(x0, cons(0, x1)))) IF3(false, cons(x0, cons(0, x1))) -> c(MAX(cons(x0, cons(0, x1)))) We considered the (Usable) Rules:none And the Tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) MAX(cons(s(s(z0)), cons(s(s(0)), x2))) -> c2(IF1(true, s(s(z0)), s(s(0)), x2), GE(s(s(z0)), s(s(0)))) MAX(cons(s(s(0)), cons(s(s(s(z0))), x2))) -> c2(IF1(false, s(s(0)), s(s(s(z0))), x2), GE(s(s(0)), s(s(s(z0))))) MAX(cons(s(s(x0)), cons(s(s(x1)), x2))) -> c2(GE(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c6(IF2(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c6(IF2(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c6(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c6(EQ(s(s(0)), s(s(0)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(if1(ge(z1, x1), z1, x1, x2), cons(z1, cons(x1, x2))), MAX(cons(z1, cons(x1, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2)))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(max(cons(z0, cons(z1, z2))), z0), if1(ge(z0, z1), z0, z1, z2), z0, cons(z1, z2)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), z0, z1, z2), z0), max(cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2))))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2))))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) MAX(cons(s(s(s(z0))), cons(s(s(s(0))), x2))) -> c2(IF1(true, s(s(s(z0))), s(s(s(0))), x2), GE(s(s(s(z0))), s(s(s(0))))) MAX(cons(s(s(s(0))), cons(s(s(s(s(z0)))), x2))) -> c2(IF1(false, s(s(s(0))), s(s(s(s(z0)))), x2), GE(s(s(s(0))), s(s(s(s(z0)))))) MAX(cons(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2))) -> c2(IF1(ge(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), GE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MAX(cons(s(s(s(x0))), cons(s(s(s(x1))), x2))) -> c2(GE(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c6(IF2(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c6(IF2(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c6(IF2(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c6(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c6(EQ(s(s(s(0))), s(s(s(0))))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(max(cons(z0, cons(0, x2))), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(max(cons(0, cons(s(z0), x2))), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(max(cons(s(z0), cons(s(z1), x2))), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), max(cons(z0, cons(0, x2))), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), max(cons(0, cons(s(z0), x2))), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), max(cons(s(z0), cons(s(z1), x2))), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(ge(z0, 0), z0, 0, x2), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(ge(0, s(z0)), 0, s(z0), x2), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), if1(ge(z0, 0), z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), if1(ge(0, s(z0)), 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z1, cons(0, x1))) -> c15(SORT(if2(eq(if1(true, z1, 0, x1), z1), if1(true, z1, 0, x1), z1, cons(0, x1))), DEL(if1(ge(z1, 0), z1, 0, x1), cons(z1, cons(0, x1))), MAX(cons(z1, cons(0, x1)))) IF3(false, cons(z0, cons(0, z2))) -> c15(SORT(del(max(cons(z0, z2)), cons(z0, cons(0, z2)))), DEL(if1(ge(z0, 0), z0, 0, z2), cons(z0, cons(0, z2))), MAX(cons(z0, cons(0, z2)))) IF3(false, cons(x0, cons(0, x1))) -> c(DEL(if1(ge(x0, 0), x0, 0, x1), cons(x0, cons(0, x1)))) IF3(false, cons(x0, cons(0, x1))) -> c(MAX(cons(x0, cons(0, x1)))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(DEL(x_1, x_2)) = 0 POL(EQ(x_1, x_2)) = 0 POL(GE(x_1, x_2)) = 0 POL(IF1(x_1, x_2, x_3, x_4)) = 0 POL(IF2(x_1, x_2, x_3, x_4)) = x_3 POL(IF3(x_1, x_2)) = [1] + x_1 POL(MAX(x_1)) = 0 POL(SORT(x_1)) = [1] POL(c(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c13(x_1)) = x_1 POL(c15(x_1)) = x_1 POL(c15(x_1, x_2)) = x_1 + x_2 POL(c15(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c2(x_1)) = x_1 POL(c2(x_1, x_2)) = x_1 + x_2 POL(c20(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c4(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c6(x_1, x_2)) = x_1 + x_2 POL(c8(x_1)) = x_1 POL(cons(x_1, x_2)) = 0 POL(del(x_1, x_2)) = x_2 POL(eq(x_1, x_2)) = 0 POL(false) = 0 POL(ge(x_1, x_2)) = 0 POL(if1(x_1, x_2, x_3, x_4)) = 0 POL(if2(x_1, x_2, x_3, x_4)) = [1] + x_4 POL(max(x_1)) = 0 POL(nil) = 0 POL(s(x_1)) = 0 POL(true) = 0 ---------------------------------------- (158) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil max(cons(z0, nil)) -> z0 max(cons(z0, cons(z1, z2))) -> if1(ge(z0, z1), z0, z1, z2) if2(true, z0, z1, z2) -> z2 if2(false, z0, z1, z2) -> cons(z1, del(z0, z2)) eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) if1(true, z0, z1, z2) -> max(cons(z0, z2)) if1(false, z0, z1, z2) -> max(cons(z1, z2)) Tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) MAX(cons(s(s(z0)), cons(s(s(0)), x2))) -> c2(IF1(true, s(s(z0)), s(s(0)), x2), GE(s(s(z0)), s(s(0)))) MAX(cons(s(s(0)), cons(s(s(s(z0))), x2))) -> c2(IF1(false, s(s(0)), s(s(s(z0))), x2), GE(s(s(0)), s(s(s(z0))))) MAX(cons(s(s(x0)), cons(s(s(x1)), x2))) -> c2(GE(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c6(IF2(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c6(IF2(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c6(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c6(EQ(s(s(0)), s(s(0)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(if1(ge(z1, x1), z1, x1, x2), cons(z1, cons(x1, x2))), MAX(cons(z1, cons(x1, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2)))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(max(cons(z0, cons(z1, z2))), z0), if1(ge(z0, z1), z0, z1, z2), z0, cons(z1, z2)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), z0, z1, z2), z0), max(cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2))))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2))))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) MAX(cons(s(s(s(z0))), cons(s(s(s(0))), x2))) -> c2(IF1(true, s(s(s(z0))), s(s(s(0))), x2), GE(s(s(s(z0))), s(s(s(0))))) MAX(cons(s(s(s(0))), cons(s(s(s(s(z0)))), x2))) -> c2(IF1(false, s(s(s(0))), s(s(s(s(z0)))), x2), GE(s(s(s(0))), s(s(s(s(z0)))))) MAX(cons(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2))) -> c2(IF1(ge(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), GE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MAX(cons(s(s(s(x0))), cons(s(s(s(x1))), x2))) -> c2(GE(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c6(IF2(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c6(IF2(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c6(IF2(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c6(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c6(EQ(s(s(s(0))), s(s(s(0))))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(max(cons(z0, cons(0, x2))), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(max(cons(0, cons(s(z0), x2))), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(max(cons(s(z0), cons(s(z1), x2))), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), max(cons(z0, cons(0, x2))), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), max(cons(0, cons(s(z0), x2))), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), max(cons(s(z0), cons(s(z1), x2))), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(ge(z0, 0), z0, 0, x2), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(ge(0, s(z0)), 0, s(z0), x2), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), if1(ge(z0, 0), z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), if1(ge(0, s(z0)), 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z1, cons(0, x1))) -> c15(SORT(if2(eq(if1(true, z1, 0, x1), z1), if1(true, z1, 0, x1), z1, cons(0, x1))), DEL(if1(ge(z1, 0), z1, 0, x1), cons(z1, cons(0, x1))), MAX(cons(z1, cons(0, x1)))) IF3(false, cons(z0, cons(0, z2))) -> c15(SORT(del(max(cons(z0, z2)), cons(z0, cons(0, z2)))), DEL(if1(ge(z0, 0), z0, 0, z2), cons(z0, cons(0, z2))), MAX(cons(z0, cons(0, z2)))) IF3(false, cons(x0, cons(0, x1))) -> c(DEL(if1(ge(x0, 0), x0, 0, x1), cons(x0, cons(0, x1)))) IF3(false, cons(x0, cons(0, x1))) -> c(MAX(cons(x0, cons(0, x1)))) S tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) MAX(cons(s(s(z0)), cons(s(s(0)), x2))) -> c2(IF1(true, s(s(z0)), s(s(0)), x2), GE(s(s(z0)), s(s(0)))) MAX(cons(s(s(0)), cons(s(s(s(z0))), x2))) -> c2(IF1(false, s(s(0)), s(s(s(z0))), x2), GE(s(s(0)), s(s(s(z0))))) MAX(cons(s(s(x0)), cons(s(s(x1)), x2))) -> c2(GE(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c6(IF2(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c6(IF2(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c6(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c6(EQ(s(s(0)), s(s(0)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2)))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(max(cons(z0, cons(z1, z2))), z0), if1(ge(z0, z1), z0, z1, z2), z0, cons(z1, z2)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), z0, z1, z2), z0), max(cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2))))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2))))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) MAX(cons(s(s(s(z0))), cons(s(s(s(0))), x2))) -> c2(IF1(true, s(s(s(z0))), s(s(s(0))), x2), GE(s(s(s(z0))), s(s(s(0))))) MAX(cons(s(s(s(0))), cons(s(s(s(s(z0)))), x2))) -> c2(IF1(false, s(s(s(0))), s(s(s(s(z0)))), x2), GE(s(s(s(0))), s(s(s(s(z0)))))) MAX(cons(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2))) -> c2(IF1(ge(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), GE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MAX(cons(s(s(s(x0))), cons(s(s(s(x1))), x2))) -> c2(GE(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c6(IF2(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c6(IF2(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c6(IF2(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c6(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c6(EQ(s(s(s(0))), s(s(s(0))))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(max(cons(z0, cons(0, x2))), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(max(cons(0, cons(s(z0), x2))), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(max(cons(s(z0), cons(s(z1), x2))), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), max(cons(z0, cons(0, x2))), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), max(cons(0, cons(s(z0), x2))), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), max(cons(s(z0), cons(s(z1), x2))), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(ge(z0, 0), z0, 0, x2), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(ge(0, s(z0)), 0, s(z0), x2), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), if1(ge(z0, 0), z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), if1(ge(0, s(z0)), 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z1, cons(0, x1))) -> c15(SORT(if2(eq(if1(true, z1, 0, x1), z1), if1(true, z1, 0, x1), z1, cons(0, x1))), DEL(if1(ge(z1, 0), z1, 0, x1), cons(z1, cons(0, x1))), MAX(cons(z1, cons(0, x1)))) IF3(false, cons(z0, cons(0, z2))) -> c15(SORT(del(max(cons(z0, z2)), cons(z0, cons(0, z2)))), DEL(if1(ge(z0, 0), z0, 0, z2), cons(z0, cons(0, z2))), MAX(cons(z0, cons(0, z2)))) K tuples: IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) IF3(false, cons(x0, cons(0, x1))) -> c(DEL(if1(ge(x0, 0), x0, 0, x1), cons(x0, cons(0, x1)))) IF3(false, cons(x0, cons(0, x1))) -> c(MAX(cons(x0, cons(0, x1)))) Defined Rule Symbols: del_2, max_1, if2_4, eq_2, ge_2, if1_4 Defined Pair Symbols: IF1_4, IF2_4, EQ_2, GE_2, MAX_1, DEL_2, SORT_1, IF3_2 Compound Symbols: c3_1, c4_1, c8_1, c12_1, c20_1, c2_1, c6_1, c13_1, c2_2, c6_2, c15_1, c15_3, c15_2, c_1 ---------------------------------------- (159) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2)))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) by IF3(false, cons(0, cons(s(x0), x1))) -> c15(SORT(if2(eq(if1(false, 0, s(x0), x1), 0), if1(false, 0, s(x0), x1), 0, cons(s(x0), x1))), DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1))), MAX(cons(0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), z2))) -> c15(SORT(del(max(cons(s(x0), z2)), cons(0, cons(s(x0), z2)))), DEL(if1(ge(0, s(x0)), 0, s(x0), z2), cons(0, cons(s(x0), z2))), MAX(cons(0, cons(s(x0), z2)))) IF3(false, cons(0, cons(s(x0), x1))) -> c15(DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1))), MAX(cons(0, cons(s(x0), x1)))) ---------------------------------------- (160) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil max(cons(z0, nil)) -> z0 max(cons(z0, cons(z1, z2))) -> if1(ge(z0, z1), z0, z1, z2) if2(true, z0, z1, z2) -> z2 if2(false, z0, z1, z2) -> cons(z1, del(z0, z2)) eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) if1(true, z0, z1, z2) -> max(cons(z0, z2)) if1(false, z0, z1, z2) -> max(cons(z1, z2)) Tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) MAX(cons(s(s(z0)), cons(s(s(0)), x2))) -> c2(IF1(true, s(s(z0)), s(s(0)), x2), GE(s(s(z0)), s(s(0)))) MAX(cons(s(s(0)), cons(s(s(s(z0))), x2))) -> c2(IF1(false, s(s(0)), s(s(s(z0))), x2), GE(s(s(0)), s(s(s(z0))))) MAX(cons(s(s(x0)), cons(s(s(x1)), x2))) -> c2(GE(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c6(IF2(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c6(IF2(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c6(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c6(EQ(s(s(0)), s(s(0)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(if1(ge(z1, x1), z1, x1, x2), cons(z1, cons(x1, x2))), MAX(cons(z1, cons(x1, x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(max(cons(z0, cons(z1, z2))), z0), if1(ge(z0, z1), z0, z1, z2), z0, cons(z1, z2)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), z0, z1, z2), z0), max(cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2))))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2))))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) MAX(cons(s(s(s(z0))), cons(s(s(s(0))), x2))) -> c2(IF1(true, s(s(s(z0))), s(s(s(0))), x2), GE(s(s(s(z0))), s(s(s(0))))) MAX(cons(s(s(s(0))), cons(s(s(s(s(z0)))), x2))) -> c2(IF1(false, s(s(s(0))), s(s(s(s(z0)))), x2), GE(s(s(s(0))), s(s(s(s(z0)))))) MAX(cons(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2))) -> c2(IF1(ge(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), GE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MAX(cons(s(s(s(x0))), cons(s(s(s(x1))), x2))) -> c2(GE(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c6(IF2(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c6(IF2(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c6(IF2(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c6(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c6(EQ(s(s(s(0))), s(s(s(0))))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(max(cons(z0, cons(0, x2))), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(max(cons(0, cons(s(z0), x2))), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(max(cons(s(z0), cons(s(z1), x2))), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), max(cons(z0, cons(0, x2))), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), max(cons(0, cons(s(z0), x2))), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), max(cons(s(z0), cons(s(z1), x2))), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(ge(z0, 0), z0, 0, x2), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(ge(0, s(z0)), 0, s(z0), x2), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), if1(ge(z0, 0), z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), if1(ge(0, s(z0)), 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z1, cons(0, x1))) -> c15(SORT(if2(eq(if1(true, z1, 0, x1), z1), if1(true, z1, 0, x1), z1, cons(0, x1))), DEL(if1(ge(z1, 0), z1, 0, x1), cons(z1, cons(0, x1))), MAX(cons(z1, cons(0, x1)))) IF3(false, cons(z0, cons(0, z2))) -> c15(SORT(del(max(cons(z0, z2)), cons(z0, cons(0, z2)))), DEL(if1(ge(z0, 0), z0, 0, z2), cons(z0, cons(0, z2))), MAX(cons(z0, cons(0, z2)))) IF3(false, cons(x0, cons(0, x1))) -> c(DEL(if1(ge(x0, 0), x0, 0, x1), cons(x0, cons(0, x1)))) IF3(false, cons(x0, cons(0, x1))) -> c(MAX(cons(x0, cons(0, x1)))) IF3(false, cons(0, cons(s(x0), x1))) -> c15(SORT(if2(eq(if1(false, 0, s(x0), x1), 0), if1(false, 0, s(x0), x1), 0, cons(s(x0), x1))), DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1))), MAX(cons(0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), z2))) -> c15(SORT(del(max(cons(s(x0), z2)), cons(0, cons(s(x0), z2)))), DEL(if1(ge(0, s(x0)), 0, s(x0), z2), cons(0, cons(s(x0), z2))), MAX(cons(0, cons(s(x0), z2)))) IF3(false, cons(0, cons(s(x0), x1))) -> c15(DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1))), MAX(cons(0, cons(s(x0), x1)))) S tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) MAX(cons(s(s(z0)), cons(s(s(0)), x2))) -> c2(IF1(true, s(s(z0)), s(s(0)), x2), GE(s(s(z0)), s(s(0)))) MAX(cons(s(s(0)), cons(s(s(s(z0))), x2))) -> c2(IF1(false, s(s(0)), s(s(s(z0))), x2), GE(s(s(0)), s(s(s(z0))))) MAX(cons(s(s(x0)), cons(s(s(x1)), x2))) -> c2(GE(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c6(IF2(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c6(IF2(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c6(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c6(EQ(s(s(0)), s(s(0)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(max(cons(z0, cons(z1, z2))), z0), if1(ge(z0, z1), z0, z1, z2), z0, cons(z1, z2)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), z0, z1, z2), z0), max(cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2))))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2))))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) MAX(cons(s(s(s(z0))), cons(s(s(s(0))), x2))) -> c2(IF1(true, s(s(s(z0))), s(s(s(0))), x2), GE(s(s(s(z0))), s(s(s(0))))) MAX(cons(s(s(s(0))), cons(s(s(s(s(z0)))), x2))) -> c2(IF1(false, s(s(s(0))), s(s(s(s(z0)))), x2), GE(s(s(s(0))), s(s(s(s(z0)))))) MAX(cons(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2))) -> c2(IF1(ge(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), GE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MAX(cons(s(s(s(x0))), cons(s(s(s(x1))), x2))) -> c2(GE(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c6(IF2(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c6(IF2(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c6(IF2(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c6(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c6(EQ(s(s(s(0))), s(s(s(0))))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(max(cons(z0, cons(0, x2))), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(max(cons(0, cons(s(z0), x2))), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(max(cons(s(z0), cons(s(z1), x2))), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), max(cons(z0, cons(0, x2))), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), max(cons(0, cons(s(z0), x2))), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), max(cons(s(z0), cons(s(z1), x2))), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(ge(z0, 0), z0, 0, x2), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(ge(0, s(z0)), 0, s(z0), x2), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), if1(ge(z0, 0), z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), if1(ge(0, s(z0)), 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z1, cons(0, x1))) -> c15(SORT(if2(eq(if1(true, z1, 0, x1), z1), if1(true, z1, 0, x1), z1, cons(0, x1))), DEL(if1(ge(z1, 0), z1, 0, x1), cons(z1, cons(0, x1))), MAX(cons(z1, cons(0, x1)))) IF3(false, cons(z0, cons(0, z2))) -> c15(SORT(del(max(cons(z0, z2)), cons(z0, cons(0, z2)))), DEL(if1(ge(z0, 0), z0, 0, z2), cons(z0, cons(0, z2))), MAX(cons(z0, cons(0, z2)))) IF3(false, cons(0, cons(s(x0), x1))) -> c15(SORT(if2(eq(if1(false, 0, s(x0), x1), 0), if1(false, 0, s(x0), x1), 0, cons(s(x0), x1))), DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1))), MAX(cons(0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), z2))) -> c15(SORT(del(max(cons(s(x0), z2)), cons(0, cons(s(x0), z2)))), DEL(if1(ge(0, s(x0)), 0, s(x0), z2), cons(0, cons(s(x0), z2))), MAX(cons(0, cons(s(x0), z2)))) IF3(false, cons(0, cons(s(x0), x1))) -> c15(DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1))), MAX(cons(0, cons(s(x0), x1)))) K tuples: IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) IF3(false, cons(x0, cons(0, x1))) -> c(DEL(if1(ge(x0, 0), x0, 0, x1), cons(x0, cons(0, x1)))) IF3(false, cons(x0, cons(0, x1))) -> c(MAX(cons(x0, cons(0, x1)))) Defined Rule Symbols: del_2, max_1, if2_4, eq_2, ge_2, if1_4 Defined Pair Symbols: IF1_4, IF2_4, EQ_2, GE_2, MAX_1, DEL_2, SORT_1, IF3_2 Compound Symbols: c3_1, c4_1, c8_1, c12_1, c20_1, c2_1, c6_1, c13_1, c2_2, c6_2, c15_1, c15_3, c15_2, c_1 ---------------------------------------- (161) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (162) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil max(cons(z0, nil)) -> z0 max(cons(z0, cons(z1, z2))) -> if1(ge(z0, z1), z0, z1, z2) if2(true, z0, z1, z2) -> z2 if2(false, z0, z1, z2) -> cons(z1, del(z0, z2)) eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) if1(true, z0, z1, z2) -> max(cons(z0, z2)) if1(false, z0, z1, z2) -> max(cons(z1, z2)) Tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) MAX(cons(s(s(z0)), cons(s(s(0)), x2))) -> c2(IF1(true, s(s(z0)), s(s(0)), x2), GE(s(s(z0)), s(s(0)))) MAX(cons(s(s(0)), cons(s(s(s(z0))), x2))) -> c2(IF1(false, s(s(0)), s(s(s(z0))), x2), GE(s(s(0)), s(s(s(z0))))) MAX(cons(s(s(x0)), cons(s(s(x1)), x2))) -> c2(GE(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c6(IF2(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c6(IF2(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c6(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c6(EQ(s(s(0)), s(s(0)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(if1(ge(z1, x1), z1, x1, x2), cons(z1, cons(x1, x2))), MAX(cons(z1, cons(x1, x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(max(cons(z0, cons(z1, z2))), z0), if1(ge(z0, z1), z0, z1, z2), z0, cons(z1, z2)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), z0, z1, z2), z0), max(cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2))))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2))))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) MAX(cons(s(s(s(z0))), cons(s(s(s(0))), x2))) -> c2(IF1(true, s(s(s(z0))), s(s(s(0))), x2), GE(s(s(s(z0))), s(s(s(0))))) MAX(cons(s(s(s(0))), cons(s(s(s(s(z0)))), x2))) -> c2(IF1(false, s(s(s(0))), s(s(s(s(z0)))), x2), GE(s(s(s(0))), s(s(s(s(z0)))))) MAX(cons(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2))) -> c2(IF1(ge(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), GE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MAX(cons(s(s(s(x0))), cons(s(s(s(x1))), x2))) -> c2(GE(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c6(IF2(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c6(IF2(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c6(IF2(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c6(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c6(EQ(s(s(s(0))), s(s(s(0))))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(max(cons(z0, cons(0, x2))), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(max(cons(0, cons(s(z0), x2))), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(max(cons(s(z0), cons(s(z1), x2))), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), max(cons(z0, cons(0, x2))), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), max(cons(0, cons(s(z0), x2))), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), max(cons(s(z0), cons(s(z1), x2))), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(ge(z0, 0), z0, 0, x2), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(ge(0, s(z0)), 0, s(z0), x2), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), if1(ge(z0, 0), z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), if1(ge(0, s(z0)), 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z1, cons(0, x1))) -> c15(SORT(if2(eq(if1(true, z1, 0, x1), z1), if1(true, z1, 0, x1), z1, cons(0, x1))), DEL(if1(ge(z1, 0), z1, 0, x1), cons(z1, cons(0, x1))), MAX(cons(z1, cons(0, x1)))) IF3(false, cons(z0, cons(0, z2))) -> c15(SORT(del(max(cons(z0, z2)), cons(z0, cons(0, z2)))), DEL(if1(ge(z0, 0), z0, 0, z2), cons(z0, cons(0, z2))), MAX(cons(z0, cons(0, z2)))) IF3(false, cons(x0, cons(0, x1))) -> c(DEL(if1(ge(x0, 0), x0, 0, x1), cons(x0, cons(0, x1)))) IF3(false, cons(x0, cons(0, x1))) -> c(MAX(cons(x0, cons(0, x1)))) IF3(false, cons(0, cons(s(x0), x1))) -> c15(SORT(if2(eq(if1(false, 0, s(x0), x1), 0), if1(false, 0, s(x0), x1), 0, cons(s(x0), x1))), DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1))), MAX(cons(0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), z2))) -> c15(SORT(del(max(cons(s(x0), z2)), cons(0, cons(s(x0), z2)))), DEL(if1(ge(0, s(x0)), 0, s(x0), z2), cons(0, cons(s(x0), z2))), MAX(cons(0, cons(s(x0), z2)))) IF3(false, cons(0, cons(s(x0), x1))) -> c1(DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), x1))) -> c1(MAX(cons(0, cons(s(x0), x1)))) S tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) MAX(cons(s(s(z0)), cons(s(s(0)), x2))) -> c2(IF1(true, s(s(z0)), s(s(0)), x2), GE(s(s(z0)), s(s(0)))) MAX(cons(s(s(0)), cons(s(s(s(z0))), x2))) -> c2(IF1(false, s(s(0)), s(s(s(z0))), x2), GE(s(s(0)), s(s(s(z0))))) MAX(cons(s(s(x0)), cons(s(s(x1)), x2))) -> c2(GE(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c6(IF2(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c6(IF2(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c6(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c6(EQ(s(s(0)), s(s(0)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(max(cons(z0, cons(z1, z2))), z0), if1(ge(z0, z1), z0, z1, z2), z0, cons(z1, z2)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), z0, z1, z2), z0), max(cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2))))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2))))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) MAX(cons(s(s(s(z0))), cons(s(s(s(0))), x2))) -> c2(IF1(true, s(s(s(z0))), s(s(s(0))), x2), GE(s(s(s(z0))), s(s(s(0))))) MAX(cons(s(s(s(0))), cons(s(s(s(s(z0)))), x2))) -> c2(IF1(false, s(s(s(0))), s(s(s(s(z0)))), x2), GE(s(s(s(0))), s(s(s(s(z0)))))) MAX(cons(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2))) -> c2(IF1(ge(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), GE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MAX(cons(s(s(s(x0))), cons(s(s(s(x1))), x2))) -> c2(GE(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c6(IF2(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c6(IF2(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c6(IF2(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c6(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c6(EQ(s(s(s(0))), s(s(s(0))))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(max(cons(z0, cons(0, x2))), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(max(cons(0, cons(s(z0), x2))), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(max(cons(s(z0), cons(s(z1), x2))), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), max(cons(z0, cons(0, x2))), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), max(cons(0, cons(s(z0), x2))), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), max(cons(s(z0), cons(s(z1), x2))), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(ge(z0, 0), z0, 0, x2), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(ge(0, s(z0)), 0, s(z0), x2), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), if1(ge(z0, 0), z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), if1(ge(0, s(z0)), 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z1, cons(0, x1))) -> c15(SORT(if2(eq(if1(true, z1, 0, x1), z1), if1(true, z1, 0, x1), z1, cons(0, x1))), DEL(if1(ge(z1, 0), z1, 0, x1), cons(z1, cons(0, x1))), MAX(cons(z1, cons(0, x1)))) IF3(false, cons(z0, cons(0, z2))) -> c15(SORT(del(max(cons(z0, z2)), cons(z0, cons(0, z2)))), DEL(if1(ge(z0, 0), z0, 0, z2), cons(z0, cons(0, z2))), MAX(cons(z0, cons(0, z2)))) IF3(false, cons(0, cons(s(x0), x1))) -> c15(SORT(if2(eq(if1(false, 0, s(x0), x1), 0), if1(false, 0, s(x0), x1), 0, cons(s(x0), x1))), DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1))), MAX(cons(0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), z2))) -> c15(SORT(del(max(cons(s(x0), z2)), cons(0, cons(s(x0), z2)))), DEL(if1(ge(0, s(x0)), 0, s(x0), z2), cons(0, cons(s(x0), z2))), MAX(cons(0, cons(s(x0), z2)))) IF3(false, cons(0, cons(s(x0), x1))) -> c1(DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), x1))) -> c1(MAX(cons(0, cons(s(x0), x1)))) K tuples: IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) IF3(false, cons(x0, cons(0, x1))) -> c(DEL(if1(ge(x0, 0), x0, 0, x1), cons(x0, cons(0, x1)))) IF3(false, cons(x0, cons(0, x1))) -> c(MAX(cons(x0, cons(0, x1)))) Defined Rule Symbols: del_2, max_1, if2_4, eq_2, ge_2, if1_4 Defined Pair Symbols: IF1_4, IF2_4, EQ_2, GE_2, MAX_1, DEL_2, SORT_1, IF3_2 Compound Symbols: c3_1, c4_1, c8_1, c12_1, c20_1, c2_1, c6_1, c13_1, c2_2, c6_2, c15_1, c15_3, c15_2, c_1, c1_1 ---------------------------------------- (163) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. IF3(false, cons(0, cons(s(x0), x1))) -> c1(DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), x1))) -> c1(MAX(cons(0, cons(s(x0), x1)))) We considered the (Usable) Rules:none And the Tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) MAX(cons(s(s(z0)), cons(s(s(0)), x2))) -> c2(IF1(true, s(s(z0)), s(s(0)), x2), GE(s(s(z0)), s(s(0)))) MAX(cons(s(s(0)), cons(s(s(s(z0))), x2))) -> c2(IF1(false, s(s(0)), s(s(s(z0))), x2), GE(s(s(0)), s(s(s(z0))))) MAX(cons(s(s(x0)), cons(s(s(x1)), x2))) -> c2(GE(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c6(IF2(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c6(IF2(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c6(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c6(EQ(s(s(0)), s(s(0)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(if1(ge(z1, x1), z1, x1, x2), cons(z1, cons(x1, x2))), MAX(cons(z1, cons(x1, x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(max(cons(z0, cons(z1, z2))), z0), if1(ge(z0, z1), z0, z1, z2), z0, cons(z1, z2)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), z0, z1, z2), z0), max(cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2))))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2))))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) MAX(cons(s(s(s(z0))), cons(s(s(s(0))), x2))) -> c2(IF1(true, s(s(s(z0))), s(s(s(0))), x2), GE(s(s(s(z0))), s(s(s(0))))) MAX(cons(s(s(s(0))), cons(s(s(s(s(z0)))), x2))) -> c2(IF1(false, s(s(s(0))), s(s(s(s(z0)))), x2), GE(s(s(s(0))), s(s(s(s(z0)))))) MAX(cons(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2))) -> c2(IF1(ge(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), GE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MAX(cons(s(s(s(x0))), cons(s(s(s(x1))), x2))) -> c2(GE(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c6(IF2(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c6(IF2(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c6(IF2(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c6(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c6(EQ(s(s(s(0))), s(s(s(0))))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(max(cons(z0, cons(0, x2))), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(max(cons(0, cons(s(z0), x2))), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(max(cons(s(z0), cons(s(z1), x2))), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), max(cons(z0, cons(0, x2))), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), max(cons(0, cons(s(z0), x2))), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), max(cons(s(z0), cons(s(z1), x2))), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(ge(z0, 0), z0, 0, x2), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(ge(0, s(z0)), 0, s(z0), x2), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), if1(ge(z0, 0), z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), if1(ge(0, s(z0)), 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z1, cons(0, x1))) -> c15(SORT(if2(eq(if1(true, z1, 0, x1), z1), if1(true, z1, 0, x1), z1, cons(0, x1))), DEL(if1(ge(z1, 0), z1, 0, x1), cons(z1, cons(0, x1))), MAX(cons(z1, cons(0, x1)))) IF3(false, cons(z0, cons(0, z2))) -> c15(SORT(del(max(cons(z0, z2)), cons(z0, cons(0, z2)))), DEL(if1(ge(z0, 0), z0, 0, z2), cons(z0, cons(0, z2))), MAX(cons(z0, cons(0, z2)))) IF3(false, cons(x0, cons(0, x1))) -> c(DEL(if1(ge(x0, 0), x0, 0, x1), cons(x0, cons(0, x1)))) IF3(false, cons(x0, cons(0, x1))) -> c(MAX(cons(x0, cons(0, x1)))) IF3(false, cons(0, cons(s(x0), x1))) -> c15(SORT(if2(eq(if1(false, 0, s(x0), x1), 0), if1(false, 0, s(x0), x1), 0, cons(s(x0), x1))), DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1))), MAX(cons(0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), z2))) -> c15(SORT(del(max(cons(s(x0), z2)), cons(0, cons(s(x0), z2)))), DEL(if1(ge(0, s(x0)), 0, s(x0), z2), cons(0, cons(s(x0), z2))), MAX(cons(0, cons(s(x0), z2)))) IF3(false, cons(0, cons(s(x0), x1))) -> c1(DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), x1))) -> c1(MAX(cons(0, cons(s(x0), x1)))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(DEL(x_1, x_2)) = 0 POL(EQ(x_1, x_2)) = 0 POL(GE(x_1, x_2)) = 0 POL(IF1(x_1, x_2, x_3, x_4)) = 0 POL(IF2(x_1, x_2, x_3, x_4)) = x_3 POL(IF3(x_1, x_2)) = [1] + x_1 POL(MAX(x_1)) = 0 POL(SORT(x_1)) = [1] POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c13(x_1)) = x_1 POL(c15(x_1)) = x_1 POL(c15(x_1, x_2)) = x_1 + x_2 POL(c15(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c2(x_1)) = x_1 POL(c2(x_1, x_2)) = x_1 + x_2 POL(c20(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c4(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c6(x_1, x_2)) = x_1 + x_2 POL(c8(x_1)) = x_1 POL(cons(x_1, x_2)) = 0 POL(del(x_1, x_2)) = x_2 POL(eq(x_1, x_2)) = 0 POL(false) = 0 POL(ge(x_1, x_2)) = 0 POL(if1(x_1, x_2, x_3, x_4)) = 0 POL(if2(x_1, x_2, x_3, x_4)) = [1] + x_4 POL(max(x_1)) = 0 POL(nil) = 0 POL(s(x_1)) = 0 POL(true) = 0 ---------------------------------------- (164) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil max(cons(z0, nil)) -> z0 max(cons(z0, cons(z1, z2))) -> if1(ge(z0, z1), z0, z1, z2) if2(true, z0, z1, z2) -> z2 if2(false, z0, z1, z2) -> cons(z1, del(z0, z2)) eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) if1(true, z0, z1, z2) -> max(cons(z0, z2)) if1(false, z0, z1, z2) -> max(cons(z1, z2)) Tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) MAX(cons(s(s(z0)), cons(s(s(0)), x2))) -> c2(IF1(true, s(s(z0)), s(s(0)), x2), GE(s(s(z0)), s(s(0)))) MAX(cons(s(s(0)), cons(s(s(s(z0))), x2))) -> c2(IF1(false, s(s(0)), s(s(s(z0))), x2), GE(s(s(0)), s(s(s(z0))))) MAX(cons(s(s(x0)), cons(s(s(x1)), x2))) -> c2(GE(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c6(IF2(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c6(IF2(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c6(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c6(EQ(s(s(0)), s(s(0)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(if1(ge(z1, x1), z1, x1, x2), cons(z1, cons(x1, x2))), MAX(cons(z1, cons(x1, x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(max(cons(z0, cons(z1, z2))), z0), if1(ge(z0, z1), z0, z1, z2), z0, cons(z1, z2)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), z0, z1, z2), z0), max(cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2))))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2))))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) MAX(cons(s(s(s(z0))), cons(s(s(s(0))), x2))) -> c2(IF1(true, s(s(s(z0))), s(s(s(0))), x2), GE(s(s(s(z0))), s(s(s(0))))) MAX(cons(s(s(s(0))), cons(s(s(s(s(z0)))), x2))) -> c2(IF1(false, s(s(s(0))), s(s(s(s(z0)))), x2), GE(s(s(s(0))), s(s(s(s(z0)))))) MAX(cons(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2))) -> c2(IF1(ge(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), GE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MAX(cons(s(s(s(x0))), cons(s(s(s(x1))), x2))) -> c2(GE(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c6(IF2(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c6(IF2(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c6(IF2(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c6(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c6(EQ(s(s(s(0))), s(s(s(0))))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(max(cons(z0, cons(0, x2))), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(max(cons(0, cons(s(z0), x2))), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(max(cons(s(z0), cons(s(z1), x2))), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), max(cons(z0, cons(0, x2))), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), max(cons(0, cons(s(z0), x2))), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), max(cons(s(z0), cons(s(z1), x2))), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(ge(z0, 0), z0, 0, x2), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(ge(0, s(z0)), 0, s(z0), x2), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), if1(ge(z0, 0), z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), if1(ge(0, s(z0)), 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z1, cons(0, x1))) -> c15(SORT(if2(eq(if1(true, z1, 0, x1), z1), if1(true, z1, 0, x1), z1, cons(0, x1))), DEL(if1(ge(z1, 0), z1, 0, x1), cons(z1, cons(0, x1))), MAX(cons(z1, cons(0, x1)))) IF3(false, cons(z0, cons(0, z2))) -> c15(SORT(del(max(cons(z0, z2)), cons(z0, cons(0, z2)))), DEL(if1(ge(z0, 0), z0, 0, z2), cons(z0, cons(0, z2))), MAX(cons(z0, cons(0, z2)))) IF3(false, cons(x0, cons(0, x1))) -> c(DEL(if1(ge(x0, 0), x0, 0, x1), cons(x0, cons(0, x1)))) IF3(false, cons(x0, cons(0, x1))) -> c(MAX(cons(x0, cons(0, x1)))) IF3(false, cons(0, cons(s(x0), x1))) -> c15(SORT(if2(eq(if1(false, 0, s(x0), x1), 0), if1(false, 0, s(x0), x1), 0, cons(s(x0), x1))), DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1))), MAX(cons(0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), z2))) -> c15(SORT(del(max(cons(s(x0), z2)), cons(0, cons(s(x0), z2)))), DEL(if1(ge(0, s(x0)), 0, s(x0), z2), cons(0, cons(s(x0), z2))), MAX(cons(0, cons(s(x0), z2)))) IF3(false, cons(0, cons(s(x0), x1))) -> c1(DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), x1))) -> c1(MAX(cons(0, cons(s(x0), x1)))) S tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) MAX(cons(s(s(z0)), cons(s(s(0)), x2))) -> c2(IF1(true, s(s(z0)), s(s(0)), x2), GE(s(s(z0)), s(s(0)))) MAX(cons(s(s(0)), cons(s(s(s(z0))), x2))) -> c2(IF1(false, s(s(0)), s(s(s(z0))), x2), GE(s(s(0)), s(s(s(z0))))) MAX(cons(s(s(x0)), cons(s(s(x1)), x2))) -> c2(GE(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c6(IF2(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c6(IF2(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c6(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c6(EQ(s(s(0)), s(s(0)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(max(cons(z0, cons(z1, z2))), z0), if1(ge(z0, z1), z0, z1, z2), z0, cons(z1, z2)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), z0, z1, z2), z0), max(cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2))))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2))))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) MAX(cons(s(s(s(z0))), cons(s(s(s(0))), x2))) -> c2(IF1(true, s(s(s(z0))), s(s(s(0))), x2), GE(s(s(s(z0))), s(s(s(0))))) MAX(cons(s(s(s(0))), cons(s(s(s(s(z0)))), x2))) -> c2(IF1(false, s(s(s(0))), s(s(s(s(z0)))), x2), GE(s(s(s(0))), s(s(s(s(z0)))))) MAX(cons(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2))) -> c2(IF1(ge(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), GE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MAX(cons(s(s(s(x0))), cons(s(s(s(x1))), x2))) -> c2(GE(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c6(IF2(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c6(IF2(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c6(IF2(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c6(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c6(EQ(s(s(s(0))), s(s(s(0))))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(max(cons(z0, cons(0, x2))), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(max(cons(0, cons(s(z0), x2))), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(max(cons(s(z0), cons(s(z1), x2))), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), max(cons(z0, cons(0, x2))), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), max(cons(0, cons(s(z0), x2))), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), max(cons(s(z0), cons(s(z1), x2))), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(ge(z0, 0), z0, 0, x2), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(ge(0, s(z0)), 0, s(z0), x2), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), if1(ge(z0, 0), z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), if1(ge(0, s(z0)), 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z1, cons(0, x1))) -> c15(SORT(if2(eq(if1(true, z1, 0, x1), z1), if1(true, z1, 0, x1), z1, cons(0, x1))), DEL(if1(ge(z1, 0), z1, 0, x1), cons(z1, cons(0, x1))), MAX(cons(z1, cons(0, x1)))) IF3(false, cons(z0, cons(0, z2))) -> c15(SORT(del(max(cons(z0, z2)), cons(z0, cons(0, z2)))), DEL(if1(ge(z0, 0), z0, 0, z2), cons(z0, cons(0, z2))), MAX(cons(z0, cons(0, z2)))) IF3(false, cons(0, cons(s(x0), x1))) -> c15(SORT(if2(eq(if1(false, 0, s(x0), x1), 0), if1(false, 0, s(x0), x1), 0, cons(s(x0), x1))), DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1))), MAX(cons(0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), z2))) -> c15(SORT(del(max(cons(s(x0), z2)), cons(0, cons(s(x0), z2)))), DEL(if1(ge(0, s(x0)), 0, s(x0), z2), cons(0, cons(s(x0), z2))), MAX(cons(0, cons(s(x0), z2)))) K tuples: IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) IF3(false, cons(x0, cons(0, x1))) -> c(DEL(if1(ge(x0, 0), x0, 0, x1), cons(x0, cons(0, x1)))) IF3(false, cons(x0, cons(0, x1))) -> c(MAX(cons(x0, cons(0, x1)))) IF3(false, cons(0, cons(s(x0), x1))) -> c1(DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), x1))) -> c1(MAX(cons(0, cons(s(x0), x1)))) Defined Rule Symbols: del_2, max_1, if2_4, eq_2, ge_2, if1_4 Defined Pair Symbols: IF1_4, IF2_4, EQ_2, GE_2, MAX_1, DEL_2, SORT_1, IF3_2 Compound Symbols: c3_1, c4_1, c8_1, c12_1, c20_1, c2_1, c6_1, c13_1, c2_2, c6_2, c15_1, c15_3, c15_2, c_1, c1_1 ---------------------------------------- (165) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) by IF3(false, cons(s(x0), cons(s(x1), x2))) -> c15(SORT(if2(eq(if1(ge(x0, x1), s(x0), s(x1), x2), s(x0)), if1(ge(x0, x1), s(x0), s(x1), x2), s(x0), cons(s(x1), x2))), DEL(if1(ge(s(x0), s(x1)), s(x0), s(x1), x2), cons(s(x0), cons(s(x1), x2))), MAX(cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(s(z0), cons(s(0), x2))) -> c15(SORT(del(if1(true, s(z0), s(0), x2), cons(s(z0), cons(s(0), x2)))), DEL(if1(ge(s(z0), s(0)), s(z0), s(0), x2), cons(s(z0), cons(s(0), x2))), MAX(cons(s(z0), cons(s(0), x2)))) IF3(false, cons(s(0), cons(s(s(z0)), x2))) -> c15(SORT(del(if1(false, s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2)))), DEL(if1(ge(s(0), s(s(z0))), s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2))), MAX(cons(s(0), cons(s(s(z0)), x2)))) IF3(false, cons(s(s(z0)), cons(s(s(z1)), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2)))), DEL(if1(ge(s(s(z0)), s(s(z1))), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2))), MAX(cons(s(s(z0)), cons(s(s(z1)), x2)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c15(DEL(if1(ge(s(x0), s(x1)), s(x0), s(x1), x2), cons(s(x0), cons(s(x1), x2))), MAX(cons(s(x0), cons(s(x1), x2)))) ---------------------------------------- (166) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil max(cons(z0, nil)) -> z0 max(cons(z0, cons(z1, z2))) -> if1(ge(z0, z1), z0, z1, z2) if2(true, z0, z1, z2) -> z2 if2(false, z0, z1, z2) -> cons(z1, del(z0, z2)) eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) if1(true, z0, z1, z2) -> max(cons(z0, z2)) if1(false, z0, z1, z2) -> max(cons(z1, z2)) Tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) MAX(cons(s(s(z0)), cons(s(s(0)), x2))) -> c2(IF1(true, s(s(z0)), s(s(0)), x2), GE(s(s(z0)), s(s(0)))) MAX(cons(s(s(0)), cons(s(s(s(z0))), x2))) -> c2(IF1(false, s(s(0)), s(s(s(z0))), x2), GE(s(s(0)), s(s(s(z0))))) MAX(cons(s(s(x0)), cons(s(s(x1)), x2))) -> c2(GE(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c6(IF2(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c6(IF2(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c6(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c6(EQ(s(s(0)), s(s(0)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(if1(ge(z1, x1), z1, x1, x2), cons(z1, cons(x1, x2))), MAX(cons(z1, cons(x1, x2)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(max(cons(z0, cons(z1, z2))), z0), if1(ge(z0, z1), z0, z1, z2), z0, cons(z1, z2)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), z0, z1, z2), z0), max(cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2))))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2))))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) MAX(cons(s(s(s(z0))), cons(s(s(s(0))), x2))) -> c2(IF1(true, s(s(s(z0))), s(s(s(0))), x2), GE(s(s(s(z0))), s(s(s(0))))) MAX(cons(s(s(s(0))), cons(s(s(s(s(z0)))), x2))) -> c2(IF1(false, s(s(s(0))), s(s(s(s(z0)))), x2), GE(s(s(s(0))), s(s(s(s(z0)))))) MAX(cons(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2))) -> c2(IF1(ge(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), GE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MAX(cons(s(s(s(x0))), cons(s(s(s(x1))), x2))) -> c2(GE(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c6(IF2(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c6(IF2(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c6(IF2(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c6(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c6(EQ(s(s(s(0))), s(s(s(0))))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(max(cons(z0, cons(0, x2))), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(max(cons(0, cons(s(z0), x2))), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(max(cons(s(z0), cons(s(z1), x2))), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), max(cons(z0, cons(0, x2))), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), max(cons(0, cons(s(z0), x2))), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), max(cons(s(z0), cons(s(z1), x2))), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(ge(z0, 0), z0, 0, x2), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(ge(0, s(z0)), 0, s(z0), x2), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), if1(ge(z0, 0), z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), if1(ge(0, s(z0)), 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z1, cons(0, x1))) -> c15(SORT(if2(eq(if1(true, z1, 0, x1), z1), if1(true, z1, 0, x1), z1, cons(0, x1))), DEL(if1(ge(z1, 0), z1, 0, x1), cons(z1, cons(0, x1))), MAX(cons(z1, cons(0, x1)))) IF3(false, cons(z0, cons(0, z2))) -> c15(SORT(del(max(cons(z0, z2)), cons(z0, cons(0, z2)))), DEL(if1(ge(z0, 0), z0, 0, z2), cons(z0, cons(0, z2))), MAX(cons(z0, cons(0, z2)))) IF3(false, cons(x0, cons(0, x1))) -> c(DEL(if1(ge(x0, 0), x0, 0, x1), cons(x0, cons(0, x1)))) IF3(false, cons(x0, cons(0, x1))) -> c(MAX(cons(x0, cons(0, x1)))) IF3(false, cons(0, cons(s(x0), x1))) -> c15(SORT(if2(eq(if1(false, 0, s(x0), x1), 0), if1(false, 0, s(x0), x1), 0, cons(s(x0), x1))), DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1))), MAX(cons(0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), z2))) -> c15(SORT(del(max(cons(s(x0), z2)), cons(0, cons(s(x0), z2)))), DEL(if1(ge(0, s(x0)), 0, s(x0), z2), cons(0, cons(s(x0), z2))), MAX(cons(0, cons(s(x0), z2)))) IF3(false, cons(0, cons(s(x0), x1))) -> c1(DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), x1))) -> c1(MAX(cons(0, cons(s(x0), x1)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c15(SORT(if2(eq(if1(ge(x0, x1), s(x0), s(x1), x2), s(x0)), if1(ge(x0, x1), s(x0), s(x1), x2), s(x0), cons(s(x1), x2))), DEL(if1(ge(s(x0), s(x1)), s(x0), s(x1), x2), cons(s(x0), cons(s(x1), x2))), MAX(cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(s(z0), cons(s(0), x2))) -> c15(SORT(del(if1(true, s(z0), s(0), x2), cons(s(z0), cons(s(0), x2)))), DEL(if1(ge(s(z0), s(0)), s(z0), s(0), x2), cons(s(z0), cons(s(0), x2))), MAX(cons(s(z0), cons(s(0), x2)))) IF3(false, cons(s(0), cons(s(s(z0)), x2))) -> c15(SORT(del(if1(false, s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2)))), DEL(if1(ge(s(0), s(s(z0))), s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2))), MAX(cons(s(0), cons(s(s(z0)), x2)))) IF3(false, cons(s(s(z0)), cons(s(s(z1)), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2)))), DEL(if1(ge(s(s(z0)), s(s(z1))), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2))), MAX(cons(s(s(z0)), cons(s(s(z1)), x2)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c15(DEL(if1(ge(s(x0), s(x1)), s(x0), s(x1), x2), cons(s(x0), cons(s(x1), x2))), MAX(cons(s(x0), cons(s(x1), x2)))) S tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) MAX(cons(s(s(z0)), cons(s(s(0)), x2))) -> c2(IF1(true, s(s(z0)), s(s(0)), x2), GE(s(s(z0)), s(s(0)))) MAX(cons(s(s(0)), cons(s(s(s(z0))), x2))) -> c2(IF1(false, s(s(0)), s(s(s(z0))), x2), GE(s(s(0)), s(s(s(z0))))) MAX(cons(s(s(x0)), cons(s(s(x1)), x2))) -> c2(GE(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c6(IF2(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c6(IF2(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c6(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c6(EQ(s(s(0)), s(s(0)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(max(cons(z0, cons(z1, z2))), z0), if1(ge(z0, z1), z0, z1, z2), z0, cons(z1, z2)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), z0, z1, z2), z0), max(cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2))))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2))))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) MAX(cons(s(s(s(z0))), cons(s(s(s(0))), x2))) -> c2(IF1(true, s(s(s(z0))), s(s(s(0))), x2), GE(s(s(s(z0))), s(s(s(0))))) MAX(cons(s(s(s(0))), cons(s(s(s(s(z0)))), x2))) -> c2(IF1(false, s(s(s(0))), s(s(s(s(z0)))), x2), GE(s(s(s(0))), s(s(s(s(z0)))))) MAX(cons(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2))) -> c2(IF1(ge(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), GE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MAX(cons(s(s(s(x0))), cons(s(s(s(x1))), x2))) -> c2(GE(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c6(IF2(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c6(IF2(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c6(IF2(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c6(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c6(EQ(s(s(s(0))), s(s(s(0))))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(max(cons(z0, cons(0, x2))), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(max(cons(0, cons(s(z0), x2))), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(max(cons(s(z0), cons(s(z1), x2))), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), max(cons(z0, cons(0, x2))), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), max(cons(0, cons(s(z0), x2))), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), max(cons(s(z0), cons(s(z1), x2))), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(ge(z0, 0), z0, 0, x2), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(ge(0, s(z0)), 0, s(z0), x2), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), if1(ge(z0, 0), z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), if1(ge(0, s(z0)), 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z1, cons(0, x1))) -> c15(SORT(if2(eq(if1(true, z1, 0, x1), z1), if1(true, z1, 0, x1), z1, cons(0, x1))), DEL(if1(ge(z1, 0), z1, 0, x1), cons(z1, cons(0, x1))), MAX(cons(z1, cons(0, x1)))) IF3(false, cons(z0, cons(0, z2))) -> c15(SORT(del(max(cons(z0, z2)), cons(z0, cons(0, z2)))), DEL(if1(ge(z0, 0), z0, 0, z2), cons(z0, cons(0, z2))), MAX(cons(z0, cons(0, z2)))) IF3(false, cons(0, cons(s(x0), x1))) -> c15(SORT(if2(eq(if1(false, 0, s(x0), x1), 0), if1(false, 0, s(x0), x1), 0, cons(s(x0), x1))), DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1))), MAX(cons(0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), z2))) -> c15(SORT(del(max(cons(s(x0), z2)), cons(0, cons(s(x0), z2)))), DEL(if1(ge(0, s(x0)), 0, s(x0), z2), cons(0, cons(s(x0), z2))), MAX(cons(0, cons(s(x0), z2)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c15(SORT(if2(eq(if1(ge(x0, x1), s(x0), s(x1), x2), s(x0)), if1(ge(x0, x1), s(x0), s(x1), x2), s(x0), cons(s(x1), x2))), DEL(if1(ge(s(x0), s(x1)), s(x0), s(x1), x2), cons(s(x0), cons(s(x1), x2))), MAX(cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(s(z0), cons(s(0), x2))) -> c15(SORT(del(if1(true, s(z0), s(0), x2), cons(s(z0), cons(s(0), x2)))), DEL(if1(ge(s(z0), s(0)), s(z0), s(0), x2), cons(s(z0), cons(s(0), x2))), MAX(cons(s(z0), cons(s(0), x2)))) IF3(false, cons(s(0), cons(s(s(z0)), x2))) -> c15(SORT(del(if1(false, s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2)))), DEL(if1(ge(s(0), s(s(z0))), s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2))), MAX(cons(s(0), cons(s(s(z0)), x2)))) IF3(false, cons(s(s(z0)), cons(s(s(z1)), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2)))), DEL(if1(ge(s(s(z0)), s(s(z1))), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2))), MAX(cons(s(s(z0)), cons(s(s(z1)), x2)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c15(DEL(if1(ge(s(x0), s(x1)), s(x0), s(x1), x2), cons(s(x0), cons(s(x1), x2))), MAX(cons(s(x0), cons(s(x1), x2)))) K tuples: IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) IF3(false, cons(x0, cons(0, x1))) -> c(DEL(if1(ge(x0, 0), x0, 0, x1), cons(x0, cons(0, x1)))) IF3(false, cons(x0, cons(0, x1))) -> c(MAX(cons(x0, cons(0, x1)))) IF3(false, cons(0, cons(s(x0), x1))) -> c1(DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), x1))) -> c1(MAX(cons(0, cons(s(x0), x1)))) Defined Rule Symbols: del_2, max_1, if2_4, eq_2, ge_2, if1_4 Defined Pair Symbols: IF1_4, IF2_4, EQ_2, GE_2, MAX_1, DEL_2, SORT_1, IF3_2 Compound Symbols: c3_1, c4_1, c8_1, c12_1, c20_1, c2_1, c6_1, c13_1, c2_2, c6_2, c15_1, c15_3, c15_2, c_1, c1_1 ---------------------------------------- (167) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (168) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil max(cons(z0, nil)) -> z0 max(cons(z0, cons(z1, z2))) -> if1(ge(z0, z1), z0, z1, z2) if2(true, z0, z1, z2) -> z2 if2(false, z0, z1, z2) -> cons(z1, del(z0, z2)) eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) if1(true, z0, z1, z2) -> max(cons(z0, z2)) if1(false, z0, z1, z2) -> max(cons(z1, z2)) Tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) MAX(cons(s(s(z0)), cons(s(s(0)), x2))) -> c2(IF1(true, s(s(z0)), s(s(0)), x2), GE(s(s(z0)), s(s(0)))) MAX(cons(s(s(0)), cons(s(s(s(z0))), x2))) -> c2(IF1(false, s(s(0)), s(s(s(z0))), x2), GE(s(s(0)), s(s(s(z0))))) MAX(cons(s(s(x0)), cons(s(s(x1)), x2))) -> c2(GE(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c6(IF2(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c6(IF2(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c6(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c6(EQ(s(s(0)), s(s(0)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(if1(ge(z1, x1), z1, x1, x2), cons(z1, cons(x1, x2))), MAX(cons(z1, cons(x1, x2)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(max(cons(z0, cons(z1, z2))), z0), if1(ge(z0, z1), z0, z1, z2), z0, cons(z1, z2)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), z0, z1, z2), z0), max(cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2))))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2))))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) MAX(cons(s(s(s(z0))), cons(s(s(s(0))), x2))) -> c2(IF1(true, s(s(s(z0))), s(s(s(0))), x2), GE(s(s(s(z0))), s(s(s(0))))) MAX(cons(s(s(s(0))), cons(s(s(s(s(z0)))), x2))) -> c2(IF1(false, s(s(s(0))), s(s(s(s(z0)))), x2), GE(s(s(s(0))), s(s(s(s(z0)))))) MAX(cons(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2))) -> c2(IF1(ge(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), GE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MAX(cons(s(s(s(x0))), cons(s(s(s(x1))), x2))) -> c2(GE(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c6(IF2(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c6(IF2(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c6(IF2(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c6(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c6(EQ(s(s(s(0))), s(s(s(0))))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(max(cons(z0, cons(0, x2))), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(max(cons(0, cons(s(z0), x2))), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(max(cons(s(z0), cons(s(z1), x2))), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), max(cons(z0, cons(0, x2))), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), max(cons(0, cons(s(z0), x2))), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), max(cons(s(z0), cons(s(z1), x2))), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(ge(z0, 0), z0, 0, x2), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(ge(0, s(z0)), 0, s(z0), x2), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), if1(ge(z0, 0), z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), if1(ge(0, s(z0)), 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z1, cons(0, x1))) -> c15(SORT(if2(eq(if1(true, z1, 0, x1), z1), if1(true, z1, 0, x1), z1, cons(0, x1))), DEL(if1(ge(z1, 0), z1, 0, x1), cons(z1, cons(0, x1))), MAX(cons(z1, cons(0, x1)))) IF3(false, cons(z0, cons(0, z2))) -> c15(SORT(del(max(cons(z0, z2)), cons(z0, cons(0, z2)))), DEL(if1(ge(z0, 0), z0, 0, z2), cons(z0, cons(0, z2))), MAX(cons(z0, cons(0, z2)))) IF3(false, cons(x0, cons(0, x1))) -> c(DEL(if1(ge(x0, 0), x0, 0, x1), cons(x0, cons(0, x1)))) IF3(false, cons(x0, cons(0, x1))) -> c(MAX(cons(x0, cons(0, x1)))) IF3(false, cons(0, cons(s(x0), x1))) -> c15(SORT(if2(eq(if1(false, 0, s(x0), x1), 0), if1(false, 0, s(x0), x1), 0, cons(s(x0), x1))), DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1))), MAX(cons(0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), z2))) -> c15(SORT(del(max(cons(s(x0), z2)), cons(0, cons(s(x0), z2)))), DEL(if1(ge(0, s(x0)), 0, s(x0), z2), cons(0, cons(s(x0), z2))), MAX(cons(0, cons(s(x0), z2)))) IF3(false, cons(0, cons(s(x0), x1))) -> c1(DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), x1))) -> c1(MAX(cons(0, cons(s(x0), x1)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c15(SORT(if2(eq(if1(ge(x0, x1), s(x0), s(x1), x2), s(x0)), if1(ge(x0, x1), s(x0), s(x1), x2), s(x0), cons(s(x1), x2))), DEL(if1(ge(s(x0), s(x1)), s(x0), s(x1), x2), cons(s(x0), cons(s(x1), x2))), MAX(cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(s(z0), cons(s(0), x2))) -> c15(SORT(del(if1(true, s(z0), s(0), x2), cons(s(z0), cons(s(0), x2)))), DEL(if1(ge(s(z0), s(0)), s(z0), s(0), x2), cons(s(z0), cons(s(0), x2))), MAX(cons(s(z0), cons(s(0), x2)))) IF3(false, cons(s(0), cons(s(s(z0)), x2))) -> c15(SORT(del(if1(false, s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2)))), DEL(if1(ge(s(0), s(s(z0))), s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2))), MAX(cons(s(0), cons(s(s(z0)), x2)))) IF3(false, cons(s(s(z0)), cons(s(s(z1)), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2)))), DEL(if1(ge(s(s(z0)), s(s(z1))), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2))), MAX(cons(s(s(z0)), cons(s(s(z1)), x2)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c5(DEL(if1(ge(s(x0), s(x1)), s(x0), s(x1), x2), cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c5(MAX(cons(s(x0), cons(s(x1), x2)))) S tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) MAX(cons(s(s(z0)), cons(s(s(0)), x2))) -> c2(IF1(true, s(s(z0)), s(s(0)), x2), GE(s(s(z0)), s(s(0)))) MAX(cons(s(s(0)), cons(s(s(s(z0))), x2))) -> c2(IF1(false, s(s(0)), s(s(s(z0))), x2), GE(s(s(0)), s(s(s(z0))))) MAX(cons(s(s(x0)), cons(s(s(x1)), x2))) -> c2(GE(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c6(IF2(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c6(IF2(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c6(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c6(EQ(s(s(0)), s(s(0)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(max(cons(z0, cons(z1, z2))), z0), if1(ge(z0, z1), z0, z1, z2), z0, cons(z1, z2)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), z0, z1, z2), z0), max(cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2))))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2))))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) MAX(cons(s(s(s(z0))), cons(s(s(s(0))), x2))) -> c2(IF1(true, s(s(s(z0))), s(s(s(0))), x2), GE(s(s(s(z0))), s(s(s(0))))) MAX(cons(s(s(s(0))), cons(s(s(s(s(z0)))), x2))) -> c2(IF1(false, s(s(s(0))), s(s(s(s(z0)))), x2), GE(s(s(s(0))), s(s(s(s(z0)))))) MAX(cons(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2))) -> c2(IF1(ge(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), GE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MAX(cons(s(s(s(x0))), cons(s(s(s(x1))), x2))) -> c2(GE(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c6(IF2(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c6(IF2(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c6(IF2(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c6(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c6(EQ(s(s(s(0))), s(s(s(0))))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(max(cons(z0, cons(0, x2))), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(max(cons(0, cons(s(z0), x2))), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(max(cons(s(z0), cons(s(z1), x2))), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), max(cons(z0, cons(0, x2))), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), max(cons(0, cons(s(z0), x2))), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), max(cons(s(z0), cons(s(z1), x2))), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(ge(z0, 0), z0, 0, x2), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(ge(0, s(z0)), 0, s(z0), x2), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), if1(ge(z0, 0), z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), if1(ge(0, s(z0)), 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z1, cons(0, x1))) -> c15(SORT(if2(eq(if1(true, z1, 0, x1), z1), if1(true, z1, 0, x1), z1, cons(0, x1))), DEL(if1(ge(z1, 0), z1, 0, x1), cons(z1, cons(0, x1))), MAX(cons(z1, cons(0, x1)))) IF3(false, cons(z0, cons(0, z2))) -> c15(SORT(del(max(cons(z0, z2)), cons(z0, cons(0, z2)))), DEL(if1(ge(z0, 0), z0, 0, z2), cons(z0, cons(0, z2))), MAX(cons(z0, cons(0, z2)))) IF3(false, cons(0, cons(s(x0), x1))) -> c15(SORT(if2(eq(if1(false, 0, s(x0), x1), 0), if1(false, 0, s(x0), x1), 0, cons(s(x0), x1))), DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1))), MAX(cons(0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), z2))) -> c15(SORT(del(max(cons(s(x0), z2)), cons(0, cons(s(x0), z2)))), DEL(if1(ge(0, s(x0)), 0, s(x0), z2), cons(0, cons(s(x0), z2))), MAX(cons(0, cons(s(x0), z2)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c15(SORT(if2(eq(if1(ge(x0, x1), s(x0), s(x1), x2), s(x0)), if1(ge(x0, x1), s(x0), s(x1), x2), s(x0), cons(s(x1), x2))), DEL(if1(ge(s(x0), s(x1)), s(x0), s(x1), x2), cons(s(x0), cons(s(x1), x2))), MAX(cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(s(z0), cons(s(0), x2))) -> c15(SORT(del(if1(true, s(z0), s(0), x2), cons(s(z0), cons(s(0), x2)))), DEL(if1(ge(s(z0), s(0)), s(z0), s(0), x2), cons(s(z0), cons(s(0), x2))), MAX(cons(s(z0), cons(s(0), x2)))) IF3(false, cons(s(0), cons(s(s(z0)), x2))) -> c15(SORT(del(if1(false, s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2)))), DEL(if1(ge(s(0), s(s(z0))), s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2))), MAX(cons(s(0), cons(s(s(z0)), x2)))) IF3(false, cons(s(s(z0)), cons(s(s(z1)), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2)))), DEL(if1(ge(s(s(z0)), s(s(z1))), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2))), MAX(cons(s(s(z0)), cons(s(s(z1)), x2)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c5(DEL(if1(ge(s(x0), s(x1)), s(x0), s(x1), x2), cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c5(MAX(cons(s(x0), cons(s(x1), x2)))) K tuples: IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) IF3(false, cons(x0, cons(0, x1))) -> c(DEL(if1(ge(x0, 0), x0, 0, x1), cons(x0, cons(0, x1)))) IF3(false, cons(x0, cons(0, x1))) -> c(MAX(cons(x0, cons(0, x1)))) IF3(false, cons(0, cons(s(x0), x1))) -> c1(DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), x1))) -> c1(MAX(cons(0, cons(s(x0), x1)))) Defined Rule Symbols: del_2, max_1, if2_4, eq_2, ge_2, if1_4 Defined Pair Symbols: IF1_4, IF2_4, EQ_2, GE_2, MAX_1, DEL_2, SORT_1, IF3_2 Compound Symbols: c3_1, c4_1, c8_1, c12_1, c20_1, c2_1, c6_1, c13_1, c2_2, c6_2, c15_1, c15_3, c15_2, c_1, c1_1, c5_1 ---------------------------------------- (169) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. IF3(false, cons(s(x0), cons(s(x1), x2))) -> c5(DEL(if1(ge(s(x0), s(x1)), s(x0), s(x1), x2), cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c5(MAX(cons(s(x0), cons(s(x1), x2)))) We considered the (Usable) Rules:none And the Tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) MAX(cons(s(s(z0)), cons(s(s(0)), x2))) -> c2(IF1(true, s(s(z0)), s(s(0)), x2), GE(s(s(z0)), s(s(0)))) MAX(cons(s(s(0)), cons(s(s(s(z0))), x2))) -> c2(IF1(false, s(s(0)), s(s(s(z0))), x2), GE(s(s(0)), s(s(s(z0))))) MAX(cons(s(s(x0)), cons(s(s(x1)), x2))) -> c2(GE(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c6(IF2(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c6(IF2(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c6(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c6(EQ(s(s(0)), s(s(0)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(if1(ge(z1, x1), z1, x1, x2), cons(z1, cons(x1, x2))), MAX(cons(z1, cons(x1, x2)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(max(cons(z0, cons(z1, z2))), z0), if1(ge(z0, z1), z0, z1, z2), z0, cons(z1, z2)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), z0, z1, z2), z0), max(cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2))))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2))))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) MAX(cons(s(s(s(z0))), cons(s(s(s(0))), x2))) -> c2(IF1(true, s(s(s(z0))), s(s(s(0))), x2), GE(s(s(s(z0))), s(s(s(0))))) MAX(cons(s(s(s(0))), cons(s(s(s(s(z0)))), x2))) -> c2(IF1(false, s(s(s(0))), s(s(s(s(z0)))), x2), GE(s(s(s(0))), s(s(s(s(z0)))))) MAX(cons(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2))) -> c2(IF1(ge(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), GE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MAX(cons(s(s(s(x0))), cons(s(s(s(x1))), x2))) -> c2(GE(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c6(IF2(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c6(IF2(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c6(IF2(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c6(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c6(EQ(s(s(s(0))), s(s(s(0))))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(max(cons(z0, cons(0, x2))), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(max(cons(0, cons(s(z0), x2))), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(max(cons(s(z0), cons(s(z1), x2))), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), max(cons(z0, cons(0, x2))), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), max(cons(0, cons(s(z0), x2))), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), max(cons(s(z0), cons(s(z1), x2))), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(ge(z0, 0), z0, 0, x2), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(ge(0, s(z0)), 0, s(z0), x2), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), if1(ge(z0, 0), z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), if1(ge(0, s(z0)), 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z1, cons(0, x1))) -> c15(SORT(if2(eq(if1(true, z1, 0, x1), z1), if1(true, z1, 0, x1), z1, cons(0, x1))), DEL(if1(ge(z1, 0), z1, 0, x1), cons(z1, cons(0, x1))), MAX(cons(z1, cons(0, x1)))) IF3(false, cons(z0, cons(0, z2))) -> c15(SORT(del(max(cons(z0, z2)), cons(z0, cons(0, z2)))), DEL(if1(ge(z0, 0), z0, 0, z2), cons(z0, cons(0, z2))), MAX(cons(z0, cons(0, z2)))) IF3(false, cons(x0, cons(0, x1))) -> c(DEL(if1(ge(x0, 0), x0, 0, x1), cons(x0, cons(0, x1)))) IF3(false, cons(x0, cons(0, x1))) -> c(MAX(cons(x0, cons(0, x1)))) IF3(false, cons(0, cons(s(x0), x1))) -> c15(SORT(if2(eq(if1(false, 0, s(x0), x1), 0), if1(false, 0, s(x0), x1), 0, cons(s(x0), x1))), DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1))), MAX(cons(0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), z2))) -> c15(SORT(del(max(cons(s(x0), z2)), cons(0, cons(s(x0), z2)))), DEL(if1(ge(0, s(x0)), 0, s(x0), z2), cons(0, cons(s(x0), z2))), MAX(cons(0, cons(s(x0), z2)))) IF3(false, cons(0, cons(s(x0), x1))) -> c1(DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), x1))) -> c1(MAX(cons(0, cons(s(x0), x1)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c15(SORT(if2(eq(if1(ge(x0, x1), s(x0), s(x1), x2), s(x0)), if1(ge(x0, x1), s(x0), s(x1), x2), s(x0), cons(s(x1), x2))), DEL(if1(ge(s(x0), s(x1)), s(x0), s(x1), x2), cons(s(x0), cons(s(x1), x2))), MAX(cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(s(z0), cons(s(0), x2))) -> c15(SORT(del(if1(true, s(z0), s(0), x2), cons(s(z0), cons(s(0), x2)))), DEL(if1(ge(s(z0), s(0)), s(z0), s(0), x2), cons(s(z0), cons(s(0), x2))), MAX(cons(s(z0), cons(s(0), x2)))) IF3(false, cons(s(0), cons(s(s(z0)), x2))) -> c15(SORT(del(if1(false, s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2)))), DEL(if1(ge(s(0), s(s(z0))), s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2))), MAX(cons(s(0), cons(s(s(z0)), x2)))) IF3(false, cons(s(s(z0)), cons(s(s(z1)), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2)))), DEL(if1(ge(s(s(z0)), s(s(z1))), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2))), MAX(cons(s(s(z0)), cons(s(s(z1)), x2)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c5(DEL(if1(ge(s(x0), s(x1)), s(x0), s(x1), x2), cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c5(MAX(cons(s(x0), cons(s(x1), x2)))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(DEL(x_1, x_2)) = 0 POL(EQ(x_1, x_2)) = 0 POL(GE(x_1, x_2)) = 0 POL(IF1(x_1, x_2, x_3, x_4)) = 0 POL(IF2(x_1, x_2, x_3, x_4)) = x_3 POL(IF3(x_1, x_2)) = [1] + x_1 POL(MAX(x_1)) = 0 POL(SORT(x_1)) = [1] POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c13(x_1)) = x_1 POL(c15(x_1)) = x_1 POL(c15(x_1, x_2)) = x_1 + x_2 POL(c15(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c2(x_1)) = x_1 POL(c2(x_1, x_2)) = x_1 + x_2 POL(c20(x_1)) = x_1 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(c6(x_1, x_2)) = x_1 + x_2 POL(c8(x_1)) = x_1 POL(cons(x_1, x_2)) = 0 POL(del(x_1, x_2)) = x_2 POL(eq(x_1, x_2)) = 0 POL(false) = 0 POL(ge(x_1, x_2)) = 0 POL(if1(x_1, x_2, x_3, x_4)) = 0 POL(if2(x_1, x_2, x_3, x_4)) = [1] + x_4 POL(max(x_1)) = 0 POL(nil) = 0 POL(s(x_1)) = 0 POL(true) = 0 ---------------------------------------- (170) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil max(cons(z0, nil)) -> z0 max(cons(z0, cons(z1, z2))) -> if1(ge(z0, z1), z0, z1, z2) if2(true, z0, z1, z2) -> z2 if2(false, z0, z1, z2) -> cons(z1, del(z0, z2)) eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) if1(true, z0, z1, z2) -> max(cons(z0, z2)) if1(false, z0, z1, z2) -> max(cons(z1, z2)) Tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) MAX(cons(s(s(z0)), cons(s(s(0)), x2))) -> c2(IF1(true, s(s(z0)), s(s(0)), x2), GE(s(s(z0)), s(s(0)))) MAX(cons(s(s(0)), cons(s(s(s(z0))), x2))) -> c2(IF1(false, s(s(0)), s(s(s(z0))), x2), GE(s(s(0)), s(s(s(z0))))) MAX(cons(s(s(x0)), cons(s(s(x1)), x2))) -> c2(GE(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c6(IF2(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c6(IF2(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c6(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c6(EQ(s(s(0)), s(s(0)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(if1(ge(z1, x1), z1, x1, x2), cons(z1, cons(x1, x2))), MAX(cons(z1, cons(x1, x2)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(max(cons(z0, cons(z1, z2))), z0), if1(ge(z0, z1), z0, z1, z2), z0, cons(z1, z2)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), z0, z1, z2), z0), max(cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2))))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2))))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) MAX(cons(s(s(s(z0))), cons(s(s(s(0))), x2))) -> c2(IF1(true, s(s(s(z0))), s(s(s(0))), x2), GE(s(s(s(z0))), s(s(s(0))))) MAX(cons(s(s(s(0))), cons(s(s(s(s(z0)))), x2))) -> c2(IF1(false, s(s(s(0))), s(s(s(s(z0)))), x2), GE(s(s(s(0))), s(s(s(s(z0)))))) MAX(cons(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2))) -> c2(IF1(ge(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), GE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MAX(cons(s(s(s(x0))), cons(s(s(s(x1))), x2))) -> c2(GE(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c6(IF2(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c6(IF2(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c6(IF2(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c6(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c6(EQ(s(s(s(0))), s(s(s(0))))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(max(cons(z0, cons(0, x2))), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(max(cons(0, cons(s(z0), x2))), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(max(cons(s(z0), cons(s(z1), x2))), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), max(cons(z0, cons(0, x2))), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), max(cons(0, cons(s(z0), x2))), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), max(cons(s(z0), cons(s(z1), x2))), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(ge(z0, 0), z0, 0, x2), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(ge(0, s(z0)), 0, s(z0), x2), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), if1(ge(z0, 0), z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), if1(ge(0, s(z0)), 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z1, cons(0, x1))) -> c15(SORT(if2(eq(if1(true, z1, 0, x1), z1), if1(true, z1, 0, x1), z1, cons(0, x1))), DEL(if1(ge(z1, 0), z1, 0, x1), cons(z1, cons(0, x1))), MAX(cons(z1, cons(0, x1)))) IF3(false, cons(z0, cons(0, z2))) -> c15(SORT(del(max(cons(z0, z2)), cons(z0, cons(0, z2)))), DEL(if1(ge(z0, 0), z0, 0, z2), cons(z0, cons(0, z2))), MAX(cons(z0, cons(0, z2)))) IF3(false, cons(x0, cons(0, x1))) -> c(DEL(if1(ge(x0, 0), x0, 0, x1), cons(x0, cons(0, x1)))) IF3(false, cons(x0, cons(0, x1))) -> c(MAX(cons(x0, cons(0, x1)))) IF3(false, cons(0, cons(s(x0), x1))) -> c15(SORT(if2(eq(if1(false, 0, s(x0), x1), 0), if1(false, 0, s(x0), x1), 0, cons(s(x0), x1))), DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1))), MAX(cons(0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), z2))) -> c15(SORT(del(max(cons(s(x0), z2)), cons(0, cons(s(x0), z2)))), DEL(if1(ge(0, s(x0)), 0, s(x0), z2), cons(0, cons(s(x0), z2))), MAX(cons(0, cons(s(x0), z2)))) IF3(false, cons(0, cons(s(x0), x1))) -> c1(DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), x1))) -> c1(MAX(cons(0, cons(s(x0), x1)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c15(SORT(if2(eq(if1(ge(x0, x1), s(x0), s(x1), x2), s(x0)), if1(ge(x0, x1), s(x0), s(x1), x2), s(x0), cons(s(x1), x2))), DEL(if1(ge(s(x0), s(x1)), s(x0), s(x1), x2), cons(s(x0), cons(s(x1), x2))), MAX(cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(s(z0), cons(s(0), x2))) -> c15(SORT(del(if1(true, s(z0), s(0), x2), cons(s(z0), cons(s(0), x2)))), DEL(if1(ge(s(z0), s(0)), s(z0), s(0), x2), cons(s(z0), cons(s(0), x2))), MAX(cons(s(z0), cons(s(0), x2)))) IF3(false, cons(s(0), cons(s(s(z0)), x2))) -> c15(SORT(del(if1(false, s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2)))), DEL(if1(ge(s(0), s(s(z0))), s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2))), MAX(cons(s(0), cons(s(s(z0)), x2)))) IF3(false, cons(s(s(z0)), cons(s(s(z1)), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2)))), DEL(if1(ge(s(s(z0)), s(s(z1))), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2))), MAX(cons(s(s(z0)), cons(s(s(z1)), x2)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c5(DEL(if1(ge(s(x0), s(x1)), s(x0), s(x1), x2), cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c5(MAX(cons(s(x0), cons(s(x1), x2)))) S tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) MAX(cons(s(s(z0)), cons(s(s(0)), x2))) -> c2(IF1(true, s(s(z0)), s(s(0)), x2), GE(s(s(z0)), s(s(0)))) MAX(cons(s(s(0)), cons(s(s(s(z0))), x2))) -> c2(IF1(false, s(s(0)), s(s(s(z0))), x2), GE(s(s(0)), s(s(s(z0))))) MAX(cons(s(s(x0)), cons(s(s(x1)), x2))) -> c2(GE(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c6(IF2(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c6(IF2(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c6(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c6(EQ(s(s(0)), s(s(0)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(max(cons(z0, cons(z1, z2))), z0), if1(ge(z0, z1), z0, z1, z2), z0, cons(z1, z2)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), z0, z1, z2), z0), max(cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2))))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2))))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) MAX(cons(s(s(s(z0))), cons(s(s(s(0))), x2))) -> c2(IF1(true, s(s(s(z0))), s(s(s(0))), x2), GE(s(s(s(z0))), s(s(s(0))))) MAX(cons(s(s(s(0))), cons(s(s(s(s(z0)))), x2))) -> c2(IF1(false, s(s(s(0))), s(s(s(s(z0)))), x2), GE(s(s(s(0))), s(s(s(s(z0)))))) MAX(cons(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2))) -> c2(IF1(ge(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), GE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MAX(cons(s(s(s(x0))), cons(s(s(s(x1))), x2))) -> c2(GE(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c6(IF2(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c6(IF2(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c6(IF2(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c6(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c6(EQ(s(s(s(0))), s(s(s(0))))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(max(cons(z0, cons(0, x2))), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(max(cons(0, cons(s(z0), x2))), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(max(cons(s(z0), cons(s(z1), x2))), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), max(cons(z0, cons(0, x2))), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), max(cons(0, cons(s(z0), x2))), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), max(cons(s(z0), cons(s(z1), x2))), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(ge(z0, 0), z0, 0, x2), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(ge(0, s(z0)), 0, s(z0), x2), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), if1(ge(z0, 0), z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), if1(ge(0, s(z0)), 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z1, cons(0, x1))) -> c15(SORT(if2(eq(if1(true, z1, 0, x1), z1), if1(true, z1, 0, x1), z1, cons(0, x1))), DEL(if1(ge(z1, 0), z1, 0, x1), cons(z1, cons(0, x1))), MAX(cons(z1, cons(0, x1)))) IF3(false, cons(z0, cons(0, z2))) -> c15(SORT(del(max(cons(z0, z2)), cons(z0, cons(0, z2)))), DEL(if1(ge(z0, 0), z0, 0, z2), cons(z0, cons(0, z2))), MAX(cons(z0, cons(0, z2)))) IF3(false, cons(0, cons(s(x0), x1))) -> c15(SORT(if2(eq(if1(false, 0, s(x0), x1), 0), if1(false, 0, s(x0), x1), 0, cons(s(x0), x1))), DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1))), MAX(cons(0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), z2))) -> c15(SORT(del(max(cons(s(x0), z2)), cons(0, cons(s(x0), z2)))), DEL(if1(ge(0, s(x0)), 0, s(x0), z2), cons(0, cons(s(x0), z2))), MAX(cons(0, cons(s(x0), z2)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c15(SORT(if2(eq(if1(ge(x0, x1), s(x0), s(x1), x2), s(x0)), if1(ge(x0, x1), s(x0), s(x1), x2), s(x0), cons(s(x1), x2))), DEL(if1(ge(s(x0), s(x1)), s(x0), s(x1), x2), cons(s(x0), cons(s(x1), x2))), MAX(cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(s(z0), cons(s(0), x2))) -> c15(SORT(del(if1(true, s(z0), s(0), x2), cons(s(z0), cons(s(0), x2)))), DEL(if1(ge(s(z0), s(0)), s(z0), s(0), x2), cons(s(z0), cons(s(0), x2))), MAX(cons(s(z0), cons(s(0), x2)))) IF3(false, cons(s(0), cons(s(s(z0)), x2))) -> c15(SORT(del(if1(false, s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2)))), DEL(if1(ge(s(0), s(s(z0))), s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2))), MAX(cons(s(0), cons(s(s(z0)), x2)))) IF3(false, cons(s(s(z0)), cons(s(s(z1)), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2)))), DEL(if1(ge(s(s(z0)), s(s(z1))), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2))), MAX(cons(s(s(z0)), cons(s(s(z1)), x2)))) K tuples: IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) IF3(false, cons(x0, cons(0, x1))) -> c(DEL(if1(ge(x0, 0), x0, 0, x1), cons(x0, cons(0, x1)))) IF3(false, cons(x0, cons(0, x1))) -> c(MAX(cons(x0, cons(0, x1)))) IF3(false, cons(0, cons(s(x0), x1))) -> c1(DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), x1))) -> c1(MAX(cons(0, cons(s(x0), x1)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c5(DEL(if1(ge(s(x0), s(x1)), s(x0), s(x1), x2), cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c5(MAX(cons(s(x0), cons(s(x1), x2)))) Defined Rule Symbols: del_2, max_1, if2_4, eq_2, ge_2, if1_4 Defined Pair Symbols: IF1_4, IF2_4, EQ_2, GE_2, MAX_1, DEL_2, SORT_1, IF3_2 Compound Symbols: c3_1, c4_1, c8_1, c12_1, c20_1, c2_1, c6_1, c13_1, c2_2, c6_2, c15_1, c15_3, c15_2, c_1, c1_1, c5_1 ---------------------------------------- (171) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil))) by IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil))) ---------------------------------------- (172) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil max(cons(z0, nil)) -> z0 max(cons(z0, cons(z1, z2))) -> if1(ge(z0, z1), z0, z1, z2) if2(true, z0, z1, z2) -> z2 if2(false, z0, z1, z2) -> cons(z1, del(z0, z2)) eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) if1(true, z0, z1, z2) -> max(cons(z0, z2)) if1(false, z0, z1, z2) -> max(cons(z1, z2)) Tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) MAX(cons(s(s(z0)), cons(s(s(0)), x2))) -> c2(IF1(true, s(s(z0)), s(s(0)), x2), GE(s(s(z0)), s(s(0)))) MAX(cons(s(s(0)), cons(s(s(s(z0))), x2))) -> c2(IF1(false, s(s(0)), s(s(s(z0))), x2), GE(s(s(0)), s(s(s(z0))))) MAX(cons(s(s(x0)), cons(s(s(x1)), x2))) -> c2(GE(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c6(IF2(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c6(IF2(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c6(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c6(EQ(s(s(0)), s(s(0)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(if1(ge(z1, x1), z1, x1, x2), cons(z1, cons(x1, x2))), MAX(cons(z1, cons(x1, x2)))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(max(cons(z0, cons(z1, z2))), z0), if1(ge(z0, z1), z0, z1, z2), z0, cons(z1, z2)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), z0, z1, z2), z0), max(cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2))))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2))))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) MAX(cons(s(s(s(z0))), cons(s(s(s(0))), x2))) -> c2(IF1(true, s(s(s(z0))), s(s(s(0))), x2), GE(s(s(s(z0))), s(s(s(0))))) MAX(cons(s(s(s(0))), cons(s(s(s(s(z0)))), x2))) -> c2(IF1(false, s(s(s(0))), s(s(s(s(z0)))), x2), GE(s(s(s(0))), s(s(s(s(z0)))))) MAX(cons(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2))) -> c2(IF1(ge(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), GE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MAX(cons(s(s(s(x0))), cons(s(s(s(x1))), x2))) -> c2(GE(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c6(IF2(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c6(IF2(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c6(IF2(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c6(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c6(EQ(s(s(s(0))), s(s(s(0))))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(max(cons(z0, cons(0, x2))), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(max(cons(0, cons(s(z0), x2))), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(max(cons(s(z0), cons(s(z1), x2))), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), max(cons(z0, cons(0, x2))), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), max(cons(0, cons(s(z0), x2))), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), max(cons(s(z0), cons(s(z1), x2))), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(ge(z0, 0), z0, 0, x2), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(ge(0, s(z0)), 0, s(z0), x2), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), if1(ge(z0, 0), z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), if1(ge(0, s(z0)), 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z1, cons(0, x1))) -> c15(SORT(if2(eq(if1(true, z1, 0, x1), z1), if1(true, z1, 0, x1), z1, cons(0, x1))), DEL(if1(ge(z1, 0), z1, 0, x1), cons(z1, cons(0, x1))), MAX(cons(z1, cons(0, x1)))) IF3(false, cons(z0, cons(0, z2))) -> c15(SORT(del(max(cons(z0, z2)), cons(z0, cons(0, z2)))), DEL(if1(ge(z0, 0), z0, 0, z2), cons(z0, cons(0, z2))), MAX(cons(z0, cons(0, z2)))) IF3(false, cons(x0, cons(0, x1))) -> c(DEL(if1(ge(x0, 0), x0, 0, x1), cons(x0, cons(0, x1)))) IF3(false, cons(x0, cons(0, x1))) -> c(MAX(cons(x0, cons(0, x1)))) IF3(false, cons(0, cons(s(x0), x1))) -> c15(SORT(if2(eq(if1(false, 0, s(x0), x1), 0), if1(false, 0, s(x0), x1), 0, cons(s(x0), x1))), DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1))), MAX(cons(0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), z2))) -> c15(SORT(del(max(cons(s(x0), z2)), cons(0, cons(s(x0), z2)))), DEL(if1(ge(0, s(x0)), 0, s(x0), z2), cons(0, cons(s(x0), z2))), MAX(cons(0, cons(s(x0), z2)))) IF3(false, cons(0, cons(s(x0), x1))) -> c1(DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), x1))) -> c1(MAX(cons(0, cons(s(x0), x1)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c15(SORT(if2(eq(if1(ge(x0, x1), s(x0), s(x1), x2), s(x0)), if1(ge(x0, x1), s(x0), s(x1), x2), s(x0), cons(s(x1), x2))), DEL(if1(ge(s(x0), s(x1)), s(x0), s(x1), x2), cons(s(x0), cons(s(x1), x2))), MAX(cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(s(z0), cons(s(0), x2))) -> c15(SORT(del(if1(true, s(z0), s(0), x2), cons(s(z0), cons(s(0), x2)))), DEL(if1(ge(s(z0), s(0)), s(z0), s(0), x2), cons(s(z0), cons(s(0), x2))), MAX(cons(s(z0), cons(s(0), x2)))) IF3(false, cons(s(0), cons(s(s(z0)), x2))) -> c15(SORT(del(if1(false, s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2)))), DEL(if1(ge(s(0), s(s(z0))), s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2))), MAX(cons(s(0), cons(s(s(z0)), x2)))) IF3(false, cons(s(s(z0)), cons(s(s(z1)), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2)))), DEL(if1(ge(s(s(z0)), s(s(z1))), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2))), MAX(cons(s(s(z0)), cons(s(s(z1)), x2)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c5(DEL(if1(ge(s(x0), s(x1)), s(x0), s(x1), x2), cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c5(MAX(cons(s(x0), cons(s(x1), x2)))) S tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) MAX(cons(s(s(z0)), cons(s(s(0)), x2))) -> c2(IF1(true, s(s(z0)), s(s(0)), x2), GE(s(s(z0)), s(s(0)))) MAX(cons(s(s(0)), cons(s(s(s(z0))), x2))) -> c2(IF1(false, s(s(0)), s(s(s(z0))), x2), GE(s(s(0)), s(s(s(z0))))) MAX(cons(s(s(x0)), cons(s(s(x1)), x2))) -> c2(GE(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c6(IF2(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c6(IF2(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c6(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c6(EQ(s(s(0)), s(s(0)))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(max(cons(z0, cons(z1, z2))), z0), if1(ge(z0, z1), z0, z1, z2), z0, cons(z1, z2)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), z0, z1, z2), z0), max(cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2))))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2))))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) MAX(cons(s(s(s(z0))), cons(s(s(s(0))), x2))) -> c2(IF1(true, s(s(s(z0))), s(s(s(0))), x2), GE(s(s(s(z0))), s(s(s(0))))) MAX(cons(s(s(s(0))), cons(s(s(s(s(z0)))), x2))) -> c2(IF1(false, s(s(s(0))), s(s(s(s(z0)))), x2), GE(s(s(s(0))), s(s(s(s(z0)))))) MAX(cons(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2))) -> c2(IF1(ge(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), GE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MAX(cons(s(s(s(x0))), cons(s(s(s(x1))), x2))) -> c2(GE(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c6(IF2(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c6(IF2(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c6(IF2(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c6(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c6(EQ(s(s(s(0))), s(s(s(0))))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(max(cons(z0, cons(0, x2))), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(max(cons(0, cons(s(z0), x2))), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(max(cons(s(z0), cons(s(z1), x2))), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), max(cons(z0, cons(0, x2))), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), max(cons(0, cons(s(z0), x2))), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), max(cons(s(z0), cons(s(z1), x2))), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(ge(z0, 0), z0, 0, x2), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(ge(0, s(z0)), 0, s(z0), x2), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), if1(ge(z0, 0), z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), if1(ge(0, s(z0)), 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z1, cons(0, x1))) -> c15(SORT(if2(eq(if1(true, z1, 0, x1), z1), if1(true, z1, 0, x1), z1, cons(0, x1))), DEL(if1(ge(z1, 0), z1, 0, x1), cons(z1, cons(0, x1))), MAX(cons(z1, cons(0, x1)))) IF3(false, cons(z0, cons(0, z2))) -> c15(SORT(del(max(cons(z0, z2)), cons(z0, cons(0, z2)))), DEL(if1(ge(z0, 0), z0, 0, z2), cons(z0, cons(0, z2))), MAX(cons(z0, cons(0, z2)))) IF3(false, cons(0, cons(s(x0), x1))) -> c15(SORT(if2(eq(if1(false, 0, s(x0), x1), 0), if1(false, 0, s(x0), x1), 0, cons(s(x0), x1))), DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1))), MAX(cons(0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), z2))) -> c15(SORT(del(max(cons(s(x0), z2)), cons(0, cons(s(x0), z2)))), DEL(if1(ge(0, s(x0)), 0, s(x0), z2), cons(0, cons(s(x0), z2))), MAX(cons(0, cons(s(x0), z2)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c15(SORT(if2(eq(if1(ge(x0, x1), s(x0), s(x1), x2), s(x0)), if1(ge(x0, x1), s(x0), s(x1), x2), s(x0), cons(s(x1), x2))), DEL(if1(ge(s(x0), s(x1)), s(x0), s(x1), x2), cons(s(x0), cons(s(x1), x2))), MAX(cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(s(z0), cons(s(0), x2))) -> c15(SORT(del(if1(true, s(z0), s(0), x2), cons(s(z0), cons(s(0), x2)))), DEL(if1(ge(s(z0), s(0)), s(z0), s(0), x2), cons(s(z0), cons(s(0), x2))), MAX(cons(s(z0), cons(s(0), x2)))) IF3(false, cons(s(0), cons(s(s(z0)), x2))) -> c15(SORT(del(if1(false, s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2)))), DEL(if1(ge(s(0), s(s(z0))), s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2))), MAX(cons(s(0), cons(s(s(z0)), x2)))) IF3(false, cons(s(s(z0)), cons(s(s(z1)), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2)))), DEL(if1(ge(s(s(z0)), s(s(z1))), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2))), MAX(cons(s(s(z0)), cons(s(s(z1)), x2)))) K tuples: IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) IF3(false, cons(x0, cons(0, x1))) -> c(DEL(if1(ge(x0, 0), x0, 0, x1), cons(x0, cons(0, x1)))) IF3(false, cons(x0, cons(0, x1))) -> c(MAX(cons(x0, cons(0, x1)))) IF3(false, cons(0, cons(s(x0), x1))) -> c1(DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), x1))) -> c1(MAX(cons(0, cons(s(x0), x1)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c5(DEL(if1(ge(s(x0), s(x1)), s(x0), s(x1), x2), cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c5(MAX(cons(s(x0), cons(s(x1), x2)))) Defined Rule Symbols: del_2, max_1, if2_4, eq_2, ge_2, if1_4 Defined Pair Symbols: IF1_4, IF2_4, EQ_2, GE_2, MAX_1, DEL_2, SORT_1, IF3_2 Compound Symbols: c3_1, c4_1, c8_1, c12_1, c20_1, c2_1, c6_1, c13_1, c2_2, c6_2, c15_1, c15_3, c15_2, c_1, c1_1, c5_1 ---------------------------------------- (173) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(max(cons(z0, cons(z1, z2))), z0), if1(ge(z0, z1), z0, z1, z2), z0, cons(z1, z2)))) by IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(max(cons(z0, cons(0, x2))), z0), if1(true, z0, 0, x2), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(max(cons(0, cons(s(z0), x2))), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(max(cons(s(z0), cons(s(z1), x2))), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), z0, z1, z2), z0), if1(ge(z0, z1), z0, z1, z2), z0, cons(z1, z2)))) ---------------------------------------- (174) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil max(cons(z0, nil)) -> z0 max(cons(z0, cons(z1, z2))) -> if1(ge(z0, z1), z0, z1, z2) if2(true, z0, z1, z2) -> z2 if2(false, z0, z1, z2) -> cons(z1, del(z0, z2)) eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) if1(true, z0, z1, z2) -> max(cons(z0, z2)) if1(false, z0, z1, z2) -> max(cons(z1, z2)) Tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) MAX(cons(s(s(z0)), cons(s(s(0)), x2))) -> c2(IF1(true, s(s(z0)), s(s(0)), x2), GE(s(s(z0)), s(s(0)))) MAX(cons(s(s(0)), cons(s(s(s(z0))), x2))) -> c2(IF1(false, s(s(0)), s(s(s(z0))), x2), GE(s(s(0)), s(s(s(z0))))) MAX(cons(s(s(x0)), cons(s(s(x1)), x2))) -> c2(GE(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c6(IF2(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c6(IF2(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c6(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c6(EQ(s(s(0)), s(s(0)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(if1(ge(z1, x1), z1, x1, x2), cons(z1, cons(x1, x2))), MAX(cons(z1, cons(x1, x2)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), z0, z1, z2), z0), max(cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2))))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2))))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) MAX(cons(s(s(s(z0))), cons(s(s(s(0))), x2))) -> c2(IF1(true, s(s(s(z0))), s(s(s(0))), x2), GE(s(s(s(z0))), s(s(s(0))))) MAX(cons(s(s(s(0))), cons(s(s(s(s(z0)))), x2))) -> c2(IF1(false, s(s(s(0))), s(s(s(s(z0)))), x2), GE(s(s(s(0))), s(s(s(s(z0)))))) MAX(cons(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2))) -> c2(IF1(ge(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), GE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MAX(cons(s(s(s(x0))), cons(s(s(s(x1))), x2))) -> c2(GE(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c6(IF2(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c6(IF2(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c6(IF2(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c6(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c6(EQ(s(s(s(0))), s(s(s(0))))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(max(cons(z0, cons(0, x2))), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(max(cons(0, cons(s(z0), x2))), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(max(cons(s(z0), cons(s(z1), x2))), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), max(cons(z0, cons(0, x2))), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), max(cons(0, cons(s(z0), x2))), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), max(cons(s(z0), cons(s(z1), x2))), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(ge(z0, 0), z0, 0, x2), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(ge(0, s(z0)), 0, s(z0), x2), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), if1(ge(z0, 0), z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), if1(ge(0, s(z0)), 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z1, cons(0, x1))) -> c15(SORT(if2(eq(if1(true, z1, 0, x1), z1), if1(true, z1, 0, x1), z1, cons(0, x1))), DEL(if1(ge(z1, 0), z1, 0, x1), cons(z1, cons(0, x1))), MAX(cons(z1, cons(0, x1)))) IF3(false, cons(z0, cons(0, z2))) -> c15(SORT(del(max(cons(z0, z2)), cons(z0, cons(0, z2)))), DEL(if1(ge(z0, 0), z0, 0, z2), cons(z0, cons(0, z2))), MAX(cons(z0, cons(0, z2)))) IF3(false, cons(x0, cons(0, x1))) -> c(DEL(if1(ge(x0, 0), x0, 0, x1), cons(x0, cons(0, x1)))) IF3(false, cons(x0, cons(0, x1))) -> c(MAX(cons(x0, cons(0, x1)))) IF3(false, cons(0, cons(s(x0), x1))) -> c15(SORT(if2(eq(if1(false, 0, s(x0), x1), 0), if1(false, 0, s(x0), x1), 0, cons(s(x0), x1))), DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1))), MAX(cons(0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), z2))) -> c15(SORT(del(max(cons(s(x0), z2)), cons(0, cons(s(x0), z2)))), DEL(if1(ge(0, s(x0)), 0, s(x0), z2), cons(0, cons(s(x0), z2))), MAX(cons(0, cons(s(x0), z2)))) IF3(false, cons(0, cons(s(x0), x1))) -> c1(DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), x1))) -> c1(MAX(cons(0, cons(s(x0), x1)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c15(SORT(if2(eq(if1(ge(x0, x1), s(x0), s(x1), x2), s(x0)), if1(ge(x0, x1), s(x0), s(x1), x2), s(x0), cons(s(x1), x2))), DEL(if1(ge(s(x0), s(x1)), s(x0), s(x1), x2), cons(s(x0), cons(s(x1), x2))), MAX(cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(s(z0), cons(s(0), x2))) -> c15(SORT(del(if1(true, s(z0), s(0), x2), cons(s(z0), cons(s(0), x2)))), DEL(if1(ge(s(z0), s(0)), s(z0), s(0), x2), cons(s(z0), cons(s(0), x2))), MAX(cons(s(z0), cons(s(0), x2)))) IF3(false, cons(s(0), cons(s(s(z0)), x2))) -> c15(SORT(del(if1(false, s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2)))), DEL(if1(ge(s(0), s(s(z0))), s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2))), MAX(cons(s(0), cons(s(s(z0)), x2)))) IF3(false, cons(s(s(z0)), cons(s(s(z1)), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2)))), DEL(if1(ge(s(s(z0)), s(s(z1))), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2))), MAX(cons(s(s(z0)), cons(s(s(z1)), x2)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c5(DEL(if1(ge(s(x0), s(x1)), s(x0), s(x1), x2), cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c5(MAX(cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(max(cons(z0, cons(0, x2))), z0), if1(true, z0, 0, x2), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(max(cons(0, cons(s(z0), x2))), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(max(cons(s(z0), cons(s(z1), x2))), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2)))) S tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) MAX(cons(s(s(z0)), cons(s(s(0)), x2))) -> c2(IF1(true, s(s(z0)), s(s(0)), x2), GE(s(s(z0)), s(s(0)))) MAX(cons(s(s(0)), cons(s(s(s(z0))), x2))) -> c2(IF1(false, s(s(0)), s(s(s(z0))), x2), GE(s(s(0)), s(s(s(z0))))) MAX(cons(s(s(x0)), cons(s(s(x1)), x2))) -> c2(GE(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c6(IF2(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c6(IF2(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c6(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c6(EQ(s(s(0)), s(s(0)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), z0, z1, z2), z0), max(cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2))))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2))))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) MAX(cons(s(s(s(z0))), cons(s(s(s(0))), x2))) -> c2(IF1(true, s(s(s(z0))), s(s(s(0))), x2), GE(s(s(s(z0))), s(s(s(0))))) MAX(cons(s(s(s(0))), cons(s(s(s(s(z0)))), x2))) -> c2(IF1(false, s(s(s(0))), s(s(s(s(z0)))), x2), GE(s(s(s(0))), s(s(s(s(z0)))))) MAX(cons(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2))) -> c2(IF1(ge(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), GE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MAX(cons(s(s(s(x0))), cons(s(s(s(x1))), x2))) -> c2(GE(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c6(IF2(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c6(IF2(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c6(IF2(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c6(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c6(EQ(s(s(s(0))), s(s(s(0))))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(max(cons(z0, cons(0, x2))), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(max(cons(0, cons(s(z0), x2))), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(max(cons(s(z0), cons(s(z1), x2))), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), max(cons(z0, cons(0, x2))), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), max(cons(0, cons(s(z0), x2))), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), max(cons(s(z0), cons(s(z1), x2))), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(ge(z0, 0), z0, 0, x2), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(ge(0, s(z0)), 0, s(z0), x2), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), if1(ge(z0, 0), z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), if1(ge(0, s(z0)), 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z1, cons(0, x1))) -> c15(SORT(if2(eq(if1(true, z1, 0, x1), z1), if1(true, z1, 0, x1), z1, cons(0, x1))), DEL(if1(ge(z1, 0), z1, 0, x1), cons(z1, cons(0, x1))), MAX(cons(z1, cons(0, x1)))) IF3(false, cons(z0, cons(0, z2))) -> c15(SORT(del(max(cons(z0, z2)), cons(z0, cons(0, z2)))), DEL(if1(ge(z0, 0), z0, 0, z2), cons(z0, cons(0, z2))), MAX(cons(z0, cons(0, z2)))) IF3(false, cons(0, cons(s(x0), x1))) -> c15(SORT(if2(eq(if1(false, 0, s(x0), x1), 0), if1(false, 0, s(x0), x1), 0, cons(s(x0), x1))), DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1))), MAX(cons(0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), z2))) -> c15(SORT(del(max(cons(s(x0), z2)), cons(0, cons(s(x0), z2)))), DEL(if1(ge(0, s(x0)), 0, s(x0), z2), cons(0, cons(s(x0), z2))), MAX(cons(0, cons(s(x0), z2)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c15(SORT(if2(eq(if1(ge(x0, x1), s(x0), s(x1), x2), s(x0)), if1(ge(x0, x1), s(x0), s(x1), x2), s(x0), cons(s(x1), x2))), DEL(if1(ge(s(x0), s(x1)), s(x0), s(x1), x2), cons(s(x0), cons(s(x1), x2))), MAX(cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(s(z0), cons(s(0), x2))) -> c15(SORT(del(if1(true, s(z0), s(0), x2), cons(s(z0), cons(s(0), x2)))), DEL(if1(ge(s(z0), s(0)), s(z0), s(0), x2), cons(s(z0), cons(s(0), x2))), MAX(cons(s(z0), cons(s(0), x2)))) IF3(false, cons(s(0), cons(s(s(z0)), x2))) -> c15(SORT(del(if1(false, s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2)))), DEL(if1(ge(s(0), s(s(z0))), s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2))), MAX(cons(s(0), cons(s(s(z0)), x2)))) IF3(false, cons(s(s(z0)), cons(s(s(z1)), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2)))), DEL(if1(ge(s(s(z0)), s(s(z1))), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2))), MAX(cons(s(s(z0)), cons(s(s(z1)), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(max(cons(z0, cons(0, x2))), z0), if1(true, z0, 0, x2), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(max(cons(0, cons(s(z0), x2))), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(max(cons(s(z0), cons(s(z1), x2))), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2)))) K tuples: IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) IF3(false, cons(x0, cons(0, x1))) -> c(DEL(if1(ge(x0, 0), x0, 0, x1), cons(x0, cons(0, x1)))) IF3(false, cons(x0, cons(0, x1))) -> c(MAX(cons(x0, cons(0, x1)))) IF3(false, cons(0, cons(s(x0), x1))) -> c1(DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), x1))) -> c1(MAX(cons(0, cons(s(x0), x1)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c5(DEL(if1(ge(s(x0), s(x1)), s(x0), s(x1), x2), cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c5(MAX(cons(s(x0), cons(s(x1), x2)))) Defined Rule Symbols: del_2, max_1, if2_4, eq_2, ge_2, if1_4 Defined Pair Symbols: IF1_4, IF2_4, EQ_2, GE_2, MAX_1, DEL_2, SORT_1, IF3_2 Compound Symbols: c3_1, c4_1, c8_1, c12_1, c20_1, c2_1, c6_1, c13_1, c2_2, c6_2, c15_1, c15_3, c15_2, c_1, c1_1, c5_1 ---------------------------------------- (175) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil))) by IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil))) IF3(false, cons(0, nil)) -> c15(SORT(if2(true, max(cons(0, nil)), 0, nil))) IF3(false, cons(s(z0), nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(s(z0), nil)), s(z0), nil))) ---------------------------------------- (176) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil max(cons(z0, nil)) -> z0 max(cons(z0, cons(z1, z2))) -> if1(ge(z0, z1), z0, z1, z2) if2(true, z0, z1, z2) -> z2 if2(false, z0, z1, z2) -> cons(z1, del(z0, z2)) eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) if1(true, z0, z1, z2) -> max(cons(z0, z2)) if1(false, z0, z1, z2) -> max(cons(z1, z2)) Tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) MAX(cons(s(s(z0)), cons(s(s(0)), x2))) -> c2(IF1(true, s(s(z0)), s(s(0)), x2), GE(s(s(z0)), s(s(0)))) MAX(cons(s(s(0)), cons(s(s(s(z0))), x2))) -> c2(IF1(false, s(s(0)), s(s(s(z0))), x2), GE(s(s(0)), s(s(s(z0))))) MAX(cons(s(s(x0)), cons(s(s(x1)), x2))) -> c2(GE(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c6(IF2(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c6(IF2(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c6(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c6(EQ(s(s(0)), s(s(0)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(if1(ge(z1, x1), z1, x1, x2), cons(z1, cons(x1, x2))), MAX(cons(z1, cons(x1, x2)))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), z0, z1, z2), z0), max(cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2))))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2))))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) MAX(cons(s(s(s(z0))), cons(s(s(s(0))), x2))) -> c2(IF1(true, s(s(s(z0))), s(s(s(0))), x2), GE(s(s(s(z0))), s(s(s(0))))) MAX(cons(s(s(s(0))), cons(s(s(s(s(z0)))), x2))) -> c2(IF1(false, s(s(s(0))), s(s(s(s(z0)))), x2), GE(s(s(s(0))), s(s(s(s(z0)))))) MAX(cons(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2))) -> c2(IF1(ge(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), GE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MAX(cons(s(s(s(x0))), cons(s(s(s(x1))), x2))) -> c2(GE(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c6(IF2(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c6(IF2(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c6(IF2(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c6(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c6(EQ(s(s(s(0))), s(s(s(0))))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(max(cons(z0, cons(0, x2))), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(max(cons(0, cons(s(z0), x2))), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(max(cons(s(z0), cons(s(z1), x2))), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), max(cons(z0, cons(0, x2))), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), max(cons(0, cons(s(z0), x2))), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), max(cons(s(z0), cons(s(z1), x2))), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(ge(z0, 0), z0, 0, x2), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(ge(0, s(z0)), 0, s(z0), x2), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), if1(ge(z0, 0), z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), if1(ge(0, s(z0)), 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z1, cons(0, x1))) -> c15(SORT(if2(eq(if1(true, z1, 0, x1), z1), if1(true, z1, 0, x1), z1, cons(0, x1))), DEL(if1(ge(z1, 0), z1, 0, x1), cons(z1, cons(0, x1))), MAX(cons(z1, cons(0, x1)))) IF3(false, cons(z0, cons(0, z2))) -> c15(SORT(del(max(cons(z0, z2)), cons(z0, cons(0, z2)))), DEL(if1(ge(z0, 0), z0, 0, z2), cons(z0, cons(0, z2))), MAX(cons(z0, cons(0, z2)))) IF3(false, cons(x0, cons(0, x1))) -> c(DEL(if1(ge(x0, 0), x0, 0, x1), cons(x0, cons(0, x1)))) IF3(false, cons(x0, cons(0, x1))) -> c(MAX(cons(x0, cons(0, x1)))) IF3(false, cons(0, cons(s(x0), x1))) -> c15(SORT(if2(eq(if1(false, 0, s(x0), x1), 0), if1(false, 0, s(x0), x1), 0, cons(s(x0), x1))), DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1))), MAX(cons(0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), z2))) -> c15(SORT(del(max(cons(s(x0), z2)), cons(0, cons(s(x0), z2)))), DEL(if1(ge(0, s(x0)), 0, s(x0), z2), cons(0, cons(s(x0), z2))), MAX(cons(0, cons(s(x0), z2)))) IF3(false, cons(0, cons(s(x0), x1))) -> c1(DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), x1))) -> c1(MAX(cons(0, cons(s(x0), x1)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c15(SORT(if2(eq(if1(ge(x0, x1), s(x0), s(x1), x2), s(x0)), if1(ge(x0, x1), s(x0), s(x1), x2), s(x0), cons(s(x1), x2))), DEL(if1(ge(s(x0), s(x1)), s(x0), s(x1), x2), cons(s(x0), cons(s(x1), x2))), MAX(cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(s(z0), cons(s(0), x2))) -> c15(SORT(del(if1(true, s(z0), s(0), x2), cons(s(z0), cons(s(0), x2)))), DEL(if1(ge(s(z0), s(0)), s(z0), s(0), x2), cons(s(z0), cons(s(0), x2))), MAX(cons(s(z0), cons(s(0), x2)))) IF3(false, cons(s(0), cons(s(s(z0)), x2))) -> c15(SORT(del(if1(false, s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2)))), DEL(if1(ge(s(0), s(s(z0))), s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2))), MAX(cons(s(0), cons(s(s(z0)), x2)))) IF3(false, cons(s(s(z0)), cons(s(s(z1)), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2)))), DEL(if1(ge(s(s(z0)), s(s(z1))), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2))), MAX(cons(s(s(z0)), cons(s(s(z1)), x2)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c5(DEL(if1(ge(s(x0), s(x1)), s(x0), s(x1), x2), cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c5(MAX(cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(max(cons(z0, cons(0, x2))), z0), if1(true, z0, 0, x2), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(max(cons(0, cons(s(z0), x2))), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(max(cons(s(z0), cons(s(z1), x2))), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2)))) IF3(false, cons(0, nil)) -> c15(SORT(if2(true, max(cons(0, nil)), 0, nil))) IF3(false, cons(s(z0), nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(s(z0), nil)), s(z0), nil))) S tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) MAX(cons(s(s(z0)), cons(s(s(0)), x2))) -> c2(IF1(true, s(s(z0)), s(s(0)), x2), GE(s(s(z0)), s(s(0)))) MAX(cons(s(s(0)), cons(s(s(s(z0))), x2))) -> c2(IF1(false, s(s(0)), s(s(s(z0))), x2), GE(s(s(0)), s(s(s(z0))))) MAX(cons(s(s(x0)), cons(s(s(x1)), x2))) -> c2(GE(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c6(IF2(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c6(IF2(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c6(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c6(EQ(s(s(0)), s(s(0)))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), z0, z1, z2), z0), max(cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2))))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2))))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) MAX(cons(s(s(s(z0))), cons(s(s(s(0))), x2))) -> c2(IF1(true, s(s(s(z0))), s(s(s(0))), x2), GE(s(s(s(z0))), s(s(s(0))))) MAX(cons(s(s(s(0))), cons(s(s(s(s(z0)))), x2))) -> c2(IF1(false, s(s(s(0))), s(s(s(s(z0)))), x2), GE(s(s(s(0))), s(s(s(s(z0)))))) MAX(cons(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2))) -> c2(IF1(ge(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), GE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MAX(cons(s(s(s(x0))), cons(s(s(s(x1))), x2))) -> c2(GE(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c6(IF2(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c6(IF2(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c6(IF2(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c6(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c6(EQ(s(s(s(0))), s(s(s(0))))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(max(cons(z0, cons(0, x2))), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(max(cons(0, cons(s(z0), x2))), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(max(cons(s(z0), cons(s(z1), x2))), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), max(cons(z0, cons(0, x2))), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), max(cons(0, cons(s(z0), x2))), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), max(cons(s(z0), cons(s(z1), x2))), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(ge(z0, 0), z0, 0, x2), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(ge(0, s(z0)), 0, s(z0), x2), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), if1(ge(z0, 0), z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), if1(ge(0, s(z0)), 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z1, cons(0, x1))) -> c15(SORT(if2(eq(if1(true, z1, 0, x1), z1), if1(true, z1, 0, x1), z1, cons(0, x1))), DEL(if1(ge(z1, 0), z1, 0, x1), cons(z1, cons(0, x1))), MAX(cons(z1, cons(0, x1)))) IF3(false, cons(z0, cons(0, z2))) -> c15(SORT(del(max(cons(z0, z2)), cons(z0, cons(0, z2)))), DEL(if1(ge(z0, 0), z0, 0, z2), cons(z0, cons(0, z2))), MAX(cons(z0, cons(0, z2)))) IF3(false, cons(0, cons(s(x0), x1))) -> c15(SORT(if2(eq(if1(false, 0, s(x0), x1), 0), if1(false, 0, s(x0), x1), 0, cons(s(x0), x1))), DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1))), MAX(cons(0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), z2))) -> c15(SORT(del(max(cons(s(x0), z2)), cons(0, cons(s(x0), z2)))), DEL(if1(ge(0, s(x0)), 0, s(x0), z2), cons(0, cons(s(x0), z2))), MAX(cons(0, cons(s(x0), z2)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c15(SORT(if2(eq(if1(ge(x0, x1), s(x0), s(x1), x2), s(x0)), if1(ge(x0, x1), s(x0), s(x1), x2), s(x0), cons(s(x1), x2))), DEL(if1(ge(s(x0), s(x1)), s(x0), s(x1), x2), cons(s(x0), cons(s(x1), x2))), MAX(cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(s(z0), cons(s(0), x2))) -> c15(SORT(del(if1(true, s(z0), s(0), x2), cons(s(z0), cons(s(0), x2)))), DEL(if1(ge(s(z0), s(0)), s(z0), s(0), x2), cons(s(z0), cons(s(0), x2))), MAX(cons(s(z0), cons(s(0), x2)))) IF3(false, cons(s(0), cons(s(s(z0)), x2))) -> c15(SORT(del(if1(false, s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2)))), DEL(if1(ge(s(0), s(s(z0))), s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2))), MAX(cons(s(0), cons(s(s(z0)), x2)))) IF3(false, cons(s(s(z0)), cons(s(s(z1)), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2)))), DEL(if1(ge(s(s(z0)), s(s(z1))), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2))), MAX(cons(s(s(z0)), cons(s(s(z1)), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(max(cons(z0, cons(0, x2))), z0), if1(true, z0, 0, x2), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(max(cons(0, cons(s(z0), x2))), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(max(cons(s(z0), cons(s(z1), x2))), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2)))) IF3(false, cons(0, nil)) -> c15(SORT(if2(true, max(cons(0, nil)), 0, nil))) IF3(false, cons(s(z0), nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(s(z0), nil)), s(z0), nil))) K tuples: IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) IF3(false, cons(x0, cons(0, x1))) -> c(DEL(if1(ge(x0, 0), x0, 0, x1), cons(x0, cons(0, x1)))) IF3(false, cons(x0, cons(0, x1))) -> c(MAX(cons(x0, cons(0, x1)))) IF3(false, cons(0, cons(s(x0), x1))) -> c1(DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), x1))) -> c1(MAX(cons(0, cons(s(x0), x1)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c5(DEL(if1(ge(s(x0), s(x1)), s(x0), s(x1), x2), cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c5(MAX(cons(s(x0), cons(s(x1), x2)))) Defined Rule Symbols: del_2, max_1, if2_4, eq_2, ge_2, if1_4 Defined Pair Symbols: IF1_4, IF2_4, EQ_2, GE_2, MAX_1, DEL_2, SORT_1, IF3_2 Compound Symbols: c3_1, c4_1, c8_1, c12_1, c20_1, c2_1, c6_1, c13_1, c2_2, c6_2, c15_1, c15_3, c15_2, c_1, c1_1, c5_1 ---------------------------------------- (177) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. IF3(false, cons(0, nil)) -> c15(SORT(if2(true, max(cons(0, nil)), 0, nil))) We considered the (Usable) Rules: if2(false, z0, z1, z2) -> cons(z1, del(z0, z2)) del(z0, nil) -> nil eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) eq(0, 0) -> true if2(true, z0, z1, z2) -> z2 del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) And the Tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) MAX(cons(s(s(z0)), cons(s(s(0)), x2))) -> c2(IF1(true, s(s(z0)), s(s(0)), x2), GE(s(s(z0)), s(s(0)))) MAX(cons(s(s(0)), cons(s(s(s(z0))), x2))) -> c2(IF1(false, s(s(0)), s(s(s(z0))), x2), GE(s(s(0)), s(s(s(z0))))) MAX(cons(s(s(x0)), cons(s(s(x1)), x2))) -> c2(GE(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c6(IF2(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c6(IF2(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c6(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c6(EQ(s(s(0)), s(s(0)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(if1(ge(z1, x1), z1, x1, x2), cons(z1, cons(x1, x2))), MAX(cons(z1, cons(x1, x2)))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), z0, z1, z2), z0), max(cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2))))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2))))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) MAX(cons(s(s(s(z0))), cons(s(s(s(0))), x2))) -> c2(IF1(true, s(s(s(z0))), s(s(s(0))), x2), GE(s(s(s(z0))), s(s(s(0))))) MAX(cons(s(s(s(0))), cons(s(s(s(s(z0)))), x2))) -> c2(IF1(false, s(s(s(0))), s(s(s(s(z0)))), x2), GE(s(s(s(0))), s(s(s(s(z0)))))) MAX(cons(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2))) -> c2(IF1(ge(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), GE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MAX(cons(s(s(s(x0))), cons(s(s(s(x1))), x2))) -> c2(GE(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c6(IF2(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c6(IF2(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c6(IF2(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c6(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c6(EQ(s(s(s(0))), s(s(s(0))))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(max(cons(z0, cons(0, x2))), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(max(cons(0, cons(s(z0), x2))), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(max(cons(s(z0), cons(s(z1), x2))), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), max(cons(z0, cons(0, x2))), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), max(cons(0, cons(s(z0), x2))), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), max(cons(s(z0), cons(s(z1), x2))), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(ge(z0, 0), z0, 0, x2), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(ge(0, s(z0)), 0, s(z0), x2), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), if1(ge(z0, 0), z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), if1(ge(0, s(z0)), 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z1, cons(0, x1))) -> c15(SORT(if2(eq(if1(true, z1, 0, x1), z1), if1(true, z1, 0, x1), z1, cons(0, x1))), DEL(if1(ge(z1, 0), z1, 0, x1), cons(z1, cons(0, x1))), MAX(cons(z1, cons(0, x1)))) IF3(false, cons(z0, cons(0, z2))) -> c15(SORT(del(max(cons(z0, z2)), cons(z0, cons(0, z2)))), DEL(if1(ge(z0, 0), z0, 0, z2), cons(z0, cons(0, z2))), MAX(cons(z0, cons(0, z2)))) IF3(false, cons(x0, cons(0, x1))) -> c(DEL(if1(ge(x0, 0), x0, 0, x1), cons(x0, cons(0, x1)))) IF3(false, cons(x0, cons(0, x1))) -> c(MAX(cons(x0, cons(0, x1)))) IF3(false, cons(0, cons(s(x0), x1))) -> c15(SORT(if2(eq(if1(false, 0, s(x0), x1), 0), if1(false, 0, s(x0), x1), 0, cons(s(x0), x1))), DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1))), MAX(cons(0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), z2))) -> c15(SORT(del(max(cons(s(x0), z2)), cons(0, cons(s(x0), z2)))), DEL(if1(ge(0, s(x0)), 0, s(x0), z2), cons(0, cons(s(x0), z2))), MAX(cons(0, cons(s(x0), z2)))) IF3(false, cons(0, cons(s(x0), x1))) -> c1(DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), x1))) -> c1(MAX(cons(0, cons(s(x0), x1)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c15(SORT(if2(eq(if1(ge(x0, x1), s(x0), s(x1), x2), s(x0)), if1(ge(x0, x1), s(x0), s(x1), x2), s(x0), cons(s(x1), x2))), DEL(if1(ge(s(x0), s(x1)), s(x0), s(x1), x2), cons(s(x0), cons(s(x1), x2))), MAX(cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(s(z0), cons(s(0), x2))) -> c15(SORT(del(if1(true, s(z0), s(0), x2), cons(s(z0), cons(s(0), x2)))), DEL(if1(ge(s(z0), s(0)), s(z0), s(0), x2), cons(s(z0), cons(s(0), x2))), MAX(cons(s(z0), cons(s(0), x2)))) IF3(false, cons(s(0), cons(s(s(z0)), x2))) -> c15(SORT(del(if1(false, s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2)))), DEL(if1(ge(s(0), s(s(z0))), s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2))), MAX(cons(s(0), cons(s(s(z0)), x2)))) IF3(false, cons(s(s(z0)), cons(s(s(z1)), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2)))), DEL(if1(ge(s(s(z0)), s(s(z1))), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2))), MAX(cons(s(s(z0)), cons(s(s(z1)), x2)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c5(DEL(if1(ge(s(x0), s(x1)), s(x0), s(x1), x2), cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c5(MAX(cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(max(cons(z0, cons(0, x2))), z0), if1(true, z0, 0, x2), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(max(cons(0, cons(s(z0), x2))), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(max(cons(s(z0), cons(s(z1), x2))), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2)))) IF3(false, cons(0, nil)) -> c15(SORT(if2(true, max(cons(0, nil)), 0, nil))) IF3(false, cons(s(z0), nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(s(z0), nil)), s(z0), nil))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = [1] POL(DEL(x_1, x_2)) = 0 POL(EQ(x_1, x_2)) = 0 POL(GE(x_1, x_2)) = 0 POL(IF1(x_1, x_2, x_3, x_4)) = 0 POL(IF2(x_1, x_2, x_3, x_4)) = 0 POL(IF3(x_1, x_2)) = x_2 POL(MAX(x_1)) = 0 POL(SORT(x_1)) = x_1 POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c13(x_1)) = x_1 POL(c15(x_1)) = x_1 POL(c15(x_1, x_2)) = x_1 + x_2 POL(c15(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c2(x_1)) = x_1 POL(c2(x_1, x_2)) = x_1 + x_2 POL(c20(x_1)) = x_1 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(c6(x_1, x_2)) = x_1 + x_2 POL(c8(x_1)) = x_1 POL(cons(x_1, x_2)) = [1] + x_1 + x_2 POL(del(x_1, x_2)) = x_2 POL(eq(x_1, x_2)) = [1] POL(false) = [1] POL(ge(x_1, x_2)) = 0 POL(if1(x_1, x_2, x_3, x_4)) = 0 POL(if2(x_1, x_2, x_3, x_4)) = x_1 + x_3 + x_4 POL(max(x_1)) = 0 POL(nil) = 0 POL(s(x_1)) = [1] + x_1 POL(true) = 0 ---------------------------------------- (178) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil max(cons(z0, nil)) -> z0 max(cons(z0, cons(z1, z2))) -> if1(ge(z0, z1), z0, z1, z2) if2(true, z0, z1, z2) -> z2 if2(false, z0, z1, z2) -> cons(z1, del(z0, z2)) eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) if1(true, z0, z1, z2) -> max(cons(z0, z2)) if1(false, z0, z1, z2) -> max(cons(z1, z2)) Tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) MAX(cons(s(s(z0)), cons(s(s(0)), x2))) -> c2(IF1(true, s(s(z0)), s(s(0)), x2), GE(s(s(z0)), s(s(0)))) MAX(cons(s(s(0)), cons(s(s(s(z0))), x2))) -> c2(IF1(false, s(s(0)), s(s(s(z0))), x2), GE(s(s(0)), s(s(s(z0))))) MAX(cons(s(s(x0)), cons(s(s(x1)), x2))) -> c2(GE(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c6(IF2(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c6(IF2(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c6(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c6(EQ(s(s(0)), s(s(0)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(if1(ge(z1, x1), z1, x1, x2), cons(z1, cons(x1, x2))), MAX(cons(z1, cons(x1, x2)))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), z0, z1, z2), z0), max(cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2))))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2))))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) MAX(cons(s(s(s(z0))), cons(s(s(s(0))), x2))) -> c2(IF1(true, s(s(s(z0))), s(s(s(0))), x2), GE(s(s(s(z0))), s(s(s(0))))) MAX(cons(s(s(s(0))), cons(s(s(s(s(z0)))), x2))) -> c2(IF1(false, s(s(s(0))), s(s(s(s(z0)))), x2), GE(s(s(s(0))), s(s(s(s(z0)))))) MAX(cons(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2))) -> c2(IF1(ge(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), GE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MAX(cons(s(s(s(x0))), cons(s(s(s(x1))), x2))) -> c2(GE(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c6(IF2(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c6(IF2(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c6(IF2(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c6(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c6(EQ(s(s(s(0))), s(s(s(0))))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(max(cons(z0, cons(0, x2))), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(max(cons(0, cons(s(z0), x2))), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(max(cons(s(z0), cons(s(z1), x2))), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), max(cons(z0, cons(0, x2))), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), max(cons(0, cons(s(z0), x2))), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), max(cons(s(z0), cons(s(z1), x2))), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(ge(z0, 0), z0, 0, x2), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(ge(0, s(z0)), 0, s(z0), x2), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), if1(ge(z0, 0), z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), if1(ge(0, s(z0)), 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z1, cons(0, x1))) -> c15(SORT(if2(eq(if1(true, z1, 0, x1), z1), if1(true, z1, 0, x1), z1, cons(0, x1))), DEL(if1(ge(z1, 0), z1, 0, x1), cons(z1, cons(0, x1))), MAX(cons(z1, cons(0, x1)))) IF3(false, cons(z0, cons(0, z2))) -> c15(SORT(del(max(cons(z0, z2)), cons(z0, cons(0, z2)))), DEL(if1(ge(z0, 0), z0, 0, z2), cons(z0, cons(0, z2))), MAX(cons(z0, cons(0, z2)))) IF3(false, cons(x0, cons(0, x1))) -> c(DEL(if1(ge(x0, 0), x0, 0, x1), cons(x0, cons(0, x1)))) IF3(false, cons(x0, cons(0, x1))) -> c(MAX(cons(x0, cons(0, x1)))) IF3(false, cons(0, cons(s(x0), x1))) -> c15(SORT(if2(eq(if1(false, 0, s(x0), x1), 0), if1(false, 0, s(x0), x1), 0, cons(s(x0), x1))), DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1))), MAX(cons(0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), z2))) -> c15(SORT(del(max(cons(s(x0), z2)), cons(0, cons(s(x0), z2)))), DEL(if1(ge(0, s(x0)), 0, s(x0), z2), cons(0, cons(s(x0), z2))), MAX(cons(0, cons(s(x0), z2)))) IF3(false, cons(0, cons(s(x0), x1))) -> c1(DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), x1))) -> c1(MAX(cons(0, cons(s(x0), x1)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c15(SORT(if2(eq(if1(ge(x0, x1), s(x0), s(x1), x2), s(x0)), if1(ge(x0, x1), s(x0), s(x1), x2), s(x0), cons(s(x1), x2))), DEL(if1(ge(s(x0), s(x1)), s(x0), s(x1), x2), cons(s(x0), cons(s(x1), x2))), MAX(cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(s(z0), cons(s(0), x2))) -> c15(SORT(del(if1(true, s(z0), s(0), x2), cons(s(z0), cons(s(0), x2)))), DEL(if1(ge(s(z0), s(0)), s(z0), s(0), x2), cons(s(z0), cons(s(0), x2))), MAX(cons(s(z0), cons(s(0), x2)))) IF3(false, cons(s(0), cons(s(s(z0)), x2))) -> c15(SORT(del(if1(false, s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2)))), DEL(if1(ge(s(0), s(s(z0))), s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2))), MAX(cons(s(0), cons(s(s(z0)), x2)))) IF3(false, cons(s(s(z0)), cons(s(s(z1)), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2)))), DEL(if1(ge(s(s(z0)), s(s(z1))), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2))), MAX(cons(s(s(z0)), cons(s(s(z1)), x2)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c5(DEL(if1(ge(s(x0), s(x1)), s(x0), s(x1), x2), cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c5(MAX(cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(max(cons(z0, cons(0, x2))), z0), if1(true, z0, 0, x2), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(max(cons(0, cons(s(z0), x2))), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(max(cons(s(z0), cons(s(z1), x2))), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2)))) IF3(false, cons(0, nil)) -> c15(SORT(if2(true, max(cons(0, nil)), 0, nil))) IF3(false, cons(s(z0), nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(s(z0), nil)), s(z0), nil))) S tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) MAX(cons(s(s(z0)), cons(s(s(0)), x2))) -> c2(IF1(true, s(s(z0)), s(s(0)), x2), GE(s(s(z0)), s(s(0)))) MAX(cons(s(s(0)), cons(s(s(s(z0))), x2))) -> c2(IF1(false, s(s(0)), s(s(s(z0))), x2), GE(s(s(0)), s(s(s(z0))))) MAX(cons(s(s(x0)), cons(s(s(x1)), x2))) -> c2(GE(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c6(IF2(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c6(IF2(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c6(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c6(EQ(s(s(0)), s(s(0)))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), z0, z1, z2), z0), max(cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2))))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2))))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) MAX(cons(s(s(s(z0))), cons(s(s(s(0))), x2))) -> c2(IF1(true, s(s(s(z0))), s(s(s(0))), x2), GE(s(s(s(z0))), s(s(s(0))))) MAX(cons(s(s(s(0))), cons(s(s(s(s(z0)))), x2))) -> c2(IF1(false, s(s(s(0))), s(s(s(s(z0)))), x2), GE(s(s(s(0))), s(s(s(s(z0)))))) MAX(cons(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2))) -> c2(IF1(ge(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), GE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MAX(cons(s(s(s(x0))), cons(s(s(s(x1))), x2))) -> c2(GE(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c6(IF2(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c6(IF2(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c6(IF2(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c6(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c6(EQ(s(s(s(0))), s(s(s(0))))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(max(cons(z0, cons(0, x2))), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(max(cons(0, cons(s(z0), x2))), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(max(cons(s(z0), cons(s(z1), x2))), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), max(cons(z0, cons(0, x2))), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), max(cons(0, cons(s(z0), x2))), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), max(cons(s(z0), cons(s(z1), x2))), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(ge(z0, 0), z0, 0, x2), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(ge(0, s(z0)), 0, s(z0), x2), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), if1(ge(z0, 0), z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), if1(ge(0, s(z0)), 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z1, cons(0, x1))) -> c15(SORT(if2(eq(if1(true, z1, 0, x1), z1), if1(true, z1, 0, x1), z1, cons(0, x1))), DEL(if1(ge(z1, 0), z1, 0, x1), cons(z1, cons(0, x1))), MAX(cons(z1, cons(0, x1)))) IF3(false, cons(z0, cons(0, z2))) -> c15(SORT(del(max(cons(z0, z2)), cons(z0, cons(0, z2)))), DEL(if1(ge(z0, 0), z0, 0, z2), cons(z0, cons(0, z2))), MAX(cons(z0, cons(0, z2)))) IF3(false, cons(0, cons(s(x0), x1))) -> c15(SORT(if2(eq(if1(false, 0, s(x0), x1), 0), if1(false, 0, s(x0), x1), 0, cons(s(x0), x1))), DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1))), MAX(cons(0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), z2))) -> c15(SORT(del(max(cons(s(x0), z2)), cons(0, cons(s(x0), z2)))), DEL(if1(ge(0, s(x0)), 0, s(x0), z2), cons(0, cons(s(x0), z2))), MAX(cons(0, cons(s(x0), z2)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c15(SORT(if2(eq(if1(ge(x0, x1), s(x0), s(x1), x2), s(x0)), if1(ge(x0, x1), s(x0), s(x1), x2), s(x0), cons(s(x1), x2))), DEL(if1(ge(s(x0), s(x1)), s(x0), s(x1), x2), cons(s(x0), cons(s(x1), x2))), MAX(cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(s(z0), cons(s(0), x2))) -> c15(SORT(del(if1(true, s(z0), s(0), x2), cons(s(z0), cons(s(0), x2)))), DEL(if1(ge(s(z0), s(0)), s(z0), s(0), x2), cons(s(z0), cons(s(0), x2))), MAX(cons(s(z0), cons(s(0), x2)))) IF3(false, cons(s(0), cons(s(s(z0)), x2))) -> c15(SORT(del(if1(false, s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2)))), DEL(if1(ge(s(0), s(s(z0))), s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2))), MAX(cons(s(0), cons(s(s(z0)), x2)))) IF3(false, cons(s(s(z0)), cons(s(s(z1)), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2)))), DEL(if1(ge(s(s(z0)), s(s(z1))), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2))), MAX(cons(s(s(z0)), cons(s(s(z1)), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(max(cons(z0, cons(0, x2))), z0), if1(true, z0, 0, x2), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(max(cons(0, cons(s(z0), x2))), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(max(cons(s(z0), cons(s(z1), x2))), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2)))) IF3(false, cons(s(z0), nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(s(z0), nil)), s(z0), nil))) K tuples: IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) IF3(false, cons(x0, cons(0, x1))) -> c(DEL(if1(ge(x0, 0), x0, 0, x1), cons(x0, cons(0, x1)))) IF3(false, cons(x0, cons(0, x1))) -> c(MAX(cons(x0, cons(0, x1)))) IF3(false, cons(0, cons(s(x0), x1))) -> c1(DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), x1))) -> c1(MAX(cons(0, cons(s(x0), x1)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c5(DEL(if1(ge(s(x0), s(x1)), s(x0), s(x1), x2), cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c5(MAX(cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(0, nil)) -> c15(SORT(if2(true, max(cons(0, nil)), 0, nil))) Defined Rule Symbols: del_2, max_1, if2_4, eq_2, ge_2, if1_4 Defined Pair Symbols: IF1_4, IF2_4, EQ_2, GE_2, MAX_1, DEL_2, SORT_1, IF3_2 Compound Symbols: c3_1, c4_1, c8_1, c12_1, c20_1, c2_1, c6_1, c13_1, c2_2, c6_2, c15_1, c15_3, c15_2, c_1, c1_1, c5_1 ---------------------------------------- (179) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), z0, z1, z2), z0), max(cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) by IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), z0, z1, z2), z0), if1(ge(z0, z1), z0, z1, z2), z0, cons(z1, z2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), max(cons(z0, cons(0, x2))), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), max(cons(0, cons(s(z0), x2))), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), max(cons(s(z0), cons(s(z1), x2))), s(z0), cons(s(z1), x2)))) ---------------------------------------- (180) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil max(cons(z0, nil)) -> z0 max(cons(z0, cons(z1, z2))) -> if1(ge(z0, z1), z0, z1, z2) if2(true, z0, z1, z2) -> z2 if2(false, z0, z1, z2) -> cons(z1, del(z0, z2)) eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) if1(true, z0, z1, z2) -> max(cons(z0, z2)) if1(false, z0, z1, z2) -> max(cons(z1, z2)) Tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) MAX(cons(s(s(z0)), cons(s(s(0)), x2))) -> c2(IF1(true, s(s(z0)), s(s(0)), x2), GE(s(s(z0)), s(s(0)))) MAX(cons(s(s(0)), cons(s(s(s(z0))), x2))) -> c2(IF1(false, s(s(0)), s(s(s(z0))), x2), GE(s(s(0)), s(s(s(z0))))) MAX(cons(s(s(x0)), cons(s(s(x1)), x2))) -> c2(GE(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c6(IF2(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c6(IF2(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c6(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c6(EQ(s(s(0)), s(s(0)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(if1(ge(z1, x1), z1, x1, x2), cons(z1, cons(x1, x2))), MAX(cons(z1, cons(x1, x2)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2))))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2))))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) MAX(cons(s(s(s(z0))), cons(s(s(s(0))), x2))) -> c2(IF1(true, s(s(s(z0))), s(s(s(0))), x2), GE(s(s(s(z0))), s(s(s(0))))) MAX(cons(s(s(s(0))), cons(s(s(s(s(z0)))), x2))) -> c2(IF1(false, s(s(s(0))), s(s(s(s(z0)))), x2), GE(s(s(s(0))), s(s(s(s(z0)))))) MAX(cons(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2))) -> c2(IF1(ge(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), GE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MAX(cons(s(s(s(x0))), cons(s(s(s(x1))), x2))) -> c2(GE(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c6(IF2(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c6(IF2(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c6(IF2(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c6(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c6(EQ(s(s(s(0))), s(s(s(0))))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(max(cons(z0, cons(0, x2))), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(max(cons(0, cons(s(z0), x2))), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(max(cons(s(z0), cons(s(z1), x2))), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), max(cons(z0, cons(0, x2))), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), max(cons(0, cons(s(z0), x2))), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), max(cons(s(z0), cons(s(z1), x2))), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(ge(z0, 0), z0, 0, x2), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(ge(0, s(z0)), 0, s(z0), x2), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), if1(ge(z0, 0), z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), if1(ge(0, s(z0)), 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z1, cons(0, x1))) -> c15(SORT(if2(eq(if1(true, z1, 0, x1), z1), if1(true, z1, 0, x1), z1, cons(0, x1))), DEL(if1(ge(z1, 0), z1, 0, x1), cons(z1, cons(0, x1))), MAX(cons(z1, cons(0, x1)))) IF3(false, cons(z0, cons(0, z2))) -> c15(SORT(del(max(cons(z0, z2)), cons(z0, cons(0, z2)))), DEL(if1(ge(z0, 0), z0, 0, z2), cons(z0, cons(0, z2))), MAX(cons(z0, cons(0, z2)))) IF3(false, cons(x0, cons(0, x1))) -> c(DEL(if1(ge(x0, 0), x0, 0, x1), cons(x0, cons(0, x1)))) IF3(false, cons(x0, cons(0, x1))) -> c(MAX(cons(x0, cons(0, x1)))) IF3(false, cons(0, cons(s(x0), x1))) -> c15(SORT(if2(eq(if1(false, 0, s(x0), x1), 0), if1(false, 0, s(x0), x1), 0, cons(s(x0), x1))), DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1))), MAX(cons(0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), z2))) -> c15(SORT(del(max(cons(s(x0), z2)), cons(0, cons(s(x0), z2)))), DEL(if1(ge(0, s(x0)), 0, s(x0), z2), cons(0, cons(s(x0), z2))), MAX(cons(0, cons(s(x0), z2)))) IF3(false, cons(0, cons(s(x0), x1))) -> c1(DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), x1))) -> c1(MAX(cons(0, cons(s(x0), x1)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c15(SORT(if2(eq(if1(ge(x0, x1), s(x0), s(x1), x2), s(x0)), if1(ge(x0, x1), s(x0), s(x1), x2), s(x0), cons(s(x1), x2))), DEL(if1(ge(s(x0), s(x1)), s(x0), s(x1), x2), cons(s(x0), cons(s(x1), x2))), MAX(cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(s(z0), cons(s(0), x2))) -> c15(SORT(del(if1(true, s(z0), s(0), x2), cons(s(z0), cons(s(0), x2)))), DEL(if1(ge(s(z0), s(0)), s(z0), s(0), x2), cons(s(z0), cons(s(0), x2))), MAX(cons(s(z0), cons(s(0), x2)))) IF3(false, cons(s(0), cons(s(s(z0)), x2))) -> c15(SORT(del(if1(false, s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2)))), DEL(if1(ge(s(0), s(s(z0))), s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2))), MAX(cons(s(0), cons(s(s(z0)), x2)))) IF3(false, cons(s(s(z0)), cons(s(s(z1)), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2)))), DEL(if1(ge(s(s(z0)), s(s(z1))), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2))), MAX(cons(s(s(z0)), cons(s(s(z1)), x2)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c5(DEL(if1(ge(s(x0), s(x1)), s(x0), s(x1), x2), cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c5(MAX(cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(max(cons(z0, cons(0, x2))), z0), if1(true, z0, 0, x2), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(max(cons(0, cons(s(z0), x2))), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(max(cons(s(z0), cons(s(z1), x2))), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2)))) IF3(false, cons(0, nil)) -> c15(SORT(if2(true, max(cons(0, nil)), 0, nil))) IF3(false, cons(s(z0), nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(s(z0), nil)), s(z0), nil))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), max(cons(z0, cons(0, x2))), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), max(cons(0, cons(s(z0), x2))), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), max(cons(s(z0), cons(s(z1), x2))), s(z0), cons(s(z1), x2)))) S tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) MAX(cons(s(s(z0)), cons(s(s(0)), x2))) -> c2(IF1(true, s(s(z0)), s(s(0)), x2), GE(s(s(z0)), s(s(0)))) MAX(cons(s(s(0)), cons(s(s(s(z0))), x2))) -> c2(IF1(false, s(s(0)), s(s(s(z0))), x2), GE(s(s(0)), s(s(s(z0))))) MAX(cons(s(s(x0)), cons(s(s(x1)), x2))) -> c2(GE(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c6(IF2(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c6(IF2(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c6(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c6(EQ(s(s(0)), s(s(0)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2))))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2))))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) MAX(cons(s(s(s(z0))), cons(s(s(s(0))), x2))) -> c2(IF1(true, s(s(s(z0))), s(s(s(0))), x2), GE(s(s(s(z0))), s(s(s(0))))) MAX(cons(s(s(s(0))), cons(s(s(s(s(z0)))), x2))) -> c2(IF1(false, s(s(s(0))), s(s(s(s(z0)))), x2), GE(s(s(s(0))), s(s(s(s(z0)))))) MAX(cons(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2))) -> c2(IF1(ge(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), GE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MAX(cons(s(s(s(x0))), cons(s(s(s(x1))), x2))) -> c2(GE(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c6(IF2(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c6(IF2(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c6(IF2(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c6(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c6(EQ(s(s(s(0))), s(s(s(0))))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(max(cons(z0, cons(0, x2))), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(max(cons(0, cons(s(z0), x2))), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(max(cons(s(z0), cons(s(z1), x2))), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), max(cons(z0, cons(0, x2))), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), max(cons(0, cons(s(z0), x2))), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), max(cons(s(z0), cons(s(z1), x2))), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(ge(z0, 0), z0, 0, x2), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(ge(0, s(z0)), 0, s(z0), x2), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), if1(ge(z0, 0), z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), if1(ge(0, s(z0)), 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z1, cons(0, x1))) -> c15(SORT(if2(eq(if1(true, z1, 0, x1), z1), if1(true, z1, 0, x1), z1, cons(0, x1))), DEL(if1(ge(z1, 0), z1, 0, x1), cons(z1, cons(0, x1))), MAX(cons(z1, cons(0, x1)))) IF3(false, cons(z0, cons(0, z2))) -> c15(SORT(del(max(cons(z0, z2)), cons(z0, cons(0, z2)))), DEL(if1(ge(z0, 0), z0, 0, z2), cons(z0, cons(0, z2))), MAX(cons(z0, cons(0, z2)))) IF3(false, cons(0, cons(s(x0), x1))) -> c15(SORT(if2(eq(if1(false, 0, s(x0), x1), 0), if1(false, 0, s(x0), x1), 0, cons(s(x0), x1))), DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1))), MAX(cons(0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), z2))) -> c15(SORT(del(max(cons(s(x0), z2)), cons(0, cons(s(x0), z2)))), DEL(if1(ge(0, s(x0)), 0, s(x0), z2), cons(0, cons(s(x0), z2))), MAX(cons(0, cons(s(x0), z2)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c15(SORT(if2(eq(if1(ge(x0, x1), s(x0), s(x1), x2), s(x0)), if1(ge(x0, x1), s(x0), s(x1), x2), s(x0), cons(s(x1), x2))), DEL(if1(ge(s(x0), s(x1)), s(x0), s(x1), x2), cons(s(x0), cons(s(x1), x2))), MAX(cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(s(z0), cons(s(0), x2))) -> c15(SORT(del(if1(true, s(z0), s(0), x2), cons(s(z0), cons(s(0), x2)))), DEL(if1(ge(s(z0), s(0)), s(z0), s(0), x2), cons(s(z0), cons(s(0), x2))), MAX(cons(s(z0), cons(s(0), x2)))) IF3(false, cons(s(0), cons(s(s(z0)), x2))) -> c15(SORT(del(if1(false, s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2)))), DEL(if1(ge(s(0), s(s(z0))), s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2))), MAX(cons(s(0), cons(s(s(z0)), x2)))) IF3(false, cons(s(s(z0)), cons(s(s(z1)), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2)))), DEL(if1(ge(s(s(z0)), s(s(z1))), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2))), MAX(cons(s(s(z0)), cons(s(s(z1)), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(max(cons(z0, cons(0, x2))), z0), if1(true, z0, 0, x2), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(max(cons(0, cons(s(z0), x2))), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(max(cons(s(z0), cons(s(z1), x2))), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2)))) IF3(false, cons(s(z0), nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(s(z0), nil)), s(z0), nil))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), max(cons(z0, cons(0, x2))), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), max(cons(0, cons(s(z0), x2))), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), max(cons(s(z0), cons(s(z1), x2))), s(z0), cons(s(z1), x2)))) K tuples: IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) IF3(false, cons(x0, cons(0, x1))) -> c(DEL(if1(ge(x0, 0), x0, 0, x1), cons(x0, cons(0, x1)))) IF3(false, cons(x0, cons(0, x1))) -> c(MAX(cons(x0, cons(0, x1)))) IF3(false, cons(0, cons(s(x0), x1))) -> c1(DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), x1))) -> c1(MAX(cons(0, cons(s(x0), x1)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c5(DEL(if1(ge(s(x0), s(x1)), s(x0), s(x1), x2), cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c5(MAX(cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(0, nil)) -> c15(SORT(if2(true, max(cons(0, nil)), 0, nil))) Defined Rule Symbols: del_2, max_1, if2_4, eq_2, ge_2, if1_4 Defined Pair Symbols: IF1_4, IF2_4, EQ_2, GE_2, MAX_1, DEL_2, SORT_1, IF3_2 Compound Symbols: c3_1, c4_1, c8_1, c12_1, c20_1, c2_1, c6_1, c13_1, c2_2, c6_2, c15_1, c15_3, c15_2, c_1, c1_1, c5_1 ---------------------------------------- (181) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2)))) by IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(ge(z0, 0), z0, 0, x2), z0), if1(true, z0, 0, x2), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(ge(0, s(z0)), 0, s(z0), x2), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), if1(ge(z0, 0), z0, 0, x2), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), if1(ge(0, s(z0)), 0, s(z0), x2), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0), cons(s(z1), x2)))) ---------------------------------------- (182) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil max(cons(z0, nil)) -> z0 max(cons(z0, cons(z1, z2))) -> if1(ge(z0, z1), z0, z1, z2) if2(true, z0, z1, z2) -> z2 if2(false, z0, z1, z2) -> cons(z1, del(z0, z2)) eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) if1(true, z0, z1, z2) -> max(cons(z0, z2)) if1(false, z0, z1, z2) -> max(cons(z1, z2)) Tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) MAX(cons(s(s(z0)), cons(s(s(0)), x2))) -> c2(IF1(true, s(s(z0)), s(s(0)), x2), GE(s(s(z0)), s(s(0)))) MAX(cons(s(s(0)), cons(s(s(s(z0))), x2))) -> c2(IF1(false, s(s(0)), s(s(s(z0))), x2), GE(s(s(0)), s(s(s(z0))))) MAX(cons(s(s(x0)), cons(s(s(x1)), x2))) -> c2(GE(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c6(IF2(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c6(IF2(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c6(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c6(EQ(s(s(0)), s(s(0)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(if1(ge(z1, x1), z1, x1, x2), cons(z1, cons(x1, x2))), MAX(cons(z1, cons(x1, x2)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2))))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2))))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) MAX(cons(s(s(s(z0))), cons(s(s(s(0))), x2))) -> c2(IF1(true, s(s(s(z0))), s(s(s(0))), x2), GE(s(s(s(z0))), s(s(s(0))))) MAX(cons(s(s(s(0))), cons(s(s(s(s(z0)))), x2))) -> c2(IF1(false, s(s(s(0))), s(s(s(s(z0)))), x2), GE(s(s(s(0))), s(s(s(s(z0)))))) MAX(cons(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2))) -> c2(IF1(ge(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), GE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MAX(cons(s(s(s(x0))), cons(s(s(s(x1))), x2))) -> c2(GE(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c6(IF2(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c6(IF2(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c6(IF2(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c6(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c6(EQ(s(s(s(0))), s(s(s(0))))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(max(cons(z0, cons(0, x2))), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(max(cons(0, cons(s(z0), x2))), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(max(cons(s(z0), cons(s(z1), x2))), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), max(cons(z0, cons(0, x2))), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), max(cons(0, cons(s(z0), x2))), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), max(cons(s(z0), cons(s(z1), x2))), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(ge(z0, 0), z0, 0, x2), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(ge(0, s(z0)), 0, s(z0), x2), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), if1(ge(z0, 0), z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), if1(ge(0, s(z0)), 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z1, cons(0, x1))) -> c15(SORT(if2(eq(if1(true, z1, 0, x1), z1), if1(true, z1, 0, x1), z1, cons(0, x1))), DEL(if1(ge(z1, 0), z1, 0, x1), cons(z1, cons(0, x1))), MAX(cons(z1, cons(0, x1)))) IF3(false, cons(z0, cons(0, z2))) -> c15(SORT(del(max(cons(z0, z2)), cons(z0, cons(0, z2)))), DEL(if1(ge(z0, 0), z0, 0, z2), cons(z0, cons(0, z2))), MAX(cons(z0, cons(0, z2)))) IF3(false, cons(x0, cons(0, x1))) -> c(DEL(if1(ge(x0, 0), x0, 0, x1), cons(x0, cons(0, x1)))) IF3(false, cons(x0, cons(0, x1))) -> c(MAX(cons(x0, cons(0, x1)))) IF3(false, cons(0, cons(s(x0), x1))) -> c15(SORT(if2(eq(if1(false, 0, s(x0), x1), 0), if1(false, 0, s(x0), x1), 0, cons(s(x0), x1))), DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1))), MAX(cons(0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), z2))) -> c15(SORT(del(max(cons(s(x0), z2)), cons(0, cons(s(x0), z2)))), DEL(if1(ge(0, s(x0)), 0, s(x0), z2), cons(0, cons(s(x0), z2))), MAX(cons(0, cons(s(x0), z2)))) IF3(false, cons(0, cons(s(x0), x1))) -> c1(DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), x1))) -> c1(MAX(cons(0, cons(s(x0), x1)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c15(SORT(if2(eq(if1(ge(x0, x1), s(x0), s(x1), x2), s(x0)), if1(ge(x0, x1), s(x0), s(x1), x2), s(x0), cons(s(x1), x2))), DEL(if1(ge(s(x0), s(x1)), s(x0), s(x1), x2), cons(s(x0), cons(s(x1), x2))), MAX(cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(s(z0), cons(s(0), x2))) -> c15(SORT(del(if1(true, s(z0), s(0), x2), cons(s(z0), cons(s(0), x2)))), DEL(if1(ge(s(z0), s(0)), s(z0), s(0), x2), cons(s(z0), cons(s(0), x2))), MAX(cons(s(z0), cons(s(0), x2)))) IF3(false, cons(s(0), cons(s(s(z0)), x2))) -> c15(SORT(del(if1(false, s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2)))), DEL(if1(ge(s(0), s(s(z0))), s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2))), MAX(cons(s(0), cons(s(s(z0)), x2)))) IF3(false, cons(s(s(z0)), cons(s(s(z1)), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2)))), DEL(if1(ge(s(s(z0)), s(s(z1))), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2))), MAX(cons(s(s(z0)), cons(s(s(z1)), x2)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c5(DEL(if1(ge(s(x0), s(x1)), s(x0), s(x1), x2), cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c5(MAX(cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(max(cons(z0, cons(0, x2))), z0), if1(true, z0, 0, x2), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(max(cons(0, cons(s(z0), x2))), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(max(cons(s(z0), cons(s(z1), x2))), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2)))) IF3(false, cons(0, nil)) -> c15(SORT(if2(true, max(cons(0, nil)), 0, nil))) IF3(false, cons(s(z0), nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(s(z0), nil)), s(z0), nil))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), max(cons(z0, cons(0, x2))), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), max(cons(0, cons(s(z0), x2))), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), max(cons(s(z0), cons(s(z1), x2))), s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(ge(z0, 0), z0, 0, x2), z0), if1(true, z0, 0, x2), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(ge(0, s(z0)), 0, s(z0), x2), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), if1(ge(z0, 0), z0, 0, x2), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), if1(ge(0, s(z0)), 0, s(z0), x2), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0), cons(s(z1), x2)))) S tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) MAX(cons(s(s(z0)), cons(s(s(0)), x2))) -> c2(IF1(true, s(s(z0)), s(s(0)), x2), GE(s(s(z0)), s(s(0)))) MAX(cons(s(s(0)), cons(s(s(s(z0))), x2))) -> c2(IF1(false, s(s(0)), s(s(s(z0))), x2), GE(s(s(0)), s(s(s(z0))))) MAX(cons(s(s(x0)), cons(s(s(x1)), x2))) -> c2(GE(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c6(IF2(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c6(IF2(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c6(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c6(EQ(s(s(0)), s(s(0)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2))))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2))))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) MAX(cons(s(s(s(z0))), cons(s(s(s(0))), x2))) -> c2(IF1(true, s(s(s(z0))), s(s(s(0))), x2), GE(s(s(s(z0))), s(s(s(0))))) MAX(cons(s(s(s(0))), cons(s(s(s(s(z0)))), x2))) -> c2(IF1(false, s(s(s(0))), s(s(s(s(z0)))), x2), GE(s(s(s(0))), s(s(s(s(z0)))))) MAX(cons(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2))) -> c2(IF1(ge(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), GE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MAX(cons(s(s(s(x0))), cons(s(s(s(x1))), x2))) -> c2(GE(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c6(IF2(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c6(IF2(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c6(IF2(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c6(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c6(EQ(s(s(s(0))), s(s(s(0))))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(max(cons(z0, cons(0, x2))), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(max(cons(0, cons(s(z0), x2))), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(max(cons(s(z0), cons(s(z1), x2))), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), max(cons(z0, cons(0, x2))), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), max(cons(0, cons(s(z0), x2))), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), max(cons(s(z0), cons(s(z1), x2))), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(ge(z0, 0), z0, 0, x2), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(ge(0, s(z0)), 0, s(z0), x2), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), if1(ge(z0, 0), z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), if1(ge(0, s(z0)), 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z1, cons(0, x1))) -> c15(SORT(if2(eq(if1(true, z1, 0, x1), z1), if1(true, z1, 0, x1), z1, cons(0, x1))), DEL(if1(ge(z1, 0), z1, 0, x1), cons(z1, cons(0, x1))), MAX(cons(z1, cons(0, x1)))) IF3(false, cons(z0, cons(0, z2))) -> c15(SORT(del(max(cons(z0, z2)), cons(z0, cons(0, z2)))), DEL(if1(ge(z0, 0), z0, 0, z2), cons(z0, cons(0, z2))), MAX(cons(z0, cons(0, z2)))) IF3(false, cons(0, cons(s(x0), x1))) -> c15(SORT(if2(eq(if1(false, 0, s(x0), x1), 0), if1(false, 0, s(x0), x1), 0, cons(s(x0), x1))), DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1))), MAX(cons(0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), z2))) -> c15(SORT(del(max(cons(s(x0), z2)), cons(0, cons(s(x0), z2)))), DEL(if1(ge(0, s(x0)), 0, s(x0), z2), cons(0, cons(s(x0), z2))), MAX(cons(0, cons(s(x0), z2)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c15(SORT(if2(eq(if1(ge(x0, x1), s(x0), s(x1), x2), s(x0)), if1(ge(x0, x1), s(x0), s(x1), x2), s(x0), cons(s(x1), x2))), DEL(if1(ge(s(x0), s(x1)), s(x0), s(x1), x2), cons(s(x0), cons(s(x1), x2))), MAX(cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(s(z0), cons(s(0), x2))) -> c15(SORT(del(if1(true, s(z0), s(0), x2), cons(s(z0), cons(s(0), x2)))), DEL(if1(ge(s(z0), s(0)), s(z0), s(0), x2), cons(s(z0), cons(s(0), x2))), MAX(cons(s(z0), cons(s(0), x2)))) IF3(false, cons(s(0), cons(s(s(z0)), x2))) -> c15(SORT(del(if1(false, s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2)))), DEL(if1(ge(s(0), s(s(z0))), s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2))), MAX(cons(s(0), cons(s(s(z0)), x2)))) IF3(false, cons(s(s(z0)), cons(s(s(z1)), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2)))), DEL(if1(ge(s(s(z0)), s(s(z1))), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2))), MAX(cons(s(s(z0)), cons(s(s(z1)), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(max(cons(z0, cons(0, x2))), z0), if1(true, z0, 0, x2), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(max(cons(0, cons(s(z0), x2))), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(max(cons(s(z0), cons(s(z1), x2))), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2)))) IF3(false, cons(s(z0), nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(s(z0), nil)), s(z0), nil))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), max(cons(z0, cons(0, x2))), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), max(cons(0, cons(s(z0), x2))), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), max(cons(s(z0), cons(s(z1), x2))), s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(ge(z0, 0), z0, 0, x2), z0), if1(true, z0, 0, x2), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(ge(0, s(z0)), 0, s(z0), x2), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), if1(ge(z0, 0), z0, 0, x2), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), if1(ge(0, s(z0)), 0, s(z0), x2), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0), cons(s(z1), x2)))) K tuples: IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) IF3(false, cons(x0, cons(0, x1))) -> c(DEL(if1(ge(x0, 0), x0, 0, x1), cons(x0, cons(0, x1)))) IF3(false, cons(x0, cons(0, x1))) -> c(MAX(cons(x0, cons(0, x1)))) IF3(false, cons(0, cons(s(x0), x1))) -> c1(DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), x1))) -> c1(MAX(cons(0, cons(s(x0), x1)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c5(DEL(if1(ge(s(x0), s(x1)), s(x0), s(x1), x2), cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c5(MAX(cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(0, nil)) -> c15(SORT(if2(true, max(cons(0, nil)), 0, nil))) Defined Rule Symbols: del_2, max_1, if2_4, eq_2, ge_2, if1_4 Defined Pair Symbols: IF1_4, IF2_4, EQ_2, GE_2, MAX_1, DEL_2, SORT_1, IF3_2 Compound Symbols: c3_1, c4_1, c8_1, c12_1, c20_1, c2_1, c6_1, c13_1, c2_2, c6_2, c15_1, c15_3, c15_2, c_1, c1_1, c5_1 ---------------------------------------- (183) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2))))) by IF3(false, cons(z1, cons(0, x1))) -> c15(SORT(if2(eq(if1(true, z1, 0, x1), z1), if1(true, z1, 0, x1), z1, cons(0, x1)))) IF3(false, cons(z0, cons(0, z2))) -> c15(SORT(del(max(cons(z0, z2)), cons(z0, cons(0, z2))))) ---------------------------------------- (184) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil max(cons(z0, nil)) -> z0 max(cons(z0, cons(z1, z2))) -> if1(ge(z0, z1), z0, z1, z2) if2(true, z0, z1, z2) -> z2 if2(false, z0, z1, z2) -> cons(z1, del(z0, z2)) eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) if1(true, z0, z1, z2) -> max(cons(z0, z2)) if1(false, z0, z1, z2) -> max(cons(z1, z2)) Tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) MAX(cons(s(s(z0)), cons(s(s(0)), x2))) -> c2(IF1(true, s(s(z0)), s(s(0)), x2), GE(s(s(z0)), s(s(0)))) MAX(cons(s(s(0)), cons(s(s(s(z0))), x2))) -> c2(IF1(false, s(s(0)), s(s(s(z0))), x2), GE(s(s(0)), s(s(s(z0))))) MAX(cons(s(s(x0)), cons(s(s(x1)), x2))) -> c2(GE(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c6(IF2(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c6(IF2(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c6(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c6(EQ(s(s(0)), s(s(0)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(if1(ge(z1, x1), z1, x1, x2), cons(z1, cons(x1, x2))), MAX(cons(z1, cons(x1, x2)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2))))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) MAX(cons(s(s(s(z0))), cons(s(s(s(0))), x2))) -> c2(IF1(true, s(s(s(z0))), s(s(s(0))), x2), GE(s(s(s(z0))), s(s(s(0))))) MAX(cons(s(s(s(0))), cons(s(s(s(s(z0)))), x2))) -> c2(IF1(false, s(s(s(0))), s(s(s(s(z0)))), x2), GE(s(s(s(0))), s(s(s(s(z0)))))) MAX(cons(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2))) -> c2(IF1(ge(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), GE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MAX(cons(s(s(s(x0))), cons(s(s(s(x1))), x2))) -> c2(GE(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c6(IF2(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c6(IF2(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c6(IF2(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c6(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c6(EQ(s(s(s(0))), s(s(s(0))))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(max(cons(z0, cons(0, x2))), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(max(cons(0, cons(s(z0), x2))), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(max(cons(s(z0), cons(s(z1), x2))), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), max(cons(z0, cons(0, x2))), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), max(cons(0, cons(s(z0), x2))), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), max(cons(s(z0), cons(s(z1), x2))), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(ge(z0, 0), z0, 0, x2), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(ge(0, s(z0)), 0, s(z0), x2), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), if1(ge(z0, 0), z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), if1(ge(0, s(z0)), 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z1, cons(0, x1))) -> c15(SORT(if2(eq(if1(true, z1, 0, x1), z1), if1(true, z1, 0, x1), z1, cons(0, x1))), DEL(if1(ge(z1, 0), z1, 0, x1), cons(z1, cons(0, x1))), MAX(cons(z1, cons(0, x1)))) IF3(false, cons(z0, cons(0, z2))) -> c15(SORT(del(max(cons(z0, z2)), cons(z0, cons(0, z2)))), DEL(if1(ge(z0, 0), z0, 0, z2), cons(z0, cons(0, z2))), MAX(cons(z0, cons(0, z2)))) IF3(false, cons(x0, cons(0, x1))) -> c(DEL(if1(ge(x0, 0), x0, 0, x1), cons(x0, cons(0, x1)))) IF3(false, cons(x0, cons(0, x1))) -> c(MAX(cons(x0, cons(0, x1)))) IF3(false, cons(0, cons(s(x0), x1))) -> c15(SORT(if2(eq(if1(false, 0, s(x0), x1), 0), if1(false, 0, s(x0), x1), 0, cons(s(x0), x1))), DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1))), MAX(cons(0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), z2))) -> c15(SORT(del(max(cons(s(x0), z2)), cons(0, cons(s(x0), z2)))), DEL(if1(ge(0, s(x0)), 0, s(x0), z2), cons(0, cons(s(x0), z2))), MAX(cons(0, cons(s(x0), z2)))) IF3(false, cons(0, cons(s(x0), x1))) -> c1(DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), x1))) -> c1(MAX(cons(0, cons(s(x0), x1)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c15(SORT(if2(eq(if1(ge(x0, x1), s(x0), s(x1), x2), s(x0)), if1(ge(x0, x1), s(x0), s(x1), x2), s(x0), cons(s(x1), x2))), DEL(if1(ge(s(x0), s(x1)), s(x0), s(x1), x2), cons(s(x0), cons(s(x1), x2))), MAX(cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(s(z0), cons(s(0), x2))) -> c15(SORT(del(if1(true, s(z0), s(0), x2), cons(s(z0), cons(s(0), x2)))), DEL(if1(ge(s(z0), s(0)), s(z0), s(0), x2), cons(s(z0), cons(s(0), x2))), MAX(cons(s(z0), cons(s(0), x2)))) IF3(false, cons(s(0), cons(s(s(z0)), x2))) -> c15(SORT(del(if1(false, s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2)))), DEL(if1(ge(s(0), s(s(z0))), s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2))), MAX(cons(s(0), cons(s(s(z0)), x2)))) IF3(false, cons(s(s(z0)), cons(s(s(z1)), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2)))), DEL(if1(ge(s(s(z0)), s(s(z1))), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2))), MAX(cons(s(s(z0)), cons(s(s(z1)), x2)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c5(DEL(if1(ge(s(x0), s(x1)), s(x0), s(x1), x2), cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c5(MAX(cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(max(cons(z0, cons(0, x2))), z0), if1(true, z0, 0, x2), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(max(cons(0, cons(s(z0), x2))), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(max(cons(s(z0), cons(s(z1), x2))), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2)))) IF3(false, cons(0, nil)) -> c15(SORT(if2(true, max(cons(0, nil)), 0, nil))) IF3(false, cons(s(z0), nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(s(z0), nil)), s(z0), nil))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), max(cons(z0, cons(0, x2))), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), max(cons(0, cons(s(z0), x2))), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), max(cons(s(z0), cons(s(z1), x2))), s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(ge(z0, 0), z0, 0, x2), z0), if1(true, z0, 0, x2), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(ge(0, s(z0)), 0, s(z0), x2), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), if1(ge(z0, 0), z0, 0, x2), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), if1(ge(0, s(z0)), 0, s(z0), x2), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0), cons(s(z1), x2)))) IF3(false, cons(z1, cons(0, x1))) -> c15(SORT(if2(eq(if1(true, z1, 0, x1), z1), if1(true, z1, 0, x1), z1, cons(0, x1)))) IF3(false, cons(z0, cons(0, z2))) -> c15(SORT(del(max(cons(z0, z2)), cons(z0, cons(0, z2))))) S tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) MAX(cons(s(s(z0)), cons(s(s(0)), x2))) -> c2(IF1(true, s(s(z0)), s(s(0)), x2), GE(s(s(z0)), s(s(0)))) MAX(cons(s(s(0)), cons(s(s(s(z0))), x2))) -> c2(IF1(false, s(s(0)), s(s(s(z0))), x2), GE(s(s(0)), s(s(s(z0))))) MAX(cons(s(s(x0)), cons(s(s(x1)), x2))) -> c2(GE(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c6(IF2(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c6(IF2(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c6(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c6(EQ(s(s(0)), s(s(0)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2))))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) MAX(cons(s(s(s(z0))), cons(s(s(s(0))), x2))) -> c2(IF1(true, s(s(s(z0))), s(s(s(0))), x2), GE(s(s(s(z0))), s(s(s(0))))) MAX(cons(s(s(s(0))), cons(s(s(s(s(z0)))), x2))) -> c2(IF1(false, s(s(s(0))), s(s(s(s(z0)))), x2), GE(s(s(s(0))), s(s(s(s(z0)))))) MAX(cons(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2))) -> c2(IF1(ge(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), GE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MAX(cons(s(s(s(x0))), cons(s(s(s(x1))), x2))) -> c2(GE(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c6(IF2(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c6(IF2(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c6(IF2(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c6(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c6(EQ(s(s(s(0))), s(s(s(0))))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(max(cons(z0, cons(0, x2))), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(max(cons(0, cons(s(z0), x2))), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(max(cons(s(z0), cons(s(z1), x2))), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), max(cons(z0, cons(0, x2))), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), max(cons(0, cons(s(z0), x2))), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), max(cons(s(z0), cons(s(z1), x2))), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(ge(z0, 0), z0, 0, x2), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(ge(0, s(z0)), 0, s(z0), x2), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), if1(ge(z0, 0), z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), if1(ge(0, s(z0)), 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z1, cons(0, x1))) -> c15(SORT(if2(eq(if1(true, z1, 0, x1), z1), if1(true, z1, 0, x1), z1, cons(0, x1))), DEL(if1(ge(z1, 0), z1, 0, x1), cons(z1, cons(0, x1))), MAX(cons(z1, cons(0, x1)))) IF3(false, cons(z0, cons(0, z2))) -> c15(SORT(del(max(cons(z0, z2)), cons(z0, cons(0, z2)))), DEL(if1(ge(z0, 0), z0, 0, z2), cons(z0, cons(0, z2))), MAX(cons(z0, cons(0, z2)))) IF3(false, cons(0, cons(s(x0), x1))) -> c15(SORT(if2(eq(if1(false, 0, s(x0), x1), 0), if1(false, 0, s(x0), x1), 0, cons(s(x0), x1))), DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1))), MAX(cons(0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), z2))) -> c15(SORT(del(max(cons(s(x0), z2)), cons(0, cons(s(x0), z2)))), DEL(if1(ge(0, s(x0)), 0, s(x0), z2), cons(0, cons(s(x0), z2))), MAX(cons(0, cons(s(x0), z2)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c15(SORT(if2(eq(if1(ge(x0, x1), s(x0), s(x1), x2), s(x0)), if1(ge(x0, x1), s(x0), s(x1), x2), s(x0), cons(s(x1), x2))), DEL(if1(ge(s(x0), s(x1)), s(x0), s(x1), x2), cons(s(x0), cons(s(x1), x2))), MAX(cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(s(z0), cons(s(0), x2))) -> c15(SORT(del(if1(true, s(z0), s(0), x2), cons(s(z0), cons(s(0), x2)))), DEL(if1(ge(s(z0), s(0)), s(z0), s(0), x2), cons(s(z0), cons(s(0), x2))), MAX(cons(s(z0), cons(s(0), x2)))) IF3(false, cons(s(0), cons(s(s(z0)), x2))) -> c15(SORT(del(if1(false, s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2)))), DEL(if1(ge(s(0), s(s(z0))), s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2))), MAX(cons(s(0), cons(s(s(z0)), x2)))) IF3(false, cons(s(s(z0)), cons(s(s(z1)), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2)))), DEL(if1(ge(s(s(z0)), s(s(z1))), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2))), MAX(cons(s(s(z0)), cons(s(s(z1)), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(max(cons(z0, cons(0, x2))), z0), if1(true, z0, 0, x2), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(max(cons(0, cons(s(z0), x2))), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(max(cons(s(z0), cons(s(z1), x2))), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2)))) IF3(false, cons(s(z0), nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(s(z0), nil)), s(z0), nil))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), max(cons(z0, cons(0, x2))), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), max(cons(0, cons(s(z0), x2))), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), max(cons(s(z0), cons(s(z1), x2))), s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(ge(z0, 0), z0, 0, x2), z0), if1(true, z0, 0, x2), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(ge(0, s(z0)), 0, s(z0), x2), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), if1(ge(z0, 0), z0, 0, x2), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), if1(ge(0, s(z0)), 0, s(z0), x2), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0), cons(s(z1), x2)))) IF3(false, cons(z1, cons(0, x1))) -> c15(SORT(if2(eq(if1(true, z1, 0, x1), z1), if1(true, z1, 0, x1), z1, cons(0, x1)))) IF3(false, cons(z0, cons(0, z2))) -> c15(SORT(del(max(cons(z0, z2)), cons(z0, cons(0, z2))))) K tuples: IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) IF3(false, cons(x0, cons(0, x1))) -> c(DEL(if1(ge(x0, 0), x0, 0, x1), cons(x0, cons(0, x1)))) IF3(false, cons(x0, cons(0, x1))) -> c(MAX(cons(x0, cons(0, x1)))) IF3(false, cons(0, cons(s(x0), x1))) -> c1(DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), x1))) -> c1(MAX(cons(0, cons(s(x0), x1)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c5(DEL(if1(ge(s(x0), s(x1)), s(x0), s(x1), x2), cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c5(MAX(cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(0, nil)) -> c15(SORT(if2(true, max(cons(0, nil)), 0, nil))) Defined Rule Symbols: del_2, max_1, if2_4, eq_2, ge_2, if1_4 Defined Pair Symbols: IF1_4, IF2_4, EQ_2, GE_2, MAX_1, DEL_2, SORT_1, IF3_2 Compound Symbols: c3_1, c4_1, c8_1, c12_1, c20_1, c2_1, c6_1, c13_1, c2_2, c6_2, c15_1, c15_3, c15_2, c_1, c1_1, c5_1 ---------------------------------------- (185) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2))))) by IF3(false, cons(0, cons(s(x0), x1))) -> c15(SORT(if2(eq(if1(false, 0, s(x0), x1), 0), if1(false, 0, s(x0), x1), 0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), z2))) -> c15(SORT(del(max(cons(s(x0), z2)), cons(0, cons(s(x0), z2))))) ---------------------------------------- (186) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil max(cons(z0, nil)) -> z0 max(cons(z0, cons(z1, z2))) -> if1(ge(z0, z1), z0, z1, z2) if2(true, z0, z1, z2) -> z2 if2(false, z0, z1, z2) -> cons(z1, del(z0, z2)) eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) if1(true, z0, z1, z2) -> max(cons(z0, z2)) if1(false, z0, z1, z2) -> max(cons(z1, z2)) Tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) MAX(cons(s(s(z0)), cons(s(s(0)), x2))) -> c2(IF1(true, s(s(z0)), s(s(0)), x2), GE(s(s(z0)), s(s(0)))) MAX(cons(s(s(0)), cons(s(s(s(z0))), x2))) -> c2(IF1(false, s(s(0)), s(s(s(z0))), x2), GE(s(s(0)), s(s(s(z0))))) MAX(cons(s(s(x0)), cons(s(s(x1)), x2))) -> c2(GE(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c6(IF2(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c6(IF2(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c6(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c6(EQ(s(s(0)), s(s(0)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(if1(ge(z1, x1), z1, x1, x2), cons(z1, cons(x1, x2))), MAX(cons(z1, cons(x1, x2)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) MAX(cons(s(s(s(z0))), cons(s(s(s(0))), x2))) -> c2(IF1(true, s(s(s(z0))), s(s(s(0))), x2), GE(s(s(s(z0))), s(s(s(0))))) MAX(cons(s(s(s(0))), cons(s(s(s(s(z0)))), x2))) -> c2(IF1(false, s(s(s(0))), s(s(s(s(z0)))), x2), GE(s(s(s(0))), s(s(s(s(z0)))))) MAX(cons(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2))) -> c2(IF1(ge(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), GE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MAX(cons(s(s(s(x0))), cons(s(s(s(x1))), x2))) -> c2(GE(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c6(IF2(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c6(IF2(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c6(IF2(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c6(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c6(EQ(s(s(s(0))), s(s(s(0))))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(max(cons(z0, cons(0, x2))), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(max(cons(0, cons(s(z0), x2))), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(max(cons(s(z0), cons(s(z1), x2))), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), max(cons(z0, cons(0, x2))), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), max(cons(0, cons(s(z0), x2))), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), max(cons(s(z0), cons(s(z1), x2))), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(ge(z0, 0), z0, 0, x2), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(ge(0, s(z0)), 0, s(z0), x2), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), if1(ge(z0, 0), z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), if1(ge(0, s(z0)), 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z1, cons(0, x1))) -> c15(SORT(if2(eq(if1(true, z1, 0, x1), z1), if1(true, z1, 0, x1), z1, cons(0, x1))), DEL(if1(ge(z1, 0), z1, 0, x1), cons(z1, cons(0, x1))), MAX(cons(z1, cons(0, x1)))) IF3(false, cons(z0, cons(0, z2))) -> c15(SORT(del(max(cons(z0, z2)), cons(z0, cons(0, z2)))), DEL(if1(ge(z0, 0), z0, 0, z2), cons(z0, cons(0, z2))), MAX(cons(z0, cons(0, z2)))) IF3(false, cons(x0, cons(0, x1))) -> c(DEL(if1(ge(x0, 0), x0, 0, x1), cons(x0, cons(0, x1)))) IF3(false, cons(x0, cons(0, x1))) -> c(MAX(cons(x0, cons(0, x1)))) IF3(false, cons(0, cons(s(x0), x1))) -> c15(SORT(if2(eq(if1(false, 0, s(x0), x1), 0), if1(false, 0, s(x0), x1), 0, cons(s(x0), x1))), DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1))), MAX(cons(0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), z2))) -> c15(SORT(del(max(cons(s(x0), z2)), cons(0, cons(s(x0), z2)))), DEL(if1(ge(0, s(x0)), 0, s(x0), z2), cons(0, cons(s(x0), z2))), MAX(cons(0, cons(s(x0), z2)))) IF3(false, cons(0, cons(s(x0), x1))) -> c1(DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), x1))) -> c1(MAX(cons(0, cons(s(x0), x1)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c15(SORT(if2(eq(if1(ge(x0, x1), s(x0), s(x1), x2), s(x0)), if1(ge(x0, x1), s(x0), s(x1), x2), s(x0), cons(s(x1), x2))), DEL(if1(ge(s(x0), s(x1)), s(x0), s(x1), x2), cons(s(x0), cons(s(x1), x2))), MAX(cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(s(z0), cons(s(0), x2))) -> c15(SORT(del(if1(true, s(z0), s(0), x2), cons(s(z0), cons(s(0), x2)))), DEL(if1(ge(s(z0), s(0)), s(z0), s(0), x2), cons(s(z0), cons(s(0), x2))), MAX(cons(s(z0), cons(s(0), x2)))) IF3(false, cons(s(0), cons(s(s(z0)), x2))) -> c15(SORT(del(if1(false, s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2)))), DEL(if1(ge(s(0), s(s(z0))), s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2))), MAX(cons(s(0), cons(s(s(z0)), x2)))) IF3(false, cons(s(s(z0)), cons(s(s(z1)), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2)))), DEL(if1(ge(s(s(z0)), s(s(z1))), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2))), MAX(cons(s(s(z0)), cons(s(s(z1)), x2)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c5(DEL(if1(ge(s(x0), s(x1)), s(x0), s(x1), x2), cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c5(MAX(cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(max(cons(z0, cons(0, x2))), z0), if1(true, z0, 0, x2), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(max(cons(0, cons(s(z0), x2))), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(max(cons(s(z0), cons(s(z1), x2))), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2)))) IF3(false, cons(0, nil)) -> c15(SORT(if2(true, max(cons(0, nil)), 0, nil))) IF3(false, cons(s(z0), nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(s(z0), nil)), s(z0), nil))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), max(cons(z0, cons(0, x2))), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), max(cons(0, cons(s(z0), x2))), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), max(cons(s(z0), cons(s(z1), x2))), s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(ge(z0, 0), z0, 0, x2), z0), if1(true, z0, 0, x2), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(ge(0, s(z0)), 0, s(z0), x2), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), if1(ge(z0, 0), z0, 0, x2), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), if1(ge(0, s(z0)), 0, s(z0), x2), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0), cons(s(z1), x2)))) IF3(false, cons(z1, cons(0, x1))) -> c15(SORT(if2(eq(if1(true, z1, 0, x1), z1), if1(true, z1, 0, x1), z1, cons(0, x1)))) IF3(false, cons(z0, cons(0, z2))) -> c15(SORT(del(max(cons(z0, z2)), cons(z0, cons(0, z2))))) IF3(false, cons(0, cons(s(x0), x1))) -> c15(SORT(if2(eq(if1(false, 0, s(x0), x1), 0), if1(false, 0, s(x0), x1), 0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), z2))) -> c15(SORT(del(max(cons(s(x0), z2)), cons(0, cons(s(x0), z2))))) S tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) MAX(cons(s(s(z0)), cons(s(s(0)), x2))) -> c2(IF1(true, s(s(z0)), s(s(0)), x2), GE(s(s(z0)), s(s(0)))) MAX(cons(s(s(0)), cons(s(s(s(z0))), x2))) -> c2(IF1(false, s(s(0)), s(s(s(z0))), x2), GE(s(s(0)), s(s(s(z0))))) MAX(cons(s(s(x0)), cons(s(s(x1)), x2))) -> c2(GE(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c6(IF2(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c6(IF2(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c6(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c6(EQ(s(s(0)), s(s(0)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) MAX(cons(s(s(s(z0))), cons(s(s(s(0))), x2))) -> c2(IF1(true, s(s(s(z0))), s(s(s(0))), x2), GE(s(s(s(z0))), s(s(s(0))))) MAX(cons(s(s(s(0))), cons(s(s(s(s(z0)))), x2))) -> c2(IF1(false, s(s(s(0))), s(s(s(s(z0)))), x2), GE(s(s(s(0))), s(s(s(s(z0)))))) MAX(cons(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2))) -> c2(IF1(ge(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), GE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MAX(cons(s(s(s(x0))), cons(s(s(s(x1))), x2))) -> c2(GE(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c6(IF2(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c6(IF2(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c6(IF2(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c6(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c6(EQ(s(s(s(0))), s(s(s(0))))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(max(cons(z0, cons(0, x2))), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(max(cons(0, cons(s(z0), x2))), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(max(cons(s(z0), cons(s(z1), x2))), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), max(cons(z0, cons(0, x2))), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), max(cons(0, cons(s(z0), x2))), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), max(cons(s(z0), cons(s(z1), x2))), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(ge(z0, 0), z0, 0, x2), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(ge(0, s(z0)), 0, s(z0), x2), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), if1(ge(z0, 0), z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), if1(ge(0, s(z0)), 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z1, cons(0, x1))) -> c15(SORT(if2(eq(if1(true, z1, 0, x1), z1), if1(true, z1, 0, x1), z1, cons(0, x1))), DEL(if1(ge(z1, 0), z1, 0, x1), cons(z1, cons(0, x1))), MAX(cons(z1, cons(0, x1)))) IF3(false, cons(z0, cons(0, z2))) -> c15(SORT(del(max(cons(z0, z2)), cons(z0, cons(0, z2)))), DEL(if1(ge(z0, 0), z0, 0, z2), cons(z0, cons(0, z2))), MAX(cons(z0, cons(0, z2)))) IF3(false, cons(0, cons(s(x0), x1))) -> c15(SORT(if2(eq(if1(false, 0, s(x0), x1), 0), if1(false, 0, s(x0), x1), 0, cons(s(x0), x1))), DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1))), MAX(cons(0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), z2))) -> c15(SORT(del(max(cons(s(x0), z2)), cons(0, cons(s(x0), z2)))), DEL(if1(ge(0, s(x0)), 0, s(x0), z2), cons(0, cons(s(x0), z2))), MAX(cons(0, cons(s(x0), z2)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c15(SORT(if2(eq(if1(ge(x0, x1), s(x0), s(x1), x2), s(x0)), if1(ge(x0, x1), s(x0), s(x1), x2), s(x0), cons(s(x1), x2))), DEL(if1(ge(s(x0), s(x1)), s(x0), s(x1), x2), cons(s(x0), cons(s(x1), x2))), MAX(cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(s(z0), cons(s(0), x2))) -> c15(SORT(del(if1(true, s(z0), s(0), x2), cons(s(z0), cons(s(0), x2)))), DEL(if1(ge(s(z0), s(0)), s(z0), s(0), x2), cons(s(z0), cons(s(0), x2))), MAX(cons(s(z0), cons(s(0), x2)))) IF3(false, cons(s(0), cons(s(s(z0)), x2))) -> c15(SORT(del(if1(false, s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2)))), DEL(if1(ge(s(0), s(s(z0))), s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2))), MAX(cons(s(0), cons(s(s(z0)), x2)))) IF3(false, cons(s(s(z0)), cons(s(s(z1)), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2)))), DEL(if1(ge(s(s(z0)), s(s(z1))), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2))), MAX(cons(s(s(z0)), cons(s(s(z1)), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(max(cons(z0, cons(0, x2))), z0), if1(true, z0, 0, x2), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(max(cons(0, cons(s(z0), x2))), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(max(cons(s(z0), cons(s(z1), x2))), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2)))) IF3(false, cons(s(z0), nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(s(z0), nil)), s(z0), nil))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), max(cons(z0, cons(0, x2))), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), max(cons(0, cons(s(z0), x2))), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), max(cons(s(z0), cons(s(z1), x2))), s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(ge(z0, 0), z0, 0, x2), z0), if1(true, z0, 0, x2), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(ge(0, s(z0)), 0, s(z0), x2), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), if1(ge(z0, 0), z0, 0, x2), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), if1(ge(0, s(z0)), 0, s(z0), x2), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0), cons(s(z1), x2)))) IF3(false, cons(z1, cons(0, x1))) -> c15(SORT(if2(eq(if1(true, z1, 0, x1), z1), if1(true, z1, 0, x1), z1, cons(0, x1)))) IF3(false, cons(z0, cons(0, z2))) -> c15(SORT(del(max(cons(z0, z2)), cons(z0, cons(0, z2))))) IF3(false, cons(0, cons(s(x0), x1))) -> c15(SORT(if2(eq(if1(false, 0, s(x0), x1), 0), if1(false, 0, s(x0), x1), 0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), z2))) -> c15(SORT(del(max(cons(s(x0), z2)), cons(0, cons(s(x0), z2))))) K tuples: IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) IF3(false, cons(x0, cons(0, x1))) -> c(DEL(if1(ge(x0, 0), x0, 0, x1), cons(x0, cons(0, x1)))) IF3(false, cons(x0, cons(0, x1))) -> c(MAX(cons(x0, cons(0, x1)))) IF3(false, cons(0, cons(s(x0), x1))) -> c1(DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), x1))) -> c1(MAX(cons(0, cons(s(x0), x1)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c5(DEL(if1(ge(s(x0), s(x1)), s(x0), s(x1), x2), cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c5(MAX(cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(0, nil)) -> c15(SORT(if2(true, max(cons(0, nil)), 0, nil))) Defined Rule Symbols: del_2, max_1, if2_4, eq_2, ge_2, if1_4 Defined Pair Symbols: IF1_4, IF2_4, EQ_2, GE_2, MAX_1, DEL_2, SORT_1, IF3_2 Compound Symbols: c3_1, c4_1, c8_1, c12_1, c20_1, c2_1, c6_1, c13_1, c2_2, c6_2, c15_1, c15_3, c15_2, c_1, c1_1, c5_1 ---------------------------------------- (187) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))))) by IF3(false, cons(s(x0), cons(s(x1), x2))) -> c15(SORT(if2(eq(if1(ge(x0, x1), s(x0), s(x1), x2), s(x0)), if1(ge(x0, x1), s(x0), s(x1), x2), s(x0), cons(s(x1), x2)))) IF3(false, cons(s(z0), cons(s(0), x2))) -> c15(SORT(del(if1(true, s(z0), s(0), x2), cons(s(z0), cons(s(0), x2))))) IF3(false, cons(s(0), cons(s(s(z0)), x2))) -> c15(SORT(del(if1(false, s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2))))) IF3(false, cons(s(s(z0)), cons(s(s(z1)), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2))))) ---------------------------------------- (188) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil max(cons(z0, nil)) -> z0 max(cons(z0, cons(z1, z2))) -> if1(ge(z0, z1), z0, z1, z2) if2(true, z0, z1, z2) -> z2 if2(false, z0, z1, z2) -> cons(z1, del(z0, z2)) eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) if1(true, z0, z1, z2) -> max(cons(z0, z2)) if1(false, z0, z1, z2) -> max(cons(z1, z2)) Tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) MAX(cons(s(s(z0)), cons(s(s(0)), x2))) -> c2(IF1(true, s(s(z0)), s(s(0)), x2), GE(s(s(z0)), s(s(0)))) MAX(cons(s(s(0)), cons(s(s(s(z0))), x2))) -> c2(IF1(false, s(s(0)), s(s(s(z0))), x2), GE(s(s(0)), s(s(s(z0))))) MAX(cons(s(s(x0)), cons(s(s(x1)), x2))) -> c2(GE(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c6(IF2(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c6(IF2(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c6(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c6(EQ(s(s(0)), s(s(0)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(if1(ge(z1, x1), z1, x1, x2), cons(z1, cons(x1, x2))), MAX(cons(z1, cons(x1, x2)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) MAX(cons(s(s(s(z0))), cons(s(s(s(0))), x2))) -> c2(IF1(true, s(s(s(z0))), s(s(s(0))), x2), GE(s(s(s(z0))), s(s(s(0))))) MAX(cons(s(s(s(0))), cons(s(s(s(s(z0)))), x2))) -> c2(IF1(false, s(s(s(0))), s(s(s(s(z0)))), x2), GE(s(s(s(0))), s(s(s(s(z0)))))) MAX(cons(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2))) -> c2(IF1(ge(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), GE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MAX(cons(s(s(s(x0))), cons(s(s(s(x1))), x2))) -> c2(GE(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c6(IF2(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c6(IF2(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c6(IF2(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c6(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c6(EQ(s(s(s(0))), s(s(s(0))))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(max(cons(z0, cons(0, x2))), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(max(cons(0, cons(s(z0), x2))), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(max(cons(s(z0), cons(s(z1), x2))), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), max(cons(z0, cons(0, x2))), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), max(cons(0, cons(s(z0), x2))), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), max(cons(s(z0), cons(s(z1), x2))), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(ge(z0, 0), z0, 0, x2), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(ge(0, s(z0)), 0, s(z0), x2), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), if1(ge(z0, 0), z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), if1(ge(0, s(z0)), 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z1, cons(0, x1))) -> c15(SORT(if2(eq(if1(true, z1, 0, x1), z1), if1(true, z1, 0, x1), z1, cons(0, x1))), DEL(if1(ge(z1, 0), z1, 0, x1), cons(z1, cons(0, x1))), MAX(cons(z1, cons(0, x1)))) IF3(false, cons(z0, cons(0, z2))) -> c15(SORT(del(max(cons(z0, z2)), cons(z0, cons(0, z2)))), DEL(if1(ge(z0, 0), z0, 0, z2), cons(z0, cons(0, z2))), MAX(cons(z0, cons(0, z2)))) IF3(false, cons(x0, cons(0, x1))) -> c(DEL(if1(ge(x0, 0), x0, 0, x1), cons(x0, cons(0, x1)))) IF3(false, cons(x0, cons(0, x1))) -> c(MAX(cons(x0, cons(0, x1)))) IF3(false, cons(0, cons(s(x0), x1))) -> c15(SORT(if2(eq(if1(false, 0, s(x0), x1), 0), if1(false, 0, s(x0), x1), 0, cons(s(x0), x1))), DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1))), MAX(cons(0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), z2))) -> c15(SORT(del(max(cons(s(x0), z2)), cons(0, cons(s(x0), z2)))), DEL(if1(ge(0, s(x0)), 0, s(x0), z2), cons(0, cons(s(x0), z2))), MAX(cons(0, cons(s(x0), z2)))) IF3(false, cons(0, cons(s(x0), x1))) -> c1(DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), x1))) -> c1(MAX(cons(0, cons(s(x0), x1)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c15(SORT(if2(eq(if1(ge(x0, x1), s(x0), s(x1), x2), s(x0)), if1(ge(x0, x1), s(x0), s(x1), x2), s(x0), cons(s(x1), x2))), DEL(if1(ge(s(x0), s(x1)), s(x0), s(x1), x2), cons(s(x0), cons(s(x1), x2))), MAX(cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(s(z0), cons(s(0), x2))) -> c15(SORT(del(if1(true, s(z0), s(0), x2), cons(s(z0), cons(s(0), x2)))), DEL(if1(ge(s(z0), s(0)), s(z0), s(0), x2), cons(s(z0), cons(s(0), x2))), MAX(cons(s(z0), cons(s(0), x2)))) IF3(false, cons(s(0), cons(s(s(z0)), x2))) -> c15(SORT(del(if1(false, s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2)))), DEL(if1(ge(s(0), s(s(z0))), s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2))), MAX(cons(s(0), cons(s(s(z0)), x2)))) IF3(false, cons(s(s(z0)), cons(s(s(z1)), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2)))), DEL(if1(ge(s(s(z0)), s(s(z1))), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2))), MAX(cons(s(s(z0)), cons(s(s(z1)), x2)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c5(DEL(if1(ge(s(x0), s(x1)), s(x0), s(x1), x2), cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c5(MAX(cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(max(cons(z0, cons(0, x2))), z0), if1(true, z0, 0, x2), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(max(cons(0, cons(s(z0), x2))), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(max(cons(s(z0), cons(s(z1), x2))), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2)))) IF3(false, cons(0, nil)) -> c15(SORT(if2(true, max(cons(0, nil)), 0, nil))) IF3(false, cons(s(z0), nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(s(z0), nil)), s(z0), nil))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), max(cons(z0, cons(0, x2))), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), max(cons(0, cons(s(z0), x2))), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), max(cons(s(z0), cons(s(z1), x2))), s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(ge(z0, 0), z0, 0, x2), z0), if1(true, z0, 0, x2), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(ge(0, s(z0)), 0, s(z0), x2), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), if1(ge(z0, 0), z0, 0, x2), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), if1(ge(0, s(z0)), 0, s(z0), x2), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0), cons(s(z1), x2)))) IF3(false, cons(z1, cons(0, x1))) -> c15(SORT(if2(eq(if1(true, z1, 0, x1), z1), if1(true, z1, 0, x1), z1, cons(0, x1)))) IF3(false, cons(z0, cons(0, z2))) -> c15(SORT(del(max(cons(z0, z2)), cons(z0, cons(0, z2))))) IF3(false, cons(0, cons(s(x0), x1))) -> c15(SORT(if2(eq(if1(false, 0, s(x0), x1), 0), if1(false, 0, s(x0), x1), 0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), z2))) -> c15(SORT(del(max(cons(s(x0), z2)), cons(0, cons(s(x0), z2))))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c15(SORT(if2(eq(if1(ge(x0, x1), s(x0), s(x1), x2), s(x0)), if1(ge(x0, x1), s(x0), s(x1), x2), s(x0), cons(s(x1), x2)))) IF3(false, cons(s(z0), cons(s(0), x2))) -> c15(SORT(del(if1(true, s(z0), s(0), x2), cons(s(z0), cons(s(0), x2))))) IF3(false, cons(s(0), cons(s(s(z0)), x2))) -> c15(SORT(del(if1(false, s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2))))) IF3(false, cons(s(s(z0)), cons(s(s(z1)), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2))))) S tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) MAX(cons(s(s(z0)), cons(s(s(0)), x2))) -> c2(IF1(true, s(s(z0)), s(s(0)), x2), GE(s(s(z0)), s(s(0)))) MAX(cons(s(s(0)), cons(s(s(s(z0))), x2))) -> c2(IF1(false, s(s(0)), s(s(s(z0))), x2), GE(s(s(0)), s(s(s(z0))))) MAX(cons(s(s(x0)), cons(s(s(x1)), x2))) -> c2(GE(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c6(IF2(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c6(IF2(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c6(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c6(EQ(s(s(0)), s(s(0)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) MAX(cons(s(s(s(z0))), cons(s(s(s(0))), x2))) -> c2(IF1(true, s(s(s(z0))), s(s(s(0))), x2), GE(s(s(s(z0))), s(s(s(0))))) MAX(cons(s(s(s(0))), cons(s(s(s(s(z0)))), x2))) -> c2(IF1(false, s(s(s(0))), s(s(s(s(z0)))), x2), GE(s(s(s(0))), s(s(s(s(z0)))))) MAX(cons(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2))) -> c2(IF1(ge(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), GE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MAX(cons(s(s(s(x0))), cons(s(s(s(x1))), x2))) -> c2(GE(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c6(IF2(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c6(IF2(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c6(IF2(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c6(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c6(EQ(s(s(s(0))), s(s(s(0))))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(max(cons(z0, cons(0, x2))), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(max(cons(0, cons(s(z0), x2))), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(max(cons(s(z0), cons(s(z1), x2))), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), max(cons(z0, cons(0, x2))), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), max(cons(0, cons(s(z0), x2))), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), max(cons(s(z0), cons(s(z1), x2))), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(ge(z0, 0), z0, 0, x2), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(ge(0, s(z0)), 0, s(z0), x2), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), if1(ge(z0, 0), z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), if1(ge(0, s(z0)), 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z1, cons(0, x1))) -> c15(SORT(if2(eq(if1(true, z1, 0, x1), z1), if1(true, z1, 0, x1), z1, cons(0, x1))), DEL(if1(ge(z1, 0), z1, 0, x1), cons(z1, cons(0, x1))), MAX(cons(z1, cons(0, x1)))) IF3(false, cons(z0, cons(0, z2))) -> c15(SORT(del(max(cons(z0, z2)), cons(z0, cons(0, z2)))), DEL(if1(ge(z0, 0), z0, 0, z2), cons(z0, cons(0, z2))), MAX(cons(z0, cons(0, z2)))) IF3(false, cons(0, cons(s(x0), x1))) -> c15(SORT(if2(eq(if1(false, 0, s(x0), x1), 0), if1(false, 0, s(x0), x1), 0, cons(s(x0), x1))), DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1))), MAX(cons(0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), z2))) -> c15(SORT(del(max(cons(s(x0), z2)), cons(0, cons(s(x0), z2)))), DEL(if1(ge(0, s(x0)), 0, s(x0), z2), cons(0, cons(s(x0), z2))), MAX(cons(0, cons(s(x0), z2)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c15(SORT(if2(eq(if1(ge(x0, x1), s(x0), s(x1), x2), s(x0)), if1(ge(x0, x1), s(x0), s(x1), x2), s(x0), cons(s(x1), x2))), DEL(if1(ge(s(x0), s(x1)), s(x0), s(x1), x2), cons(s(x0), cons(s(x1), x2))), MAX(cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(s(z0), cons(s(0), x2))) -> c15(SORT(del(if1(true, s(z0), s(0), x2), cons(s(z0), cons(s(0), x2)))), DEL(if1(ge(s(z0), s(0)), s(z0), s(0), x2), cons(s(z0), cons(s(0), x2))), MAX(cons(s(z0), cons(s(0), x2)))) IF3(false, cons(s(0), cons(s(s(z0)), x2))) -> c15(SORT(del(if1(false, s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2)))), DEL(if1(ge(s(0), s(s(z0))), s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2))), MAX(cons(s(0), cons(s(s(z0)), x2)))) IF3(false, cons(s(s(z0)), cons(s(s(z1)), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2)))), DEL(if1(ge(s(s(z0)), s(s(z1))), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2))), MAX(cons(s(s(z0)), cons(s(s(z1)), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(max(cons(z0, cons(0, x2))), z0), if1(true, z0, 0, x2), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(max(cons(0, cons(s(z0), x2))), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(max(cons(s(z0), cons(s(z1), x2))), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2)))) IF3(false, cons(s(z0), nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(s(z0), nil)), s(z0), nil))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), max(cons(z0, cons(0, x2))), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), max(cons(0, cons(s(z0), x2))), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), max(cons(s(z0), cons(s(z1), x2))), s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(ge(z0, 0), z0, 0, x2), z0), if1(true, z0, 0, x2), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(ge(0, s(z0)), 0, s(z0), x2), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), if1(ge(z0, 0), z0, 0, x2), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), if1(ge(0, s(z0)), 0, s(z0), x2), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0), cons(s(z1), x2)))) IF3(false, cons(z1, cons(0, x1))) -> c15(SORT(if2(eq(if1(true, z1, 0, x1), z1), if1(true, z1, 0, x1), z1, cons(0, x1)))) IF3(false, cons(z0, cons(0, z2))) -> c15(SORT(del(max(cons(z0, z2)), cons(z0, cons(0, z2))))) IF3(false, cons(0, cons(s(x0), x1))) -> c15(SORT(if2(eq(if1(false, 0, s(x0), x1), 0), if1(false, 0, s(x0), x1), 0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), z2))) -> c15(SORT(del(max(cons(s(x0), z2)), cons(0, cons(s(x0), z2))))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c15(SORT(if2(eq(if1(ge(x0, x1), s(x0), s(x1), x2), s(x0)), if1(ge(x0, x1), s(x0), s(x1), x2), s(x0), cons(s(x1), x2)))) IF3(false, cons(s(z0), cons(s(0), x2))) -> c15(SORT(del(if1(true, s(z0), s(0), x2), cons(s(z0), cons(s(0), x2))))) IF3(false, cons(s(0), cons(s(s(z0)), x2))) -> c15(SORT(del(if1(false, s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2))))) IF3(false, cons(s(s(z0)), cons(s(s(z1)), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2))))) K tuples: IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) IF3(false, cons(x0, cons(0, x1))) -> c(DEL(if1(ge(x0, 0), x0, 0, x1), cons(x0, cons(0, x1)))) IF3(false, cons(x0, cons(0, x1))) -> c(MAX(cons(x0, cons(0, x1)))) IF3(false, cons(0, cons(s(x0), x1))) -> c1(DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), x1))) -> c1(MAX(cons(0, cons(s(x0), x1)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c5(DEL(if1(ge(s(x0), s(x1)), s(x0), s(x1), x2), cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c5(MAX(cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(0, nil)) -> c15(SORT(if2(true, max(cons(0, nil)), 0, nil))) Defined Rule Symbols: del_2, max_1, if2_4, eq_2, ge_2, if1_4 Defined Pair Symbols: IF1_4, IF2_4, EQ_2, GE_2, MAX_1, DEL_2, SORT_1, IF3_2 Compound Symbols: c3_1, c4_1, c8_1, c12_1, c20_1, c2_1, c6_1, c13_1, c2_2, c6_2, c15_1, c15_3, c15_2, c_1, c1_1, c5_1 ---------------------------------------- (189) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil))) by IF3(false, cons(0, nil)) -> c15(SORT(if2(true, 0, 0, nil))) IF3(false, cons(s(z0), nil)) -> c15(SORT(if2(eq(z0, z0), s(z0), s(z0), nil))) ---------------------------------------- (190) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil max(cons(z0, nil)) -> z0 max(cons(z0, cons(z1, z2))) -> if1(ge(z0, z1), z0, z1, z2) if2(true, z0, z1, z2) -> z2 if2(false, z0, z1, z2) -> cons(z1, del(z0, z2)) eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) if1(true, z0, z1, z2) -> max(cons(z0, z2)) if1(false, z0, z1, z2) -> max(cons(z1, z2)) Tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) MAX(cons(s(s(z0)), cons(s(s(0)), x2))) -> c2(IF1(true, s(s(z0)), s(s(0)), x2), GE(s(s(z0)), s(s(0)))) MAX(cons(s(s(0)), cons(s(s(s(z0))), x2))) -> c2(IF1(false, s(s(0)), s(s(s(z0))), x2), GE(s(s(0)), s(s(s(z0))))) MAX(cons(s(s(x0)), cons(s(s(x1)), x2))) -> c2(GE(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c6(IF2(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c6(IF2(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c6(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c6(EQ(s(s(0)), s(s(0)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(if1(ge(z1, x1), z1, x1, x2), cons(z1, cons(x1, x2))), MAX(cons(z1, cons(x1, x2)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) MAX(cons(s(s(s(z0))), cons(s(s(s(0))), x2))) -> c2(IF1(true, s(s(s(z0))), s(s(s(0))), x2), GE(s(s(s(z0))), s(s(s(0))))) MAX(cons(s(s(s(0))), cons(s(s(s(s(z0)))), x2))) -> c2(IF1(false, s(s(s(0))), s(s(s(s(z0)))), x2), GE(s(s(s(0))), s(s(s(s(z0)))))) MAX(cons(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2))) -> c2(IF1(ge(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), GE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MAX(cons(s(s(s(x0))), cons(s(s(s(x1))), x2))) -> c2(GE(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c6(IF2(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c6(IF2(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c6(IF2(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c6(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c6(EQ(s(s(s(0))), s(s(s(0))))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(max(cons(z0, cons(0, x2))), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(max(cons(0, cons(s(z0), x2))), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(max(cons(s(z0), cons(s(z1), x2))), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), max(cons(z0, cons(0, x2))), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), max(cons(0, cons(s(z0), x2))), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), max(cons(s(z0), cons(s(z1), x2))), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(ge(z0, 0), z0, 0, x2), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(ge(0, s(z0)), 0, s(z0), x2), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), if1(ge(z0, 0), z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), if1(ge(0, s(z0)), 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z1, cons(0, x1))) -> c15(SORT(if2(eq(if1(true, z1, 0, x1), z1), if1(true, z1, 0, x1), z1, cons(0, x1))), DEL(if1(ge(z1, 0), z1, 0, x1), cons(z1, cons(0, x1))), MAX(cons(z1, cons(0, x1)))) IF3(false, cons(z0, cons(0, z2))) -> c15(SORT(del(max(cons(z0, z2)), cons(z0, cons(0, z2)))), DEL(if1(ge(z0, 0), z0, 0, z2), cons(z0, cons(0, z2))), MAX(cons(z0, cons(0, z2)))) IF3(false, cons(x0, cons(0, x1))) -> c(DEL(if1(ge(x0, 0), x0, 0, x1), cons(x0, cons(0, x1)))) IF3(false, cons(x0, cons(0, x1))) -> c(MAX(cons(x0, cons(0, x1)))) IF3(false, cons(0, cons(s(x0), x1))) -> c15(SORT(if2(eq(if1(false, 0, s(x0), x1), 0), if1(false, 0, s(x0), x1), 0, cons(s(x0), x1))), DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1))), MAX(cons(0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), z2))) -> c15(SORT(del(max(cons(s(x0), z2)), cons(0, cons(s(x0), z2)))), DEL(if1(ge(0, s(x0)), 0, s(x0), z2), cons(0, cons(s(x0), z2))), MAX(cons(0, cons(s(x0), z2)))) IF3(false, cons(0, cons(s(x0), x1))) -> c1(DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), x1))) -> c1(MAX(cons(0, cons(s(x0), x1)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c15(SORT(if2(eq(if1(ge(x0, x1), s(x0), s(x1), x2), s(x0)), if1(ge(x0, x1), s(x0), s(x1), x2), s(x0), cons(s(x1), x2))), DEL(if1(ge(s(x0), s(x1)), s(x0), s(x1), x2), cons(s(x0), cons(s(x1), x2))), MAX(cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(s(z0), cons(s(0), x2))) -> c15(SORT(del(if1(true, s(z0), s(0), x2), cons(s(z0), cons(s(0), x2)))), DEL(if1(ge(s(z0), s(0)), s(z0), s(0), x2), cons(s(z0), cons(s(0), x2))), MAX(cons(s(z0), cons(s(0), x2)))) IF3(false, cons(s(0), cons(s(s(z0)), x2))) -> c15(SORT(del(if1(false, s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2)))), DEL(if1(ge(s(0), s(s(z0))), s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2))), MAX(cons(s(0), cons(s(s(z0)), x2)))) IF3(false, cons(s(s(z0)), cons(s(s(z1)), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2)))), DEL(if1(ge(s(s(z0)), s(s(z1))), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2))), MAX(cons(s(s(z0)), cons(s(s(z1)), x2)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c5(DEL(if1(ge(s(x0), s(x1)), s(x0), s(x1), x2), cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c5(MAX(cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(max(cons(z0, cons(0, x2))), z0), if1(true, z0, 0, x2), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(max(cons(0, cons(s(z0), x2))), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(max(cons(s(z0), cons(s(z1), x2))), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2)))) IF3(false, cons(0, nil)) -> c15(SORT(if2(true, max(cons(0, nil)), 0, nil))) IF3(false, cons(s(z0), nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(s(z0), nil)), s(z0), nil))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), max(cons(z0, cons(0, x2))), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), max(cons(0, cons(s(z0), x2))), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), max(cons(s(z0), cons(s(z1), x2))), s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(ge(z0, 0), z0, 0, x2), z0), if1(true, z0, 0, x2), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(ge(0, s(z0)), 0, s(z0), x2), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), if1(ge(z0, 0), z0, 0, x2), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), if1(ge(0, s(z0)), 0, s(z0), x2), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0), cons(s(z1), x2)))) IF3(false, cons(z1, cons(0, x1))) -> c15(SORT(if2(eq(if1(true, z1, 0, x1), z1), if1(true, z1, 0, x1), z1, cons(0, x1)))) IF3(false, cons(z0, cons(0, z2))) -> c15(SORT(del(max(cons(z0, z2)), cons(z0, cons(0, z2))))) IF3(false, cons(0, cons(s(x0), x1))) -> c15(SORT(if2(eq(if1(false, 0, s(x0), x1), 0), if1(false, 0, s(x0), x1), 0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), z2))) -> c15(SORT(del(max(cons(s(x0), z2)), cons(0, cons(s(x0), z2))))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c15(SORT(if2(eq(if1(ge(x0, x1), s(x0), s(x1), x2), s(x0)), if1(ge(x0, x1), s(x0), s(x1), x2), s(x0), cons(s(x1), x2)))) IF3(false, cons(s(z0), cons(s(0), x2))) -> c15(SORT(del(if1(true, s(z0), s(0), x2), cons(s(z0), cons(s(0), x2))))) IF3(false, cons(s(0), cons(s(s(z0)), x2))) -> c15(SORT(del(if1(false, s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2))))) IF3(false, cons(s(s(z0)), cons(s(s(z1)), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2))))) IF3(false, cons(0, nil)) -> c15(SORT(if2(true, 0, 0, nil))) IF3(false, cons(s(z0), nil)) -> c15(SORT(if2(eq(z0, z0), s(z0), s(z0), nil))) S tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) MAX(cons(s(s(z0)), cons(s(s(0)), x2))) -> c2(IF1(true, s(s(z0)), s(s(0)), x2), GE(s(s(z0)), s(s(0)))) MAX(cons(s(s(0)), cons(s(s(s(z0))), x2))) -> c2(IF1(false, s(s(0)), s(s(s(z0))), x2), GE(s(s(0)), s(s(s(z0))))) MAX(cons(s(s(x0)), cons(s(s(x1)), x2))) -> c2(GE(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c6(IF2(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c6(IF2(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c6(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c6(EQ(s(s(0)), s(s(0)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) MAX(cons(s(s(s(z0))), cons(s(s(s(0))), x2))) -> c2(IF1(true, s(s(s(z0))), s(s(s(0))), x2), GE(s(s(s(z0))), s(s(s(0))))) MAX(cons(s(s(s(0))), cons(s(s(s(s(z0)))), x2))) -> c2(IF1(false, s(s(s(0))), s(s(s(s(z0)))), x2), GE(s(s(s(0))), s(s(s(s(z0)))))) MAX(cons(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2))) -> c2(IF1(ge(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), GE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MAX(cons(s(s(s(x0))), cons(s(s(s(x1))), x2))) -> c2(GE(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c6(IF2(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c6(IF2(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c6(IF2(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c6(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c6(EQ(s(s(s(0))), s(s(s(0))))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(max(cons(z0, cons(0, x2))), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(max(cons(0, cons(s(z0), x2))), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(max(cons(s(z0), cons(s(z1), x2))), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), max(cons(z0, cons(0, x2))), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), max(cons(0, cons(s(z0), x2))), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), max(cons(s(z0), cons(s(z1), x2))), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(ge(z0, 0), z0, 0, x2), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(ge(0, s(z0)), 0, s(z0), x2), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), if1(ge(z0, 0), z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), if1(ge(0, s(z0)), 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z1, cons(0, x1))) -> c15(SORT(if2(eq(if1(true, z1, 0, x1), z1), if1(true, z1, 0, x1), z1, cons(0, x1))), DEL(if1(ge(z1, 0), z1, 0, x1), cons(z1, cons(0, x1))), MAX(cons(z1, cons(0, x1)))) IF3(false, cons(z0, cons(0, z2))) -> c15(SORT(del(max(cons(z0, z2)), cons(z0, cons(0, z2)))), DEL(if1(ge(z0, 0), z0, 0, z2), cons(z0, cons(0, z2))), MAX(cons(z0, cons(0, z2)))) IF3(false, cons(0, cons(s(x0), x1))) -> c15(SORT(if2(eq(if1(false, 0, s(x0), x1), 0), if1(false, 0, s(x0), x1), 0, cons(s(x0), x1))), DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1))), MAX(cons(0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), z2))) -> c15(SORT(del(max(cons(s(x0), z2)), cons(0, cons(s(x0), z2)))), DEL(if1(ge(0, s(x0)), 0, s(x0), z2), cons(0, cons(s(x0), z2))), MAX(cons(0, cons(s(x0), z2)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c15(SORT(if2(eq(if1(ge(x0, x1), s(x0), s(x1), x2), s(x0)), if1(ge(x0, x1), s(x0), s(x1), x2), s(x0), cons(s(x1), x2))), DEL(if1(ge(s(x0), s(x1)), s(x0), s(x1), x2), cons(s(x0), cons(s(x1), x2))), MAX(cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(s(z0), cons(s(0), x2))) -> c15(SORT(del(if1(true, s(z0), s(0), x2), cons(s(z0), cons(s(0), x2)))), DEL(if1(ge(s(z0), s(0)), s(z0), s(0), x2), cons(s(z0), cons(s(0), x2))), MAX(cons(s(z0), cons(s(0), x2)))) IF3(false, cons(s(0), cons(s(s(z0)), x2))) -> c15(SORT(del(if1(false, s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2)))), DEL(if1(ge(s(0), s(s(z0))), s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2))), MAX(cons(s(0), cons(s(s(z0)), x2)))) IF3(false, cons(s(s(z0)), cons(s(s(z1)), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2)))), DEL(if1(ge(s(s(z0)), s(s(z1))), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2))), MAX(cons(s(s(z0)), cons(s(s(z1)), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(max(cons(z0, cons(0, x2))), z0), if1(true, z0, 0, x2), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(max(cons(0, cons(s(z0), x2))), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(max(cons(s(z0), cons(s(z1), x2))), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2)))) IF3(false, cons(s(z0), nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(s(z0), nil)), s(z0), nil))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), max(cons(z0, cons(0, x2))), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), max(cons(0, cons(s(z0), x2))), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), max(cons(s(z0), cons(s(z1), x2))), s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(ge(z0, 0), z0, 0, x2), z0), if1(true, z0, 0, x2), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(ge(0, s(z0)), 0, s(z0), x2), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), if1(ge(z0, 0), z0, 0, x2), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), if1(ge(0, s(z0)), 0, s(z0), x2), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0), cons(s(z1), x2)))) IF3(false, cons(z1, cons(0, x1))) -> c15(SORT(if2(eq(if1(true, z1, 0, x1), z1), if1(true, z1, 0, x1), z1, cons(0, x1)))) IF3(false, cons(z0, cons(0, z2))) -> c15(SORT(del(max(cons(z0, z2)), cons(z0, cons(0, z2))))) IF3(false, cons(0, cons(s(x0), x1))) -> c15(SORT(if2(eq(if1(false, 0, s(x0), x1), 0), if1(false, 0, s(x0), x1), 0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), z2))) -> c15(SORT(del(max(cons(s(x0), z2)), cons(0, cons(s(x0), z2))))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c15(SORT(if2(eq(if1(ge(x0, x1), s(x0), s(x1), x2), s(x0)), if1(ge(x0, x1), s(x0), s(x1), x2), s(x0), cons(s(x1), x2)))) IF3(false, cons(s(z0), cons(s(0), x2))) -> c15(SORT(del(if1(true, s(z0), s(0), x2), cons(s(z0), cons(s(0), x2))))) IF3(false, cons(s(0), cons(s(s(z0)), x2))) -> c15(SORT(del(if1(false, s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2))))) IF3(false, cons(s(s(z0)), cons(s(s(z1)), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2))))) IF3(false, cons(0, nil)) -> c15(SORT(if2(true, 0, 0, nil))) IF3(false, cons(s(z0), nil)) -> c15(SORT(if2(eq(z0, z0), s(z0), s(z0), nil))) K tuples: IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) IF3(false, cons(x0, cons(0, x1))) -> c(DEL(if1(ge(x0, 0), x0, 0, x1), cons(x0, cons(0, x1)))) IF3(false, cons(x0, cons(0, x1))) -> c(MAX(cons(x0, cons(0, x1)))) IF3(false, cons(0, cons(s(x0), x1))) -> c1(DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), x1))) -> c1(MAX(cons(0, cons(s(x0), x1)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c5(DEL(if1(ge(s(x0), s(x1)), s(x0), s(x1), x2), cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c5(MAX(cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(0, nil)) -> c15(SORT(if2(true, max(cons(0, nil)), 0, nil))) Defined Rule Symbols: del_2, max_1, if2_4, eq_2, ge_2, if1_4 Defined Pair Symbols: IF1_4, IF2_4, EQ_2, GE_2, MAX_1, DEL_2, SORT_1, IF3_2 Compound Symbols: c3_1, c4_1, c8_1, c12_1, c20_1, c2_1, c6_1, c13_1, c2_2, c6_2, c15_1, c15_3, c15_2, c_1, c1_1, c5_1 ---------------------------------------- (191) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. IF3(false, cons(0, nil)) -> c15(SORT(if2(true, 0, 0, nil))) We considered the (Usable) Rules: if2(false, z0, z1, z2) -> cons(z1, del(z0, z2)) del(z0, nil) -> nil eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) eq(0, 0) -> true if2(true, z0, z1, z2) -> z2 del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) And the Tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) MAX(cons(s(s(z0)), cons(s(s(0)), x2))) -> c2(IF1(true, s(s(z0)), s(s(0)), x2), GE(s(s(z0)), s(s(0)))) MAX(cons(s(s(0)), cons(s(s(s(z0))), x2))) -> c2(IF1(false, s(s(0)), s(s(s(z0))), x2), GE(s(s(0)), s(s(s(z0))))) MAX(cons(s(s(x0)), cons(s(s(x1)), x2))) -> c2(GE(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c6(IF2(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c6(IF2(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c6(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c6(EQ(s(s(0)), s(s(0)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(if1(ge(z1, x1), z1, x1, x2), cons(z1, cons(x1, x2))), MAX(cons(z1, cons(x1, x2)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) MAX(cons(s(s(s(z0))), cons(s(s(s(0))), x2))) -> c2(IF1(true, s(s(s(z0))), s(s(s(0))), x2), GE(s(s(s(z0))), s(s(s(0))))) MAX(cons(s(s(s(0))), cons(s(s(s(s(z0)))), x2))) -> c2(IF1(false, s(s(s(0))), s(s(s(s(z0)))), x2), GE(s(s(s(0))), s(s(s(s(z0)))))) MAX(cons(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2))) -> c2(IF1(ge(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), GE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MAX(cons(s(s(s(x0))), cons(s(s(s(x1))), x2))) -> c2(GE(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c6(IF2(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c6(IF2(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c6(IF2(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c6(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c6(EQ(s(s(s(0))), s(s(s(0))))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(max(cons(z0, cons(0, x2))), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(max(cons(0, cons(s(z0), x2))), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(max(cons(s(z0), cons(s(z1), x2))), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), max(cons(z0, cons(0, x2))), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), max(cons(0, cons(s(z0), x2))), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), max(cons(s(z0), cons(s(z1), x2))), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(ge(z0, 0), z0, 0, x2), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(ge(0, s(z0)), 0, s(z0), x2), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), if1(ge(z0, 0), z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), if1(ge(0, s(z0)), 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z1, cons(0, x1))) -> c15(SORT(if2(eq(if1(true, z1, 0, x1), z1), if1(true, z1, 0, x1), z1, cons(0, x1))), DEL(if1(ge(z1, 0), z1, 0, x1), cons(z1, cons(0, x1))), MAX(cons(z1, cons(0, x1)))) IF3(false, cons(z0, cons(0, z2))) -> c15(SORT(del(max(cons(z0, z2)), cons(z0, cons(0, z2)))), DEL(if1(ge(z0, 0), z0, 0, z2), cons(z0, cons(0, z2))), MAX(cons(z0, cons(0, z2)))) IF3(false, cons(x0, cons(0, x1))) -> c(DEL(if1(ge(x0, 0), x0, 0, x1), cons(x0, cons(0, x1)))) IF3(false, cons(x0, cons(0, x1))) -> c(MAX(cons(x0, cons(0, x1)))) IF3(false, cons(0, cons(s(x0), x1))) -> c15(SORT(if2(eq(if1(false, 0, s(x0), x1), 0), if1(false, 0, s(x0), x1), 0, cons(s(x0), x1))), DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1))), MAX(cons(0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), z2))) -> c15(SORT(del(max(cons(s(x0), z2)), cons(0, cons(s(x0), z2)))), DEL(if1(ge(0, s(x0)), 0, s(x0), z2), cons(0, cons(s(x0), z2))), MAX(cons(0, cons(s(x0), z2)))) IF3(false, cons(0, cons(s(x0), x1))) -> c1(DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), x1))) -> c1(MAX(cons(0, cons(s(x0), x1)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c15(SORT(if2(eq(if1(ge(x0, x1), s(x0), s(x1), x2), s(x0)), if1(ge(x0, x1), s(x0), s(x1), x2), s(x0), cons(s(x1), x2))), DEL(if1(ge(s(x0), s(x1)), s(x0), s(x1), x2), cons(s(x0), cons(s(x1), x2))), MAX(cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(s(z0), cons(s(0), x2))) -> c15(SORT(del(if1(true, s(z0), s(0), x2), cons(s(z0), cons(s(0), x2)))), DEL(if1(ge(s(z0), s(0)), s(z0), s(0), x2), cons(s(z0), cons(s(0), x2))), MAX(cons(s(z0), cons(s(0), x2)))) IF3(false, cons(s(0), cons(s(s(z0)), x2))) -> c15(SORT(del(if1(false, s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2)))), DEL(if1(ge(s(0), s(s(z0))), s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2))), MAX(cons(s(0), cons(s(s(z0)), x2)))) IF3(false, cons(s(s(z0)), cons(s(s(z1)), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2)))), DEL(if1(ge(s(s(z0)), s(s(z1))), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2))), MAX(cons(s(s(z0)), cons(s(s(z1)), x2)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c5(DEL(if1(ge(s(x0), s(x1)), s(x0), s(x1), x2), cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c5(MAX(cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(max(cons(z0, cons(0, x2))), z0), if1(true, z0, 0, x2), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(max(cons(0, cons(s(z0), x2))), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(max(cons(s(z0), cons(s(z1), x2))), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2)))) IF3(false, cons(0, nil)) -> c15(SORT(if2(true, max(cons(0, nil)), 0, nil))) IF3(false, cons(s(z0), nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(s(z0), nil)), s(z0), nil))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), max(cons(z0, cons(0, x2))), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), max(cons(0, cons(s(z0), x2))), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), max(cons(s(z0), cons(s(z1), x2))), s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(ge(z0, 0), z0, 0, x2), z0), if1(true, z0, 0, x2), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(ge(0, s(z0)), 0, s(z0), x2), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), if1(ge(z0, 0), z0, 0, x2), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), if1(ge(0, s(z0)), 0, s(z0), x2), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0), cons(s(z1), x2)))) IF3(false, cons(z1, cons(0, x1))) -> c15(SORT(if2(eq(if1(true, z1, 0, x1), z1), if1(true, z1, 0, x1), z1, cons(0, x1)))) IF3(false, cons(z0, cons(0, z2))) -> c15(SORT(del(max(cons(z0, z2)), cons(z0, cons(0, z2))))) IF3(false, cons(0, cons(s(x0), x1))) -> c15(SORT(if2(eq(if1(false, 0, s(x0), x1), 0), if1(false, 0, s(x0), x1), 0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), z2))) -> c15(SORT(del(max(cons(s(x0), z2)), cons(0, cons(s(x0), z2))))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c15(SORT(if2(eq(if1(ge(x0, x1), s(x0), s(x1), x2), s(x0)), if1(ge(x0, x1), s(x0), s(x1), x2), s(x0), cons(s(x1), x2)))) IF3(false, cons(s(z0), cons(s(0), x2))) -> c15(SORT(del(if1(true, s(z0), s(0), x2), cons(s(z0), cons(s(0), x2))))) IF3(false, cons(s(0), cons(s(s(z0)), x2))) -> c15(SORT(del(if1(false, s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2))))) IF3(false, cons(s(s(z0)), cons(s(s(z1)), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2))))) IF3(false, cons(0, nil)) -> c15(SORT(if2(true, 0, 0, nil))) IF3(false, cons(s(z0), nil)) -> c15(SORT(if2(eq(z0, z0), s(z0), s(z0), nil))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = [1] POL(DEL(x_1, x_2)) = 0 POL(EQ(x_1, x_2)) = 0 POL(GE(x_1, x_2)) = 0 POL(IF1(x_1, x_2, x_3, x_4)) = 0 POL(IF2(x_1, x_2, x_3, x_4)) = 0 POL(IF3(x_1, x_2)) = x_2 POL(MAX(x_1)) = 0 POL(SORT(x_1)) = x_1 POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c13(x_1)) = x_1 POL(c15(x_1)) = x_1 POL(c15(x_1, x_2)) = x_1 + x_2 POL(c15(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c2(x_1)) = x_1 POL(c2(x_1, x_2)) = x_1 + x_2 POL(c20(x_1)) = x_1 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(c6(x_1, x_2)) = x_1 + x_2 POL(c8(x_1)) = x_1 POL(cons(x_1, x_2)) = [1] + x_1 + x_2 POL(del(x_1, x_2)) = x_2 POL(eq(x_1, x_2)) = [1] POL(false) = [1] POL(ge(x_1, x_2)) = 0 POL(if1(x_1, x_2, x_3, x_4)) = 0 POL(if2(x_1, x_2, x_3, x_4)) = x_1 + x_3 + x_4 POL(max(x_1)) = 0 POL(nil) = 0 POL(s(x_1)) = [1] + x_1 POL(true) = 0 ---------------------------------------- (192) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil max(cons(z0, nil)) -> z0 max(cons(z0, cons(z1, z2))) -> if1(ge(z0, z1), z0, z1, z2) if2(true, z0, z1, z2) -> z2 if2(false, z0, z1, z2) -> cons(z1, del(z0, z2)) eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) if1(true, z0, z1, z2) -> max(cons(z0, z2)) if1(false, z0, z1, z2) -> max(cons(z1, z2)) Tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) MAX(cons(s(s(z0)), cons(s(s(0)), x2))) -> c2(IF1(true, s(s(z0)), s(s(0)), x2), GE(s(s(z0)), s(s(0)))) MAX(cons(s(s(0)), cons(s(s(s(z0))), x2))) -> c2(IF1(false, s(s(0)), s(s(s(z0))), x2), GE(s(s(0)), s(s(s(z0))))) MAX(cons(s(s(x0)), cons(s(s(x1)), x2))) -> c2(GE(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c6(IF2(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c6(IF2(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c6(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c6(EQ(s(s(0)), s(s(0)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(if1(ge(z1, x1), z1, x1, x2), cons(z1, cons(x1, x2))), MAX(cons(z1, cons(x1, x2)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) MAX(cons(s(s(s(z0))), cons(s(s(s(0))), x2))) -> c2(IF1(true, s(s(s(z0))), s(s(s(0))), x2), GE(s(s(s(z0))), s(s(s(0))))) MAX(cons(s(s(s(0))), cons(s(s(s(s(z0)))), x2))) -> c2(IF1(false, s(s(s(0))), s(s(s(s(z0)))), x2), GE(s(s(s(0))), s(s(s(s(z0)))))) MAX(cons(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2))) -> c2(IF1(ge(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), GE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MAX(cons(s(s(s(x0))), cons(s(s(s(x1))), x2))) -> c2(GE(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c6(IF2(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c6(IF2(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c6(IF2(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c6(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c6(EQ(s(s(s(0))), s(s(s(0))))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(max(cons(z0, cons(0, x2))), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(max(cons(0, cons(s(z0), x2))), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(max(cons(s(z0), cons(s(z1), x2))), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), max(cons(z0, cons(0, x2))), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), max(cons(0, cons(s(z0), x2))), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), max(cons(s(z0), cons(s(z1), x2))), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(ge(z0, 0), z0, 0, x2), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(ge(0, s(z0)), 0, s(z0), x2), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), if1(ge(z0, 0), z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), if1(ge(0, s(z0)), 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z1, cons(0, x1))) -> c15(SORT(if2(eq(if1(true, z1, 0, x1), z1), if1(true, z1, 0, x1), z1, cons(0, x1))), DEL(if1(ge(z1, 0), z1, 0, x1), cons(z1, cons(0, x1))), MAX(cons(z1, cons(0, x1)))) IF3(false, cons(z0, cons(0, z2))) -> c15(SORT(del(max(cons(z0, z2)), cons(z0, cons(0, z2)))), DEL(if1(ge(z0, 0), z0, 0, z2), cons(z0, cons(0, z2))), MAX(cons(z0, cons(0, z2)))) IF3(false, cons(x0, cons(0, x1))) -> c(DEL(if1(ge(x0, 0), x0, 0, x1), cons(x0, cons(0, x1)))) IF3(false, cons(x0, cons(0, x1))) -> c(MAX(cons(x0, cons(0, x1)))) IF3(false, cons(0, cons(s(x0), x1))) -> c15(SORT(if2(eq(if1(false, 0, s(x0), x1), 0), if1(false, 0, s(x0), x1), 0, cons(s(x0), x1))), DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1))), MAX(cons(0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), z2))) -> c15(SORT(del(max(cons(s(x0), z2)), cons(0, cons(s(x0), z2)))), DEL(if1(ge(0, s(x0)), 0, s(x0), z2), cons(0, cons(s(x0), z2))), MAX(cons(0, cons(s(x0), z2)))) IF3(false, cons(0, cons(s(x0), x1))) -> c1(DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), x1))) -> c1(MAX(cons(0, cons(s(x0), x1)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c15(SORT(if2(eq(if1(ge(x0, x1), s(x0), s(x1), x2), s(x0)), if1(ge(x0, x1), s(x0), s(x1), x2), s(x0), cons(s(x1), x2))), DEL(if1(ge(s(x0), s(x1)), s(x0), s(x1), x2), cons(s(x0), cons(s(x1), x2))), MAX(cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(s(z0), cons(s(0), x2))) -> c15(SORT(del(if1(true, s(z0), s(0), x2), cons(s(z0), cons(s(0), x2)))), DEL(if1(ge(s(z0), s(0)), s(z0), s(0), x2), cons(s(z0), cons(s(0), x2))), MAX(cons(s(z0), cons(s(0), x2)))) IF3(false, cons(s(0), cons(s(s(z0)), x2))) -> c15(SORT(del(if1(false, s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2)))), DEL(if1(ge(s(0), s(s(z0))), s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2))), MAX(cons(s(0), cons(s(s(z0)), x2)))) IF3(false, cons(s(s(z0)), cons(s(s(z1)), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2)))), DEL(if1(ge(s(s(z0)), s(s(z1))), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2))), MAX(cons(s(s(z0)), cons(s(s(z1)), x2)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c5(DEL(if1(ge(s(x0), s(x1)), s(x0), s(x1), x2), cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c5(MAX(cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(max(cons(z0, cons(0, x2))), z0), if1(true, z0, 0, x2), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(max(cons(0, cons(s(z0), x2))), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(max(cons(s(z0), cons(s(z1), x2))), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2)))) IF3(false, cons(0, nil)) -> c15(SORT(if2(true, max(cons(0, nil)), 0, nil))) IF3(false, cons(s(z0), nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(s(z0), nil)), s(z0), nil))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), max(cons(z0, cons(0, x2))), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), max(cons(0, cons(s(z0), x2))), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), max(cons(s(z0), cons(s(z1), x2))), s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(ge(z0, 0), z0, 0, x2), z0), if1(true, z0, 0, x2), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(ge(0, s(z0)), 0, s(z0), x2), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), if1(ge(z0, 0), z0, 0, x2), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), if1(ge(0, s(z0)), 0, s(z0), x2), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0), cons(s(z1), x2)))) IF3(false, cons(z1, cons(0, x1))) -> c15(SORT(if2(eq(if1(true, z1, 0, x1), z1), if1(true, z1, 0, x1), z1, cons(0, x1)))) IF3(false, cons(z0, cons(0, z2))) -> c15(SORT(del(max(cons(z0, z2)), cons(z0, cons(0, z2))))) IF3(false, cons(0, cons(s(x0), x1))) -> c15(SORT(if2(eq(if1(false, 0, s(x0), x1), 0), if1(false, 0, s(x0), x1), 0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), z2))) -> c15(SORT(del(max(cons(s(x0), z2)), cons(0, cons(s(x0), z2))))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c15(SORT(if2(eq(if1(ge(x0, x1), s(x0), s(x1), x2), s(x0)), if1(ge(x0, x1), s(x0), s(x1), x2), s(x0), cons(s(x1), x2)))) IF3(false, cons(s(z0), cons(s(0), x2))) -> c15(SORT(del(if1(true, s(z0), s(0), x2), cons(s(z0), cons(s(0), x2))))) IF3(false, cons(s(0), cons(s(s(z0)), x2))) -> c15(SORT(del(if1(false, s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2))))) IF3(false, cons(s(s(z0)), cons(s(s(z1)), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2))))) IF3(false, cons(0, nil)) -> c15(SORT(if2(true, 0, 0, nil))) IF3(false, cons(s(z0), nil)) -> c15(SORT(if2(eq(z0, z0), s(z0), s(z0), nil))) S tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) MAX(cons(s(s(z0)), cons(s(s(0)), x2))) -> c2(IF1(true, s(s(z0)), s(s(0)), x2), GE(s(s(z0)), s(s(0)))) MAX(cons(s(s(0)), cons(s(s(s(z0))), x2))) -> c2(IF1(false, s(s(0)), s(s(s(z0))), x2), GE(s(s(0)), s(s(s(z0))))) MAX(cons(s(s(x0)), cons(s(s(x1)), x2))) -> c2(GE(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c6(IF2(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c6(IF2(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c6(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c6(EQ(s(s(0)), s(s(0)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) MAX(cons(s(s(s(z0))), cons(s(s(s(0))), x2))) -> c2(IF1(true, s(s(s(z0))), s(s(s(0))), x2), GE(s(s(s(z0))), s(s(s(0))))) MAX(cons(s(s(s(0))), cons(s(s(s(s(z0)))), x2))) -> c2(IF1(false, s(s(s(0))), s(s(s(s(z0)))), x2), GE(s(s(s(0))), s(s(s(s(z0)))))) MAX(cons(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2))) -> c2(IF1(ge(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), GE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MAX(cons(s(s(s(x0))), cons(s(s(s(x1))), x2))) -> c2(GE(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c6(IF2(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c6(IF2(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c6(IF2(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c6(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c6(EQ(s(s(s(0))), s(s(s(0))))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(max(cons(z0, cons(0, x2))), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(max(cons(0, cons(s(z0), x2))), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(max(cons(s(z0), cons(s(z1), x2))), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), max(cons(z0, cons(0, x2))), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), max(cons(0, cons(s(z0), x2))), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), max(cons(s(z0), cons(s(z1), x2))), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(ge(z0, 0), z0, 0, x2), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(ge(0, s(z0)), 0, s(z0), x2), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), if1(ge(z0, 0), z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), if1(ge(0, s(z0)), 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z1, cons(0, x1))) -> c15(SORT(if2(eq(if1(true, z1, 0, x1), z1), if1(true, z1, 0, x1), z1, cons(0, x1))), DEL(if1(ge(z1, 0), z1, 0, x1), cons(z1, cons(0, x1))), MAX(cons(z1, cons(0, x1)))) IF3(false, cons(z0, cons(0, z2))) -> c15(SORT(del(max(cons(z0, z2)), cons(z0, cons(0, z2)))), DEL(if1(ge(z0, 0), z0, 0, z2), cons(z0, cons(0, z2))), MAX(cons(z0, cons(0, z2)))) IF3(false, cons(0, cons(s(x0), x1))) -> c15(SORT(if2(eq(if1(false, 0, s(x0), x1), 0), if1(false, 0, s(x0), x1), 0, cons(s(x0), x1))), DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1))), MAX(cons(0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), z2))) -> c15(SORT(del(max(cons(s(x0), z2)), cons(0, cons(s(x0), z2)))), DEL(if1(ge(0, s(x0)), 0, s(x0), z2), cons(0, cons(s(x0), z2))), MAX(cons(0, cons(s(x0), z2)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c15(SORT(if2(eq(if1(ge(x0, x1), s(x0), s(x1), x2), s(x0)), if1(ge(x0, x1), s(x0), s(x1), x2), s(x0), cons(s(x1), x2))), DEL(if1(ge(s(x0), s(x1)), s(x0), s(x1), x2), cons(s(x0), cons(s(x1), x2))), MAX(cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(s(z0), cons(s(0), x2))) -> c15(SORT(del(if1(true, s(z0), s(0), x2), cons(s(z0), cons(s(0), x2)))), DEL(if1(ge(s(z0), s(0)), s(z0), s(0), x2), cons(s(z0), cons(s(0), x2))), MAX(cons(s(z0), cons(s(0), x2)))) IF3(false, cons(s(0), cons(s(s(z0)), x2))) -> c15(SORT(del(if1(false, s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2)))), DEL(if1(ge(s(0), s(s(z0))), s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2))), MAX(cons(s(0), cons(s(s(z0)), x2)))) IF3(false, cons(s(s(z0)), cons(s(s(z1)), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2)))), DEL(if1(ge(s(s(z0)), s(s(z1))), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2))), MAX(cons(s(s(z0)), cons(s(s(z1)), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(max(cons(z0, cons(0, x2))), z0), if1(true, z0, 0, x2), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(max(cons(0, cons(s(z0), x2))), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(max(cons(s(z0), cons(s(z1), x2))), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2)))) IF3(false, cons(s(z0), nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(s(z0), nil)), s(z0), nil))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), max(cons(z0, cons(0, x2))), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), max(cons(0, cons(s(z0), x2))), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), max(cons(s(z0), cons(s(z1), x2))), s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(ge(z0, 0), z0, 0, x2), z0), if1(true, z0, 0, x2), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(ge(0, s(z0)), 0, s(z0), x2), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), if1(ge(z0, 0), z0, 0, x2), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), if1(ge(0, s(z0)), 0, s(z0), x2), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0), cons(s(z1), x2)))) IF3(false, cons(z1, cons(0, x1))) -> c15(SORT(if2(eq(if1(true, z1, 0, x1), z1), if1(true, z1, 0, x1), z1, cons(0, x1)))) IF3(false, cons(z0, cons(0, z2))) -> c15(SORT(del(max(cons(z0, z2)), cons(z0, cons(0, z2))))) IF3(false, cons(0, cons(s(x0), x1))) -> c15(SORT(if2(eq(if1(false, 0, s(x0), x1), 0), if1(false, 0, s(x0), x1), 0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), z2))) -> c15(SORT(del(max(cons(s(x0), z2)), cons(0, cons(s(x0), z2))))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c15(SORT(if2(eq(if1(ge(x0, x1), s(x0), s(x1), x2), s(x0)), if1(ge(x0, x1), s(x0), s(x1), x2), s(x0), cons(s(x1), x2)))) IF3(false, cons(s(z0), cons(s(0), x2))) -> c15(SORT(del(if1(true, s(z0), s(0), x2), cons(s(z0), cons(s(0), x2))))) IF3(false, cons(s(0), cons(s(s(z0)), x2))) -> c15(SORT(del(if1(false, s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2))))) IF3(false, cons(s(s(z0)), cons(s(s(z1)), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2))))) IF3(false, cons(s(z0), nil)) -> c15(SORT(if2(eq(z0, z0), s(z0), s(z0), nil))) K tuples: IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) IF3(false, cons(x0, cons(0, x1))) -> c(DEL(if1(ge(x0, 0), x0, 0, x1), cons(x0, cons(0, x1)))) IF3(false, cons(x0, cons(0, x1))) -> c(MAX(cons(x0, cons(0, x1)))) IF3(false, cons(0, cons(s(x0), x1))) -> c1(DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), x1))) -> c1(MAX(cons(0, cons(s(x0), x1)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c5(DEL(if1(ge(s(x0), s(x1)), s(x0), s(x1), x2), cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c5(MAX(cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(0, nil)) -> c15(SORT(if2(true, max(cons(0, nil)), 0, nil))) IF3(false, cons(0, nil)) -> c15(SORT(if2(true, 0, 0, nil))) Defined Rule Symbols: del_2, max_1, if2_4, eq_2, ge_2, if1_4 Defined Pair Symbols: IF1_4, IF2_4, EQ_2, GE_2, MAX_1, DEL_2, SORT_1, IF3_2 Compound Symbols: c3_1, c4_1, c8_1, c12_1, c20_1, c2_1, c6_1, c13_1, c2_2, c6_2, c15_1, c15_3, c15_2, c_1, c1_1, c5_1 ---------------------------------------- (193) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF3(false, cons(0, nil)) -> c15(SORT(if2(true, 0, 0, nil))) by IF3(false, cons(0, nil)) -> c15(SORT(nil)) ---------------------------------------- (194) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil max(cons(z0, nil)) -> z0 max(cons(z0, cons(z1, z2))) -> if1(ge(z0, z1), z0, z1, z2) if2(true, z0, z1, z2) -> z2 if2(false, z0, z1, z2) -> cons(z1, del(z0, z2)) eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) if1(true, z0, z1, z2) -> max(cons(z0, z2)) if1(false, z0, z1, z2) -> max(cons(z1, z2)) Tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) MAX(cons(s(s(z0)), cons(s(s(0)), x2))) -> c2(IF1(true, s(s(z0)), s(s(0)), x2), GE(s(s(z0)), s(s(0)))) MAX(cons(s(s(0)), cons(s(s(s(z0))), x2))) -> c2(IF1(false, s(s(0)), s(s(s(z0))), x2), GE(s(s(0)), s(s(s(z0))))) MAX(cons(s(s(x0)), cons(s(s(x1)), x2))) -> c2(GE(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c6(IF2(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c6(IF2(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c6(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c6(EQ(s(s(0)), s(s(0)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(if1(ge(z1, x1), z1, x1, x2), cons(z1, cons(x1, x2))), MAX(cons(z1, cons(x1, x2)))) IF3(false, cons(z1, cons(x1, x2))) -> c15(SORT(if2(eq(if1(ge(z1, x1), z1, x1, x2), z1), if1(ge(z1, x1), z1, x1, x2), z1, cons(x1, x2)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) MAX(cons(s(s(s(z0))), cons(s(s(s(0))), x2))) -> c2(IF1(true, s(s(s(z0))), s(s(s(0))), x2), GE(s(s(s(z0))), s(s(s(0))))) MAX(cons(s(s(s(0))), cons(s(s(s(s(z0)))), x2))) -> c2(IF1(false, s(s(s(0))), s(s(s(s(z0)))), x2), GE(s(s(s(0))), s(s(s(s(z0)))))) MAX(cons(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2))) -> c2(IF1(ge(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), GE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MAX(cons(s(s(s(x0))), cons(s(s(s(x1))), x2))) -> c2(GE(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c6(IF2(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c6(IF2(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c6(IF2(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c6(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c6(EQ(s(s(s(0))), s(s(s(0))))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(max(cons(z0, cons(0, x2))), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(max(cons(0, cons(s(z0), x2))), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(max(cons(s(z0), cons(s(z1), x2))), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), max(cons(z0, cons(0, x2))), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), max(cons(0, cons(s(z0), x2))), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), max(cons(s(z0), cons(s(z1), x2))), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(ge(z0, 0), z0, 0, x2), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(ge(0, s(z0)), 0, s(z0), x2), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), if1(ge(z0, 0), z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), if1(ge(0, s(z0)), 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z1, cons(0, x1))) -> c15(SORT(if2(eq(if1(true, z1, 0, x1), z1), if1(true, z1, 0, x1), z1, cons(0, x1))), DEL(if1(ge(z1, 0), z1, 0, x1), cons(z1, cons(0, x1))), MAX(cons(z1, cons(0, x1)))) IF3(false, cons(z0, cons(0, z2))) -> c15(SORT(del(max(cons(z0, z2)), cons(z0, cons(0, z2)))), DEL(if1(ge(z0, 0), z0, 0, z2), cons(z0, cons(0, z2))), MAX(cons(z0, cons(0, z2)))) IF3(false, cons(x0, cons(0, x1))) -> c(DEL(if1(ge(x0, 0), x0, 0, x1), cons(x0, cons(0, x1)))) IF3(false, cons(x0, cons(0, x1))) -> c(MAX(cons(x0, cons(0, x1)))) IF3(false, cons(0, cons(s(x0), x1))) -> c15(SORT(if2(eq(if1(false, 0, s(x0), x1), 0), if1(false, 0, s(x0), x1), 0, cons(s(x0), x1))), DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1))), MAX(cons(0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), z2))) -> c15(SORT(del(max(cons(s(x0), z2)), cons(0, cons(s(x0), z2)))), DEL(if1(ge(0, s(x0)), 0, s(x0), z2), cons(0, cons(s(x0), z2))), MAX(cons(0, cons(s(x0), z2)))) IF3(false, cons(0, cons(s(x0), x1))) -> c1(DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), x1))) -> c1(MAX(cons(0, cons(s(x0), x1)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c15(SORT(if2(eq(if1(ge(x0, x1), s(x0), s(x1), x2), s(x0)), if1(ge(x0, x1), s(x0), s(x1), x2), s(x0), cons(s(x1), x2))), DEL(if1(ge(s(x0), s(x1)), s(x0), s(x1), x2), cons(s(x0), cons(s(x1), x2))), MAX(cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(s(z0), cons(s(0), x2))) -> c15(SORT(del(if1(true, s(z0), s(0), x2), cons(s(z0), cons(s(0), x2)))), DEL(if1(ge(s(z0), s(0)), s(z0), s(0), x2), cons(s(z0), cons(s(0), x2))), MAX(cons(s(z0), cons(s(0), x2)))) IF3(false, cons(s(0), cons(s(s(z0)), x2))) -> c15(SORT(del(if1(false, s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2)))), DEL(if1(ge(s(0), s(s(z0))), s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2))), MAX(cons(s(0), cons(s(s(z0)), x2)))) IF3(false, cons(s(s(z0)), cons(s(s(z1)), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2)))), DEL(if1(ge(s(s(z0)), s(s(z1))), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2))), MAX(cons(s(s(z0)), cons(s(s(z1)), x2)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c5(DEL(if1(ge(s(x0), s(x1)), s(x0), s(x1), x2), cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c5(MAX(cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(max(cons(z0, cons(0, x2))), z0), if1(true, z0, 0, x2), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(max(cons(0, cons(s(z0), x2))), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(max(cons(s(z0), cons(s(z1), x2))), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2)))) IF3(false, cons(0, nil)) -> c15(SORT(if2(true, max(cons(0, nil)), 0, nil))) IF3(false, cons(s(z0), nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(s(z0), nil)), s(z0), nil))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), max(cons(z0, cons(0, x2))), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), max(cons(0, cons(s(z0), x2))), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), max(cons(s(z0), cons(s(z1), x2))), s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(ge(z0, 0), z0, 0, x2), z0), if1(true, z0, 0, x2), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(ge(0, s(z0)), 0, s(z0), x2), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), if1(ge(z0, 0), z0, 0, x2), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), if1(ge(0, s(z0)), 0, s(z0), x2), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0), cons(s(z1), x2)))) IF3(false, cons(z1, cons(0, x1))) -> c15(SORT(if2(eq(if1(true, z1, 0, x1), z1), if1(true, z1, 0, x1), z1, cons(0, x1)))) IF3(false, cons(z0, cons(0, z2))) -> c15(SORT(del(max(cons(z0, z2)), cons(z0, cons(0, z2))))) IF3(false, cons(0, cons(s(x0), x1))) -> c15(SORT(if2(eq(if1(false, 0, s(x0), x1), 0), if1(false, 0, s(x0), x1), 0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), z2))) -> c15(SORT(del(max(cons(s(x0), z2)), cons(0, cons(s(x0), z2))))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c15(SORT(if2(eq(if1(ge(x0, x1), s(x0), s(x1), x2), s(x0)), if1(ge(x0, x1), s(x0), s(x1), x2), s(x0), cons(s(x1), x2)))) IF3(false, cons(s(z0), cons(s(0), x2))) -> c15(SORT(del(if1(true, s(z0), s(0), x2), cons(s(z0), cons(s(0), x2))))) IF3(false, cons(s(0), cons(s(s(z0)), x2))) -> c15(SORT(del(if1(false, s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2))))) IF3(false, cons(s(s(z0)), cons(s(s(z1)), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2))))) IF3(false, cons(s(z0), nil)) -> c15(SORT(if2(eq(z0, z0), s(z0), s(z0), nil))) IF3(false, cons(0, nil)) -> c15(SORT(nil)) S tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) MAX(cons(s(s(z0)), cons(s(s(0)), x2))) -> c2(IF1(true, s(s(z0)), s(s(0)), x2), GE(s(s(z0)), s(s(0)))) MAX(cons(s(s(0)), cons(s(s(s(z0))), x2))) -> c2(IF1(false, s(s(0)), s(s(s(z0))), x2), GE(s(s(0)), s(s(s(z0))))) MAX(cons(s(s(x0)), cons(s(s(x1)), x2))) -> c2(GE(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c6(IF2(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c6(IF2(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c6(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c6(EQ(s(s(0)), s(s(0)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) MAX(cons(s(s(s(z0))), cons(s(s(s(0))), x2))) -> c2(IF1(true, s(s(s(z0))), s(s(s(0))), x2), GE(s(s(s(z0))), s(s(s(0))))) MAX(cons(s(s(s(0))), cons(s(s(s(s(z0)))), x2))) -> c2(IF1(false, s(s(s(0))), s(s(s(s(z0)))), x2), GE(s(s(s(0))), s(s(s(s(z0)))))) MAX(cons(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2))) -> c2(IF1(ge(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), GE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MAX(cons(s(s(s(x0))), cons(s(s(s(x1))), x2))) -> c2(GE(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c6(IF2(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c6(IF2(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c6(IF2(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c6(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c6(EQ(s(s(s(0))), s(s(s(0))))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(max(cons(z0, cons(0, x2))), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(max(cons(0, cons(s(z0), x2))), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(max(cons(s(z0), cons(s(z1), x2))), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), max(cons(z0, cons(0, x2))), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), max(cons(0, cons(s(z0), x2))), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), max(cons(s(z0), cons(s(z1), x2))), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(ge(z0, 0), z0, 0, x2), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(ge(0, s(z0)), 0, s(z0), x2), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), if1(ge(z0, 0), z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), if1(ge(0, s(z0)), 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z1, cons(0, x1))) -> c15(SORT(if2(eq(if1(true, z1, 0, x1), z1), if1(true, z1, 0, x1), z1, cons(0, x1))), DEL(if1(ge(z1, 0), z1, 0, x1), cons(z1, cons(0, x1))), MAX(cons(z1, cons(0, x1)))) IF3(false, cons(z0, cons(0, z2))) -> c15(SORT(del(max(cons(z0, z2)), cons(z0, cons(0, z2)))), DEL(if1(ge(z0, 0), z0, 0, z2), cons(z0, cons(0, z2))), MAX(cons(z0, cons(0, z2)))) IF3(false, cons(0, cons(s(x0), x1))) -> c15(SORT(if2(eq(if1(false, 0, s(x0), x1), 0), if1(false, 0, s(x0), x1), 0, cons(s(x0), x1))), DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1))), MAX(cons(0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), z2))) -> c15(SORT(del(max(cons(s(x0), z2)), cons(0, cons(s(x0), z2)))), DEL(if1(ge(0, s(x0)), 0, s(x0), z2), cons(0, cons(s(x0), z2))), MAX(cons(0, cons(s(x0), z2)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c15(SORT(if2(eq(if1(ge(x0, x1), s(x0), s(x1), x2), s(x0)), if1(ge(x0, x1), s(x0), s(x1), x2), s(x0), cons(s(x1), x2))), DEL(if1(ge(s(x0), s(x1)), s(x0), s(x1), x2), cons(s(x0), cons(s(x1), x2))), MAX(cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(s(z0), cons(s(0), x2))) -> c15(SORT(del(if1(true, s(z0), s(0), x2), cons(s(z0), cons(s(0), x2)))), DEL(if1(ge(s(z0), s(0)), s(z0), s(0), x2), cons(s(z0), cons(s(0), x2))), MAX(cons(s(z0), cons(s(0), x2)))) IF3(false, cons(s(0), cons(s(s(z0)), x2))) -> c15(SORT(del(if1(false, s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2)))), DEL(if1(ge(s(0), s(s(z0))), s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2))), MAX(cons(s(0), cons(s(s(z0)), x2)))) IF3(false, cons(s(s(z0)), cons(s(s(z1)), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2)))), DEL(if1(ge(s(s(z0)), s(s(z1))), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2))), MAX(cons(s(s(z0)), cons(s(s(z1)), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(max(cons(z0, cons(0, x2))), z0), if1(true, z0, 0, x2), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(max(cons(0, cons(s(z0), x2))), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(max(cons(s(z0), cons(s(z1), x2))), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2)))) IF3(false, cons(s(z0), nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(s(z0), nil)), s(z0), nil))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), max(cons(z0, cons(0, x2))), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), max(cons(0, cons(s(z0), x2))), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), max(cons(s(z0), cons(s(z1), x2))), s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(ge(z0, 0), z0, 0, x2), z0), if1(true, z0, 0, x2), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(ge(0, s(z0)), 0, s(z0), x2), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), if1(ge(z0, 0), z0, 0, x2), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), if1(ge(0, s(z0)), 0, s(z0), x2), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0), cons(s(z1), x2)))) IF3(false, cons(z1, cons(0, x1))) -> c15(SORT(if2(eq(if1(true, z1, 0, x1), z1), if1(true, z1, 0, x1), z1, cons(0, x1)))) IF3(false, cons(z0, cons(0, z2))) -> c15(SORT(del(max(cons(z0, z2)), cons(z0, cons(0, z2))))) IF3(false, cons(0, cons(s(x0), x1))) -> c15(SORT(if2(eq(if1(false, 0, s(x0), x1), 0), if1(false, 0, s(x0), x1), 0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), z2))) -> c15(SORT(del(max(cons(s(x0), z2)), cons(0, cons(s(x0), z2))))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c15(SORT(if2(eq(if1(ge(x0, x1), s(x0), s(x1), x2), s(x0)), if1(ge(x0, x1), s(x0), s(x1), x2), s(x0), cons(s(x1), x2)))) IF3(false, cons(s(z0), cons(s(0), x2))) -> c15(SORT(del(if1(true, s(z0), s(0), x2), cons(s(z0), cons(s(0), x2))))) IF3(false, cons(s(0), cons(s(s(z0)), x2))) -> c15(SORT(del(if1(false, s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2))))) IF3(false, cons(s(s(z0)), cons(s(s(z1)), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2))))) IF3(false, cons(s(z0), nil)) -> c15(SORT(if2(eq(z0, z0), s(z0), s(z0), nil))) K tuples: IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) IF3(false, cons(x0, cons(0, x1))) -> c(DEL(if1(ge(x0, 0), x0, 0, x1), cons(x0, cons(0, x1)))) IF3(false, cons(x0, cons(0, x1))) -> c(MAX(cons(x0, cons(0, x1)))) IF3(false, cons(0, cons(s(x0), x1))) -> c1(DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), x1))) -> c1(MAX(cons(0, cons(s(x0), x1)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c5(DEL(if1(ge(s(x0), s(x1)), s(x0), s(x1), x2), cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c5(MAX(cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(0, nil)) -> c15(SORT(if2(true, max(cons(0, nil)), 0, nil))) IF3(false, cons(0, nil)) -> c15(SORT(if2(true, 0, 0, nil))) Defined Rule Symbols: del_2, max_1, if2_4, eq_2, ge_2, if1_4 Defined Pair Symbols: IF1_4, IF2_4, EQ_2, GE_2, MAX_1, DEL_2, SORT_1, IF3_2 Compound Symbols: c3_1, c4_1, c8_1, c12_1, c20_1, c2_1, c6_1, c13_1, c2_2, c6_2, c15_1, c15_3, c15_2, c_1, c1_1, c5_1 ---------------------------------------- (195) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: IF3(false, cons(0, nil)) -> c15(SORT(nil)) ---------------------------------------- (196) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil max(cons(z0, nil)) -> z0 max(cons(z0, cons(z1, z2))) -> if1(ge(z0, z1), z0, z1, z2) if2(true, z0, z1, z2) -> z2 if2(false, z0, z1, z2) -> cons(z1, del(z0, z2)) eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) if1(true, z0, z1, z2) -> max(cons(z0, z2)) if1(false, z0, z1, z2) -> max(cons(z1, z2)) Tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) MAX(cons(s(s(z0)), cons(s(s(0)), x2))) -> c2(IF1(true, s(s(z0)), s(s(0)), x2), GE(s(s(z0)), s(s(0)))) MAX(cons(s(s(0)), cons(s(s(s(z0))), x2))) -> c2(IF1(false, s(s(0)), s(s(s(z0))), x2), GE(s(s(0)), s(s(s(z0))))) MAX(cons(s(s(x0)), cons(s(s(x1)), x2))) -> c2(GE(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c6(IF2(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c6(IF2(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c6(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c6(EQ(s(s(0)), s(s(0)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) MAX(cons(s(s(s(z0))), cons(s(s(s(0))), x2))) -> c2(IF1(true, s(s(s(z0))), s(s(s(0))), x2), GE(s(s(s(z0))), s(s(s(0))))) MAX(cons(s(s(s(0))), cons(s(s(s(s(z0)))), x2))) -> c2(IF1(false, s(s(s(0))), s(s(s(s(z0)))), x2), GE(s(s(s(0))), s(s(s(s(z0)))))) MAX(cons(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2))) -> c2(IF1(ge(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), GE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MAX(cons(s(s(s(x0))), cons(s(s(s(x1))), x2))) -> c2(GE(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c6(IF2(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c6(IF2(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c6(IF2(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c6(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c6(EQ(s(s(s(0))), s(s(s(0))))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(max(cons(z0, cons(0, x2))), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(max(cons(0, cons(s(z0), x2))), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(max(cons(s(z0), cons(s(z1), x2))), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), z0, z1, z2), z0), if1(ge(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), max(cons(z0, cons(0, x2))), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), max(cons(0, cons(s(z0), x2))), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), max(cons(s(z0), cons(s(z1), x2))), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(ge(z0, 0), z0, 0, x2), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(ge(0, s(z0)), 0, s(z0), x2), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), if1(ge(z0, 0), z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), if1(ge(0, s(z0)), 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z1, cons(0, x1))) -> c15(SORT(if2(eq(if1(true, z1, 0, x1), z1), if1(true, z1, 0, x1), z1, cons(0, x1))), DEL(if1(ge(z1, 0), z1, 0, x1), cons(z1, cons(0, x1))), MAX(cons(z1, cons(0, x1)))) IF3(false, cons(z0, cons(0, z2))) -> c15(SORT(del(max(cons(z0, z2)), cons(z0, cons(0, z2)))), DEL(if1(ge(z0, 0), z0, 0, z2), cons(z0, cons(0, z2))), MAX(cons(z0, cons(0, z2)))) IF3(false, cons(x0, cons(0, x1))) -> c(DEL(if1(ge(x0, 0), x0, 0, x1), cons(x0, cons(0, x1)))) IF3(false, cons(x0, cons(0, x1))) -> c(MAX(cons(x0, cons(0, x1)))) IF3(false, cons(0, cons(s(x0), x1))) -> c15(SORT(if2(eq(if1(false, 0, s(x0), x1), 0), if1(false, 0, s(x0), x1), 0, cons(s(x0), x1))), DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1))), MAX(cons(0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), z2))) -> c15(SORT(del(max(cons(s(x0), z2)), cons(0, cons(s(x0), z2)))), DEL(if1(ge(0, s(x0)), 0, s(x0), z2), cons(0, cons(s(x0), z2))), MAX(cons(0, cons(s(x0), z2)))) IF3(false, cons(0, cons(s(x0), x1))) -> c1(DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), x1))) -> c1(MAX(cons(0, cons(s(x0), x1)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c15(SORT(if2(eq(if1(ge(x0, x1), s(x0), s(x1), x2), s(x0)), if1(ge(x0, x1), s(x0), s(x1), x2), s(x0), cons(s(x1), x2))), DEL(if1(ge(s(x0), s(x1)), s(x0), s(x1), x2), cons(s(x0), cons(s(x1), x2))), MAX(cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(s(z0), cons(s(0), x2))) -> c15(SORT(del(if1(true, s(z0), s(0), x2), cons(s(z0), cons(s(0), x2)))), DEL(if1(ge(s(z0), s(0)), s(z0), s(0), x2), cons(s(z0), cons(s(0), x2))), MAX(cons(s(z0), cons(s(0), x2)))) IF3(false, cons(s(0), cons(s(s(z0)), x2))) -> c15(SORT(del(if1(false, s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2)))), DEL(if1(ge(s(0), s(s(z0))), s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2))), MAX(cons(s(0), cons(s(s(z0)), x2)))) IF3(false, cons(s(s(z0)), cons(s(s(z1)), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2)))), DEL(if1(ge(s(s(z0)), s(s(z1))), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2))), MAX(cons(s(s(z0)), cons(s(s(z1)), x2)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c5(DEL(if1(ge(s(x0), s(x1)), s(x0), s(x1), x2), cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c5(MAX(cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(max(cons(z0, cons(0, x2))), z0), if1(true, z0, 0, x2), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(max(cons(0, cons(s(z0), x2))), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(max(cons(s(z0), cons(s(z1), x2))), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), z0, z1, z2), z0), if1(ge(z0, z1), z0, z1, z2), z0, cons(z1, z2)))) IF3(false, cons(0, nil)) -> c15(SORT(if2(true, max(cons(0, nil)), 0, nil))) IF3(false, cons(s(z0), nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(s(z0), nil)), s(z0), nil))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), max(cons(z0, cons(0, x2))), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), max(cons(0, cons(s(z0), x2))), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), max(cons(s(z0), cons(s(z1), x2))), s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(ge(z0, 0), z0, 0, x2), z0), if1(true, z0, 0, x2), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(ge(0, s(z0)), 0, s(z0), x2), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), if1(ge(z0, 0), z0, 0, x2), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), if1(ge(0, s(z0)), 0, s(z0), x2), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0), cons(s(z1), x2)))) IF3(false, cons(z1, cons(0, x1))) -> c15(SORT(if2(eq(if1(true, z1, 0, x1), z1), if1(true, z1, 0, x1), z1, cons(0, x1)))) IF3(false, cons(z0, cons(0, z2))) -> c15(SORT(del(max(cons(z0, z2)), cons(z0, cons(0, z2))))) IF3(false, cons(0, cons(s(x0), x1))) -> c15(SORT(if2(eq(if1(false, 0, s(x0), x1), 0), if1(false, 0, s(x0), x1), 0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), z2))) -> c15(SORT(del(max(cons(s(x0), z2)), cons(0, cons(s(x0), z2))))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c15(SORT(if2(eq(if1(ge(x0, x1), s(x0), s(x1), x2), s(x0)), if1(ge(x0, x1), s(x0), s(x1), x2), s(x0), cons(s(x1), x2)))) IF3(false, cons(s(z0), cons(s(0), x2))) -> c15(SORT(del(if1(true, s(z0), s(0), x2), cons(s(z0), cons(s(0), x2))))) IF3(false, cons(s(0), cons(s(s(z0)), x2))) -> c15(SORT(del(if1(false, s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2))))) IF3(false, cons(s(s(z0)), cons(s(s(z1)), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2))))) IF3(false, cons(s(z0), nil)) -> c15(SORT(if2(eq(z0, z0), s(z0), s(z0), nil))) S tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) MAX(cons(s(s(z0)), cons(s(s(0)), x2))) -> c2(IF1(true, s(s(z0)), s(s(0)), x2), GE(s(s(z0)), s(s(0)))) MAX(cons(s(s(0)), cons(s(s(s(z0))), x2))) -> c2(IF1(false, s(s(0)), s(s(s(z0))), x2), GE(s(s(0)), s(s(s(z0))))) MAX(cons(s(s(x0)), cons(s(s(x1)), x2))) -> c2(GE(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c6(IF2(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c6(IF2(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c6(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c6(EQ(s(s(0)), s(s(0)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) MAX(cons(s(s(s(z0))), cons(s(s(s(0))), x2))) -> c2(IF1(true, s(s(s(z0))), s(s(s(0))), x2), GE(s(s(s(z0))), s(s(s(0))))) MAX(cons(s(s(s(0))), cons(s(s(s(s(z0)))), x2))) -> c2(IF1(false, s(s(s(0))), s(s(s(s(z0)))), x2), GE(s(s(s(0))), s(s(s(s(z0)))))) MAX(cons(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2))) -> c2(IF1(ge(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), GE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MAX(cons(s(s(s(x0))), cons(s(s(s(x1))), x2))) -> c2(GE(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c6(IF2(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c6(IF2(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c6(IF2(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c6(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c6(EQ(s(s(s(0))), s(s(s(0))))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(max(cons(z0, cons(0, x2))), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(max(cons(0, cons(s(z0), x2))), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(max(cons(s(z0), cons(s(z1), x2))), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), max(cons(z0, cons(0, x2))), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), max(cons(0, cons(s(z0), x2))), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), max(cons(s(z0), cons(s(z1), x2))), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(ge(z0, 0), z0, 0, x2), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(ge(0, s(z0)), 0, s(z0), x2), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), if1(ge(z0, 0), z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), if1(ge(0, s(z0)), 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z1, cons(0, x1))) -> c15(SORT(if2(eq(if1(true, z1, 0, x1), z1), if1(true, z1, 0, x1), z1, cons(0, x1))), DEL(if1(ge(z1, 0), z1, 0, x1), cons(z1, cons(0, x1))), MAX(cons(z1, cons(0, x1)))) IF3(false, cons(z0, cons(0, z2))) -> c15(SORT(del(max(cons(z0, z2)), cons(z0, cons(0, z2)))), DEL(if1(ge(z0, 0), z0, 0, z2), cons(z0, cons(0, z2))), MAX(cons(z0, cons(0, z2)))) IF3(false, cons(0, cons(s(x0), x1))) -> c15(SORT(if2(eq(if1(false, 0, s(x0), x1), 0), if1(false, 0, s(x0), x1), 0, cons(s(x0), x1))), DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1))), MAX(cons(0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), z2))) -> c15(SORT(del(max(cons(s(x0), z2)), cons(0, cons(s(x0), z2)))), DEL(if1(ge(0, s(x0)), 0, s(x0), z2), cons(0, cons(s(x0), z2))), MAX(cons(0, cons(s(x0), z2)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c15(SORT(if2(eq(if1(ge(x0, x1), s(x0), s(x1), x2), s(x0)), if1(ge(x0, x1), s(x0), s(x1), x2), s(x0), cons(s(x1), x2))), DEL(if1(ge(s(x0), s(x1)), s(x0), s(x1), x2), cons(s(x0), cons(s(x1), x2))), MAX(cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(s(z0), cons(s(0), x2))) -> c15(SORT(del(if1(true, s(z0), s(0), x2), cons(s(z0), cons(s(0), x2)))), DEL(if1(ge(s(z0), s(0)), s(z0), s(0), x2), cons(s(z0), cons(s(0), x2))), MAX(cons(s(z0), cons(s(0), x2)))) IF3(false, cons(s(0), cons(s(s(z0)), x2))) -> c15(SORT(del(if1(false, s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2)))), DEL(if1(ge(s(0), s(s(z0))), s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2))), MAX(cons(s(0), cons(s(s(z0)), x2)))) IF3(false, cons(s(s(z0)), cons(s(s(z1)), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2)))), DEL(if1(ge(s(s(z0)), s(s(z1))), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2))), MAX(cons(s(s(z0)), cons(s(s(z1)), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(max(cons(z0, cons(0, x2))), z0), if1(true, z0, 0, x2), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(max(cons(0, cons(s(z0), x2))), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(max(cons(s(z0), cons(s(z1), x2))), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2)))) IF3(false, cons(s(z0), nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(s(z0), nil)), s(z0), nil))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), max(cons(z0, cons(0, x2))), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), max(cons(0, cons(s(z0), x2))), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), max(cons(s(z0), cons(s(z1), x2))), s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(ge(z0, 0), z0, 0, x2), z0), if1(true, z0, 0, x2), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(ge(0, s(z0)), 0, s(z0), x2), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), if1(ge(z0, 0), z0, 0, x2), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), if1(ge(0, s(z0)), 0, s(z0), x2), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0), cons(s(z1), x2)))) IF3(false, cons(z1, cons(0, x1))) -> c15(SORT(if2(eq(if1(true, z1, 0, x1), z1), if1(true, z1, 0, x1), z1, cons(0, x1)))) IF3(false, cons(z0, cons(0, z2))) -> c15(SORT(del(max(cons(z0, z2)), cons(z0, cons(0, z2))))) IF3(false, cons(0, cons(s(x0), x1))) -> c15(SORT(if2(eq(if1(false, 0, s(x0), x1), 0), if1(false, 0, s(x0), x1), 0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), z2))) -> c15(SORT(del(max(cons(s(x0), z2)), cons(0, cons(s(x0), z2))))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c15(SORT(if2(eq(if1(ge(x0, x1), s(x0), s(x1), x2), s(x0)), if1(ge(x0, x1), s(x0), s(x1), x2), s(x0), cons(s(x1), x2)))) IF3(false, cons(s(z0), cons(s(0), x2))) -> c15(SORT(del(if1(true, s(z0), s(0), x2), cons(s(z0), cons(s(0), x2))))) IF3(false, cons(s(0), cons(s(s(z0)), x2))) -> c15(SORT(del(if1(false, s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2))))) IF3(false, cons(s(s(z0)), cons(s(s(z1)), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2))))) IF3(false, cons(s(z0), nil)) -> c15(SORT(if2(eq(z0, z0), s(z0), s(z0), nil))) K tuples: IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) IF3(false, cons(x0, cons(0, x1))) -> c(DEL(if1(ge(x0, 0), x0, 0, x1), cons(x0, cons(0, x1)))) IF3(false, cons(x0, cons(0, x1))) -> c(MAX(cons(x0, cons(0, x1)))) IF3(false, cons(0, cons(s(x0), x1))) -> c1(DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), x1))) -> c1(MAX(cons(0, cons(s(x0), x1)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c5(DEL(if1(ge(s(x0), s(x1)), s(x0), s(x1), x2), cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c5(MAX(cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(0, nil)) -> c15(SORT(if2(true, max(cons(0, nil)), 0, nil))) Defined Rule Symbols: del_2, max_1, if2_4, eq_2, ge_2, if1_4 Defined Pair Symbols: IF1_4, IF2_4, EQ_2, GE_2, MAX_1, DEL_2, SORT_1, IF3_2 Compound Symbols: c3_1, c4_1, c8_1, c12_1, c20_1, c2_1, c6_1, c13_1, c2_2, c6_2, c15_1, c15_2, c15_3, c_1, c1_1, c5_1 ---------------------------------------- (197) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) by IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) ---------------------------------------- (198) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil max(cons(z0, nil)) -> z0 max(cons(z0, cons(z1, z2))) -> if1(ge(z0, z1), z0, z1, z2) if2(true, z0, z1, z2) -> z2 if2(false, z0, z1, z2) -> cons(z1, del(z0, z2)) eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) if1(true, z0, z1, z2) -> max(cons(z0, z2)) if1(false, z0, z1, z2) -> max(cons(z1, z2)) Tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) MAX(cons(s(s(z0)), cons(s(s(0)), x2))) -> c2(IF1(true, s(s(z0)), s(s(0)), x2), GE(s(s(z0)), s(s(0)))) MAX(cons(s(s(0)), cons(s(s(s(z0))), x2))) -> c2(IF1(false, s(s(0)), s(s(s(z0))), x2), GE(s(s(0)), s(s(s(z0))))) MAX(cons(s(s(x0)), cons(s(s(x1)), x2))) -> c2(GE(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c6(IF2(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c6(IF2(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c6(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c6(EQ(s(s(0)), s(s(0)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) MAX(cons(s(s(s(z0))), cons(s(s(s(0))), x2))) -> c2(IF1(true, s(s(s(z0))), s(s(s(0))), x2), GE(s(s(s(z0))), s(s(s(0))))) MAX(cons(s(s(s(0))), cons(s(s(s(s(z0)))), x2))) -> c2(IF1(false, s(s(s(0))), s(s(s(s(z0)))), x2), GE(s(s(s(0))), s(s(s(s(z0)))))) MAX(cons(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2))) -> c2(IF1(ge(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), GE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MAX(cons(s(s(s(x0))), cons(s(s(s(x1))), x2))) -> c2(GE(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c6(IF2(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c6(IF2(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c6(IF2(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c6(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c6(EQ(s(s(s(0))), s(s(s(0))))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(max(cons(z0, cons(0, x2))), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(max(cons(0, cons(s(z0), x2))), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(max(cons(s(z0), cons(s(z1), x2))), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), z0, z1, z2), z0), if1(ge(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), max(cons(z0, cons(0, x2))), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), max(cons(0, cons(s(z0), x2))), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), max(cons(s(z0), cons(s(z1), x2))), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(ge(z0, 0), z0, 0, x2), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(ge(0, s(z0)), 0, s(z0), x2), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), if1(ge(z0, 0), z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), if1(ge(0, s(z0)), 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z1, cons(0, x1))) -> c15(SORT(if2(eq(if1(true, z1, 0, x1), z1), if1(true, z1, 0, x1), z1, cons(0, x1))), DEL(if1(ge(z1, 0), z1, 0, x1), cons(z1, cons(0, x1))), MAX(cons(z1, cons(0, x1)))) IF3(false, cons(z0, cons(0, z2))) -> c15(SORT(del(max(cons(z0, z2)), cons(z0, cons(0, z2)))), DEL(if1(ge(z0, 0), z0, 0, z2), cons(z0, cons(0, z2))), MAX(cons(z0, cons(0, z2)))) IF3(false, cons(x0, cons(0, x1))) -> c(DEL(if1(ge(x0, 0), x0, 0, x1), cons(x0, cons(0, x1)))) IF3(false, cons(x0, cons(0, x1))) -> c(MAX(cons(x0, cons(0, x1)))) IF3(false, cons(0, cons(s(x0), x1))) -> c15(SORT(if2(eq(if1(false, 0, s(x0), x1), 0), if1(false, 0, s(x0), x1), 0, cons(s(x0), x1))), DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1))), MAX(cons(0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), z2))) -> c15(SORT(del(max(cons(s(x0), z2)), cons(0, cons(s(x0), z2)))), DEL(if1(ge(0, s(x0)), 0, s(x0), z2), cons(0, cons(s(x0), z2))), MAX(cons(0, cons(s(x0), z2)))) IF3(false, cons(0, cons(s(x0), x1))) -> c1(DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), x1))) -> c1(MAX(cons(0, cons(s(x0), x1)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c15(SORT(if2(eq(if1(ge(x0, x1), s(x0), s(x1), x2), s(x0)), if1(ge(x0, x1), s(x0), s(x1), x2), s(x0), cons(s(x1), x2))), DEL(if1(ge(s(x0), s(x1)), s(x0), s(x1), x2), cons(s(x0), cons(s(x1), x2))), MAX(cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(s(z0), cons(s(0), x2))) -> c15(SORT(del(if1(true, s(z0), s(0), x2), cons(s(z0), cons(s(0), x2)))), DEL(if1(ge(s(z0), s(0)), s(z0), s(0), x2), cons(s(z0), cons(s(0), x2))), MAX(cons(s(z0), cons(s(0), x2)))) IF3(false, cons(s(0), cons(s(s(z0)), x2))) -> c15(SORT(del(if1(false, s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2)))), DEL(if1(ge(s(0), s(s(z0))), s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2))), MAX(cons(s(0), cons(s(s(z0)), x2)))) IF3(false, cons(s(s(z0)), cons(s(s(z1)), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2)))), DEL(if1(ge(s(s(z0)), s(s(z1))), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2))), MAX(cons(s(s(z0)), cons(s(s(z1)), x2)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c5(DEL(if1(ge(s(x0), s(x1)), s(x0), s(x1), x2), cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c5(MAX(cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(max(cons(z0, cons(0, x2))), z0), if1(true, z0, 0, x2), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(max(cons(0, cons(s(z0), x2))), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(max(cons(s(z0), cons(s(z1), x2))), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), z0, z1, z2), z0), if1(ge(z0, z1), z0, z1, z2), z0, cons(z1, z2)))) IF3(false, cons(0, nil)) -> c15(SORT(if2(true, max(cons(0, nil)), 0, nil))) IF3(false, cons(s(z0), nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(s(z0), nil)), s(z0), nil))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), max(cons(z0, cons(0, x2))), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), max(cons(0, cons(s(z0), x2))), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), max(cons(s(z0), cons(s(z1), x2))), s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(ge(z0, 0), z0, 0, x2), z0), if1(true, z0, 0, x2), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(ge(0, s(z0)), 0, s(z0), x2), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), if1(ge(z0, 0), z0, 0, x2), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), if1(ge(0, s(z0)), 0, s(z0), x2), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0), cons(s(z1), x2)))) IF3(false, cons(z1, cons(0, x1))) -> c15(SORT(if2(eq(if1(true, z1, 0, x1), z1), if1(true, z1, 0, x1), z1, cons(0, x1)))) IF3(false, cons(z0, cons(0, z2))) -> c15(SORT(del(max(cons(z0, z2)), cons(z0, cons(0, z2))))) IF3(false, cons(0, cons(s(x0), x1))) -> c15(SORT(if2(eq(if1(false, 0, s(x0), x1), 0), if1(false, 0, s(x0), x1), 0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), z2))) -> c15(SORT(del(max(cons(s(x0), z2)), cons(0, cons(s(x0), z2))))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c15(SORT(if2(eq(if1(ge(x0, x1), s(x0), s(x1), x2), s(x0)), if1(ge(x0, x1), s(x0), s(x1), x2), s(x0), cons(s(x1), x2)))) IF3(false, cons(s(z0), cons(s(0), x2))) -> c15(SORT(del(if1(true, s(z0), s(0), x2), cons(s(z0), cons(s(0), x2))))) IF3(false, cons(s(0), cons(s(s(z0)), x2))) -> c15(SORT(del(if1(false, s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2))))) IF3(false, cons(s(s(z0)), cons(s(s(z1)), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2))))) IF3(false, cons(s(z0), nil)) -> c15(SORT(if2(eq(z0, z0), s(z0), s(z0), nil))) S tuples: IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) MAX(cons(s(s(z0)), cons(s(s(0)), x2))) -> c2(IF1(true, s(s(z0)), s(s(0)), x2), GE(s(s(z0)), s(s(0)))) MAX(cons(s(s(0)), cons(s(s(s(z0))), x2))) -> c2(IF1(false, s(s(0)), s(s(s(z0))), x2), GE(s(s(0)), s(s(s(z0))))) MAX(cons(s(s(x0)), cons(s(s(x1)), x2))) -> c2(GE(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c6(IF2(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c6(IF2(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c6(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c6(EQ(s(s(0)), s(s(0)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) MAX(cons(s(s(s(z0))), cons(s(s(s(0))), x2))) -> c2(IF1(true, s(s(s(z0))), s(s(s(0))), x2), GE(s(s(s(z0))), s(s(s(0))))) MAX(cons(s(s(s(0))), cons(s(s(s(s(z0)))), x2))) -> c2(IF1(false, s(s(s(0))), s(s(s(s(z0)))), x2), GE(s(s(s(0))), s(s(s(s(z0)))))) MAX(cons(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2))) -> c2(IF1(ge(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), GE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MAX(cons(s(s(s(x0))), cons(s(s(s(x1))), x2))) -> c2(GE(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c6(IF2(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c6(IF2(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c6(IF2(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c6(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c6(EQ(s(s(s(0))), s(s(s(0))))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(max(cons(z0, cons(0, x2))), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(max(cons(0, cons(s(z0), x2))), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(max(cons(s(z0), cons(s(z1), x2))), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), max(cons(z0, cons(0, x2))), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), max(cons(0, cons(s(z0), x2))), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), max(cons(s(z0), cons(s(z1), x2))), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(ge(z0, 0), z0, 0, x2), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(ge(0, s(z0)), 0, s(z0), x2), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), if1(ge(z0, 0), z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), if1(ge(0, s(z0)), 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z1, cons(0, x1))) -> c15(SORT(if2(eq(if1(true, z1, 0, x1), z1), if1(true, z1, 0, x1), z1, cons(0, x1))), DEL(if1(ge(z1, 0), z1, 0, x1), cons(z1, cons(0, x1))), MAX(cons(z1, cons(0, x1)))) IF3(false, cons(z0, cons(0, z2))) -> c15(SORT(del(max(cons(z0, z2)), cons(z0, cons(0, z2)))), DEL(if1(ge(z0, 0), z0, 0, z2), cons(z0, cons(0, z2))), MAX(cons(z0, cons(0, z2)))) IF3(false, cons(0, cons(s(x0), x1))) -> c15(SORT(if2(eq(if1(false, 0, s(x0), x1), 0), if1(false, 0, s(x0), x1), 0, cons(s(x0), x1))), DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1))), MAX(cons(0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), z2))) -> c15(SORT(del(max(cons(s(x0), z2)), cons(0, cons(s(x0), z2)))), DEL(if1(ge(0, s(x0)), 0, s(x0), z2), cons(0, cons(s(x0), z2))), MAX(cons(0, cons(s(x0), z2)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c15(SORT(if2(eq(if1(ge(x0, x1), s(x0), s(x1), x2), s(x0)), if1(ge(x0, x1), s(x0), s(x1), x2), s(x0), cons(s(x1), x2))), DEL(if1(ge(s(x0), s(x1)), s(x0), s(x1), x2), cons(s(x0), cons(s(x1), x2))), MAX(cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(s(z0), cons(s(0), x2))) -> c15(SORT(del(if1(true, s(z0), s(0), x2), cons(s(z0), cons(s(0), x2)))), DEL(if1(ge(s(z0), s(0)), s(z0), s(0), x2), cons(s(z0), cons(s(0), x2))), MAX(cons(s(z0), cons(s(0), x2)))) IF3(false, cons(s(0), cons(s(s(z0)), x2))) -> c15(SORT(del(if1(false, s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2)))), DEL(if1(ge(s(0), s(s(z0))), s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2))), MAX(cons(s(0), cons(s(s(z0)), x2)))) IF3(false, cons(s(s(z0)), cons(s(s(z1)), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2)))), DEL(if1(ge(s(s(z0)), s(s(z1))), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2))), MAX(cons(s(s(z0)), cons(s(s(z1)), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(max(cons(z0, cons(0, x2))), z0), if1(true, z0, 0, x2), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(max(cons(0, cons(s(z0), x2))), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(max(cons(s(z0), cons(s(z1), x2))), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2)))) IF3(false, cons(s(z0), nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(s(z0), nil)), s(z0), nil))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), max(cons(z0, cons(0, x2))), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), max(cons(0, cons(s(z0), x2))), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), max(cons(s(z0), cons(s(z1), x2))), s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(ge(z0, 0), z0, 0, x2), z0), if1(true, z0, 0, x2), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(ge(0, s(z0)), 0, s(z0), x2), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), if1(ge(z0, 0), z0, 0, x2), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), if1(ge(0, s(z0)), 0, s(z0), x2), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0), cons(s(z1), x2)))) IF3(false, cons(z1, cons(0, x1))) -> c15(SORT(if2(eq(if1(true, z1, 0, x1), z1), if1(true, z1, 0, x1), z1, cons(0, x1)))) IF3(false, cons(z0, cons(0, z2))) -> c15(SORT(del(max(cons(z0, z2)), cons(z0, cons(0, z2))))) IF3(false, cons(0, cons(s(x0), x1))) -> c15(SORT(if2(eq(if1(false, 0, s(x0), x1), 0), if1(false, 0, s(x0), x1), 0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), z2))) -> c15(SORT(del(max(cons(s(x0), z2)), cons(0, cons(s(x0), z2))))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c15(SORT(if2(eq(if1(ge(x0, x1), s(x0), s(x1), x2), s(x0)), if1(ge(x0, x1), s(x0), s(x1), x2), s(x0), cons(s(x1), x2)))) IF3(false, cons(s(z0), cons(s(0), x2))) -> c15(SORT(del(if1(true, s(z0), s(0), x2), cons(s(z0), cons(s(0), x2))))) IF3(false, cons(s(0), cons(s(s(z0)), x2))) -> c15(SORT(del(if1(false, s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2))))) IF3(false, cons(s(s(z0)), cons(s(s(z1)), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2))))) IF3(false, cons(s(z0), nil)) -> c15(SORT(if2(eq(z0, z0), s(z0), s(z0), nil))) K tuples: IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) IF3(false, cons(x0, cons(0, x1))) -> c(DEL(if1(ge(x0, 0), x0, 0, x1), cons(x0, cons(0, x1)))) IF3(false, cons(x0, cons(0, x1))) -> c(MAX(cons(x0, cons(0, x1)))) IF3(false, cons(0, cons(s(x0), x1))) -> c1(DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), x1))) -> c1(MAX(cons(0, cons(s(x0), x1)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c5(DEL(if1(ge(s(x0), s(x1)), s(x0), s(x1), x2), cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c5(MAX(cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(0, nil)) -> c15(SORT(if2(true, max(cons(0, nil)), 0, nil))) Defined Rule Symbols: del_2, max_1, if2_4, eq_2, ge_2, if1_4 Defined Pair Symbols: IF1_4, IF2_4, EQ_2, GE_2, MAX_1, DEL_2, SORT_1, IF3_2 Compound Symbols: c3_1, c4_1, c8_1, c12_1, c20_1, c2_1, c6_1, c13_1, c2_2, c6_2, c15_1, c15_2, c15_3, c_1, c1_1, c5_1 ---------------------------------------- (199) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IF1(true, z0, z1, z2) -> c3(MAX(cons(z0, z2))) by IF1(true, x0, 0, x1) -> c3(MAX(cons(x0, x1))) IF1(true, s(x0), s(0), x1) -> c3(MAX(cons(s(x0), x1))) IF1(true, s(s(x0)), s(s(0)), x1) -> c3(MAX(cons(s(s(x0)), x1))) IF1(true, s(s(s(x0))), s(s(s(0))), x1) -> c3(MAX(cons(s(s(s(x0))), x1))) IF1(true, s(s(s(s(x0)))), s(s(s(s(x1)))), x2) -> c3(MAX(cons(s(s(s(s(x0)))), x2))) ---------------------------------------- (200) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil max(cons(z0, nil)) -> z0 max(cons(z0, cons(z1, z2))) -> if1(ge(z0, z1), z0, z1, z2) if2(true, z0, z1, z2) -> z2 if2(false, z0, z1, z2) -> cons(z1, del(z0, z2)) eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) if1(true, z0, z1, z2) -> max(cons(z0, z2)) if1(false, z0, z1, z2) -> max(cons(z1, z2)) Tuples: IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) MAX(cons(s(s(z0)), cons(s(s(0)), x2))) -> c2(IF1(true, s(s(z0)), s(s(0)), x2), GE(s(s(z0)), s(s(0)))) MAX(cons(s(s(0)), cons(s(s(s(z0))), x2))) -> c2(IF1(false, s(s(0)), s(s(s(z0))), x2), GE(s(s(0)), s(s(s(z0))))) MAX(cons(s(s(x0)), cons(s(s(x1)), x2))) -> c2(GE(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c6(IF2(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c6(IF2(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c6(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c6(EQ(s(s(0)), s(s(0)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) MAX(cons(s(s(s(z0))), cons(s(s(s(0))), x2))) -> c2(IF1(true, s(s(s(z0))), s(s(s(0))), x2), GE(s(s(s(z0))), s(s(s(0))))) MAX(cons(s(s(s(0))), cons(s(s(s(s(z0)))), x2))) -> c2(IF1(false, s(s(s(0))), s(s(s(s(z0)))), x2), GE(s(s(s(0))), s(s(s(s(z0)))))) MAX(cons(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2))) -> c2(IF1(ge(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), GE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MAX(cons(s(s(s(x0))), cons(s(s(s(x1))), x2))) -> c2(GE(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c6(IF2(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c6(IF2(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c6(IF2(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c6(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c6(EQ(s(s(s(0))), s(s(s(0))))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(max(cons(z0, cons(0, x2))), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(max(cons(0, cons(s(z0), x2))), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(max(cons(s(z0), cons(s(z1), x2))), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), z0, z1, z2), z0), if1(ge(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), max(cons(z0, cons(0, x2))), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), max(cons(0, cons(s(z0), x2))), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), max(cons(s(z0), cons(s(z1), x2))), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(ge(z0, 0), z0, 0, x2), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(ge(0, s(z0)), 0, s(z0), x2), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), if1(ge(z0, 0), z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), if1(ge(0, s(z0)), 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z1, cons(0, x1))) -> c15(SORT(if2(eq(if1(true, z1, 0, x1), z1), if1(true, z1, 0, x1), z1, cons(0, x1))), DEL(if1(ge(z1, 0), z1, 0, x1), cons(z1, cons(0, x1))), MAX(cons(z1, cons(0, x1)))) IF3(false, cons(z0, cons(0, z2))) -> c15(SORT(del(max(cons(z0, z2)), cons(z0, cons(0, z2)))), DEL(if1(ge(z0, 0), z0, 0, z2), cons(z0, cons(0, z2))), MAX(cons(z0, cons(0, z2)))) IF3(false, cons(x0, cons(0, x1))) -> c(DEL(if1(ge(x0, 0), x0, 0, x1), cons(x0, cons(0, x1)))) IF3(false, cons(x0, cons(0, x1))) -> c(MAX(cons(x0, cons(0, x1)))) IF3(false, cons(0, cons(s(x0), x1))) -> c15(SORT(if2(eq(if1(false, 0, s(x0), x1), 0), if1(false, 0, s(x0), x1), 0, cons(s(x0), x1))), DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1))), MAX(cons(0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), z2))) -> c15(SORT(del(max(cons(s(x0), z2)), cons(0, cons(s(x0), z2)))), DEL(if1(ge(0, s(x0)), 0, s(x0), z2), cons(0, cons(s(x0), z2))), MAX(cons(0, cons(s(x0), z2)))) IF3(false, cons(0, cons(s(x0), x1))) -> c1(DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), x1))) -> c1(MAX(cons(0, cons(s(x0), x1)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c15(SORT(if2(eq(if1(ge(x0, x1), s(x0), s(x1), x2), s(x0)), if1(ge(x0, x1), s(x0), s(x1), x2), s(x0), cons(s(x1), x2))), DEL(if1(ge(s(x0), s(x1)), s(x0), s(x1), x2), cons(s(x0), cons(s(x1), x2))), MAX(cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(s(z0), cons(s(0), x2))) -> c15(SORT(del(if1(true, s(z0), s(0), x2), cons(s(z0), cons(s(0), x2)))), DEL(if1(ge(s(z0), s(0)), s(z0), s(0), x2), cons(s(z0), cons(s(0), x2))), MAX(cons(s(z0), cons(s(0), x2)))) IF3(false, cons(s(0), cons(s(s(z0)), x2))) -> c15(SORT(del(if1(false, s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2)))), DEL(if1(ge(s(0), s(s(z0))), s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2))), MAX(cons(s(0), cons(s(s(z0)), x2)))) IF3(false, cons(s(s(z0)), cons(s(s(z1)), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2)))), DEL(if1(ge(s(s(z0)), s(s(z1))), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2))), MAX(cons(s(s(z0)), cons(s(s(z1)), x2)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c5(DEL(if1(ge(s(x0), s(x1)), s(x0), s(x1), x2), cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c5(MAX(cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(max(cons(z0, cons(0, x2))), z0), if1(true, z0, 0, x2), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(max(cons(0, cons(s(z0), x2))), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(max(cons(s(z0), cons(s(z1), x2))), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), z0, z1, z2), z0), if1(ge(z0, z1), z0, z1, z2), z0, cons(z1, z2)))) IF3(false, cons(0, nil)) -> c15(SORT(if2(true, max(cons(0, nil)), 0, nil))) IF3(false, cons(s(z0), nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(s(z0), nil)), s(z0), nil))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), max(cons(z0, cons(0, x2))), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), max(cons(0, cons(s(z0), x2))), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), max(cons(s(z0), cons(s(z1), x2))), s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(ge(z0, 0), z0, 0, x2), z0), if1(true, z0, 0, x2), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(ge(0, s(z0)), 0, s(z0), x2), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), if1(ge(z0, 0), z0, 0, x2), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), if1(ge(0, s(z0)), 0, s(z0), x2), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0), cons(s(z1), x2)))) IF3(false, cons(z1, cons(0, x1))) -> c15(SORT(if2(eq(if1(true, z1, 0, x1), z1), if1(true, z1, 0, x1), z1, cons(0, x1)))) IF3(false, cons(z0, cons(0, z2))) -> c15(SORT(del(max(cons(z0, z2)), cons(z0, cons(0, z2))))) IF3(false, cons(0, cons(s(x0), x1))) -> c15(SORT(if2(eq(if1(false, 0, s(x0), x1), 0), if1(false, 0, s(x0), x1), 0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), z2))) -> c15(SORT(del(max(cons(s(x0), z2)), cons(0, cons(s(x0), z2))))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c15(SORT(if2(eq(if1(ge(x0, x1), s(x0), s(x1), x2), s(x0)), if1(ge(x0, x1), s(x0), s(x1), x2), s(x0), cons(s(x1), x2)))) IF3(false, cons(s(z0), cons(s(0), x2))) -> c15(SORT(del(if1(true, s(z0), s(0), x2), cons(s(z0), cons(s(0), x2))))) IF3(false, cons(s(0), cons(s(s(z0)), x2))) -> c15(SORT(del(if1(false, s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2))))) IF3(false, cons(s(s(z0)), cons(s(s(z1)), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2))))) IF3(false, cons(s(z0), nil)) -> c15(SORT(if2(eq(z0, z0), s(z0), s(z0), nil))) IF1(true, x0, 0, x1) -> c3(MAX(cons(x0, x1))) IF1(true, s(x0), s(0), x1) -> c3(MAX(cons(s(x0), x1))) IF1(true, s(s(x0)), s(s(0)), x1) -> c3(MAX(cons(s(s(x0)), x1))) IF1(true, s(s(s(x0))), s(s(s(0))), x1) -> c3(MAX(cons(s(s(s(x0))), x1))) IF1(true, s(s(s(s(x0)))), s(s(s(s(x1)))), x2) -> c3(MAX(cons(s(s(s(s(x0)))), x2))) S tuples: IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) MAX(cons(s(s(z0)), cons(s(s(0)), x2))) -> c2(IF1(true, s(s(z0)), s(s(0)), x2), GE(s(s(z0)), s(s(0)))) MAX(cons(s(s(0)), cons(s(s(s(z0))), x2))) -> c2(IF1(false, s(s(0)), s(s(s(z0))), x2), GE(s(s(0)), s(s(s(z0))))) MAX(cons(s(s(x0)), cons(s(s(x1)), x2))) -> c2(GE(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c6(IF2(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c6(IF2(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c6(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c6(EQ(s(s(0)), s(s(0)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) MAX(cons(s(s(s(z0))), cons(s(s(s(0))), x2))) -> c2(IF1(true, s(s(s(z0))), s(s(s(0))), x2), GE(s(s(s(z0))), s(s(s(0))))) MAX(cons(s(s(s(0))), cons(s(s(s(s(z0)))), x2))) -> c2(IF1(false, s(s(s(0))), s(s(s(s(z0)))), x2), GE(s(s(s(0))), s(s(s(s(z0)))))) MAX(cons(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2))) -> c2(IF1(ge(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), GE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MAX(cons(s(s(s(x0))), cons(s(s(s(x1))), x2))) -> c2(GE(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c6(IF2(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c6(IF2(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c6(IF2(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c6(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c6(EQ(s(s(s(0))), s(s(s(0))))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(max(cons(z0, cons(0, x2))), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(max(cons(0, cons(s(z0), x2))), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(max(cons(s(z0), cons(s(z1), x2))), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), max(cons(z0, cons(0, x2))), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), max(cons(0, cons(s(z0), x2))), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), max(cons(s(z0), cons(s(z1), x2))), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(ge(z0, 0), z0, 0, x2), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(ge(0, s(z0)), 0, s(z0), x2), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), if1(ge(z0, 0), z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), if1(ge(0, s(z0)), 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z1, cons(0, x1))) -> c15(SORT(if2(eq(if1(true, z1, 0, x1), z1), if1(true, z1, 0, x1), z1, cons(0, x1))), DEL(if1(ge(z1, 0), z1, 0, x1), cons(z1, cons(0, x1))), MAX(cons(z1, cons(0, x1)))) IF3(false, cons(z0, cons(0, z2))) -> c15(SORT(del(max(cons(z0, z2)), cons(z0, cons(0, z2)))), DEL(if1(ge(z0, 0), z0, 0, z2), cons(z0, cons(0, z2))), MAX(cons(z0, cons(0, z2)))) IF3(false, cons(0, cons(s(x0), x1))) -> c15(SORT(if2(eq(if1(false, 0, s(x0), x1), 0), if1(false, 0, s(x0), x1), 0, cons(s(x0), x1))), DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1))), MAX(cons(0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), z2))) -> c15(SORT(del(max(cons(s(x0), z2)), cons(0, cons(s(x0), z2)))), DEL(if1(ge(0, s(x0)), 0, s(x0), z2), cons(0, cons(s(x0), z2))), MAX(cons(0, cons(s(x0), z2)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c15(SORT(if2(eq(if1(ge(x0, x1), s(x0), s(x1), x2), s(x0)), if1(ge(x0, x1), s(x0), s(x1), x2), s(x0), cons(s(x1), x2))), DEL(if1(ge(s(x0), s(x1)), s(x0), s(x1), x2), cons(s(x0), cons(s(x1), x2))), MAX(cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(s(z0), cons(s(0), x2))) -> c15(SORT(del(if1(true, s(z0), s(0), x2), cons(s(z0), cons(s(0), x2)))), DEL(if1(ge(s(z0), s(0)), s(z0), s(0), x2), cons(s(z0), cons(s(0), x2))), MAX(cons(s(z0), cons(s(0), x2)))) IF3(false, cons(s(0), cons(s(s(z0)), x2))) -> c15(SORT(del(if1(false, s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2)))), DEL(if1(ge(s(0), s(s(z0))), s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2))), MAX(cons(s(0), cons(s(s(z0)), x2)))) IF3(false, cons(s(s(z0)), cons(s(s(z1)), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2)))), DEL(if1(ge(s(s(z0)), s(s(z1))), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2))), MAX(cons(s(s(z0)), cons(s(s(z1)), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(max(cons(z0, cons(0, x2))), z0), if1(true, z0, 0, x2), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(max(cons(0, cons(s(z0), x2))), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(max(cons(s(z0), cons(s(z1), x2))), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2)))) IF3(false, cons(s(z0), nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(s(z0), nil)), s(z0), nil))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), max(cons(z0, cons(0, x2))), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), max(cons(0, cons(s(z0), x2))), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), max(cons(s(z0), cons(s(z1), x2))), s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(ge(z0, 0), z0, 0, x2), z0), if1(true, z0, 0, x2), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(ge(0, s(z0)), 0, s(z0), x2), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), if1(ge(z0, 0), z0, 0, x2), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), if1(ge(0, s(z0)), 0, s(z0), x2), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0), cons(s(z1), x2)))) IF3(false, cons(z1, cons(0, x1))) -> c15(SORT(if2(eq(if1(true, z1, 0, x1), z1), if1(true, z1, 0, x1), z1, cons(0, x1)))) IF3(false, cons(z0, cons(0, z2))) -> c15(SORT(del(max(cons(z0, z2)), cons(z0, cons(0, z2))))) IF3(false, cons(0, cons(s(x0), x1))) -> c15(SORT(if2(eq(if1(false, 0, s(x0), x1), 0), if1(false, 0, s(x0), x1), 0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), z2))) -> c15(SORT(del(max(cons(s(x0), z2)), cons(0, cons(s(x0), z2))))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c15(SORT(if2(eq(if1(ge(x0, x1), s(x0), s(x1), x2), s(x0)), if1(ge(x0, x1), s(x0), s(x1), x2), s(x0), cons(s(x1), x2)))) IF3(false, cons(s(z0), cons(s(0), x2))) -> c15(SORT(del(if1(true, s(z0), s(0), x2), cons(s(z0), cons(s(0), x2))))) IF3(false, cons(s(0), cons(s(s(z0)), x2))) -> c15(SORT(del(if1(false, s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2))))) IF3(false, cons(s(s(z0)), cons(s(s(z1)), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2))))) IF3(false, cons(s(z0), nil)) -> c15(SORT(if2(eq(z0, z0), s(z0), s(z0), nil))) IF1(true, x0, 0, x1) -> c3(MAX(cons(x0, x1))) IF1(true, s(x0), s(0), x1) -> c3(MAX(cons(s(x0), x1))) IF1(true, s(s(x0)), s(s(0)), x1) -> c3(MAX(cons(s(s(x0)), x1))) IF1(true, s(s(s(x0))), s(s(s(0))), x1) -> c3(MAX(cons(s(s(s(x0))), x1))) IF1(true, s(s(s(s(x0)))), s(s(s(s(x1)))), x2) -> c3(MAX(cons(s(s(s(s(x0)))), x2))) K tuples: IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) IF3(false, cons(x0, cons(0, x1))) -> c(DEL(if1(ge(x0, 0), x0, 0, x1), cons(x0, cons(0, x1)))) IF3(false, cons(x0, cons(0, x1))) -> c(MAX(cons(x0, cons(0, x1)))) IF3(false, cons(0, cons(s(x0), x1))) -> c1(DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), x1))) -> c1(MAX(cons(0, cons(s(x0), x1)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c5(DEL(if1(ge(s(x0), s(x1)), s(x0), s(x1), x2), cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c5(MAX(cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(0, nil)) -> c15(SORT(if2(true, max(cons(0, nil)), 0, nil))) Defined Rule Symbols: del_2, max_1, if2_4, eq_2, ge_2, if1_4 Defined Pair Symbols: IF1_4, IF2_4, EQ_2, GE_2, MAX_1, DEL_2, SORT_1, IF3_2 Compound Symbols: c4_1, c8_1, c12_1, c20_1, c2_1, c6_1, c13_1, c2_2, c6_2, c15_1, c15_2, c15_3, c_1, c1_1, c5_1, c3_1 ---------------------------------------- (201) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) by IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) ---------------------------------------- (202) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil max(cons(z0, nil)) -> z0 max(cons(z0, cons(z1, z2))) -> if1(ge(z0, z1), z0, z1, z2) if2(true, z0, z1, z2) -> z2 if2(false, z0, z1, z2) -> cons(z1, del(z0, z2)) eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) if1(true, z0, z1, z2) -> max(cons(z0, z2)) if1(false, z0, z1, z2) -> max(cons(z1, z2)) Tuples: IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) MAX(cons(s(s(z0)), cons(s(s(0)), x2))) -> c2(IF1(true, s(s(z0)), s(s(0)), x2), GE(s(s(z0)), s(s(0)))) MAX(cons(s(s(0)), cons(s(s(s(z0))), x2))) -> c2(IF1(false, s(s(0)), s(s(s(z0))), x2), GE(s(s(0)), s(s(s(z0))))) MAX(cons(s(s(x0)), cons(s(s(x1)), x2))) -> c2(GE(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c6(IF2(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c6(IF2(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c6(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c6(EQ(s(s(0)), s(s(0)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) MAX(cons(s(s(s(z0))), cons(s(s(s(0))), x2))) -> c2(IF1(true, s(s(s(z0))), s(s(s(0))), x2), GE(s(s(s(z0))), s(s(s(0))))) MAX(cons(s(s(s(0))), cons(s(s(s(s(z0)))), x2))) -> c2(IF1(false, s(s(s(0))), s(s(s(s(z0)))), x2), GE(s(s(s(0))), s(s(s(s(z0)))))) MAX(cons(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2))) -> c2(IF1(ge(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), GE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MAX(cons(s(s(s(x0))), cons(s(s(s(x1))), x2))) -> c2(GE(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c6(IF2(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c6(IF2(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c6(IF2(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c6(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c6(EQ(s(s(s(0))), s(s(s(0))))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(max(cons(z0, cons(0, x2))), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(max(cons(0, cons(s(z0), x2))), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(max(cons(s(z0), cons(s(z1), x2))), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), z0, z1, z2), z0), if1(ge(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), max(cons(z0, cons(0, x2))), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), max(cons(0, cons(s(z0), x2))), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), max(cons(s(z0), cons(s(z1), x2))), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(ge(z0, 0), z0, 0, x2), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(ge(0, s(z0)), 0, s(z0), x2), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), if1(ge(z0, 0), z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), if1(ge(0, s(z0)), 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z1, cons(0, x1))) -> c15(SORT(if2(eq(if1(true, z1, 0, x1), z1), if1(true, z1, 0, x1), z1, cons(0, x1))), DEL(if1(ge(z1, 0), z1, 0, x1), cons(z1, cons(0, x1))), MAX(cons(z1, cons(0, x1)))) IF3(false, cons(z0, cons(0, z2))) -> c15(SORT(del(max(cons(z0, z2)), cons(z0, cons(0, z2)))), DEL(if1(ge(z0, 0), z0, 0, z2), cons(z0, cons(0, z2))), MAX(cons(z0, cons(0, z2)))) IF3(false, cons(x0, cons(0, x1))) -> c(DEL(if1(ge(x0, 0), x0, 0, x1), cons(x0, cons(0, x1)))) IF3(false, cons(x0, cons(0, x1))) -> c(MAX(cons(x0, cons(0, x1)))) IF3(false, cons(0, cons(s(x0), x1))) -> c15(SORT(if2(eq(if1(false, 0, s(x0), x1), 0), if1(false, 0, s(x0), x1), 0, cons(s(x0), x1))), DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1))), MAX(cons(0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), z2))) -> c15(SORT(del(max(cons(s(x0), z2)), cons(0, cons(s(x0), z2)))), DEL(if1(ge(0, s(x0)), 0, s(x0), z2), cons(0, cons(s(x0), z2))), MAX(cons(0, cons(s(x0), z2)))) IF3(false, cons(0, cons(s(x0), x1))) -> c1(DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), x1))) -> c1(MAX(cons(0, cons(s(x0), x1)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c15(SORT(if2(eq(if1(ge(x0, x1), s(x0), s(x1), x2), s(x0)), if1(ge(x0, x1), s(x0), s(x1), x2), s(x0), cons(s(x1), x2))), DEL(if1(ge(s(x0), s(x1)), s(x0), s(x1), x2), cons(s(x0), cons(s(x1), x2))), MAX(cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(s(z0), cons(s(0), x2))) -> c15(SORT(del(if1(true, s(z0), s(0), x2), cons(s(z0), cons(s(0), x2)))), DEL(if1(ge(s(z0), s(0)), s(z0), s(0), x2), cons(s(z0), cons(s(0), x2))), MAX(cons(s(z0), cons(s(0), x2)))) IF3(false, cons(s(0), cons(s(s(z0)), x2))) -> c15(SORT(del(if1(false, s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2)))), DEL(if1(ge(s(0), s(s(z0))), s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2))), MAX(cons(s(0), cons(s(s(z0)), x2)))) IF3(false, cons(s(s(z0)), cons(s(s(z1)), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2)))), DEL(if1(ge(s(s(z0)), s(s(z1))), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2))), MAX(cons(s(s(z0)), cons(s(s(z1)), x2)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c5(DEL(if1(ge(s(x0), s(x1)), s(x0), s(x1), x2), cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c5(MAX(cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(max(cons(z0, cons(0, x2))), z0), if1(true, z0, 0, x2), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(max(cons(0, cons(s(z0), x2))), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(max(cons(s(z0), cons(s(z1), x2))), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), z0, z1, z2), z0), if1(ge(z0, z1), z0, z1, z2), z0, cons(z1, z2)))) IF3(false, cons(0, nil)) -> c15(SORT(if2(true, max(cons(0, nil)), 0, nil))) IF3(false, cons(s(z0), nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(s(z0), nil)), s(z0), nil))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), max(cons(z0, cons(0, x2))), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), max(cons(0, cons(s(z0), x2))), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), max(cons(s(z0), cons(s(z1), x2))), s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(ge(z0, 0), z0, 0, x2), z0), if1(true, z0, 0, x2), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(ge(0, s(z0)), 0, s(z0), x2), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), if1(ge(z0, 0), z0, 0, x2), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), if1(ge(0, s(z0)), 0, s(z0), x2), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0), cons(s(z1), x2)))) IF3(false, cons(z1, cons(0, x1))) -> c15(SORT(if2(eq(if1(true, z1, 0, x1), z1), if1(true, z1, 0, x1), z1, cons(0, x1)))) IF3(false, cons(z0, cons(0, z2))) -> c15(SORT(del(max(cons(z0, z2)), cons(z0, cons(0, z2))))) IF3(false, cons(0, cons(s(x0), x1))) -> c15(SORT(if2(eq(if1(false, 0, s(x0), x1), 0), if1(false, 0, s(x0), x1), 0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), z2))) -> c15(SORT(del(max(cons(s(x0), z2)), cons(0, cons(s(x0), z2))))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c15(SORT(if2(eq(if1(ge(x0, x1), s(x0), s(x1), x2), s(x0)), if1(ge(x0, x1), s(x0), s(x1), x2), s(x0), cons(s(x1), x2)))) IF3(false, cons(s(z0), cons(s(0), x2))) -> c15(SORT(del(if1(true, s(z0), s(0), x2), cons(s(z0), cons(s(0), x2))))) IF3(false, cons(s(0), cons(s(s(z0)), x2))) -> c15(SORT(del(if1(false, s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2))))) IF3(false, cons(s(s(z0)), cons(s(s(z1)), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2))))) IF3(false, cons(s(z0), nil)) -> c15(SORT(if2(eq(z0, z0), s(z0), s(z0), nil))) IF1(true, x0, 0, x1) -> c3(MAX(cons(x0, x1))) IF1(true, s(x0), s(0), x1) -> c3(MAX(cons(s(x0), x1))) IF1(true, s(s(x0)), s(s(0)), x1) -> c3(MAX(cons(s(s(x0)), x1))) IF1(true, s(s(s(x0))), s(s(s(0))), x1) -> c3(MAX(cons(s(s(s(x0))), x1))) IF1(true, s(s(s(s(x0)))), s(s(s(s(x1)))), x2) -> c3(MAX(cons(s(s(s(s(x0)))), x2))) S tuples: IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) MAX(cons(s(s(z0)), cons(s(s(0)), x2))) -> c2(IF1(true, s(s(z0)), s(s(0)), x2), GE(s(s(z0)), s(s(0)))) MAX(cons(s(s(0)), cons(s(s(s(z0))), x2))) -> c2(IF1(false, s(s(0)), s(s(s(z0))), x2), GE(s(s(0)), s(s(s(z0))))) MAX(cons(s(s(x0)), cons(s(s(x1)), x2))) -> c2(GE(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c6(IF2(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c6(IF2(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c6(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c6(EQ(s(s(0)), s(s(0)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) MAX(cons(s(s(s(z0))), cons(s(s(s(0))), x2))) -> c2(IF1(true, s(s(s(z0))), s(s(s(0))), x2), GE(s(s(s(z0))), s(s(s(0))))) MAX(cons(s(s(s(0))), cons(s(s(s(s(z0)))), x2))) -> c2(IF1(false, s(s(s(0))), s(s(s(s(z0)))), x2), GE(s(s(s(0))), s(s(s(s(z0)))))) MAX(cons(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2))) -> c2(IF1(ge(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), GE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MAX(cons(s(s(s(x0))), cons(s(s(s(x1))), x2))) -> c2(GE(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c6(IF2(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c6(IF2(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c6(IF2(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c6(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c6(EQ(s(s(s(0))), s(s(s(0))))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(max(cons(z0, cons(0, x2))), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(max(cons(0, cons(s(z0), x2))), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(max(cons(s(z0), cons(s(z1), x2))), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), max(cons(z0, cons(0, x2))), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), max(cons(0, cons(s(z0), x2))), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), max(cons(s(z0), cons(s(z1), x2))), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(ge(z0, 0), z0, 0, x2), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(ge(0, s(z0)), 0, s(z0), x2), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), if1(ge(z0, 0), z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), if1(ge(0, s(z0)), 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z1, cons(0, x1))) -> c15(SORT(if2(eq(if1(true, z1, 0, x1), z1), if1(true, z1, 0, x1), z1, cons(0, x1))), DEL(if1(ge(z1, 0), z1, 0, x1), cons(z1, cons(0, x1))), MAX(cons(z1, cons(0, x1)))) IF3(false, cons(z0, cons(0, z2))) -> c15(SORT(del(max(cons(z0, z2)), cons(z0, cons(0, z2)))), DEL(if1(ge(z0, 0), z0, 0, z2), cons(z0, cons(0, z2))), MAX(cons(z0, cons(0, z2)))) IF3(false, cons(0, cons(s(x0), x1))) -> c15(SORT(if2(eq(if1(false, 0, s(x0), x1), 0), if1(false, 0, s(x0), x1), 0, cons(s(x0), x1))), DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1))), MAX(cons(0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), z2))) -> c15(SORT(del(max(cons(s(x0), z2)), cons(0, cons(s(x0), z2)))), DEL(if1(ge(0, s(x0)), 0, s(x0), z2), cons(0, cons(s(x0), z2))), MAX(cons(0, cons(s(x0), z2)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c15(SORT(if2(eq(if1(ge(x0, x1), s(x0), s(x1), x2), s(x0)), if1(ge(x0, x1), s(x0), s(x1), x2), s(x0), cons(s(x1), x2))), DEL(if1(ge(s(x0), s(x1)), s(x0), s(x1), x2), cons(s(x0), cons(s(x1), x2))), MAX(cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(s(z0), cons(s(0), x2))) -> c15(SORT(del(if1(true, s(z0), s(0), x2), cons(s(z0), cons(s(0), x2)))), DEL(if1(ge(s(z0), s(0)), s(z0), s(0), x2), cons(s(z0), cons(s(0), x2))), MAX(cons(s(z0), cons(s(0), x2)))) IF3(false, cons(s(0), cons(s(s(z0)), x2))) -> c15(SORT(del(if1(false, s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2)))), DEL(if1(ge(s(0), s(s(z0))), s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2))), MAX(cons(s(0), cons(s(s(z0)), x2)))) IF3(false, cons(s(s(z0)), cons(s(s(z1)), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2)))), DEL(if1(ge(s(s(z0)), s(s(z1))), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2))), MAX(cons(s(s(z0)), cons(s(s(z1)), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(max(cons(z0, cons(0, x2))), z0), if1(true, z0, 0, x2), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(max(cons(0, cons(s(z0), x2))), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(max(cons(s(z0), cons(s(z1), x2))), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2)))) IF3(false, cons(s(z0), nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(s(z0), nil)), s(z0), nil))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), max(cons(z0, cons(0, x2))), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), max(cons(0, cons(s(z0), x2))), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), max(cons(s(z0), cons(s(z1), x2))), s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(ge(z0, 0), z0, 0, x2), z0), if1(true, z0, 0, x2), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(ge(0, s(z0)), 0, s(z0), x2), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), if1(ge(z0, 0), z0, 0, x2), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), if1(ge(0, s(z0)), 0, s(z0), x2), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0), cons(s(z1), x2)))) IF3(false, cons(z1, cons(0, x1))) -> c15(SORT(if2(eq(if1(true, z1, 0, x1), z1), if1(true, z1, 0, x1), z1, cons(0, x1)))) IF3(false, cons(z0, cons(0, z2))) -> c15(SORT(del(max(cons(z0, z2)), cons(z0, cons(0, z2))))) IF3(false, cons(0, cons(s(x0), x1))) -> c15(SORT(if2(eq(if1(false, 0, s(x0), x1), 0), if1(false, 0, s(x0), x1), 0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), z2))) -> c15(SORT(del(max(cons(s(x0), z2)), cons(0, cons(s(x0), z2))))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c15(SORT(if2(eq(if1(ge(x0, x1), s(x0), s(x1), x2), s(x0)), if1(ge(x0, x1), s(x0), s(x1), x2), s(x0), cons(s(x1), x2)))) IF3(false, cons(s(z0), cons(s(0), x2))) -> c15(SORT(del(if1(true, s(z0), s(0), x2), cons(s(z0), cons(s(0), x2))))) IF3(false, cons(s(0), cons(s(s(z0)), x2))) -> c15(SORT(del(if1(false, s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2))))) IF3(false, cons(s(s(z0)), cons(s(s(z1)), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2))))) IF3(false, cons(s(z0), nil)) -> c15(SORT(if2(eq(z0, z0), s(z0), s(z0), nil))) IF1(true, x0, 0, x1) -> c3(MAX(cons(x0, x1))) IF1(true, s(x0), s(0), x1) -> c3(MAX(cons(s(x0), x1))) IF1(true, s(s(x0)), s(s(0)), x1) -> c3(MAX(cons(s(s(x0)), x1))) IF1(true, s(s(s(x0))), s(s(s(0))), x1) -> c3(MAX(cons(s(s(s(x0))), x1))) IF1(true, s(s(s(s(x0)))), s(s(s(s(x1)))), x2) -> c3(MAX(cons(s(s(s(s(x0)))), x2))) K tuples: IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) IF3(false, cons(x0, cons(0, x1))) -> c(DEL(if1(ge(x0, 0), x0, 0, x1), cons(x0, cons(0, x1)))) IF3(false, cons(x0, cons(0, x1))) -> c(MAX(cons(x0, cons(0, x1)))) IF3(false, cons(0, cons(s(x0), x1))) -> c1(DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), x1))) -> c1(MAX(cons(0, cons(s(x0), x1)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c5(DEL(if1(ge(s(x0), s(x1)), s(x0), s(x1), x2), cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c5(MAX(cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(0, nil)) -> c15(SORT(if2(true, max(cons(0, nil)), 0, nil))) Defined Rule Symbols: del_2, max_1, if2_4, eq_2, ge_2, if1_4 Defined Pair Symbols: IF1_4, IF2_4, EQ_2, GE_2, MAX_1, DEL_2, SORT_1, IF3_2 Compound Symbols: c4_1, c8_1, c12_1, c20_1, c2_1, c6_1, c13_1, c2_2, c6_2, c15_1, c15_2, c15_3, c_1, c1_1, c5_1, c3_1 ---------------------------------------- (203) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IF1(false, z0, z1, z2) -> c4(MAX(cons(z1, z2))) by IF1(false, 0, s(x0), x1) -> c4(MAX(cons(s(x0), x1))) IF1(false, s(0), s(s(x0)), x1) -> c4(MAX(cons(s(s(x0)), x1))) IF1(false, s(s(0)), s(s(s(x0))), x1) -> c4(MAX(cons(s(s(s(x0))), x1))) IF1(false, s(s(s(0))), s(s(s(s(x0)))), x1) -> c4(MAX(cons(s(s(s(s(x0)))), x1))) IF1(false, s(s(s(s(x0)))), s(s(s(s(x1)))), x2) -> c4(MAX(cons(s(s(s(s(x1)))), x2))) ---------------------------------------- (204) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil max(cons(z0, nil)) -> z0 max(cons(z0, cons(z1, z2))) -> if1(ge(z0, z1), z0, z1, z2) if2(true, z0, z1, z2) -> z2 if2(false, z0, z1, z2) -> cons(z1, del(z0, z2)) eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) if1(true, z0, z1, z2) -> max(cons(z0, z2)) if1(false, z0, z1, z2) -> max(cons(z1, z2)) Tuples: IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) MAX(cons(s(s(z0)), cons(s(s(0)), x2))) -> c2(IF1(true, s(s(z0)), s(s(0)), x2), GE(s(s(z0)), s(s(0)))) MAX(cons(s(s(0)), cons(s(s(s(z0))), x2))) -> c2(IF1(false, s(s(0)), s(s(s(z0))), x2), GE(s(s(0)), s(s(s(z0))))) MAX(cons(s(s(x0)), cons(s(s(x1)), x2))) -> c2(GE(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c6(IF2(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c6(IF2(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c6(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c6(EQ(s(s(0)), s(s(0)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) MAX(cons(s(s(s(z0))), cons(s(s(s(0))), x2))) -> c2(IF1(true, s(s(s(z0))), s(s(s(0))), x2), GE(s(s(s(z0))), s(s(s(0))))) MAX(cons(s(s(s(0))), cons(s(s(s(s(z0)))), x2))) -> c2(IF1(false, s(s(s(0))), s(s(s(s(z0)))), x2), GE(s(s(s(0))), s(s(s(s(z0)))))) MAX(cons(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2))) -> c2(IF1(ge(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), GE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MAX(cons(s(s(s(x0))), cons(s(s(s(x1))), x2))) -> c2(GE(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c6(IF2(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c6(IF2(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c6(IF2(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c6(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c6(EQ(s(s(s(0))), s(s(s(0))))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(max(cons(z0, cons(0, x2))), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(max(cons(0, cons(s(z0), x2))), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(max(cons(s(z0), cons(s(z1), x2))), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), z0, z1, z2), z0), if1(ge(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), max(cons(z0, cons(0, x2))), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), max(cons(0, cons(s(z0), x2))), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), max(cons(s(z0), cons(s(z1), x2))), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(ge(z0, 0), z0, 0, x2), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(ge(0, s(z0)), 0, s(z0), x2), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), if1(ge(z0, 0), z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), if1(ge(0, s(z0)), 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z1, cons(0, x1))) -> c15(SORT(if2(eq(if1(true, z1, 0, x1), z1), if1(true, z1, 0, x1), z1, cons(0, x1))), DEL(if1(ge(z1, 0), z1, 0, x1), cons(z1, cons(0, x1))), MAX(cons(z1, cons(0, x1)))) IF3(false, cons(z0, cons(0, z2))) -> c15(SORT(del(max(cons(z0, z2)), cons(z0, cons(0, z2)))), DEL(if1(ge(z0, 0), z0, 0, z2), cons(z0, cons(0, z2))), MAX(cons(z0, cons(0, z2)))) IF3(false, cons(x0, cons(0, x1))) -> c(DEL(if1(ge(x0, 0), x0, 0, x1), cons(x0, cons(0, x1)))) IF3(false, cons(x0, cons(0, x1))) -> c(MAX(cons(x0, cons(0, x1)))) IF3(false, cons(0, cons(s(x0), x1))) -> c15(SORT(if2(eq(if1(false, 0, s(x0), x1), 0), if1(false, 0, s(x0), x1), 0, cons(s(x0), x1))), DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1))), MAX(cons(0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), z2))) -> c15(SORT(del(max(cons(s(x0), z2)), cons(0, cons(s(x0), z2)))), DEL(if1(ge(0, s(x0)), 0, s(x0), z2), cons(0, cons(s(x0), z2))), MAX(cons(0, cons(s(x0), z2)))) IF3(false, cons(0, cons(s(x0), x1))) -> c1(DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), x1))) -> c1(MAX(cons(0, cons(s(x0), x1)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c15(SORT(if2(eq(if1(ge(x0, x1), s(x0), s(x1), x2), s(x0)), if1(ge(x0, x1), s(x0), s(x1), x2), s(x0), cons(s(x1), x2))), DEL(if1(ge(s(x0), s(x1)), s(x0), s(x1), x2), cons(s(x0), cons(s(x1), x2))), MAX(cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(s(z0), cons(s(0), x2))) -> c15(SORT(del(if1(true, s(z0), s(0), x2), cons(s(z0), cons(s(0), x2)))), DEL(if1(ge(s(z0), s(0)), s(z0), s(0), x2), cons(s(z0), cons(s(0), x2))), MAX(cons(s(z0), cons(s(0), x2)))) IF3(false, cons(s(0), cons(s(s(z0)), x2))) -> c15(SORT(del(if1(false, s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2)))), DEL(if1(ge(s(0), s(s(z0))), s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2))), MAX(cons(s(0), cons(s(s(z0)), x2)))) IF3(false, cons(s(s(z0)), cons(s(s(z1)), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2)))), DEL(if1(ge(s(s(z0)), s(s(z1))), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2))), MAX(cons(s(s(z0)), cons(s(s(z1)), x2)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c5(DEL(if1(ge(s(x0), s(x1)), s(x0), s(x1), x2), cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c5(MAX(cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(max(cons(z0, cons(0, x2))), z0), if1(true, z0, 0, x2), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(max(cons(0, cons(s(z0), x2))), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(max(cons(s(z0), cons(s(z1), x2))), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), z0, z1, z2), z0), if1(ge(z0, z1), z0, z1, z2), z0, cons(z1, z2)))) IF3(false, cons(0, nil)) -> c15(SORT(if2(true, max(cons(0, nil)), 0, nil))) IF3(false, cons(s(z0), nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(s(z0), nil)), s(z0), nil))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), max(cons(z0, cons(0, x2))), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), max(cons(0, cons(s(z0), x2))), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), max(cons(s(z0), cons(s(z1), x2))), s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(ge(z0, 0), z0, 0, x2), z0), if1(true, z0, 0, x2), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(ge(0, s(z0)), 0, s(z0), x2), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), if1(ge(z0, 0), z0, 0, x2), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), if1(ge(0, s(z0)), 0, s(z0), x2), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0), cons(s(z1), x2)))) IF3(false, cons(z1, cons(0, x1))) -> c15(SORT(if2(eq(if1(true, z1, 0, x1), z1), if1(true, z1, 0, x1), z1, cons(0, x1)))) IF3(false, cons(z0, cons(0, z2))) -> c15(SORT(del(max(cons(z0, z2)), cons(z0, cons(0, z2))))) IF3(false, cons(0, cons(s(x0), x1))) -> c15(SORT(if2(eq(if1(false, 0, s(x0), x1), 0), if1(false, 0, s(x0), x1), 0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), z2))) -> c15(SORT(del(max(cons(s(x0), z2)), cons(0, cons(s(x0), z2))))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c15(SORT(if2(eq(if1(ge(x0, x1), s(x0), s(x1), x2), s(x0)), if1(ge(x0, x1), s(x0), s(x1), x2), s(x0), cons(s(x1), x2)))) IF3(false, cons(s(z0), cons(s(0), x2))) -> c15(SORT(del(if1(true, s(z0), s(0), x2), cons(s(z0), cons(s(0), x2))))) IF3(false, cons(s(0), cons(s(s(z0)), x2))) -> c15(SORT(del(if1(false, s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2))))) IF3(false, cons(s(s(z0)), cons(s(s(z1)), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2))))) IF3(false, cons(s(z0), nil)) -> c15(SORT(if2(eq(z0, z0), s(z0), s(z0), nil))) IF1(true, x0, 0, x1) -> c3(MAX(cons(x0, x1))) IF1(true, s(x0), s(0), x1) -> c3(MAX(cons(s(x0), x1))) IF1(true, s(s(x0)), s(s(0)), x1) -> c3(MAX(cons(s(s(x0)), x1))) IF1(true, s(s(s(x0))), s(s(s(0))), x1) -> c3(MAX(cons(s(s(s(x0))), x1))) IF1(true, s(s(s(s(x0)))), s(s(s(s(x1)))), x2) -> c3(MAX(cons(s(s(s(s(x0)))), x2))) IF1(false, 0, s(x0), x1) -> c4(MAX(cons(s(x0), x1))) IF1(false, s(0), s(s(x0)), x1) -> c4(MAX(cons(s(s(x0)), x1))) IF1(false, s(s(0)), s(s(s(x0))), x1) -> c4(MAX(cons(s(s(s(x0))), x1))) IF1(false, s(s(s(0))), s(s(s(s(x0)))), x1) -> c4(MAX(cons(s(s(s(s(x0)))), x1))) IF1(false, s(s(s(s(x0)))), s(s(s(s(x1)))), x2) -> c4(MAX(cons(s(s(s(s(x1)))), x2))) S tuples: IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) MAX(cons(s(s(z0)), cons(s(s(0)), x2))) -> c2(IF1(true, s(s(z0)), s(s(0)), x2), GE(s(s(z0)), s(s(0)))) MAX(cons(s(s(0)), cons(s(s(s(z0))), x2))) -> c2(IF1(false, s(s(0)), s(s(s(z0))), x2), GE(s(s(0)), s(s(s(z0))))) MAX(cons(s(s(x0)), cons(s(s(x1)), x2))) -> c2(GE(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c6(IF2(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c6(IF2(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c6(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c6(EQ(s(s(0)), s(s(0)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) MAX(cons(s(s(s(z0))), cons(s(s(s(0))), x2))) -> c2(IF1(true, s(s(s(z0))), s(s(s(0))), x2), GE(s(s(s(z0))), s(s(s(0))))) MAX(cons(s(s(s(0))), cons(s(s(s(s(z0)))), x2))) -> c2(IF1(false, s(s(s(0))), s(s(s(s(z0)))), x2), GE(s(s(s(0))), s(s(s(s(z0)))))) MAX(cons(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2))) -> c2(IF1(ge(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), GE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MAX(cons(s(s(s(x0))), cons(s(s(s(x1))), x2))) -> c2(GE(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c6(IF2(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c6(IF2(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c6(IF2(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c6(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c6(EQ(s(s(s(0))), s(s(s(0))))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(max(cons(z0, cons(0, x2))), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(max(cons(0, cons(s(z0), x2))), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(max(cons(s(z0), cons(s(z1), x2))), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), max(cons(z0, cons(0, x2))), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), max(cons(0, cons(s(z0), x2))), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), max(cons(s(z0), cons(s(z1), x2))), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(ge(z0, 0), z0, 0, x2), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(ge(0, s(z0)), 0, s(z0), x2), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), if1(ge(z0, 0), z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), if1(ge(0, s(z0)), 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z1, cons(0, x1))) -> c15(SORT(if2(eq(if1(true, z1, 0, x1), z1), if1(true, z1, 0, x1), z1, cons(0, x1))), DEL(if1(ge(z1, 0), z1, 0, x1), cons(z1, cons(0, x1))), MAX(cons(z1, cons(0, x1)))) IF3(false, cons(z0, cons(0, z2))) -> c15(SORT(del(max(cons(z0, z2)), cons(z0, cons(0, z2)))), DEL(if1(ge(z0, 0), z0, 0, z2), cons(z0, cons(0, z2))), MAX(cons(z0, cons(0, z2)))) IF3(false, cons(0, cons(s(x0), x1))) -> c15(SORT(if2(eq(if1(false, 0, s(x0), x1), 0), if1(false, 0, s(x0), x1), 0, cons(s(x0), x1))), DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1))), MAX(cons(0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), z2))) -> c15(SORT(del(max(cons(s(x0), z2)), cons(0, cons(s(x0), z2)))), DEL(if1(ge(0, s(x0)), 0, s(x0), z2), cons(0, cons(s(x0), z2))), MAX(cons(0, cons(s(x0), z2)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c15(SORT(if2(eq(if1(ge(x0, x1), s(x0), s(x1), x2), s(x0)), if1(ge(x0, x1), s(x0), s(x1), x2), s(x0), cons(s(x1), x2))), DEL(if1(ge(s(x0), s(x1)), s(x0), s(x1), x2), cons(s(x0), cons(s(x1), x2))), MAX(cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(s(z0), cons(s(0), x2))) -> c15(SORT(del(if1(true, s(z0), s(0), x2), cons(s(z0), cons(s(0), x2)))), DEL(if1(ge(s(z0), s(0)), s(z0), s(0), x2), cons(s(z0), cons(s(0), x2))), MAX(cons(s(z0), cons(s(0), x2)))) IF3(false, cons(s(0), cons(s(s(z0)), x2))) -> c15(SORT(del(if1(false, s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2)))), DEL(if1(ge(s(0), s(s(z0))), s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2))), MAX(cons(s(0), cons(s(s(z0)), x2)))) IF3(false, cons(s(s(z0)), cons(s(s(z1)), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2)))), DEL(if1(ge(s(s(z0)), s(s(z1))), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2))), MAX(cons(s(s(z0)), cons(s(s(z1)), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(max(cons(z0, cons(0, x2))), z0), if1(true, z0, 0, x2), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(max(cons(0, cons(s(z0), x2))), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(max(cons(s(z0), cons(s(z1), x2))), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2)))) IF3(false, cons(s(z0), nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(s(z0), nil)), s(z0), nil))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), max(cons(z0, cons(0, x2))), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), max(cons(0, cons(s(z0), x2))), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), max(cons(s(z0), cons(s(z1), x2))), s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(ge(z0, 0), z0, 0, x2), z0), if1(true, z0, 0, x2), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(ge(0, s(z0)), 0, s(z0), x2), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), if1(ge(z0, 0), z0, 0, x2), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), if1(ge(0, s(z0)), 0, s(z0), x2), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0), cons(s(z1), x2)))) IF3(false, cons(z1, cons(0, x1))) -> c15(SORT(if2(eq(if1(true, z1, 0, x1), z1), if1(true, z1, 0, x1), z1, cons(0, x1)))) IF3(false, cons(z0, cons(0, z2))) -> c15(SORT(del(max(cons(z0, z2)), cons(z0, cons(0, z2))))) IF3(false, cons(0, cons(s(x0), x1))) -> c15(SORT(if2(eq(if1(false, 0, s(x0), x1), 0), if1(false, 0, s(x0), x1), 0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), z2))) -> c15(SORT(del(max(cons(s(x0), z2)), cons(0, cons(s(x0), z2))))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c15(SORT(if2(eq(if1(ge(x0, x1), s(x0), s(x1), x2), s(x0)), if1(ge(x0, x1), s(x0), s(x1), x2), s(x0), cons(s(x1), x2)))) IF3(false, cons(s(z0), cons(s(0), x2))) -> c15(SORT(del(if1(true, s(z0), s(0), x2), cons(s(z0), cons(s(0), x2))))) IF3(false, cons(s(0), cons(s(s(z0)), x2))) -> c15(SORT(del(if1(false, s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2))))) IF3(false, cons(s(s(z0)), cons(s(s(z1)), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2))))) IF3(false, cons(s(z0), nil)) -> c15(SORT(if2(eq(z0, z0), s(z0), s(z0), nil))) IF1(true, x0, 0, x1) -> c3(MAX(cons(x0, x1))) IF1(true, s(x0), s(0), x1) -> c3(MAX(cons(s(x0), x1))) IF1(true, s(s(x0)), s(s(0)), x1) -> c3(MAX(cons(s(s(x0)), x1))) IF1(true, s(s(s(x0))), s(s(s(0))), x1) -> c3(MAX(cons(s(s(s(x0))), x1))) IF1(true, s(s(s(s(x0)))), s(s(s(s(x1)))), x2) -> c3(MAX(cons(s(s(s(s(x0)))), x2))) IF1(false, 0, s(x0), x1) -> c4(MAX(cons(s(x0), x1))) IF1(false, s(0), s(s(x0)), x1) -> c4(MAX(cons(s(s(x0)), x1))) IF1(false, s(s(0)), s(s(s(x0))), x1) -> c4(MAX(cons(s(s(s(x0))), x1))) IF1(false, s(s(s(0))), s(s(s(s(x0)))), x1) -> c4(MAX(cons(s(s(s(s(x0)))), x1))) IF1(false, s(s(s(s(x0)))), s(s(s(s(x1)))), x2) -> c4(MAX(cons(s(s(s(s(x1)))), x2))) K tuples: IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) IF3(false, cons(x0, cons(0, x1))) -> c(DEL(if1(ge(x0, 0), x0, 0, x1), cons(x0, cons(0, x1)))) IF3(false, cons(x0, cons(0, x1))) -> c(MAX(cons(x0, cons(0, x1)))) IF3(false, cons(0, cons(s(x0), x1))) -> c1(DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), x1))) -> c1(MAX(cons(0, cons(s(x0), x1)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c5(DEL(if1(ge(s(x0), s(x1)), s(x0), s(x1), x2), cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c5(MAX(cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(0, nil)) -> c15(SORT(if2(true, max(cons(0, nil)), 0, nil))) Defined Rule Symbols: del_2, max_1, if2_4, eq_2, ge_2, if1_4 Defined Pair Symbols: IF2_4, EQ_2, GE_2, MAX_1, DEL_2, SORT_1, IF3_2, IF1_4 Compound Symbols: c8_1, c12_1, c20_1, c2_1, c6_1, c13_1, c2_2, c6_2, c15_1, c15_2, c15_3, c_1, c1_1, c5_1, c3_1, c4_1 ---------------------------------------- (205) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) by IF2(false, 0, s(x0), x1) -> c8(DEL(0, x1)) IF2(false, s(x0), 0, x1) -> c8(DEL(s(x0), x1)) IF2(false, s(0), s(s(x0)), x1) -> c8(DEL(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c8(DEL(s(s(x0)), x1)) IF2(false, s(s(0)), s(s(s(x0))), x1) -> c8(DEL(s(s(0)), x1)) IF2(false, s(s(s(x0))), s(s(0)), x1) -> c8(DEL(s(s(s(x0))), x1)) IF2(false, s(s(s(0))), s(s(s(s(x0)))), x1) -> c8(DEL(s(s(s(0))), x1)) IF2(false, s(s(s(s(x0)))), s(s(s(0))), x1) -> c8(DEL(s(s(s(s(x0)))), x1)) IF2(false, s(s(s(s(x0)))), s(s(s(s(x1)))), x2) -> c8(DEL(s(s(s(s(x0)))), x2)) ---------------------------------------- (206) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil max(cons(z0, nil)) -> z0 max(cons(z0, cons(z1, z2))) -> if1(ge(z0, z1), z0, z1, z2) if2(true, z0, z1, z2) -> z2 if2(false, z0, z1, z2) -> cons(z1, del(z0, z2)) eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) if1(true, z0, z1, z2) -> max(cons(z0, z2)) if1(false, z0, z1, z2) -> max(cons(z1, z2)) Tuples: EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) MAX(cons(s(s(z0)), cons(s(s(0)), x2))) -> c2(IF1(true, s(s(z0)), s(s(0)), x2), GE(s(s(z0)), s(s(0)))) MAX(cons(s(s(0)), cons(s(s(s(z0))), x2))) -> c2(IF1(false, s(s(0)), s(s(s(z0))), x2), GE(s(s(0)), s(s(s(z0))))) MAX(cons(s(s(x0)), cons(s(s(x1)), x2))) -> c2(GE(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c6(IF2(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c6(IF2(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c6(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c6(EQ(s(s(0)), s(s(0)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) MAX(cons(s(s(s(z0))), cons(s(s(s(0))), x2))) -> c2(IF1(true, s(s(s(z0))), s(s(s(0))), x2), GE(s(s(s(z0))), s(s(s(0))))) MAX(cons(s(s(s(0))), cons(s(s(s(s(z0)))), x2))) -> c2(IF1(false, s(s(s(0))), s(s(s(s(z0)))), x2), GE(s(s(s(0))), s(s(s(s(z0)))))) MAX(cons(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2))) -> c2(IF1(ge(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), GE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MAX(cons(s(s(s(x0))), cons(s(s(s(x1))), x2))) -> c2(GE(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c6(IF2(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c6(IF2(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c6(IF2(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c6(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c6(EQ(s(s(s(0))), s(s(s(0))))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(max(cons(z0, cons(0, x2))), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(max(cons(0, cons(s(z0), x2))), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(max(cons(s(z0), cons(s(z1), x2))), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), z0, z1, z2), z0), if1(ge(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), max(cons(z0, cons(0, x2))), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), max(cons(0, cons(s(z0), x2))), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), max(cons(s(z0), cons(s(z1), x2))), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(ge(z0, 0), z0, 0, x2), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(ge(0, s(z0)), 0, s(z0), x2), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), if1(ge(z0, 0), z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), if1(ge(0, s(z0)), 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z1, cons(0, x1))) -> c15(SORT(if2(eq(if1(true, z1, 0, x1), z1), if1(true, z1, 0, x1), z1, cons(0, x1))), DEL(if1(ge(z1, 0), z1, 0, x1), cons(z1, cons(0, x1))), MAX(cons(z1, cons(0, x1)))) IF3(false, cons(z0, cons(0, z2))) -> c15(SORT(del(max(cons(z0, z2)), cons(z0, cons(0, z2)))), DEL(if1(ge(z0, 0), z0, 0, z2), cons(z0, cons(0, z2))), MAX(cons(z0, cons(0, z2)))) IF3(false, cons(x0, cons(0, x1))) -> c(DEL(if1(ge(x0, 0), x0, 0, x1), cons(x0, cons(0, x1)))) IF3(false, cons(x0, cons(0, x1))) -> c(MAX(cons(x0, cons(0, x1)))) IF3(false, cons(0, cons(s(x0), x1))) -> c15(SORT(if2(eq(if1(false, 0, s(x0), x1), 0), if1(false, 0, s(x0), x1), 0, cons(s(x0), x1))), DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1))), MAX(cons(0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), z2))) -> c15(SORT(del(max(cons(s(x0), z2)), cons(0, cons(s(x0), z2)))), DEL(if1(ge(0, s(x0)), 0, s(x0), z2), cons(0, cons(s(x0), z2))), MAX(cons(0, cons(s(x0), z2)))) IF3(false, cons(0, cons(s(x0), x1))) -> c1(DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), x1))) -> c1(MAX(cons(0, cons(s(x0), x1)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c15(SORT(if2(eq(if1(ge(x0, x1), s(x0), s(x1), x2), s(x0)), if1(ge(x0, x1), s(x0), s(x1), x2), s(x0), cons(s(x1), x2))), DEL(if1(ge(s(x0), s(x1)), s(x0), s(x1), x2), cons(s(x0), cons(s(x1), x2))), MAX(cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(s(z0), cons(s(0), x2))) -> c15(SORT(del(if1(true, s(z0), s(0), x2), cons(s(z0), cons(s(0), x2)))), DEL(if1(ge(s(z0), s(0)), s(z0), s(0), x2), cons(s(z0), cons(s(0), x2))), MAX(cons(s(z0), cons(s(0), x2)))) IF3(false, cons(s(0), cons(s(s(z0)), x2))) -> c15(SORT(del(if1(false, s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2)))), DEL(if1(ge(s(0), s(s(z0))), s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2))), MAX(cons(s(0), cons(s(s(z0)), x2)))) IF3(false, cons(s(s(z0)), cons(s(s(z1)), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2)))), DEL(if1(ge(s(s(z0)), s(s(z1))), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2))), MAX(cons(s(s(z0)), cons(s(s(z1)), x2)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c5(DEL(if1(ge(s(x0), s(x1)), s(x0), s(x1), x2), cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c5(MAX(cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(max(cons(z0, cons(0, x2))), z0), if1(true, z0, 0, x2), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(max(cons(0, cons(s(z0), x2))), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(max(cons(s(z0), cons(s(z1), x2))), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), z0, z1, z2), z0), if1(ge(z0, z1), z0, z1, z2), z0, cons(z1, z2)))) IF3(false, cons(0, nil)) -> c15(SORT(if2(true, max(cons(0, nil)), 0, nil))) IF3(false, cons(s(z0), nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(s(z0), nil)), s(z0), nil))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), max(cons(z0, cons(0, x2))), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), max(cons(0, cons(s(z0), x2))), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), max(cons(s(z0), cons(s(z1), x2))), s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(ge(z0, 0), z0, 0, x2), z0), if1(true, z0, 0, x2), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(ge(0, s(z0)), 0, s(z0), x2), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), if1(ge(z0, 0), z0, 0, x2), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), if1(ge(0, s(z0)), 0, s(z0), x2), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0), cons(s(z1), x2)))) IF3(false, cons(z1, cons(0, x1))) -> c15(SORT(if2(eq(if1(true, z1, 0, x1), z1), if1(true, z1, 0, x1), z1, cons(0, x1)))) IF3(false, cons(z0, cons(0, z2))) -> c15(SORT(del(max(cons(z0, z2)), cons(z0, cons(0, z2))))) IF3(false, cons(0, cons(s(x0), x1))) -> c15(SORT(if2(eq(if1(false, 0, s(x0), x1), 0), if1(false, 0, s(x0), x1), 0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), z2))) -> c15(SORT(del(max(cons(s(x0), z2)), cons(0, cons(s(x0), z2))))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c15(SORT(if2(eq(if1(ge(x0, x1), s(x0), s(x1), x2), s(x0)), if1(ge(x0, x1), s(x0), s(x1), x2), s(x0), cons(s(x1), x2)))) IF3(false, cons(s(z0), cons(s(0), x2))) -> c15(SORT(del(if1(true, s(z0), s(0), x2), cons(s(z0), cons(s(0), x2))))) IF3(false, cons(s(0), cons(s(s(z0)), x2))) -> c15(SORT(del(if1(false, s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2))))) IF3(false, cons(s(s(z0)), cons(s(s(z1)), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2))))) IF3(false, cons(s(z0), nil)) -> c15(SORT(if2(eq(z0, z0), s(z0), s(z0), nil))) IF1(true, x0, 0, x1) -> c3(MAX(cons(x0, x1))) IF1(true, s(x0), s(0), x1) -> c3(MAX(cons(s(x0), x1))) IF1(true, s(s(x0)), s(s(0)), x1) -> c3(MAX(cons(s(s(x0)), x1))) IF1(true, s(s(s(x0))), s(s(s(0))), x1) -> c3(MAX(cons(s(s(s(x0))), x1))) IF1(true, s(s(s(s(x0)))), s(s(s(s(x1)))), x2) -> c3(MAX(cons(s(s(s(s(x0)))), x2))) IF1(false, 0, s(x0), x1) -> c4(MAX(cons(s(x0), x1))) IF1(false, s(0), s(s(x0)), x1) -> c4(MAX(cons(s(s(x0)), x1))) IF1(false, s(s(0)), s(s(s(x0))), x1) -> c4(MAX(cons(s(s(s(x0))), x1))) IF1(false, s(s(s(0))), s(s(s(s(x0)))), x1) -> c4(MAX(cons(s(s(s(s(x0)))), x1))) IF1(false, s(s(s(s(x0)))), s(s(s(s(x1)))), x2) -> c4(MAX(cons(s(s(s(s(x1)))), x2))) IF2(false, 0, s(x0), x1) -> c8(DEL(0, x1)) IF2(false, s(x0), 0, x1) -> c8(DEL(s(x0), x1)) IF2(false, s(0), s(s(x0)), x1) -> c8(DEL(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c8(DEL(s(s(x0)), x1)) IF2(false, s(s(0)), s(s(s(x0))), x1) -> c8(DEL(s(s(0)), x1)) IF2(false, s(s(s(x0))), s(s(0)), x1) -> c8(DEL(s(s(s(x0))), x1)) IF2(false, s(s(s(0))), s(s(s(s(x0)))), x1) -> c8(DEL(s(s(s(0))), x1)) IF2(false, s(s(s(s(x0)))), s(s(s(0))), x1) -> c8(DEL(s(s(s(s(x0)))), x1)) IF2(false, s(s(s(s(x0)))), s(s(s(s(x1)))), x2) -> c8(DEL(s(s(s(s(x0)))), x2)) S tuples: EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) MAX(cons(s(s(z0)), cons(s(s(0)), x2))) -> c2(IF1(true, s(s(z0)), s(s(0)), x2), GE(s(s(z0)), s(s(0)))) MAX(cons(s(s(0)), cons(s(s(s(z0))), x2))) -> c2(IF1(false, s(s(0)), s(s(s(z0))), x2), GE(s(s(0)), s(s(s(z0))))) MAX(cons(s(s(x0)), cons(s(s(x1)), x2))) -> c2(GE(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c6(IF2(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c6(IF2(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c6(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c6(EQ(s(s(0)), s(s(0)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) MAX(cons(s(s(s(z0))), cons(s(s(s(0))), x2))) -> c2(IF1(true, s(s(s(z0))), s(s(s(0))), x2), GE(s(s(s(z0))), s(s(s(0))))) MAX(cons(s(s(s(0))), cons(s(s(s(s(z0)))), x2))) -> c2(IF1(false, s(s(s(0))), s(s(s(s(z0)))), x2), GE(s(s(s(0))), s(s(s(s(z0)))))) MAX(cons(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2))) -> c2(IF1(ge(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), GE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MAX(cons(s(s(s(x0))), cons(s(s(s(x1))), x2))) -> c2(GE(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c6(IF2(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c6(IF2(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c6(IF2(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c6(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c6(EQ(s(s(s(0))), s(s(s(0))))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(max(cons(z0, cons(0, x2))), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(max(cons(0, cons(s(z0), x2))), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(max(cons(s(z0), cons(s(z1), x2))), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), max(cons(z0, cons(0, x2))), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), max(cons(0, cons(s(z0), x2))), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), max(cons(s(z0), cons(s(z1), x2))), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(ge(z0, 0), z0, 0, x2), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(ge(0, s(z0)), 0, s(z0), x2), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), if1(ge(z0, 0), z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), if1(ge(0, s(z0)), 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z1, cons(0, x1))) -> c15(SORT(if2(eq(if1(true, z1, 0, x1), z1), if1(true, z1, 0, x1), z1, cons(0, x1))), DEL(if1(ge(z1, 0), z1, 0, x1), cons(z1, cons(0, x1))), MAX(cons(z1, cons(0, x1)))) IF3(false, cons(z0, cons(0, z2))) -> c15(SORT(del(max(cons(z0, z2)), cons(z0, cons(0, z2)))), DEL(if1(ge(z0, 0), z0, 0, z2), cons(z0, cons(0, z2))), MAX(cons(z0, cons(0, z2)))) IF3(false, cons(0, cons(s(x0), x1))) -> c15(SORT(if2(eq(if1(false, 0, s(x0), x1), 0), if1(false, 0, s(x0), x1), 0, cons(s(x0), x1))), DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1))), MAX(cons(0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), z2))) -> c15(SORT(del(max(cons(s(x0), z2)), cons(0, cons(s(x0), z2)))), DEL(if1(ge(0, s(x0)), 0, s(x0), z2), cons(0, cons(s(x0), z2))), MAX(cons(0, cons(s(x0), z2)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c15(SORT(if2(eq(if1(ge(x0, x1), s(x0), s(x1), x2), s(x0)), if1(ge(x0, x1), s(x0), s(x1), x2), s(x0), cons(s(x1), x2))), DEL(if1(ge(s(x0), s(x1)), s(x0), s(x1), x2), cons(s(x0), cons(s(x1), x2))), MAX(cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(s(z0), cons(s(0), x2))) -> c15(SORT(del(if1(true, s(z0), s(0), x2), cons(s(z0), cons(s(0), x2)))), DEL(if1(ge(s(z0), s(0)), s(z0), s(0), x2), cons(s(z0), cons(s(0), x2))), MAX(cons(s(z0), cons(s(0), x2)))) IF3(false, cons(s(0), cons(s(s(z0)), x2))) -> c15(SORT(del(if1(false, s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2)))), DEL(if1(ge(s(0), s(s(z0))), s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2))), MAX(cons(s(0), cons(s(s(z0)), x2)))) IF3(false, cons(s(s(z0)), cons(s(s(z1)), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2)))), DEL(if1(ge(s(s(z0)), s(s(z1))), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2))), MAX(cons(s(s(z0)), cons(s(s(z1)), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(max(cons(z0, cons(0, x2))), z0), if1(true, z0, 0, x2), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(max(cons(0, cons(s(z0), x2))), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(max(cons(s(z0), cons(s(z1), x2))), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2)))) IF3(false, cons(s(z0), nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(s(z0), nil)), s(z0), nil))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), max(cons(z0, cons(0, x2))), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), max(cons(0, cons(s(z0), x2))), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), max(cons(s(z0), cons(s(z1), x2))), s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(ge(z0, 0), z0, 0, x2), z0), if1(true, z0, 0, x2), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(ge(0, s(z0)), 0, s(z0), x2), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), if1(ge(z0, 0), z0, 0, x2), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), if1(ge(0, s(z0)), 0, s(z0), x2), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0), cons(s(z1), x2)))) IF3(false, cons(z1, cons(0, x1))) -> c15(SORT(if2(eq(if1(true, z1, 0, x1), z1), if1(true, z1, 0, x1), z1, cons(0, x1)))) IF3(false, cons(z0, cons(0, z2))) -> c15(SORT(del(max(cons(z0, z2)), cons(z0, cons(0, z2))))) IF3(false, cons(0, cons(s(x0), x1))) -> c15(SORT(if2(eq(if1(false, 0, s(x0), x1), 0), if1(false, 0, s(x0), x1), 0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), z2))) -> c15(SORT(del(max(cons(s(x0), z2)), cons(0, cons(s(x0), z2))))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c15(SORT(if2(eq(if1(ge(x0, x1), s(x0), s(x1), x2), s(x0)), if1(ge(x0, x1), s(x0), s(x1), x2), s(x0), cons(s(x1), x2)))) IF3(false, cons(s(z0), cons(s(0), x2))) -> c15(SORT(del(if1(true, s(z0), s(0), x2), cons(s(z0), cons(s(0), x2))))) IF3(false, cons(s(0), cons(s(s(z0)), x2))) -> c15(SORT(del(if1(false, s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2))))) IF3(false, cons(s(s(z0)), cons(s(s(z1)), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2))))) IF3(false, cons(s(z0), nil)) -> c15(SORT(if2(eq(z0, z0), s(z0), s(z0), nil))) IF1(true, x0, 0, x1) -> c3(MAX(cons(x0, x1))) IF1(true, s(x0), s(0), x1) -> c3(MAX(cons(s(x0), x1))) IF1(true, s(s(x0)), s(s(0)), x1) -> c3(MAX(cons(s(s(x0)), x1))) IF1(true, s(s(s(x0))), s(s(s(0))), x1) -> c3(MAX(cons(s(s(s(x0))), x1))) IF1(true, s(s(s(s(x0)))), s(s(s(s(x1)))), x2) -> c3(MAX(cons(s(s(s(s(x0)))), x2))) IF1(false, 0, s(x0), x1) -> c4(MAX(cons(s(x0), x1))) IF1(false, s(0), s(s(x0)), x1) -> c4(MAX(cons(s(s(x0)), x1))) IF1(false, s(s(0)), s(s(s(x0))), x1) -> c4(MAX(cons(s(s(s(x0))), x1))) IF1(false, s(s(s(0))), s(s(s(s(x0)))), x1) -> c4(MAX(cons(s(s(s(s(x0)))), x1))) IF1(false, s(s(s(s(x0)))), s(s(s(s(x1)))), x2) -> c4(MAX(cons(s(s(s(s(x1)))), x2))) IF2(false, 0, s(x0), x1) -> c8(DEL(0, x1)) IF2(false, s(x0), 0, x1) -> c8(DEL(s(x0), x1)) IF2(false, s(0), s(s(x0)), x1) -> c8(DEL(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c8(DEL(s(s(x0)), x1)) IF2(false, s(s(0)), s(s(s(x0))), x1) -> c8(DEL(s(s(0)), x1)) IF2(false, s(s(s(x0))), s(s(0)), x1) -> c8(DEL(s(s(s(x0))), x1)) IF2(false, s(s(s(0))), s(s(s(s(x0)))), x1) -> c8(DEL(s(s(s(0))), x1)) IF2(false, s(s(s(s(x0)))), s(s(s(0))), x1) -> c8(DEL(s(s(s(s(x0)))), x1)) IF2(false, s(s(s(s(x0)))), s(s(s(s(x1)))), x2) -> c8(DEL(s(s(s(s(x0)))), x2)) K tuples: IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) IF3(false, cons(x0, cons(0, x1))) -> c(DEL(if1(ge(x0, 0), x0, 0, x1), cons(x0, cons(0, x1)))) IF3(false, cons(x0, cons(0, x1))) -> c(MAX(cons(x0, cons(0, x1)))) IF3(false, cons(0, cons(s(x0), x1))) -> c1(DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), x1))) -> c1(MAX(cons(0, cons(s(x0), x1)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c5(DEL(if1(ge(s(x0), s(x1)), s(x0), s(x1), x2), cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c5(MAX(cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(0, nil)) -> c15(SORT(if2(true, max(cons(0, nil)), 0, nil))) Defined Rule Symbols: del_2, max_1, if2_4, eq_2, ge_2, if1_4 Defined Pair Symbols: EQ_2, GE_2, MAX_1, DEL_2, SORT_1, IF3_2, IF1_4, IF2_4 Compound Symbols: c12_1, c20_1, c2_1, c6_1, c13_1, c2_2, c6_2, c15_1, c15_2, c15_3, c_1, c1_1, c5_1, c3_1, c4_1, c8_1 ---------------------------------------- (207) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) by EQ(s(s(y0)), s(s(y1))) -> c12(EQ(s(y0), s(y1))) ---------------------------------------- (208) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil max(cons(z0, nil)) -> z0 max(cons(z0, cons(z1, z2))) -> if1(ge(z0, z1), z0, z1, z2) if2(true, z0, z1, z2) -> z2 if2(false, z0, z1, z2) -> cons(z1, del(z0, z2)) eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) if1(true, z0, z1, z2) -> max(cons(z0, z2)) if1(false, z0, z1, z2) -> max(cons(z1, z2)) Tuples: GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) MAX(cons(s(s(z0)), cons(s(s(0)), x2))) -> c2(IF1(true, s(s(z0)), s(s(0)), x2), GE(s(s(z0)), s(s(0)))) MAX(cons(s(s(0)), cons(s(s(s(z0))), x2))) -> c2(IF1(false, s(s(0)), s(s(s(z0))), x2), GE(s(s(0)), s(s(s(z0))))) MAX(cons(s(s(x0)), cons(s(s(x1)), x2))) -> c2(GE(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c6(IF2(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c6(IF2(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c6(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c6(EQ(s(s(0)), s(s(0)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) MAX(cons(s(s(s(z0))), cons(s(s(s(0))), x2))) -> c2(IF1(true, s(s(s(z0))), s(s(s(0))), x2), GE(s(s(s(z0))), s(s(s(0))))) MAX(cons(s(s(s(0))), cons(s(s(s(s(z0)))), x2))) -> c2(IF1(false, s(s(s(0))), s(s(s(s(z0)))), x2), GE(s(s(s(0))), s(s(s(s(z0)))))) MAX(cons(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2))) -> c2(IF1(ge(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), GE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MAX(cons(s(s(s(x0))), cons(s(s(s(x1))), x2))) -> c2(GE(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c6(IF2(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c6(IF2(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c6(IF2(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c6(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c6(EQ(s(s(s(0))), s(s(s(0))))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(max(cons(z0, cons(0, x2))), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(max(cons(0, cons(s(z0), x2))), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(max(cons(s(z0), cons(s(z1), x2))), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), z0, z1, z2), z0), if1(ge(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), max(cons(z0, cons(0, x2))), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), max(cons(0, cons(s(z0), x2))), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), max(cons(s(z0), cons(s(z1), x2))), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(ge(z0, 0), z0, 0, x2), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(ge(0, s(z0)), 0, s(z0), x2), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), if1(ge(z0, 0), z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), if1(ge(0, s(z0)), 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z1, cons(0, x1))) -> c15(SORT(if2(eq(if1(true, z1, 0, x1), z1), if1(true, z1, 0, x1), z1, cons(0, x1))), DEL(if1(ge(z1, 0), z1, 0, x1), cons(z1, cons(0, x1))), MAX(cons(z1, cons(0, x1)))) IF3(false, cons(z0, cons(0, z2))) -> c15(SORT(del(max(cons(z0, z2)), cons(z0, cons(0, z2)))), DEL(if1(ge(z0, 0), z0, 0, z2), cons(z0, cons(0, z2))), MAX(cons(z0, cons(0, z2)))) IF3(false, cons(x0, cons(0, x1))) -> c(DEL(if1(ge(x0, 0), x0, 0, x1), cons(x0, cons(0, x1)))) IF3(false, cons(x0, cons(0, x1))) -> c(MAX(cons(x0, cons(0, x1)))) IF3(false, cons(0, cons(s(x0), x1))) -> c15(SORT(if2(eq(if1(false, 0, s(x0), x1), 0), if1(false, 0, s(x0), x1), 0, cons(s(x0), x1))), DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1))), MAX(cons(0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), z2))) -> c15(SORT(del(max(cons(s(x0), z2)), cons(0, cons(s(x0), z2)))), DEL(if1(ge(0, s(x0)), 0, s(x0), z2), cons(0, cons(s(x0), z2))), MAX(cons(0, cons(s(x0), z2)))) IF3(false, cons(0, cons(s(x0), x1))) -> c1(DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), x1))) -> c1(MAX(cons(0, cons(s(x0), x1)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c15(SORT(if2(eq(if1(ge(x0, x1), s(x0), s(x1), x2), s(x0)), if1(ge(x0, x1), s(x0), s(x1), x2), s(x0), cons(s(x1), x2))), DEL(if1(ge(s(x0), s(x1)), s(x0), s(x1), x2), cons(s(x0), cons(s(x1), x2))), MAX(cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(s(z0), cons(s(0), x2))) -> c15(SORT(del(if1(true, s(z0), s(0), x2), cons(s(z0), cons(s(0), x2)))), DEL(if1(ge(s(z0), s(0)), s(z0), s(0), x2), cons(s(z0), cons(s(0), x2))), MAX(cons(s(z0), cons(s(0), x2)))) IF3(false, cons(s(0), cons(s(s(z0)), x2))) -> c15(SORT(del(if1(false, s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2)))), DEL(if1(ge(s(0), s(s(z0))), s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2))), MAX(cons(s(0), cons(s(s(z0)), x2)))) IF3(false, cons(s(s(z0)), cons(s(s(z1)), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2)))), DEL(if1(ge(s(s(z0)), s(s(z1))), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2))), MAX(cons(s(s(z0)), cons(s(s(z1)), x2)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c5(DEL(if1(ge(s(x0), s(x1)), s(x0), s(x1), x2), cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c5(MAX(cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(max(cons(z0, cons(0, x2))), z0), if1(true, z0, 0, x2), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(max(cons(0, cons(s(z0), x2))), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(max(cons(s(z0), cons(s(z1), x2))), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), z0, z1, z2), z0), if1(ge(z0, z1), z0, z1, z2), z0, cons(z1, z2)))) IF3(false, cons(0, nil)) -> c15(SORT(if2(true, max(cons(0, nil)), 0, nil))) IF3(false, cons(s(z0), nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(s(z0), nil)), s(z0), nil))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), max(cons(z0, cons(0, x2))), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), max(cons(0, cons(s(z0), x2))), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), max(cons(s(z0), cons(s(z1), x2))), s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(ge(z0, 0), z0, 0, x2), z0), if1(true, z0, 0, x2), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(ge(0, s(z0)), 0, s(z0), x2), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), if1(ge(z0, 0), z0, 0, x2), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), if1(ge(0, s(z0)), 0, s(z0), x2), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0), cons(s(z1), x2)))) IF3(false, cons(z1, cons(0, x1))) -> c15(SORT(if2(eq(if1(true, z1, 0, x1), z1), if1(true, z1, 0, x1), z1, cons(0, x1)))) IF3(false, cons(z0, cons(0, z2))) -> c15(SORT(del(max(cons(z0, z2)), cons(z0, cons(0, z2))))) IF3(false, cons(0, cons(s(x0), x1))) -> c15(SORT(if2(eq(if1(false, 0, s(x0), x1), 0), if1(false, 0, s(x0), x1), 0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), z2))) -> c15(SORT(del(max(cons(s(x0), z2)), cons(0, cons(s(x0), z2))))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c15(SORT(if2(eq(if1(ge(x0, x1), s(x0), s(x1), x2), s(x0)), if1(ge(x0, x1), s(x0), s(x1), x2), s(x0), cons(s(x1), x2)))) IF3(false, cons(s(z0), cons(s(0), x2))) -> c15(SORT(del(if1(true, s(z0), s(0), x2), cons(s(z0), cons(s(0), x2))))) IF3(false, cons(s(0), cons(s(s(z0)), x2))) -> c15(SORT(del(if1(false, s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2))))) IF3(false, cons(s(s(z0)), cons(s(s(z1)), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2))))) IF3(false, cons(s(z0), nil)) -> c15(SORT(if2(eq(z0, z0), s(z0), s(z0), nil))) IF1(true, x0, 0, x1) -> c3(MAX(cons(x0, x1))) IF1(true, s(x0), s(0), x1) -> c3(MAX(cons(s(x0), x1))) IF1(true, s(s(x0)), s(s(0)), x1) -> c3(MAX(cons(s(s(x0)), x1))) IF1(true, s(s(s(x0))), s(s(s(0))), x1) -> c3(MAX(cons(s(s(s(x0))), x1))) IF1(true, s(s(s(s(x0)))), s(s(s(s(x1)))), x2) -> c3(MAX(cons(s(s(s(s(x0)))), x2))) IF1(false, 0, s(x0), x1) -> c4(MAX(cons(s(x0), x1))) IF1(false, s(0), s(s(x0)), x1) -> c4(MAX(cons(s(s(x0)), x1))) IF1(false, s(s(0)), s(s(s(x0))), x1) -> c4(MAX(cons(s(s(s(x0))), x1))) IF1(false, s(s(s(0))), s(s(s(s(x0)))), x1) -> c4(MAX(cons(s(s(s(s(x0)))), x1))) IF1(false, s(s(s(s(x0)))), s(s(s(s(x1)))), x2) -> c4(MAX(cons(s(s(s(s(x1)))), x2))) IF2(false, 0, s(x0), x1) -> c8(DEL(0, x1)) IF2(false, s(x0), 0, x1) -> c8(DEL(s(x0), x1)) IF2(false, s(0), s(s(x0)), x1) -> c8(DEL(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c8(DEL(s(s(x0)), x1)) IF2(false, s(s(0)), s(s(s(x0))), x1) -> c8(DEL(s(s(0)), x1)) IF2(false, s(s(s(x0))), s(s(0)), x1) -> c8(DEL(s(s(s(x0))), x1)) IF2(false, s(s(s(0))), s(s(s(s(x0)))), x1) -> c8(DEL(s(s(s(0))), x1)) IF2(false, s(s(s(s(x0)))), s(s(s(0))), x1) -> c8(DEL(s(s(s(s(x0)))), x1)) IF2(false, s(s(s(s(x0)))), s(s(s(s(x1)))), x2) -> c8(DEL(s(s(s(s(x0)))), x2)) EQ(s(s(y0)), s(s(y1))) -> c12(EQ(s(y0), s(y1))) S tuples: GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c6(EQ(s(0), s(0))) MAX(cons(s(s(z0)), cons(s(s(0)), x2))) -> c2(IF1(true, s(s(z0)), s(s(0)), x2), GE(s(s(z0)), s(s(0)))) MAX(cons(s(s(0)), cons(s(s(s(z0))), x2))) -> c2(IF1(false, s(s(0)), s(s(s(z0))), x2), GE(s(s(0)), s(s(s(z0))))) MAX(cons(s(s(x0)), cons(s(s(x1)), x2))) -> c2(GE(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c6(IF2(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c6(IF2(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c6(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c6(EQ(s(s(0)), s(s(0)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) MAX(cons(s(s(s(z0))), cons(s(s(s(0))), x2))) -> c2(IF1(true, s(s(s(z0))), s(s(s(0))), x2), GE(s(s(s(z0))), s(s(s(0))))) MAX(cons(s(s(s(0))), cons(s(s(s(s(z0)))), x2))) -> c2(IF1(false, s(s(s(0))), s(s(s(s(z0)))), x2), GE(s(s(s(0))), s(s(s(s(z0)))))) MAX(cons(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2))) -> c2(IF1(ge(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), GE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MAX(cons(s(s(s(x0))), cons(s(s(s(x1))), x2))) -> c2(GE(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c6(IF2(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c6(IF2(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c6(IF2(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c6(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c6(EQ(s(s(s(0))), s(s(s(0))))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(max(cons(z0, cons(0, x2))), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(max(cons(0, cons(s(z0), x2))), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(max(cons(s(z0), cons(s(z1), x2))), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), max(cons(z0, cons(0, x2))), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), max(cons(0, cons(s(z0), x2))), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), max(cons(s(z0), cons(s(z1), x2))), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(ge(z0, 0), z0, 0, x2), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(ge(0, s(z0)), 0, s(z0), x2), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), if1(ge(z0, 0), z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), if1(ge(0, s(z0)), 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z1, cons(0, x1))) -> c15(SORT(if2(eq(if1(true, z1, 0, x1), z1), if1(true, z1, 0, x1), z1, cons(0, x1))), DEL(if1(ge(z1, 0), z1, 0, x1), cons(z1, cons(0, x1))), MAX(cons(z1, cons(0, x1)))) IF3(false, cons(z0, cons(0, z2))) -> c15(SORT(del(max(cons(z0, z2)), cons(z0, cons(0, z2)))), DEL(if1(ge(z0, 0), z0, 0, z2), cons(z0, cons(0, z2))), MAX(cons(z0, cons(0, z2)))) IF3(false, cons(0, cons(s(x0), x1))) -> c15(SORT(if2(eq(if1(false, 0, s(x0), x1), 0), if1(false, 0, s(x0), x1), 0, cons(s(x0), x1))), DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1))), MAX(cons(0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), z2))) -> c15(SORT(del(max(cons(s(x0), z2)), cons(0, cons(s(x0), z2)))), DEL(if1(ge(0, s(x0)), 0, s(x0), z2), cons(0, cons(s(x0), z2))), MAX(cons(0, cons(s(x0), z2)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c15(SORT(if2(eq(if1(ge(x0, x1), s(x0), s(x1), x2), s(x0)), if1(ge(x0, x1), s(x0), s(x1), x2), s(x0), cons(s(x1), x2))), DEL(if1(ge(s(x0), s(x1)), s(x0), s(x1), x2), cons(s(x0), cons(s(x1), x2))), MAX(cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(s(z0), cons(s(0), x2))) -> c15(SORT(del(if1(true, s(z0), s(0), x2), cons(s(z0), cons(s(0), x2)))), DEL(if1(ge(s(z0), s(0)), s(z0), s(0), x2), cons(s(z0), cons(s(0), x2))), MAX(cons(s(z0), cons(s(0), x2)))) IF3(false, cons(s(0), cons(s(s(z0)), x2))) -> c15(SORT(del(if1(false, s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2)))), DEL(if1(ge(s(0), s(s(z0))), s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2))), MAX(cons(s(0), cons(s(s(z0)), x2)))) IF3(false, cons(s(s(z0)), cons(s(s(z1)), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2)))), DEL(if1(ge(s(s(z0)), s(s(z1))), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2))), MAX(cons(s(s(z0)), cons(s(s(z1)), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(max(cons(z0, cons(0, x2))), z0), if1(true, z0, 0, x2), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(max(cons(0, cons(s(z0), x2))), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(max(cons(s(z0), cons(s(z1), x2))), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2)))) IF3(false, cons(s(z0), nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(s(z0), nil)), s(z0), nil))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), max(cons(z0, cons(0, x2))), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), max(cons(0, cons(s(z0), x2))), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), max(cons(s(z0), cons(s(z1), x2))), s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(ge(z0, 0), z0, 0, x2), z0), if1(true, z0, 0, x2), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(ge(0, s(z0)), 0, s(z0), x2), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), if1(ge(z0, 0), z0, 0, x2), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), if1(ge(0, s(z0)), 0, s(z0), x2), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0), cons(s(z1), x2)))) IF3(false, cons(z1, cons(0, x1))) -> c15(SORT(if2(eq(if1(true, z1, 0, x1), z1), if1(true, z1, 0, x1), z1, cons(0, x1)))) IF3(false, cons(z0, cons(0, z2))) -> c15(SORT(del(max(cons(z0, z2)), cons(z0, cons(0, z2))))) IF3(false, cons(0, cons(s(x0), x1))) -> c15(SORT(if2(eq(if1(false, 0, s(x0), x1), 0), if1(false, 0, s(x0), x1), 0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), z2))) -> c15(SORT(del(max(cons(s(x0), z2)), cons(0, cons(s(x0), z2))))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c15(SORT(if2(eq(if1(ge(x0, x1), s(x0), s(x1), x2), s(x0)), if1(ge(x0, x1), s(x0), s(x1), x2), s(x0), cons(s(x1), x2)))) IF3(false, cons(s(z0), cons(s(0), x2))) -> c15(SORT(del(if1(true, s(z0), s(0), x2), cons(s(z0), cons(s(0), x2))))) IF3(false, cons(s(0), cons(s(s(z0)), x2))) -> c15(SORT(del(if1(false, s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2))))) IF3(false, cons(s(s(z0)), cons(s(s(z1)), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2))))) IF3(false, cons(s(z0), nil)) -> c15(SORT(if2(eq(z0, z0), s(z0), s(z0), nil))) IF1(true, x0, 0, x1) -> c3(MAX(cons(x0, x1))) IF1(true, s(x0), s(0), x1) -> c3(MAX(cons(s(x0), x1))) IF1(true, s(s(x0)), s(s(0)), x1) -> c3(MAX(cons(s(s(x0)), x1))) IF1(true, s(s(s(x0))), s(s(s(0))), x1) -> c3(MAX(cons(s(s(s(x0))), x1))) IF1(true, s(s(s(s(x0)))), s(s(s(s(x1)))), x2) -> c3(MAX(cons(s(s(s(s(x0)))), x2))) IF1(false, 0, s(x0), x1) -> c4(MAX(cons(s(x0), x1))) IF1(false, s(0), s(s(x0)), x1) -> c4(MAX(cons(s(s(x0)), x1))) IF1(false, s(s(0)), s(s(s(x0))), x1) -> c4(MAX(cons(s(s(s(x0))), x1))) IF1(false, s(s(s(0))), s(s(s(s(x0)))), x1) -> c4(MAX(cons(s(s(s(s(x0)))), x1))) IF1(false, s(s(s(s(x0)))), s(s(s(s(x1)))), x2) -> c4(MAX(cons(s(s(s(s(x1)))), x2))) IF2(false, 0, s(x0), x1) -> c8(DEL(0, x1)) IF2(false, s(x0), 0, x1) -> c8(DEL(s(x0), x1)) IF2(false, s(0), s(s(x0)), x1) -> c8(DEL(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c8(DEL(s(s(x0)), x1)) IF2(false, s(s(0)), s(s(s(x0))), x1) -> c8(DEL(s(s(0)), x1)) IF2(false, s(s(s(x0))), s(s(0)), x1) -> c8(DEL(s(s(s(x0))), x1)) IF2(false, s(s(s(0))), s(s(s(s(x0)))), x1) -> c8(DEL(s(s(s(0))), x1)) IF2(false, s(s(s(s(x0)))), s(s(s(0))), x1) -> c8(DEL(s(s(s(s(x0)))), x1)) IF2(false, s(s(s(s(x0)))), s(s(s(s(x1)))), x2) -> c8(DEL(s(s(s(s(x0)))), x2)) EQ(s(s(y0)), s(s(y1))) -> c12(EQ(s(y0), s(y1))) K tuples: IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) IF3(false, cons(x0, cons(0, x1))) -> c(DEL(if1(ge(x0, 0), x0, 0, x1), cons(x0, cons(0, x1)))) IF3(false, cons(x0, cons(0, x1))) -> c(MAX(cons(x0, cons(0, x1)))) IF3(false, cons(0, cons(s(x0), x1))) -> c1(DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), x1))) -> c1(MAX(cons(0, cons(s(x0), x1)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c5(DEL(if1(ge(s(x0), s(x1)), s(x0), s(x1), x2), cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c5(MAX(cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(0, nil)) -> c15(SORT(if2(true, max(cons(0, nil)), 0, nil))) Defined Rule Symbols: del_2, max_1, if2_4, eq_2, ge_2, if1_4 Defined Pair Symbols: GE_2, MAX_1, DEL_2, SORT_1, IF3_2, IF1_4, IF2_4, EQ_2 Compound Symbols: c20_1, c2_1, c6_1, c13_1, c2_2, c6_2, c15_1, c15_2, c15_3, c_1, c1_1, c5_1, c3_1, c4_1, c8_1, c12_1 ---------------------------------------- (209) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (210) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil max(cons(z0, nil)) -> z0 max(cons(z0, cons(z1, z2))) -> if1(ge(z0, z1), z0, z1, z2) if2(true, z0, z1, z2) -> z2 if2(false, z0, z1, z2) -> cons(z1, del(z0, z2)) eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) if1(true, z0, z1, z2) -> max(cons(z0, z2)) if1(false, z0, z1, z2) -> max(cons(z1, z2)) Tuples: GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) MAX(cons(s(s(z0)), cons(s(s(0)), x2))) -> c2(IF1(true, s(s(z0)), s(s(0)), x2), GE(s(s(z0)), s(s(0)))) MAX(cons(s(s(0)), cons(s(s(s(z0))), x2))) -> c2(IF1(false, s(s(0)), s(s(s(z0))), x2), GE(s(s(0)), s(s(s(z0))))) MAX(cons(s(s(x0)), cons(s(s(x1)), x2))) -> c2(GE(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c6(IF2(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c6(IF2(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c6(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c6(EQ(s(s(0)), s(s(0)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) MAX(cons(s(s(s(z0))), cons(s(s(s(0))), x2))) -> c2(IF1(true, s(s(s(z0))), s(s(s(0))), x2), GE(s(s(s(z0))), s(s(s(0))))) MAX(cons(s(s(s(0))), cons(s(s(s(s(z0)))), x2))) -> c2(IF1(false, s(s(s(0))), s(s(s(s(z0)))), x2), GE(s(s(s(0))), s(s(s(s(z0)))))) MAX(cons(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2))) -> c2(IF1(ge(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), GE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MAX(cons(s(s(s(x0))), cons(s(s(s(x1))), x2))) -> c2(GE(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c6(IF2(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c6(IF2(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c6(IF2(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c6(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c6(EQ(s(s(s(0))), s(s(s(0))))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(max(cons(z0, cons(0, x2))), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(max(cons(0, cons(s(z0), x2))), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(max(cons(s(z0), cons(s(z1), x2))), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), z0, z1, z2), z0), if1(ge(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), max(cons(z0, cons(0, x2))), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), max(cons(0, cons(s(z0), x2))), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), max(cons(s(z0), cons(s(z1), x2))), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(ge(z0, 0), z0, 0, x2), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(ge(0, s(z0)), 0, s(z0), x2), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), if1(ge(z0, 0), z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), if1(ge(0, s(z0)), 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z1, cons(0, x1))) -> c15(SORT(if2(eq(if1(true, z1, 0, x1), z1), if1(true, z1, 0, x1), z1, cons(0, x1))), DEL(if1(ge(z1, 0), z1, 0, x1), cons(z1, cons(0, x1))), MAX(cons(z1, cons(0, x1)))) IF3(false, cons(z0, cons(0, z2))) -> c15(SORT(del(max(cons(z0, z2)), cons(z0, cons(0, z2)))), DEL(if1(ge(z0, 0), z0, 0, z2), cons(z0, cons(0, z2))), MAX(cons(z0, cons(0, z2)))) IF3(false, cons(x0, cons(0, x1))) -> c(DEL(if1(ge(x0, 0), x0, 0, x1), cons(x0, cons(0, x1)))) IF3(false, cons(x0, cons(0, x1))) -> c(MAX(cons(x0, cons(0, x1)))) IF3(false, cons(0, cons(s(x0), x1))) -> c15(SORT(if2(eq(if1(false, 0, s(x0), x1), 0), if1(false, 0, s(x0), x1), 0, cons(s(x0), x1))), DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1))), MAX(cons(0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), z2))) -> c15(SORT(del(max(cons(s(x0), z2)), cons(0, cons(s(x0), z2)))), DEL(if1(ge(0, s(x0)), 0, s(x0), z2), cons(0, cons(s(x0), z2))), MAX(cons(0, cons(s(x0), z2)))) IF3(false, cons(0, cons(s(x0), x1))) -> c1(DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), x1))) -> c1(MAX(cons(0, cons(s(x0), x1)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c15(SORT(if2(eq(if1(ge(x0, x1), s(x0), s(x1), x2), s(x0)), if1(ge(x0, x1), s(x0), s(x1), x2), s(x0), cons(s(x1), x2))), DEL(if1(ge(s(x0), s(x1)), s(x0), s(x1), x2), cons(s(x0), cons(s(x1), x2))), MAX(cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(s(z0), cons(s(0), x2))) -> c15(SORT(del(if1(true, s(z0), s(0), x2), cons(s(z0), cons(s(0), x2)))), DEL(if1(ge(s(z0), s(0)), s(z0), s(0), x2), cons(s(z0), cons(s(0), x2))), MAX(cons(s(z0), cons(s(0), x2)))) IF3(false, cons(s(0), cons(s(s(z0)), x2))) -> c15(SORT(del(if1(false, s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2)))), DEL(if1(ge(s(0), s(s(z0))), s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2))), MAX(cons(s(0), cons(s(s(z0)), x2)))) IF3(false, cons(s(s(z0)), cons(s(s(z1)), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2)))), DEL(if1(ge(s(s(z0)), s(s(z1))), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2))), MAX(cons(s(s(z0)), cons(s(s(z1)), x2)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c5(DEL(if1(ge(s(x0), s(x1)), s(x0), s(x1), x2), cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c5(MAX(cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(max(cons(z0, cons(0, x2))), z0), if1(true, z0, 0, x2), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(max(cons(0, cons(s(z0), x2))), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(max(cons(s(z0), cons(s(z1), x2))), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), z0, z1, z2), z0), if1(ge(z0, z1), z0, z1, z2), z0, cons(z1, z2)))) IF3(false, cons(0, nil)) -> c15(SORT(if2(true, max(cons(0, nil)), 0, nil))) IF3(false, cons(s(z0), nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(s(z0), nil)), s(z0), nil))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), max(cons(z0, cons(0, x2))), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), max(cons(0, cons(s(z0), x2))), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), max(cons(s(z0), cons(s(z1), x2))), s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(ge(z0, 0), z0, 0, x2), z0), if1(true, z0, 0, x2), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(ge(0, s(z0)), 0, s(z0), x2), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), if1(ge(z0, 0), z0, 0, x2), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), if1(ge(0, s(z0)), 0, s(z0), x2), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0), cons(s(z1), x2)))) IF3(false, cons(z1, cons(0, x1))) -> c15(SORT(if2(eq(if1(true, z1, 0, x1), z1), if1(true, z1, 0, x1), z1, cons(0, x1)))) IF3(false, cons(z0, cons(0, z2))) -> c15(SORT(del(max(cons(z0, z2)), cons(z0, cons(0, z2))))) IF3(false, cons(0, cons(s(x0), x1))) -> c15(SORT(if2(eq(if1(false, 0, s(x0), x1), 0), if1(false, 0, s(x0), x1), 0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), z2))) -> c15(SORT(del(max(cons(s(x0), z2)), cons(0, cons(s(x0), z2))))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c15(SORT(if2(eq(if1(ge(x0, x1), s(x0), s(x1), x2), s(x0)), if1(ge(x0, x1), s(x0), s(x1), x2), s(x0), cons(s(x1), x2)))) IF3(false, cons(s(z0), cons(s(0), x2))) -> c15(SORT(del(if1(true, s(z0), s(0), x2), cons(s(z0), cons(s(0), x2))))) IF3(false, cons(s(0), cons(s(s(z0)), x2))) -> c15(SORT(del(if1(false, s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2))))) IF3(false, cons(s(s(z0)), cons(s(s(z1)), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2))))) IF3(false, cons(s(z0), nil)) -> c15(SORT(if2(eq(z0, z0), s(z0), s(z0), nil))) IF1(true, x0, 0, x1) -> c3(MAX(cons(x0, x1))) IF1(true, s(x0), s(0), x1) -> c3(MAX(cons(s(x0), x1))) IF1(true, s(s(x0)), s(s(0)), x1) -> c3(MAX(cons(s(s(x0)), x1))) IF1(true, s(s(s(x0))), s(s(s(0))), x1) -> c3(MAX(cons(s(s(s(x0))), x1))) IF1(true, s(s(s(s(x0)))), s(s(s(s(x1)))), x2) -> c3(MAX(cons(s(s(s(s(x0)))), x2))) IF1(false, 0, s(x0), x1) -> c4(MAX(cons(s(x0), x1))) IF1(false, s(0), s(s(x0)), x1) -> c4(MAX(cons(s(s(x0)), x1))) IF1(false, s(s(0)), s(s(s(x0))), x1) -> c4(MAX(cons(s(s(s(x0))), x1))) IF1(false, s(s(s(0))), s(s(s(s(x0)))), x1) -> c4(MAX(cons(s(s(s(s(x0)))), x1))) IF1(false, s(s(s(s(x0)))), s(s(s(s(x1)))), x2) -> c4(MAX(cons(s(s(s(s(x1)))), x2))) IF2(false, 0, s(x0), x1) -> c8(DEL(0, x1)) IF2(false, s(x0), 0, x1) -> c8(DEL(s(x0), x1)) IF2(false, s(0), s(s(x0)), x1) -> c8(DEL(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c8(DEL(s(s(x0)), x1)) IF2(false, s(s(0)), s(s(s(x0))), x1) -> c8(DEL(s(s(0)), x1)) IF2(false, s(s(s(x0))), s(s(0)), x1) -> c8(DEL(s(s(s(x0))), x1)) IF2(false, s(s(s(0))), s(s(s(s(x0)))), x1) -> c8(DEL(s(s(s(0))), x1)) IF2(false, s(s(s(s(x0)))), s(s(s(0))), x1) -> c8(DEL(s(s(s(s(x0)))), x1)) IF2(false, s(s(s(s(x0)))), s(s(s(s(x1)))), x2) -> c8(DEL(s(s(s(s(x0)))), x2)) EQ(s(s(y0)), s(s(y1))) -> c12(EQ(s(y0), s(y1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c6 S tuples: GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) MAX(cons(s(s(z0)), cons(s(s(0)), x2))) -> c2(IF1(true, s(s(z0)), s(s(0)), x2), GE(s(s(z0)), s(s(0)))) MAX(cons(s(s(0)), cons(s(s(s(z0))), x2))) -> c2(IF1(false, s(s(0)), s(s(s(z0))), x2), GE(s(s(0)), s(s(s(z0))))) MAX(cons(s(s(x0)), cons(s(s(x1)), x2))) -> c2(GE(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c6(IF2(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c6(IF2(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c6(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c6(EQ(s(s(0)), s(s(0)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) MAX(cons(s(s(s(z0))), cons(s(s(s(0))), x2))) -> c2(IF1(true, s(s(s(z0))), s(s(s(0))), x2), GE(s(s(s(z0))), s(s(s(0))))) MAX(cons(s(s(s(0))), cons(s(s(s(s(z0)))), x2))) -> c2(IF1(false, s(s(s(0))), s(s(s(s(z0)))), x2), GE(s(s(s(0))), s(s(s(s(z0)))))) MAX(cons(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2))) -> c2(IF1(ge(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), GE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MAX(cons(s(s(s(x0))), cons(s(s(s(x1))), x2))) -> c2(GE(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c6(IF2(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c6(IF2(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c6(IF2(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c6(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c6(EQ(s(s(s(0))), s(s(s(0))))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(max(cons(z0, cons(0, x2))), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(max(cons(0, cons(s(z0), x2))), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(max(cons(s(z0), cons(s(z1), x2))), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), max(cons(z0, cons(0, x2))), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), max(cons(0, cons(s(z0), x2))), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), max(cons(s(z0), cons(s(z1), x2))), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(ge(z0, 0), z0, 0, x2), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(ge(0, s(z0)), 0, s(z0), x2), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), if1(ge(z0, 0), z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), if1(ge(0, s(z0)), 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z1, cons(0, x1))) -> c15(SORT(if2(eq(if1(true, z1, 0, x1), z1), if1(true, z1, 0, x1), z1, cons(0, x1))), DEL(if1(ge(z1, 0), z1, 0, x1), cons(z1, cons(0, x1))), MAX(cons(z1, cons(0, x1)))) IF3(false, cons(z0, cons(0, z2))) -> c15(SORT(del(max(cons(z0, z2)), cons(z0, cons(0, z2)))), DEL(if1(ge(z0, 0), z0, 0, z2), cons(z0, cons(0, z2))), MAX(cons(z0, cons(0, z2)))) IF3(false, cons(0, cons(s(x0), x1))) -> c15(SORT(if2(eq(if1(false, 0, s(x0), x1), 0), if1(false, 0, s(x0), x1), 0, cons(s(x0), x1))), DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1))), MAX(cons(0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), z2))) -> c15(SORT(del(max(cons(s(x0), z2)), cons(0, cons(s(x0), z2)))), DEL(if1(ge(0, s(x0)), 0, s(x0), z2), cons(0, cons(s(x0), z2))), MAX(cons(0, cons(s(x0), z2)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c15(SORT(if2(eq(if1(ge(x0, x1), s(x0), s(x1), x2), s(x0)), if1(ge(x0, x1), s(x0), s(x1), x2), s(x0), cons(s(x1), x2))), DEL(if1(ge(s(x0), s(x1)), s(x0), s(x1), x2), cons(s(x0), cons(s(x1), x2))), MAX(cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(s(z0), cons(s(0), x2))) -> c15(SORT(del(if1(true, s(z0), s(0), x2), cons(s(z0), cons(s(0), x2)))), DEL(if1(ge(s(z0), s(0)), s(z0), s(0), x2), cons(s(z0), cons(s(0), x2))), MAX(cons(s(z0), cons(s(0), x2)))) IF3(false, cons(s(0), cons(s(s(z0)), x2))) -> c15(SORT(del(if1(false, s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2)))), DEL(if1(ge(s(0), s(s(z0))), s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2))), MAX(cons(s(0), cons(s(s(z0)), x2)))) IF3(false, cons(s(s(z0)), cons(s(s(z1)), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2)))), DEL(if1(ge(s(s(z0)), s(s(z1))), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2))), MAX(cons(s(s(z0)), cons(s(s(z1)), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(max(cons(z0, cons(0, x2))), z0), if1(true, z0, 0, x2), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(max(cons(0, cons(s(z0), x2))), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(max(cons(s(z0), cons(s(z1), x2))), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2)))) IF3(false, cons(s(z0), nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(s(z0), nil)), s(z0), nil))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), max(cons(z0, cons(0, x2))), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), max(cons(0, cons(s(z0), x2))), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), max(cons(s(z0), cons(s(z1), x2))), s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(ge(z0, 0), z0, 0, x2), z0), if1(true, z0, 0, x2), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(ge(0, s(z0)), 0, s(z0), x2), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), if1(ge(z0, 0), z0, 0, x2), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), if1(ge(0, s(z0)), 0, s(z0), x2), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0), cons(s(z1), x2)))) IF3(false, cons(z1, cons(0, x1))) -> c15(SORT(if2(eq(if1(true, z1, 0, x1), z1), if1(true, z1, 0, x1), z1, cons(0, x1)))) IF3(false, cons(z0, cons(0, z2))) -> c15(SORT(del(max(cons(z0, z2)), cons(z0, cons(0, z2))))) IF3(false, cons(0, cons(s(x0), x1))) -> c15(SORT(if2(eq(if1(false, 0, s(x0), x1), 0), if1(false, 0, s(x0), x1), 0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), z2))) -> c15(SORT(del(max(cons(s(x0), z2)), cons(0, cons(s(x0), z2))))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c15(SORT(if2(eq(if1(ge(x0, x1), s(x0), s(x1), x2), s(x0)), if1(ge(x0, x1), s(x0), s(x1), x2), s(x0), cons(s(x1), x2)))) IF3(false, cons(s(z0), cons(s(0), x2))) -> c15(SORT(del(if1(true, s(z0), s(0), x2), cons(s(z0), cons(s(0), x2))))) IF3(false, cons(s(0), cons(s(s(z0)), x2))) -> c15(SORT(del(if1(false, s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2))))) IF3(false, cons(s(s(z0)), cons(s(s(z1)), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2))))) IF3(false, cons(s(z0), nil)) -> c15(SORT(if2(eq(z0, z0), s(z0), s(z0), nil))) IF1(true, x0, 0, x1) -> c3(MAX(cons(x0, x1))) IF1(true, s(x0), s(0), x1) -> c3(MAX(cons(s(x0), x1))) IF1(true, s(s(x0)), s(s(0)), x1) -> c3(MAX(cons(s(s(x0)), x1))) IF1(true, s(s(s(x0))), s(s(s(0))), x1) -> c3(MAX(cons(s(s(s(x0))), x1))) IF1(true, s(s(s(s(x0)))), s(s(s(s(x1)))), x2) -> c3(MAX(cons(s(s(s(s(x0)))), x2))) IF1(false, 0, s(x0), x1) -> c4(MAX(cons(s(x0), x1))) IF1(false, s(0), s(s(x0)), x1) -> c4(MAX(cons(s(s(x0)), x1))) IF1(false, s(s(0)), s(s(s(x0))), x1) -> c4(MAX(cons(s(s(s(x0))), x1))) IF1(false, s(s(s(0))), s(s(s(s(x0)))), x1) -> c4(MAX(cons(s(s(s(s(x0)))), x1))) IF1(false, s(s(s(s(x0)))), s(s(s(s(x1)))), x2) -> c4(MAX(cons(s(s(s(s(x1)))), x2))) IF2(false, 0, s(x0), x1) -> c8(DEL(0, x1)) IF2(false, s(x0), 0, x1) -> c8(DEL(s(x0), x1)) IF2(false, s(0), s(s(x0)), x1) -> c8(DEL(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c8(DEL(s(s(x0)), x1)) IF2(false, s(s(0)), s(s(s(x0))), x1) -> c8(DEL(s(s(0)), x1)) IF2(false, s(s(s(x0))), s(s(0)), x1) -> c8(DEL(s(s(s(x0))), x1)) IF2(false, s(s(s(0))), s(s(s(s(x0)))), x1) -> c8(DEL(s(s(s(0))), x1)) IF2(false, s(s(s(s(x0)))), s(s(s(0))), x1) -> c8(DEL(s(s(s(s(x0)))), x1)) IF2(false, s(s(s(s(x0)))), s(s(s(s(x1)))), x2) -> c8(DEL(s(s(s(s(x0)))), x2)) EQ(s(s(y0)), s(s(y1))) -> c12(EQ(s(y0), s(y1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c6 K tuples: IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) IF3(false, cons(x0, cons(0, x1))) -> c(DEL(if1(ge(x0, 0), x0, 0, x1), cons(x0, cons(0, x1)))) IF3(false, cons(x0, cons(0, x1))) -> c(MAX(cons(x0, cons(0, x1)))) IF3(false, cons(0, cons(s(x0), x1))) -> c1(DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), x1))) -> c1(MAX(cons(0, cons(s(x0), x1)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c5(DEL(if1(ge(s(x0), s(x1)), s(x0), s(x1), x2), cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c5(MAX(cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(0, nil)) -> c15(SORT(if2(true, max(cons(0, nil)), 0, nil))) Defined Rule Symbols: del_2, max_1, if2_4, eq_2, ge_2, if1_4 Defined Pair Symbols: GE_2, MAX_1, DEL_2, SORT_1, IF3_2, IF1_4, IF2_4, EQ_2 Compound Symbols: c20_1, c2_1, c6_1, c13_1, c2_2, c15_1, c6_2, c15_2, c15_3, c_1, c1_1, c5_1, c3_1, c4_1, c8_1, c12_1, c6 ---------------------------------------- (211) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(max(cons(z0, cons(0, x2))), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) by IF3(false, cons(z0, cons(0, z1))) -> c15(SORT(if2(eq(max(cons(z0, cons(0, z1))), z0), if1(true, z0, 0, z1), z0, cons(0, z1))), DEL(if1(true, z0, 0, z1), cons(z0, cons(0, z1))), MAX(cons(z0, cons(0, z1)))) ---------------------------------------- (212) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil max(cons(z0, nil)) -> z0 max(cons(z0, cons(z1, z2))) -> if1(ge(z0, z1), z0, z1, z2) if2(true, z0, z1, z2) -> z2 if2(false, z0, z1, z2) -> cons(z1, del(z0, z2)) eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) if1(true, z0, z1, z2) -> max(cons(z0, z2)) if1(false, z0, z1, z2) -> max(cons(z1, z2)) Tuples: GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) MAX(cons(s(s(z0)), cons(s(s(0)), x2))) -> c2(IF1(true, s(s(z0)), s(s(0)), x2), GE(s(s(z0)), s(s(0)))) MAX(cons(s(s(0)), cons(s(s(s(z0))), x2))) -> c2(IF1(false, s(s(0)), s(s(s(z0))), x2), GE(s(s(0)), s(s(s(z0))))) MAX(cons(s(s(x0)), cons(s(s(x1)), x2))) -> c2(GE(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c6(IF2(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c6(IF2(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c6(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c6(EQ(s(s(0)), s(s(0)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) MAX(cons(s(s(s(z0))), cons(s(s(s(0))), x2))) -> c2(IF1(true, s(s(s(z0))), s(s(s(0))), x2), GE(s(s(s(z0))), s(s(s(0))))) MAX(cons(s(s(s(0))), cons(s(s(s(s(z0)))), x2))) -> c2(IF1(false, s(s(s(0))), s(s(s(s(z0)))), x2), GE(s(s(s(0))), s(s(s(s(z0)))))) MAX(cons(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2))) -> c2(IF1(ge(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), GE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MAX(cons(s(s(s(x0))), cons(s(s(s(x1))), x2))) -> c2(GE(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c6(IF2(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c6(IF2(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c6(IF2(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c6(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c6(EQ(s(s(s(0))), s(s(s(0))))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(max(cons(0, cons(s(z0), x2))), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(max(cons(s(z0), cons(s(z1), x2))), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), z0, z1, z2), z0), if1(ge(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), max(cons(z0, cons(0, x2))), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), max(cons(0, cons(s(z0), x2))), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), max(cons(s(z0), cons(s(z1), x2))), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(ge(z0, 0), z0, 0, x2), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(ge(0, s(z0)), 0, s(z0), x2), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), if1(ge(z0, 0), z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), if1(ge(0, s(z0)), 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z1, cons(0, x1))) -> c15(SORT(if2(eq(if1(true, z1, 0, x1), z1), if1(true, z1, 0, x1), z1, cons(0, x1))), DEL(if1(ge(z1, 0), z1, 0, x1), cons(z1, cons(0, x1))), MAX(cons(z1, cons(0, x1)))) IF3(false, cons(z0, cons(0, z2))) -> c15(SORT(del(max(cons(z0, z2)), cons(z0, cons(0, z2)))), DEL(if1(ge(z0, 0), z0, 0, z2), cons(z0, cons(0, z2))), MAX(cons(z0, cons(0, z2)))) IF3(false, cons(x0, cons(0, x1))) -> c(DEL(if1(ge(x0, 0), x0, 0, x1), cons(x0, cons(0, x1)))) IF3(false, cons(x0, cons(0, x1))) -> c(MAX(cons(x0, cons(0, x1)))) IF3(false, cons(0, cons(s(x0), x1))) -> c15(SORT(if2(eq(if1(false, 0, s(x0), x1), 0), if1(false, 0, s(x0), x1), 0, cons(s(x0), x1))), DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1))), MAX(cons(0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), z2))) -> c15(SORT(del(max(cons(s(x0), z2)), cons(0, cons(s(x0), z2)))), DEL(if1(ge(0, s(x0)), 0, s(x0), z2), cons(0, cons(s(x0), z2))), MAX(cons(0, cons(s(x0), z2)))) IF3(false, cons(0, cons(s(x0), x1))) -> c1(DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), x1))) -> c1(MAX(cons(0, cons(s(x0), x1)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c15(SORT(if2(eq(if1(ge(x0, x1), s(x0), s(x1), x2), s(x0)), if1(ge(x0, x1), s(x0), s(x1), x2), s(x0), cons(s(x1), x2))), DEL(if1(ge(s(x0), s(x1)), s(x0), s(x1), x2), cons(s(x0), cons(s(x1), x2))), MAX(cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(s(z0), cons(s(0), x2))) -> c15(SORT(del(if1(true, s(z0), s(0), x2), cons(s(z0), cons(s(0), x2)))), DEL(if1(ge(s(z0), s(0)), s(z0), s(0), x2), cons(s(z0), cons(s(0), x2))), MAX(cons(s(z0), cons(s(0), x2)))) IF3(false, cons(s(0), cons(s(s(z0)), x2))) -> c15(SORT(del(if1(false, s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2)))), DEL(if1(ge(s(0), s(s(z0))), s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2))), MAX(cons(s(0), cons(s(s(z0)), x2)))) IF3(false, cons(s(s(z0)), cons(s(s(z1)), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2)))), DEL(if1(ge(s(s(z0)), s(s(z1))), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2))), MAX(cons(s(s(z0)), cons(s(s(z1)), x2)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c5(DEL(if1(ge(s(x0), s(x1)), s(x0), s(x1), x2), cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c5(MAX(cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(max(cons(z0, cons(0, x2))), z0), if1(true, z0, 0, x2), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(max(cons(0, cons(s(z0), x2))), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(max(cons(s(z0), cons(s(z1), x2))), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(z1, z2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), z0, z1, z2), z0), if1(ge(z0, z1), z0, z1, z2), z0, cons(z1, z2)))) IF3(false, cons(0, nil)) -> c15(SORT(if2(true, max(cons(0, nil)), 0, nil))) IF3(false, cons(s(z0), nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(s(z0), nil)), s(z0), nil))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), max(cons(z0, cons(0, x2))), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), max(cons(0, cons(s(z0), x2))), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), max(cons(s(z0), cons(s(z1), x2))), s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(ge(z0, 0), z0, 0, x2), z0), if1(true, z0, 0, x2), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(ge(0, s(z0)), 0, s(z0), x2), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), if1(ge(z0, 0), z0, 0, x2), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), if1(ge(0, s(z0)), 0, s(z0), x2), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0), cons(s(z1), x2)))) IF3(false, cons(z1, cons(0, x1))) -> c15(SORT(if2(eq(if1(true, z1, 0, x1), z1), if1(true, z1, 0, x1), z1, cons(0, x1)))) IF3(false, cons(z0, cons(0, z2))) -> c15(SORT(del(max(cons(z0, z2)), cons(z0, cons(0, z2))))) IF3(false, cons(0, cons(s(x0), x1))) -> c15(SORT(if2(eq(if1(false, 0, s(x0), x1), 0), if1(false, 0, s(x0), x1), 0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), z2))) -> c15(SORT(del(max(cons(s(x0), z2)), cons(0, cons(s(x0), z2))))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c15(SORT(if2(eq(if1(ge(x0, x1), s(x0), s(x1), x2), s(x0)), if1(ge(x0, x1), s(x0), s(x1), x2), s(x0), cons(s(x1), x2)))) IF3(false, cons(s(z0), cons(s(0), x2))) -> c15(SORT(del(if1(true, s(z0), s(0), x2), cons(s(z0), cons(s(0), x2))))) IF3(false, cons(s(0), cons(s(s(z0)), x2))) -> c15(SORT(del(if1(false, s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2))))) IF3(false, cons(s(s(z0)), cons(s(s(z1)), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2))))) IF3(false, cons(s(z0), nil)) -> c15(SORT(if2(eq(z0, z0), s(z0), s(z0), nil))) IF1(true, x0, 0, x1) -> c3(MAX(cons(x0, x1))) IF1(true, s(x0), s(0), x1) -> c3(MAX(cons(s(x0), x1))) IF1(true, s(s(x0)), s(s(0)), x1) -> c3(MAX(cons(s(s(x0)), x1))) IF1(true, s(s(s(x0))), s(s(s(0))), x1) -> c3(MAX(cons(s(s(s(x0))), x1))) IF1(true, s(s(s(s(x0)))), s(s(s(s(x1)))), x2) -> c3(MAX(cons(s(s(s(s(x0)))), x2))) IF1(false, 0, s(x0), x1) -> c4(MAX(cons(s(x0), x1))) IF1(false, s(0), s(s(x0)), x1) -> c4(MAX(cons(s(s(x0)), x1))) IF1(false, s(s(0)), s(s(s(x0))), x1) -> c4(MAX(cons(s(s(s(x0))), x1))) IF1(false, s(s(s(0))), s(s(s(s(x0)))), x1) -> c4(MAX(cons(s(s(s(s(x0)))), x1))) IF1(false, s(s(s(s(x0)))), s(s(s(s(x1)))), x2) -> c4(MAX(cons(s(s(s(s(x1)))), x2))) IF2(false, 0, s(x0), x1) -> c8(DEL(0, x1)) IF2(false, s(x0), 0, x1) -> c8(DEL(s(x0), x1)) IF2(false, s(0), s(s(x0)), x1) -> c8(DEL(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c8(DEL(s(s(x0)), x1)) IF2(false, s(s(0)), s(s(s(x0))), x1) -> c8(DEL(s(s(0)), x1)) IF2(false, s(s(s(x0))), s(s(0)), x1) -> c8(DEL(s(s(s(x0))), x1)) IF2(false, s(s(s(0))), s(s(s(s(x0)))), x1) -> c8(DEL(s(s(s(0))), x1)) IF2(false, s(s(s(s(x0)))), s(s(s(0))), x1) -> c8(DEL(s(s(s(s(x0)))), x1)) IF2(false, s(s(s(s(x0)))), s(s(s(s(x1)))), x2) -> c8(DEL(s(s(s(s(x0)))), x2)) EQ(s(s(y0)), s(s(y1))) -> c12(EQ(s(y0), s(y1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c6 IF3(false, cons(z0, cons(0, z1))) -> c15(SORT(if2(eq(max(cons(z0, cons(0, z1))), z0), if1(true, z0, 0, z1), z0, cons(0, z1))), DEL(if1(true, z0, 0, z1), cons(z0, cons(0, z1))), MAX(cons(z0, cons(0, z1)))) S tuples: GE(s(z0), s(z1)) -> c20(GE(z0, z1)) MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) SORT(cons(z0, z1)) -> c13(IF3(false, cons(z0, z1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) MAX(cons(s(s(z0)), cons(s(s(0)), x2))) -> c2(IF1(true, s(s(z0)), s(s(0)), x2), GE(s(s(z0)), s(s(0)))) MAX(cons(s(s(0)), cons(s(s(s(z0))), x2))) -> c2(IF1(false, s(s(0)), s(s(s(z0))), x2), GE(s(s(0)), s(s(s(z0))))) MAX(cons(s(s(x0)), cons(s(s(x1)), x2))) -> c2(GE(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c6(IF2(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c6(IF2(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c6(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c6(EQ(s(s(0)), s(s(0)))) IF3(false, cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) MAX(cons(s(s(s(z0))), cons(s(s(s(0))), x2))) -> c2(IF1(true, s(s(s(z0))), s(s(s(0))), x2), GE(s(s(s(z0))), s(s(s(0))))) MAX(cons(s(s(s(0))), cons(s(s(s(s(z0)))), x2))) -> c2(IF1(false, s(s(s(0))), s(s(s(s(z0)))), x2), GE(s(s(s(0))), s(s(s(s(z0)))))) MAX(cons(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2))) -> c2(IF1(ge(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), GE(s(s(s(s(z0)))), s(s(s(s(z1)))))) MAX(cons(s(s(s(x0))), cons(s(s(s(x1))), x2))) -> c2(GE(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c6(IF2(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c6(IF2(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c6(IF2(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c6(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c6(EQ(s(s(s(0))), s(s(s(0))))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(max(cons(0, cons(s(z0), x2))), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(max(cons(s(z0), cons(s(z1), x2))), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), max(cons(z0, cons(0, x2))), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), max(cons(0, cons(s(z0), x2))), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), max(cons(s(z0), cons(s(z1), x2))), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(ge(z0, 0), z0, 0, x2), z0), if1(true, z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(ge(0, s(z0)), 0, s(z0), x2), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), if1(ge(z0, 0), z0, 0, x2), z0, cons(0, x2))), DEL(if1(ge(z0, 0), z0, 0, x2), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), if1(ge(0, s(z0)), 0, s(z0), x2), 0, cons(s(z0), x2))), DEL(if1(ge(0, s(z0)), 0, s(z0), x2), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0), cons(s(z1), x2))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) IF3(false, cons(z1, cons(0, x1))) -> c15(SORT(if2(eq(if1(true, z1, 0, x1), z1), if1(true, z1, 0, x1), z1, cons(0, x1))), DEL(if1(ge(z1, 0), z1, 0, x1), cons(z1, cons(0, x1))), MAX(cons(z1, cons(0, x1)))) IF3(false, cons(z0, cons(0, z2))) -> c15(SORT(del(max(cons(z0, z2)), cons(z0, cons(0, z2)))), DEL(if1(ge(z0, 0), z0, 0, z2), cons(z0, cons(0, z2))), MAX(cons(z0, cons(0, z2)))) IF3(false, cons(0, cons(s(x0), x1))) -> c15(SORT(if2(eq(if1(false, 0, s(x0), x1), 0), if1(false, 0, s(x0), x1), 0, cons(s(x0), x1))), DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1))), MAX(cons(0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), z2))) -> c15(SORT(del(max(cons(s(x0), z2)), cons(0, cons(s(x0), z2)))), DEL(if1(ge(0, s(x0)), 0, s(x0), z2), cons(0, cons(s(x0), z2))), MAX(cons(0, cons(s(x0), z2)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c15(SORT(if2(eq(if1(ge(x0, x1), s(x0), s(x1), x2), s(x0)), if1(ge(x0, x1), s(x0), s(x1), x2), s(x0), cons(s(x1), x2))), DEL(if1(ge(s(x0), s(x1)), s(x0), s(x1), x2), cons(s(x0), cons(s(x1), x2))), MAX(cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(s(z0), cons(s(0), x2))) -> c15(SORT(del(if1(true, s(z0), s(0), x2), cons(s(z0), cons(s(0), x2)))), DEL(if1(ge(s(z0), s(0)), s(z0), s(0), x2), cons(s(z0), cons(s(0), x2))), MAX(cons(s(z0), cons(s(0), x2)))) IF3(false, cons(s(0), cons(s(s(z0)), x2))) -> c15(SORT(del(if1(false, s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2)))), DEL(if1(ge(s(0), s(s(z0))), s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2))), MAX(cons(s(0), cons(s(s(z0)), x2)))) IF3(false, cons(s(s(z0)), cons(s(s(z1)), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2)))), DEL(if1(ge(s(s(z0)), s(s(z1))), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2))), MAX(cons(s(s(z0)), cons(s(s(z1)), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(max(cons(z0, cons(0, x2))), z0), if1(true, z0, 0, x2), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(max(cons(0, cons(s(z0), x2))), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(max(cons(s(z0), cons(s(z1), x2))), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2)))) IF3(false, cons(s(z0), nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(s(z0), nil)), s(z0), nil))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), max(cons(z0, cons(0, x2))), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), max(cons(0, cons(s(z0), x2))), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), max(cons(s(z0), cons(s(z1), x2))), s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(ge(z0, 0), z0, 0, x2), z0), if1(true, z0, 0, x2), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(ge(0, s(z0)), 0, s(z0), x2), 0), if1(false, 0, s(z0), x2), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0)), if1(ge(z0, z1), s(z0), s(z1), x2), s(z0), cons(s(z1), x2)))) IF3(false, cons(z0, cons(0, x2))) -> c15(SORT(if2(eq(if1(true, z0, 0, x2), z0), if1(ge(z0, 0), z0, 0, x2), z0, cons(0, x2)))) IF3(false, cons(0, cons(s(z0), x2))) -> c15(SORT(if2(eq(if1(false, 0, s(z0), x2), 0), if1(ge(0, s(z0)), 0, s(z0), x2), 0, cons(s(z0), x2)))) IF3(false, cons(s(z0), cons(s(z1), x2))) -> c15(SORT(if2(eq(if1(ge(z0, z1), s(z0), s(z1), x2), s(z0)), if1(ge(s(z0), s(z1)), s(z0), s(z1), x2), s(z0), cons(s(z1), x2)))) IF3(false, cons(z1, cons(0, x1))) -> c15(SORT(if2(eq(if1(true, z1, 0, x1), z1), if1(true, z1, 0, x1), z1, cons(0, x1)))) IF3(false, cons(z0, cons(0, z2))) -> c15(SORT(del(max(cons(z0, z2)), cons(z0, cons(0, z2))))) IF3(false, cons(0, cons(s(x0), x1))) -> c15(SORT(if2(eq(if1(false, 0, s(x0), x1), 0), if1(false, 0, s(x0), x1), 0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), z2))) -> c15(SORT(del(max(cons(s(x0), z2)), cons(0, cons(s(x0), z2))))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c15(SORT(if2(eq(if1(ge(x0, x1), s(x0), s(x1), x2), s(x0)), if1(ge(x0, x1), s(x0), s(x1), x2), s(x0), cons(s(x1), x2)))) IF3(false, cons(s(z0), cons(s(0), x2))) -> c15(SORT(del(if1(true, s(z0), s(0), x2), cons(s(z0), cons(s(0), x2))))) IF3(false, cons(s(0), cons(s(s(z0)), x2))) -> c15(SORT(del(if1(false, s(0), s(s(z0)), x2), cons(s(0), cons(s(s(z0)), x2))))) IF3(false, cons(s(s(z0)), cons(s(s(z1)), x2))) -> c15(SORT(del(if1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), cons(s(s(z0)), cons(s(s(z1)), x2))))) IF3(false, cons(s(z0), nil)) -> c15(SORT(if2(eq(z0, z0), s(z0), s(z0), nil))) IF1(true, x0, 0, x1) -> c3(MAX(cons(x0, x1))) IF1(true, s(x0), s(0), x1) -> c3(MAX(cons(s(x0), x1))) IF1(true, s(s(x0)), s(s(0)), x1) -> c3(MAX(cons(s(s(x0)), x1))) IF1(true, s(s(s(x0))), s(s(s(0))), x1) -> c3(MAX(cons(s(s(s(x0))), x1))) IF1(true, s(s(s(s(x0)))), s(s(s(s(x1)))), x2) -> c3(MAX(cons(s(s(s(s(x0)))), x2))) IF1(false, 0, s(x0), x1) -> c4(MAX(cons(s(x0), x1))) IF1(false, s(0), s(s(x0)), x1) -> c4(MAX(cons(s(s(x0)), x1))) IF1(false, s(s(0)), s(s(s(x0))), x1) -> c4(MAX(cons(s(s(s(x0))), x1))) IF1(false, s(s(s(0))), s(s(s(s(x0)))), x1) -> c4(MAX(cons(s(s(s(s(x0)))), x1))) IF1(false, s(s(s(s(x0)))), s(s(s(s(x1)))), x2) -> c4(MAX(cons(s(s(s(s(x1)))), x2))) IF2(false, 0, s(x0), x1) -> c8(DEL(0, x1)) IF2(false, s(x0), 0, x1) -> c8(DEL(s(x0), x1)) IF2(false, s(0), s(s(x0)), x1) -> c8(DEL(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c8(DEL(s(s(x0)), x1)) IF2(false, s(s(0)), s(s(s(x0))), x1) -> c8(DEL(s(s(0)), x1)) IF2(false, s(s(s(x0))), s(s(0)), x1) -> c8(DEL(s(s(s(x0))), x1)) IF2(false, s(s(s(0))), s(s(s(s(x0)))), x1) -> c8(DEL(s(s(s(0))), x1)) IF2(false, s(s(s(s(x0)))), s(s(s(0))), x1) -> c8(DEL(s(s(s(s(x0)))), x1)) IF2(false, s(s(s(s(x0)))), s(s(s(s(x1)))), x2) -> c8(DEL(s(s(s(s(x0)))), x2)) EQ(s(s(y0)), s(s(y1))) -> c12(EQ(s(y0), s(y1))) DEL(s(0), cons(s(s(z0)), x2)) -> c6(IF2(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c6(IF2(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c6 IF3(false, cons(z0, cons(0, z1))) -> c15(SORT(if2(eq(max(cons(z0, cons(0, z1))), z0), if1(true, z0, 0, z1), z0, cons(0, z1))), DEL(if1(true, z0, 0, z1), cons(z0, cons(0, z1))), MAX(cons(z0, cons(0, z1)))) K tuples: IF3(false, cons(x0, nil)) -> c15(DEL(x0, cons(x0, nil))) IF3(false, cons(x0, cons(0, x1))) -> c(DEL(if1(ge(x0, 0), x0, 0, x1), cons(x0, cons(0, x1)))) IF3(false, cons(x0, cons(0, x1))) -> c(MAX(cons(x0, cons(0, x1)))) IF3(false, cons(0, cons(s(x0), x1))) -> c1(DEL(if1(ge(0, s(x0)), 0, s(x0), x1), cons(0, cons(s(x0), x1)))) IF3(false, cons(0, cons(s(x0), x1))) -> c1(MAX(cons(0, cons(s(x0), x1)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c5(DEL(if1(ge(s(x0), s(x1)), s(x0), s(x1), x2), cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(s(x0), cons(s(x1), x2))) -> c5(MAX(cons(s(x0), cons(s(x1), x2)))) IF3(false, cons(0, nil)) -> c15(SORT(if2(true, max(cons(0, nil)), 0, nil))) Defined Rule Symbols: del_2, max_1, if2_4, eq_2, ge_2, if1_4 Defined Pair Symbols: GE_2, MAX_1, DEL_2, SORT_1, IF3_2, IF1_4, IF2_4, EQ_2 Compound Symbols: c20_1, c2_1, c6_1, c13_1, c2_2, c15_1, c6_2, c15_2, c15_3, c_1, c1_1, c5_1, c3_1, c4_1, c8_1, c12_1, c6