KILLED proof of input_1LkwlcuxcR.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). (0) CpxTRS (1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxTRS (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (4) CpxTRS (5) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxWeightedTrs (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (8) CpxTypedWeightedTrs (9) CompletionProof [UPPER BOUND(ID), 0 ms] (10) CpxTypedWeightedCompleteTrs (11) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (12) CpxRNTS (13) CompletionProof [UPPER BOUND(ID), 0 ms] (14) CpxTypedWeightedCompleteTrs (15) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (16) CpxTypedWeightedCompleteTrs (17) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (18) CpxRNTS (19) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (20) CpxRNTS (21) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 4 ms] (22) CpxRNTS (23) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (24) CpxRNTS (25) IntTrsBoundProof [UPPER BOUND(ID), 262 ms] (26) CpxRNTS (27) IntTrsBoundProof [UPPER BOUND(ID), 131 ms] (28) CpxRNTS (29) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (30) CpxRNTS (31) IntTrsBoundProof [UPPER BOUND(ID), 347 ms] (32) CpxRNTS (33) IntTrsBoundProof [UPPER BOUND(ID), 148 ms] (34) CpxRNTS (35) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (36) CpxRNTS (37) IntTrsBoundProof [UPPER BOUND(ID), 960 ms] (38) CpxRNTS (39) IntTrsBoundProof [UPPER BOUND(ID), 312 ms] (40) CpxRNTS (41) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (42) CpxRNTS (43) IntTrsBoundProof [UPPER BOUND(ID), 1131 ms] (44) CpxRNTS (45) IntTrsBoundProof [UPPER BOUND(ID), 554 ms] (46) CpxRNTS (47) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (48) CpxRNTS (49) IntTrsBoundProof [UPPER BOUND(ID), 1214 ms] (50) CpxRNTS (51) IntTrsBoundProof [UPPER BOUND(ID), 116 ms] (52) CpxRNTS (53) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (54) CdtProblem (55) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (56) CdtProblem (57) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (58) CdtProblem (59) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 1110 ms] (60) CdtProblem (61) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (62) CdtProblem (63) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (64) CdtProblem (65) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (66) CdtProblem (67) CdtLeafRemovalProof [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) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (74) CdtProblem (75) CdtNarrowingProof [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) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (84) CdtProblem (85) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (86) CdtProblem (87) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 68 ms] (88) CdtProblem (89) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (90) CdtProblem (91) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 84 ms] (92) CdtProblem (93) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 3 ms] (94) CdtProblem (95) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (96) CdtProblem (97) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (98) CdtProblem (99) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (100) CdtProblem (101) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (102) CdtProblem (103) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (104) CdtProblem (105) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (106) CdtProblem (107) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (108) CdtProblem (109) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (110) CdtProblem (111) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (112) CdtProblem (113) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (114) CdtProblem (115) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (116) CdtProblem (117) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (118) CdtProblem (119) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (120) CdtProblem (121) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (122) CdtProblem (123) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (124) CdtProblem (125) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (126) CdtProblem (127) CdtRewritingProof [BOTH BOUNDS(ID, ID), 5 ms] (128) CdtProblem (129) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (130) CdtProblem (131) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (132) CdtProblem (133) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (134) CdtProblem (135) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (136) CdtProblem (137) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (138) CdtProblem (139) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (140) CdtProblem (141) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (142) CdtProblem (143) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (144) CdtProblem (145) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (146) CdtProblem (147) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (148) CdtProblem (149) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (150) CdtProblem (151) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (152) CdtProblem (153) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (154) CdtProblem (155) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 2 ms] (156) CdtProblem (157) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (158) CdtProblem (159) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (160) CdtProblem (161) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (162) CdtProblem (163) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (164) CdtProblem (165) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (166) CdtProblem (167) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (168) CdtProblem (169) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (170) CdtProblem (171) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (172) 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(nil) -> nil sort(cons(x, xs)) -> cons(max(cons(x, xs)), sort(del(max(cons(x, xs)), cons(x, xs)))) 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(nil) -> nil sort(cons(x, xs)) -> cons(max(cons(x, xs)), sort(del(max(cons(x, xs)), cons(x, xs)))) 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(nil) -> nil sort(cons(x, xs)) -> cons(max(cons(x, xs)), sort(del(max(cons(x, xs)), cons(x, xs)))) 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(nil) -> nil [1] sort(cons(x, xs)) -> cons(max(cons(x, xs)), sort(del(max(cons(x, xs)), cons(x, xs)))) [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(nil) -> nil [1] sort(cons(x, xs)) -> cons(max(cons(x, xs)), sort(del(max(cons(x, xs)), cons(x, xs)))) [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 Rewrite Strategy: INNERMOST ---------------------------------------- (9) CompletionProof (UPPER BOUND(ID)) The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added: none And the following fresh constants: none ---------------------------------------- (10) Obligation: Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: 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(nil) -> nil [1] sort(cons(x, xs)) -> cons(max(cons(x, xs)), sort(del(max(cons(x, xs)), cons(x, xs)))) [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 Rewrite Strategy: INNERMOST ---------------------------------------- (11) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: nil => 0 0 => 0 true => 1 false => 0 ---------------------------------------- (12) 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 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 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 }-> 0 :|: z = 0 sort(z) -{ 1 }-> 1 + max(1 + x + xs) + sort(del(max(1 + x + xs), 1 + x + xs)) :|: z = 1 + x + xs, xs >= 0, x >= 0 Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (13) CompletionProof (UPPER BOUND(ID)) The transformation into a RNTS is sound, since: (a) The obligation is a constructor system where every type has a constant constructor, (b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols: sort_1 (c) The following functions are completely defined: del_2 max_1 eq_2 ge_2 if2_4 if1_4 Due to the following rules being added: none And the following fresh constants: none ---------------------------------------- (14) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: 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(nil) -> nil [1] sort(cons(x, xs)) -> cons(max(cons(x, xs)), sort(del(max(cons(x, xs)), cons(x, xs)))) [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 Rewrite Strategy: INNERMOST ---------------------------------------- (15) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (16) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: 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) -> nil [1] sort(cons(x, nil)) -> cons(max(cons(x, nil)), sort(del(x, cons(x, nil)))) [2] sort(cons(x, cons(y2, xs'))) -> cons(max(cons(x, cons(y2, xs'))), sort(del(if1(ge(x, y2), x, y2, xs'), cons(x, cons(y2, xs'))))) [2] 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 Rewrite Strategy: INNERMOST ---------------------------------------- (17) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: nil => 0 0 => 0 true => 1 false => 0 ---------------------------------------- (18) 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 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 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) -{ 1 }-> 0 :|: z = 0 sort(z) -{ 2 }-> 1 + max(1 + x + 0) + sort(del(x, 1 + x + 0)) :|: x >= 0, z = 1 + x + 0 sort(z) -{ 2 }-> 1 + max(1 + x + (1 + y2 + xs')) + sort(del(if1(ge(x, y2), x, y2, xs'), 1 + x + (1 + y2 + xs'))) :|: z = 1 + x + (1 + y2 + xs'), x >= 0, xs' >= 0, y2 >= 0 ---------------------------------------- (19) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: 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 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 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) -{ 1 }-> 0 :|: z = 0 sort(z) -{ 2 }-> 1 + max(1 + x + (1 + y2 + xs')) + sort(del(if1(ge(x, y2), x, y2, xs'), 1 + x + (1 + y2 + xs'))) :|: z = 1 + x + (1 + y2 + xs'), x >= 0, xs' >= 0, y2 >= 0 sort(z) -{ 2 }-> 1 + max(1 + (z - 1) + 0) + sort(del(z - 1, 1 + (z - 1) + 0)) :|: z - 1 >= 0 ---------------------------------------- (21) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { ge } { eq } { max, if1 } { del, if2 } { sort } ---------------------------------------- (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 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 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) -{ 1 }-> 0 :|: z = 0 sort(z) -{ 2 }-> 1 + max(1 + x + (1 + y2 + xs')) + sort(del(if1(ge(x, y2), x, y2, xs'), 1 + x + (1 + y2 + xs'))) :|: z = 1 + x + (1 + y2 + xs'), x >= 0, xs' >= 0, y2 >= 0 sort(z) -{ 2 }-> 1 + max(1 + (z - 1) + 0) + sort(del(z - 1, 1 + (z - 1) + 0)) :|: z - 1 >= 0 Function symbols to be analyzed: {ge}, {eq}, {max,if1}, {del,if2}, {sort} ---------------------------------------- (23) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (24) Obligation: Complexity RNTS consisting of the following rules: 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 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 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) -{ 1 }-> 0 :|: z = 0 sort(z) -{ 2 }-> 1 + max(1 + x + (1 + y2 + xs')) + sort(del(if1(ge(x, y2), x, y2, xs'), 1 + x + (1 + y2 + xs'))) :|: z = 1 + x + (1 + y2 + xs'), x >= 0, xs' >= 0, y2 >= 0 sort(z) -{ 2 }-> 1 + max(1 + (z - 1) + 0) + sort(del(z - 1, 1 + (z - 1) + 0)) :|: z - 1 >= 0 Function symbols to be analyzed: {ge}, {eq}, {max,if1}, {del,if2}, {sort} ---------------------------------------- (25) 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 ---------------------------------------- (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 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 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) -{ 1 }-> 0 :|: z = 0 sort(z) -{ 2 }-> 1 + max(1 + x + (1 + y2 + xs')) + sort(del(if1(ge(x, y2), x, y2, xs'), 1 + x + (1 + y2 + xs'))) :|: z = 1 + x + (1 + y2 + xs'), x >= 0, xs' >= 0, y2 >= 0 sort(z) -{ 2 }-> 1 + max(1 + (z - 1) + 0) + sort(del(z - 1, 1 + (z - 1) + 0)) :|: z - 1 >= 0 Function symbols to be analyzed: {ge}, {eq}, {max,if1}, {del,if2}, {sort} Previous analysis results are: ge: runtime: ?, size: O(1) [1] ---------------------------------------- (27) 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' ---------------------------------------- (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 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 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) -{ 1 }-> 0 :|: z = 0 sort(z) -{ 2 }-> 1 + max(1 + x + (1 + y2 + xs')) + sort(del(if1(ge(x, y2), x, y2, xs'), 1 + x + (1 + y2 + xs'))) :|: z = 1 + x + (1 + y2 + xs'), x >= 0, xs' >= 0, y2 >= 0 sort(z) -{ 2 }-> 1 + max(1 + (z - 1) + 0) + sort(del(z - 1, 1 + (z - 1) + 0)) :|: z - 1 >= 0 Function symbols to be analyzed: {eq}, {max,if1}, {del,if2}, {sort} Previous analysis results are: ge: runtime: O(n^1) [2 + z'], size: O(1) [1] ---------------------------------------- (29) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (30) Obligation: Complexity RNTS consisting of the following rules: 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 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') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 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 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) -{ 1 }-> 0 :|: z = 0 sort(z) -{ 4 + y2 }-> 1 + max(1 + x + (1 + y2 + xs')) + sort(del(if1(s', x, y2, xs'), 1 + x + (1 + y2 + xs'))) :|: s' >= 0, s' <= 1, z = 1 + x + (1 + y2 + xs'), x >= 0, xs' >= 0, y2 >= 0 sort(z) -{ 2 }-> 1 + max(1 + (z - 1) + 0) + sort(del(z - 1, 1 + (z - 1) + 0)) :|: z - 1 >= 0 Function symbols to be analyzed: {eq}, {max,if1}, {del,if2}, {sort} Previous analysis results are: ge: runtime: O(n^1) [2 + z'], size: O(1) [1] ---------------------------------------- (31) 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 ---------------------------------------- (32) 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 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') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 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 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) -{ 1 }-> 0 :|: z = 0 sort(z) -{ 4 + y2 }-> 1 + max(1 + x + (1 + y2 + xs')) + sort(del(if1(s', x, y2, xs'), 1 + x + (1 + y2 + xs'))) :|: s' >= 0, s' <= 1, z = 1 + x + (1 + y2 + xs'), x >= 0, xs' >= 0, y2 >= 0 sort(z) -{ 2 }-> 1 + max(1 + (z - 1) + 0) + sort(del(z - 1, 1 + (z - 1) + 0)) :|: z - 1 >= 0 Function symbols to be analyzed: {eq}, {max,if1}, {del,if2}, {sort} Previous analysis results are: ge: runtime: O(n^1) [2 + z'], size: O(1) [1] eq: runtime: ?, size: O(1) [1] ---------------------------------------- (33) 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' ---------------------------------------- (34) 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 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') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 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 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) -{ 1 }-> 0 :|: z = 0 sort(z) -{ 4 + y2 }-> 1 + max(1 + x + (1 + y2 + xs')) + sort(del(if1(s', x, y2, xs'), 1 + x + (1 + y2 + xs'))) :|: s' >= 0, s' <= 1, z = 1 + x + (1 + y2 + xs'), x >= 0, xs' >= 0, y2 >= 0 sort(z) -{ 2 }-> 1 + max(1 + (z - 1) + 0) + sort(del(z - 1, 1 + (z - 1) + 0)) :|: z - 1 >= 0 Function symbols to be analyzed: {max,if1}, {del,if2}, {sort} Previous analysis results are: ge: runtime: O(n^1) [2 + z'], size: O(1) [1] eq: runtime: O(n^1) [3 + z'], size: O(1) [1] ---------------------------------------- (35) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (36) Obligation: Complexity RNTS consisting of the following rules: del(z, z') -{ 5 + y1 }-> if2(s1, 1 + (z - 1), 1 + y1, xs) :|: s1 >= 0, s1 <= 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 eq(z, z') -{ 3 + z' }-> s2 :|: s2 >= 0, s2 <= 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' }-> s'' :|: s'' >= 0, s'' <= 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 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) -{ 1 }-> 0 :|: z = 0 sort(z) -{ 4 + y2 }-> 1 + max(1 + x + (1 + y2 + xs')) + sort(del(if1(s', x, y2, xs'), 1 + x + (1 + y2 + xs'))) :|: s' >= 0, s' <= 1, z = 1 + x + (1 + y2 + xs'), x >= 0, xs' >= 0, y2 >= 0 sort(z) -{ 2 }-> 1 + max(1 + (z - 1) + 0) + sort(del(z - 1, 1 + (z - 1) + 0)) :|: z - 1 >= 0 Function symbols to be analyzed: {max,if1}, {del,if2}, {sort} Previous analysis results are: ge: runtime: O(n^1) [2 + z'], size: O(1) [1] eq: runtime: O(n^1) [3 + z'], size: O(1) [1] ---------------------------------------- (37) 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 ---------------------------------------- (38) Obligation: Complexity RNTS consisting of the following rules: del(z, z') -{ 5 + y1 }-> if2(s1, 1 + (z - 1), 1 + y1, xs) :|: s1 >= 0, s1 <= 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 eq(z, z') -{ 3 + z' }-> s2 :|: s2 >= 0, s2 <= 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' }-> s'' :|: s'' >= 0, s'' <= 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 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) -{ 1 }-> 0 :|: z = 0 sort(z) -{ 4 + y2 }-> 1 + max(1 + x + (1 + y2 + xs')) + sort(del(if1(s', x, y2, xs'), 1 + x + (1 + y2 + xs'))) :|: s' >= 0, s' <= 1, z = 1 + x + (1 + y2 + xs'), x >= 0, xs' >= 0, y2 >= 0 sort(z) -{ 2 }-> 1 + max(1 + (z - 1) + 0) + sort(del(z - 1, 1 + (z - 1) + 0)) :|: z - 1 >= 0 Function symbols to be analyzed: {max,if1}, {del,if2}, {sort} Previous analysis results are: ge: runtime: O(n^1) [2 + z'], size: O(1) [1] eq: runtime: O(n^1) [3 + z'], size: O(1) [1] max: runtime: ?, size: O(n^1) [z] if1: runtime: ?, size: O(n^1) [1 + z' + z'' + z1] ---------------------------------------- (39) 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 ---------------------------------------- (40) Obligation: Complexity RNTS consisting of the following rules: del(z, z') -{ 5 + y1 }-> if2(s1, 1 + (z - 1), 1 + y1, xs) :|: s1 >= 0, s1 <= 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 eq(z, z') -{ 3 + z' }-> s2 :|: s2 >= 0, s2 <= 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' }-> s'' :|: s'' >= 0, s'' <= 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 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) -{ 1 }-> 0 :|: z = 0 sort(z) -{ 4 + y2 }-> 1 + max(1 + x + (1 + y2 + xs')) + sort(del(if1(s', x, y2, xs'), 1 + x + (1 + y2 + xs'))) :|: s' >= 0, s' <= 1, z = 1 + x + (1 + y2 + xs'), x >= 0, xs' >= 0, y2 >= 0 sort(z) -{ 2 }-> 1 + max(1 + (z - 1) + 0) + sort(del(z - 1, 1 + (z - 1) + 0)) :|: z - 1 >= 0 Function symbols to be analyzed: {del,if2}, {sort} Previous analysis results are: ge: runtime: O(n^1) [2 + z'], size: O(1) [1] eq: runtime: O(n^1) [3 + z'], size: O(1) [1] 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] ---------------------------------------- (41) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (42) Obligation: Complexity RNTS consisting of the following rules: del(z, z') -{ 5 + y1 }-> if2(s1, 1 + (z - 1), 1 + y1, xs) :|: s1 >= 0, s1 <= 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 eq(z, z') -{ 3 + z' }-> s2 :|: s2 >= 0, s2 <= 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' }-> s'' :|: s'' >= 0, s'' <= 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 }-> s6 :|: s6 >= 0, s6 <= 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 }-> s7 :|: s7 >= 0, s7 <= 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 max(z) -{ 54 + 11*x + 4*x*xs + 2*x^2 + 22*xs + 4*xs^2 }-> s3 :|: s3 >= 0, s3 <= 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 }-> s4 :|: s4 >= 0, s4 <= 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 }-> s5 :|: s5 >= 0, s5 <= 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) -{ 1 }-> 0 :|: z = 0 sort(z) -{ 18 + 7*z + 2*z^2 }-> 1 + s8 + sort(del(z - 1, 1 + (z - 1) + 0)) :|: s8 >= 0, s8 <= 1 + (z - 1) + 0, z - 1 >= 0 sort(z) -{ 94 + 26*x + 8*x*xs' + 4*x*y2 + 4*x^2 + 37*xs' + 8*xs'*y2 + 6*xs'^2 + 27*y2 + 4*y2^2 }-> 1 + s9 + sort(del(s10, 1 + x + (1 + y2 + xs'))) :|: s9 >= 0, s9 <= 1 + x + (1 + y2 + xs'), s10 >= 0, s10 <= y2 + xs' + 1 + x, s' >= 0, s' <= 1, z = 1 + x + (1 + y2 + xs'), x >= 0, xs' >= 0, y2 >= 0 Function symbols to be analyzed: {del,if2}, {sort} Previous analysis results are: ge: runtime: O(n^1) [2 + z'], size: O(1) [1] eq: runtime: O(n^1) [3 + z'], size: O(1) [1] 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] ---------------------------------------- (43) 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 ---------------------------------------- (44) Obligation: Complexity RNTS consisting of the following rules: del(z, z') -{ 5 + y1 }-> if2(s1, 1 + (z - 1), 1 + y1, xs) :|: s1 >= 0, s1 <= 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 eq(z, z') -{ 3 + z' }-> s2 :|: s2 >= 0, s2 <= 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' }-> s'' :|: s'' >= 0, s'' <= 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 }-> s6 :|: s6 >= 0, s6 <= 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 }-> s7 :|: s7 >= 0, s7 <= 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 max(z) -{ 54 + 11*x + 4*x*xs + 2*x^2 + 22*xs + 4*xs^2 }-> s3 :|: s3 >= 0, s3 <= 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 }-> s4 :|: s4 >= 0, s4 <= 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 }-> s5 :|: s5 >= 0, s5 <= 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) -{ 1 }-> 0 :|: z = 0 sort(z) -{ 18 + 7*z + 2*z^2 }-> 1 + s8 + sort(del(z - 1, 1 + (z - 1) + 0)) :|: s8 >= 0, s8 <= 1 + (z - 1) + 0, z - 1 >= 0 sort(z) -{ 94 + 26*x + 8*x*xs' + 4*x*y2 + 4*x^2 + 37*xs' + 8*xs'*y2 + 6*xs'^2 + 27*y2 + 4*y2^2 }-> 1 + s9 + sort(del(s10, 1 + x + (1 + y2 + xs'))) :|: s9 >= 0, s9 <= 1 + x + (1 + y2 + xs'), s10 >= 0, s10 <= y2 + xs' + 1 + x, s' >= 0, s' <= 1, z = 1 + x + (1 + y2 + xs'), x >= 0, xs' >= 0, y2 >= 0 Function symbols to be analyzed: {del,if2}, {sort} Previous analysis results are: ge: runtime: O(n^1) [2 + z'], size: O(1) [1] eq: runtime: O(n^1) [3 + z'], size: O(1) [1] 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] del: runtime: ?, size: O(n^1) [z'] if2: runtime: ?, size: O(n^1) [1 + z'' + z1] ---------------------------------------- (45) 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 ---------------------------------------- (46) Obligation: Complexity RNTS consisting of the following rules: del(z, z') -{ 5 + y1 }-> if2(s1, 1 + (z - 1), 1 + y1, xs) :|: s1 >= 0, s1 <= 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 eq(z, z') -{ 3 + z' }-> s2 :|: s2 >= 0, s2 <= 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' }-> s'' :|: s'' >= 0, s'' <= 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 }-> s6 :|: s6 >= 0, s6 <= 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 }-> s7 :|: s7 >= 0, s7 <= 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 max(z) -{ 54 + 11*x + 4*x*xs + 2*x^2 + 22*xs + 4*xs^2 }-> s3 :|: s3 >= 0, s3 <= 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 }-> s4 :|: s4 >= 0, s4 <= 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 }-> s5 :|: s5 >= 0, s5 <= 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) -{ 1 }-> 0 :|: z = 0 sort(z) -{ 18 + 7*z + 2*z^2 }-> 1 + s8 + sort(del(z - 1, 1 + (z - 1) + 0)) :|: s8 >= 0, s8 <= 1 + (z - 1) + 0, z - 1 >= 0 sort(z) -{ 94 + 26*x + 8*x*xs' + 4*x*y2 + 4*x^2 + 37*xs' + 8*xs'*y2 + 6*xs'^2 + 27*y2 + 4*y2^2 }-> 1 + s9 + sort(del(s10, 1 + x + (1 + y2 + xs'))) :|: s9 >= 0, s9 <= 1 + x + (1 + y2 + xs'), s10 >= 0, s10 <= y2 + xs' + 1 + x, s' >= 0, s' <= 1, z = 1 + x + (1 + y2 + xs'), x >= 0, xs' >= 0, y2 >= 0 Function symbols to be analyzed: {sort} Previous analysis results are: ge: runtime: O(n^1) [2 + z'], size: O(1) [1] eq: runtime: O(n^1) [3 + z'], size: O(1) [1] 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] 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] ---------------------------------------- (47) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (48) Obligation: Complexity RNTS consisting of the following rules: del(z, z') -{ 35 + 17*z' }-> s11 :|: s11 >= 0, s11 <= 0 + (z' - 1) + 1, z' - 1 >= 0, z = 0 del(z, z') -{ 52 + 17*xs }-> s12 :|: s12 >= 0, s12 <= 1 + y'' + xs + 1, xs >= 0, z' = 1 + (1 + y'') + xs, y'' >= 0, z = 0 del(z, z') -{ 35 + 17*z' }-> s13 :|: s13 >= 0, s13 <= 0 + (z' - 1) + 1, z' - 1 >= 0, z - 1 >= 0 del(z, z') -{ 55 + 17*xs + y1 }-> s14 :|: s14 >= 0, s14 <= 1 + y1 + xs + 1, s1 >= 0, s1 <= 1, y1 >= 0, xs >= 0, z' = 1 + (1 + y1) + xs, z - 1 >= 0 del(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 eq(z, z') -{ 3 + z' }-> s2 :|: s2 >= 0, s2 <= 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' }-> s'' :|: s'' >= 0, s'' <= 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 }-> s6 :|: s6 >= 0, s6 <= 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 }-> s7 :|: s7 >= 0, s7 <= 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'' + s15 :|: s15 >= 0, s15 <= z1, z1 >= 0, z' >= 0, z'' >= 0, z = 0 max(z) -{ 54 + 11*x + 4*x*xs + 2*x^2 + 22*xs + 4*xs^2 }-> s3 :|: s3 >= 0, s3 <= 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 }-> s4 :|: s4 >= 0, s4 <= 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 }-> s5 :|: s5 >= 0, s5 <= 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) -{ 1 }-> 0 :|: z = 0 sort(z) -{ 67 + 24*z + 2*z^2 }-> 1 + s8 + sort(s16) :|: s16 >= 0, s16 <= 1 + (z - 1) + 0, s8 >= 0, s8 <= 1 + (z - 1) + 0, z - 1 >= 0 sort(z) -{ 177 + 43*x + 8*x*xs' + 4*x*y2 + 4*x^2 + 54*xs' + 8*xs'*y2 + 6*xs'^2 + 44*y2 + 4*y2^2 }-> 1 + s9 + sort(s17) :|: s17 >= 0, s17 <= 1 + x + (1 + y2 + xs'), s9 >= 0, s9 <= 1 + x + (1 + y2 + xs'), s10 >= 0, s10 <= y2 + xs' + 1 + x, s' >= 0, s' <= 1, z = 1 + x + (1 + y2 + xs'), x >= 0, xs' >= 0, y2 >= 0 Function symbols to be analyzed: {sort} Previous analysis results are: ge: runtime: O(n^1) [2 + z'], size: O(1) [1] eq: runtime: O(n^1) [3 + z'], size: O(1) [1] 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] 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] ---------------------------------------- (49) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: sort after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (50) Obligation: Complexity RNTS consisting of the following rules: del(z, z') -{ 35 + 17*z' }-> s11 :|: s11 >= 0, s11 <= 0 + (z' - 1) + 1, z' - 1 >= 0, z = 0 del(z, z') -{ 52 + 17*xs }-> s12 :|: s12 >= 0, s12 <= 1 + y'' + xs + 1, xs >= 0, z' = 1 + (1 + y'') + xs, y'' >= 0, z = 0 del(z, z') -{ 35 + 17*z' }-> s13 :|: s13 >= 0, s13 <= 0 + (z' - 1) + 1, z' - 1 >= 0, z - 1 >= 0 del(z, z') -{ 55 + 17*xs + y1 }-> s14 :|: s14 >= 0, s14 <= 1 + y1 + xs + 1, s1 >= 0, s1 <= 1, y1 >= 0, xs >= 0, z' = 1 + (1 + y1) + xs, z - 1 >= 0 del(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 eq(z, z') -{ 3 + z' }-> s2 :|: s2 >= 0, s2 <= 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' }-> s'' :|: s'' >= 0, s'' <= 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 }-> s6 :|: s6 >= 0, s6 <= 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 }-> s7 :|: s7 >= 0, s7 <= 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'' + s15 :|: s15 >= 0, s15 <= z1, z1 >= 0, z' >= 0, z'' >= 0, z = 0 max(z) -{ 54 + 11*x + 4*x*xs + 2*x^2 + 22*xs + 4*xs^2 }-> s3 :|: s3 >= 0, s3 <= 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 }-> s4 :|: s4 >= 0, s4 <= 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 }-> s5 :|: s5 >= 0, s5 <= 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) -{ 1 }-> 0 :|: z = 0 sort(z) -{ 67 + 24*z + 2*z^2 }-> 1 + s8 + sort(s16) :|: s16 >= 0, s16 <= 1 + (z - 1) + 0, s8 >= 0, s8 <= 1 + (z - 1) + 0, z - 1 >= 0 sort(z) -{ 177 + 43*x + 8*x*xs' + 4*x*y2 + 4*x^2 + 54*xs' + 8*xs'*y2 + 6*xs'^2 + 44*y2 + 4*y2^2 }-> 1 + s9 + sort(s17) :|: s17 >= 0, s17 <= 1 + x + (1 + y2 + xs'), s9 >= 0, s9 <= 1 + x + (1 + y2 + xs'), s10 >= 0, s10 <= y2 + xs' + 1 + x, s' >= 0, s' <= 1, z = 1 + x + (1 + y2 + xs'), x >= 0, xs' >= 0, y2 >= 0 Function symbols to be analyzed: {sort} Previous analysis results are: ge: runtime: O(n^1) [2 + z'], size: O(1) [1] eq: runtime: O(n^1) [3 + z'], size: O(1) [1] 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] 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] sort: runtime: ?, size: INF ---------------------------------------- (51) 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: ? ---------------------------------------- (52) Obligation: Complexity RNTS consisting of the following rules: del(z, z') -{ 35 + 17*z' }-> s11 :|: s11 >= 0, s11 <= 0 + (z' - 1) + 1, z' - 1 >= 0, z = 0 del(z, z') -{ 52 + 17*xs }-> s12 :|: s12 >= 0, s12 <= 1 + y'' + xs + 1, xs >= 0, z' = 1 + (1 + y'') + xs, y'' >= 0, z = 0 del(z, z') -{ 35 + 17*z' }-> s13 :|: s13 >= 0, s13 <= 0 + (z' - 1) + 1, z' - 1 >= 0, z - 1 >= 0 del(z, z') -{ 55 + 17*xs + y1 }-> s14 :|: s14 >= 0, s14 <= 1 + y1 + xs + 1, s1 >= 0, s1 <= 1, y1 >= 0, xs >= 0, z' = 1 + (1 + y1) + xs, z - 1 >= 0 del(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 eq(z, z') -{ 3 + z' }-> s2 :|: s2 >= 0, s2 <= 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' }-> s'' :|: s'' >= 0, s'' <= 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 }-> s6 :|: s6 >= 0, s6 <= 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 }-> s7 :|: s7 >= 0, s7 <= 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'' + s15 :|: s15 >= 0, s15 <= z1, z1 >= 0, z' >= 0, z'' >= 0, z = 0 max(z) -{ 54 + 11*x + 4*x*xs + 2*x^2 + 22*xs + 4*xs^2 }-> s3 :|: s3 >= 0, s3 <= 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 }-> s4 :|: s4 >= 0, s4 <= 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 }-> s5 :|: s5 >= 0, s5 <= 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) -{ 1 }-> 0 :|: z = 0 sort(z) -{ 67 + 24*z + 2*z^2 }-> 1 + s8 + sort(s16) :|: s16 >= 0, s16 <= 1 + (z - 1) + 0, s8 >= 0, s8 <= 1 + (z - 1) + 0, z - 1 >= 0 sort(z) -{ 177 + 43*x + 8*x*xs' + 4*x*y2 + 4*x^2 + 54*xs' + 8*xs'*y2 + 6*xs'^2 + 44*y2 + 4*y2^2 }-> 1 + s9 + sort(s17) :|: s17 >= 0, s17 <= 1 + x + (1 + y2 + xs'), s9 >= 0, s9 <= 1 + x + (1 + y2 + xs'), s10 >= 0, s10 <= y2 + xs' + 1 + x, s' >= 0, s' <= 1, z = 1 + x + (1 + y2 + xs'), x >= 0, xs' >= 0, y2 >= 0 Function symbols to be analyzed: {sort} Previous analysis results are: ge: runtime: O(n^1) [2 + z'], size: O(1) [1] eq: runtime: O(n^1) [3 + z'], size: O(1) [1] 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] 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] sort: runtime: INF, size: INF ---------------------------------------- (53) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (54) 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(nil) -> nil sort(cons(z0, z1)) -> cons(max(cons(z0, z1)), sort(del(max(cons(z0, z1)), cons(z0, z1)))) 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(nil) -> c13 SORT(cons(z0, z1)) -> c14(MAX(cons(z0, z1))) SORT(cons(z0, z1)) -> c15(SORT(del(max(cons(z0, z1)), cons(z0, z1))), DEL(max(cons(z0, z1)), cons(z0, z1)), MAX(cons(z0, z1))) GE(z0, 0) -> c16 GE(0, s(z0)) -> c17 GE(s(z0), s(z1)) -> c18(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(nil) -> c13 SORT(cons(z0, z1)) -> c14(MAX(cons(z0, z1))) SORT(cons(z0, z1)) -> c15(SORT(del(max(cons(z0, z1)), cons(z0, z1))), DEL(max(cons(z0, z1)), cons(z0, z1)), MAX(cons(z0, z1))) GE(z0, 0) -> c16 GE(0, s(z0)) -> c17 GE(s(z0), s(z1)) -> c18(GE(z0, z1)) K tuples:none Defined Rule Symbols: max_1, if1_4, del_2, if2_4, eq_2, sort_1, ge_2 Defined Pair Symbols: MAX_1, IF1_4, DEL_2, IF2_4, EQ_2, SORT_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, c14_1, c15_3, c16, c17, c18_1 ---------------------------------------- (55) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 10 trailing nodes: IF2(true, z0, z1, z2) -> c7 MAX(cons(z0, nil)) -> c1 GE(z0, 0) -> c16 EQ(s(z0), 0) -> c11 MAX(nil) -> c SORT(nil) -> c13 EQ(0, s(z0)) -> c10 GE(0, s(z0)) -> c17 EQ(0, 0) -> c9 DEL(z0, nil) -> c5 ---------------------------------------- (56) 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(nil) -> nil sort(cons(z0, z1)) -> cons(max(cons(z0, z1)), sort(del(max(cons(z0, z1)), cons(z0, z1)))) 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(cons(z0, z1)) -> c14(MAX(cons(z0, z1))) SORT(cons(z0, z1)) -> c15(SORT(del(max(cons(z0, z1)), cons(z0, z1))), DEL(max(cons(z0, z1)), cons(z0, z1)), MAX(cons(z0, z1))) GE(s(z0), s(z1)) -> c18(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(cons(z0, z1)) -> c14(MAX(cons(z0, z1))) SORT(cons(z0, z1)) -> c15(SORT(del(max(cons(z0, z1)), cons(z0, z1))), DEL(max(cons(z0, z1)), cons(z0, z1)), MAX(cons(z0, z1))) GE(s(z0), s(z1)) -> c18(GE(z0, z1)) K tuples:none Defined Rule Symbols: max_1, if1_4, del_2, if2_4, eq_2, sort_1, ge_2 Defined Pair Symbols: MAX_1, IF1_4, DEL_2, IF2_4, EQ_2, SORT_1, GE_2 Compound Symbols: c2_2, c3_1, c4_1, c6_2, c8_1, c12_1, c14_1, c15_3, c18_1 ---------------------------------------- (57) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: max(nil) -> 0 sort(nil) -> nil sort(cons(z0, z1)) -> cons(max(cons(z0, z1)), sort(del(max(cons(z0, z1)), cons(z0, z1)))) ---------------------------------------- (58) 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, 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)) 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(cons(z0, z1)) -> c14(MAX(cons(z0, z1))) SORT(cons(z0, z1)) -> c15(SORT(del(max(cons(z0, z1)), cons(z0, z1))), DEL(max(cons(z0, z1)), cons(z0, z1)), MAX(cons(z0, z1))) GE(s(z0), s(z1)) -> c18(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(cons(z0, z1)) -> c14(MAX(cons(z0, z1))) SORT(cons(z0, z1)) -> c15(SORT(del(max(cons(z0, z1)), cons(z0, z1))), DEL(max(cons(z0, z1)), cons(z0, z1)), MAX(cons(z0, z1))) GE(s(z0), s(z1)) -> c18(GE(z0, z1)) K tuples:none Defined Rule Symbols: ge_2, eq_2, del_2, max_1, if1_4, if2_4 Defined Pair Symbols: MAX_1, IF1_4, DEL_2, IF2_4, EQ_2, SORT_1, GE_2 Compound Symbols: c2_2, c3_1, c4_1, c6_2, c8_1, c12_1, c14_1, c15_3, c18_1 ---------------------------------------- (59) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. SORT(cons(z0, z1)) -> c14(MAX(cons(z0, z1))) We considered the (Usable) Rules:none And the 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(cons(z0, z1)) -> c14(MAX(cons(z0, z1))) SORT(cons(z0, z1)) -> c15(SORT(del(max(cons(z0, z1)), cons(z0, z1))), DEL(max(cons(z0, z1)), cons(z0, z1)), MAX(cons(z0, z1))) GE(s(z0), s(z1)) -> c18(GE(z0, z1)) 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)) = 0 POL(MAX(x_1)) = 0 POL(SORT(x_1)) = [2] POL(c12(x_1)) = x_1 POL(c14(x_1)) = x_1 POL(c15(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c18(x_1)) = x_1 POL(c2(x_1, x_2)) = x_1 + x_2 POL(c3(x_1)) = x_1 POL(c4(x_1)) = x_1 POL(c6(x_1, x_2)) = x_1 + x_2 POL(c8(x_1)) = x_1 POL(cons(x_1, x_2)) = [2] + x_2 POL(del(x_1, x_2)) = [2] + [2]x_2^2 POL(eq(x_1, x_2)) = [1] POL(false) = [2] POL(ge(x_1, x_2)) = 0 POL(if1(x_1, x_2, x_3, x_4)) = [1] + x_2 + x_3 + x_4 + x_4^2 + x_3*x_4 + x_2*x_4 + x_3^2 + x_2*x_3 + x_2^2 POL(if2(x_1, x_2, x_3, x_4)) = [2] + x_4 + [2]x_4^2 + [2]x_1^2 POL(max(x_1)) = 0 POL(nil) = [2] POL(s(x_1)) = 0 POL(true) = 0 ---------------------------------------- (60) 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, 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)) 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(cons(z0, z1)) -> c14(MAX(cons(z0, z1))) SORT(cons(z0, z1)) -> c15(SORT(del(max(cons(z0, z1)), cons(z0, z1))), DEL(max(cons(z0, z1)), cons(z0, z1)), MAX(cons(z0, z1))) GE(s(z0), s(z1)) -> c18(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(cons(z0, z1)) -> c15(SORT(del(max(cons(z0, z1)), cons(z0, z1))), DEL(max(cons(z0, z1)), cons(z0, z1)), MAX(cons(z0, z1))) GE(s(z0), s(z1)) -> c18(GE(z0, z1)) K tuples: SORT(cons(z0, z1)) -> c14(MAX(cons(z0, z1))) Defined Rule Symbols: ge_2, eq_2, del_2, max_1, if1_4, if2_4 Defined Pair Symbols: MAX_1, IF1_4, DEL_2, IF2_4, EQ_2, SORT_1, GE_2 Compound Symbols: c2_2, c3_1, c4_1, c6_2, c8_1, c12_1, c14_1, c15_3, c18_1 ---------------------------------------- (61) 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))) ---------------------------------------- (62) 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, 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)) 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)) SORT(cons(z0, z1)) -> c14(MAX(cons(z0, z1))) SORT(cons(z0, z1)) -> c15(SORT(del(max(cons(z0, z1)), cons(z0, z1))), DEL(max(cons(z0, z1)), cons(z0, z1)), MAX(cons(z0, z1))) GE(s(z0), s(z1)) -> c18(GE(z0, z1)) 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)) SORT(cons(z0, z1)) -> c15(SORT(del(max(cons(z0, z1)), cons(z0, z1))), DEL(max(cons(z0, z1)), cons(z0, z1)), MAX(cons(z0, z1))) GE(s(z0), s(z1)) -> c18(GE(z0, z1)) 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: SORT(cons(z0, z1)) -> c14(MAX(cons(z0, z1))) Defined Rule Symbols: ge_2, eq_2, del_2, max_1, if1_4, if2_4 Defined Pair Symbols: IF1_4, DEL_2, IF2_4, EQ_2, SORT_1, GE_2, MAX_1 Compound Symbols: c3_1, c4_1, c6_2, c8_1, c12_1, c14_1, c15_3, c18_1, c2_2 ---------------------------------------- (63) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (64) 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, 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)) 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)) SORT(cons(z0, z1)) -> c14(MAX(cons(z0, z1))) SORT(cons(z0, z1)) -> c15(SORT(del(max(cons(z0, z1)), cons(z0, z1))), DEL(max(cons(z0, z1)), cons(z0, z1)), MAX(cons(z0, z1))) GE(s(z0), s(z1)) -> c18(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)) 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)) SORT(cons(z0, z1)) -> c15(SORT(del(max(cons(z0, z1)), cons(z0, z1))), DEL(max(cons(z0, z1)), cons(z0, z1)), MAX(cons(z0, z1))) GE(s(z0), s(z1)) -> c18(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)) K tuples: SORT(cons(z0, z1)) -> c14(MAX(cons(z0, z1))) Defined Rule Symbols: ge_2, eq_2, del_2, max_1, if1_4, if2_4 Defined Pair Symbols: IF1_4, DEL_2, IF2_4, EQ_2, SORT_1, GE_2, MAX_1 Compound Symbols: c3_1, c4_1, c6_2, c8_1, c12_1, c14_1, c15_3, c18_1, c2_2, c2_1 ---------------------------------------- (65) 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))) ---------------------------------------- (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, 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) 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)) SORT(cons(z0, z1)) -> c14(MAX(cons(z0, z1))) SORT(cons(z0, z1)) -> c15(SORT(del(max(cons(z0, z1)), cons(z0, z1))), DEL(max(cons(z0, z1)), cons(z0, z1)), MAX(cons(z0, z1))) GE(s(z0), s(z1)) -> c18(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(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)) SORT(cons(z0, z1)) -> c15(SORT(del(max(cons(z0, z1)), cons(z0, z1))), DEL(max(cons(z0, z1)), cons(z0, z1)), MAX(cons(z0, z1))) GE(s(z0), s(z1)) -> c18(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(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: SORT(cons(z0, z1)) -> c14(MAX(cons(z0, z1))) Defined Rule Symbols: ge_2, eq_2, del_2, max_1, if1_4, if2_4 Defined Pair Symbols: IF1_4, IF2_4, EQ_2, SORT_1, GE_2, MAX_1, DEL_2 Compound Symbols: c3_1, c4_1, c8_1, c12_1, c14_1, c15_3, c18_1, c2_2, c2_1, c6_2 ---------------------------------------- (67) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: DEL(0, cons(0, x2)) -> c6(IF2(true, 0, 0, x2), EQ(0, 0)) ---------------------------------------- (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, 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) 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)) SORT(cons(z0, z1)) -> c14(MAX(cons(z0, z1))) SORT(cons(z0, z1)) -> c15(SORT(del(max(cons(z0, z1)), cons(z0, z1))), DEL(max(cons(z0, z1)), cons(z0, z1)), MAX(cons(z0, z1))) GE(s(z0), s(z1)) -> c18(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(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)) SORT(cons(z0, z1)) -> c15(SORT(del(max(cons(z0, z1)), cons(z0, z1))), DEL(max(cons(z0, z1)), cons(z0, z1)), MAX(cons(z0, z1))) GE(s(z0), s(z1)) -> c18(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(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: SORT(cons(z0, z1)) -> c14(MAX(cons(z0, z1))) Defined Rule Symbols: ge_2, eq_2, del_2, max_1, if1_4, if2_4 Defined Pair Symbols: IF1_4, IF2_4, EQ_2, SORT_1, GE_2, MAX_1, DEL_2 Compound Symbols: c3_1, c4_1, c8_1, c12_1, c14_1, c15_3, c18_1, c2_2, c2_1, c6_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, 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) 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)) SORT(cons(z0, z1)) -> c14(MAX(cons(z0, z1))) SORT(cons(z0, z1)) -> c15(SORT(del(max(cons(z0, z1)), cons(z0, z1))), DEL(max(cons(z0, z1)), cons(z0, z1)), MAX(cons(z0, z1))) GE(s(z0), s(z1)) -> c18(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)) 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)) SORT(cons(z0, z1)) -> c15(SORT(del(max(cons(z0, z1)), cons(z0, z1))), DEL(max(cons(z0, z1)), cons(z0, z1)), MAX(cons(z0, z1))) GE(s(z0), s(z1)) -> c18(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)) K tuples: SORT(cons(z0, z1)) -> c14(MAX(cons(z0, z1))) Defined Rule Symbols: ge_2, eq_2, del_2, max_1, if1_4, if2_4 Defined Pair Symbols: IF1_4, IF2_4, EQ_2, SORT_1, GE_2, MAX_1, DEL_2 Compound Symbols: c3_1, c4_1, c8_1, c12_1, c14_1, c15_3, c18_1, c2_2, c2_1, c6_2, c6_1 ---------------------------------------- (71) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace SORT(cons(z0, z1)) -> c15(SORT(del(max(cons(z0, z1)), cons(z0, z1))), DEL(max(cons(z0, z1)), cons(z0, z1)), MAX(cons(z0, z1))) by SORT(cons(z1, z2)) -> c15(SORT(if2(eq(max(cons(z1, z2)), z1), max(cons(z1, z2)), z1, z2)), DEL(max(cons(z1, z2)), cons(z1, z2)), MAX(cons(z1, z2))) SORT(cons(z0, nil)) -> c15(SORT(del(z0, cons(z0, nil))), DEL(max(cons(z0, nil)), cons(z0, nil)), MAX(cons(z0, nil))) SORT(cons(z0, cons(z1, z2))) -> c15(SORT(del(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))), DEL(max(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) ---------------------------------------- (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, 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) 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)) SORT(cons(z0, z1)) -> c14(MAX(cons(z0, z1))) GE(s(z0), s(z1)) -> c18(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)) SORT(cons(z1, z2)) -> c15(SORT(if2(eq(max(cons(z1, z2)), z1), max(cons(z1, z2)), z1, z2)), DEL(max(cons(z1, z2)), cons(z1, z2)), MAX(cons(z1, z2))) SORT(cons(z0, nil)) -> c15(SORT(del(z0, cons(z0, nil))), DEL(max(cons(z0, nil)), cons(z0, nil)), MAX(cons(z0, nil))) SORT(cons(z0, cons(z1, z2))) -> c15(SORT(del(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))), DEL(max(cons(z0, cons(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)) -> c18(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)) SORT(cons(z1, z2)) -> c15(SORT(if2(eq(max(cons(z1, z2)), z1), max(cons(z1, z2)), z1, z2)), DEL(max(cons(z1, z2)), cons(z1, z2)), MAX(cons(z1, z2))) SORT(cons(z0, nil)) -> c15(SORT(del(z0, cons(z0, nil))), DEL(max(cons(z0, nil)), cons(z0, nil)), MAX(cons(z0, nil))) SORT(cons(z0, cons(z1, z2))) -> c15(SORT(del(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))), DEL(max(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) K tuples: SORT(cons(z0, z1)) -> c14(MAX(cons(z0, z1))) Defined Rule Symbols: ge_2, eq_2, del_2, max_1, if1_4, if2_4 Defined Pair Symbols: IF1_4, IF2_4, EQ_2, SORT_1, GE_2, MAX_1, DEL_2 Compound Symbols: c3_1, c4_1, c8_1, c12_1, c14_1, c18_1, c2_2, c2_1, c6_2, c6_1, c15_3 ---------------------------------------- (73) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (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, 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) 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)) SORT(cons(z0, z1)) -> c14(MAX(cons(z0, z1))) GE(s(z0), s(z1)) -> c18(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)) SORT(cons(z1, z2)) -> c15(SORT(if2(eq(max(cons(z1, z2)), z1), max(cons(z1, z2)), z1, z2)), DEL(max(cons(z1, z2)), cons(z1, z2)), MAX(cons(z1, z2))) SORT(cons(z0, cons(z1, z2))) -> c15(SORT(del(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))), DEL(max(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) SORT(cons(z0, nil)) -> c15(SORT(del(z0, cons(z0, nil))), DEL(max(cons(z0, nil)), 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)) -> c18(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)) SORT(cons(z1, z2)) -> c15(SORT(if2(eq(max(cons(z1, z2)), z1), max(cons(z1, z2)), z1, z2)), DEL(max(cons(z1, z2)), cons(z1, z2)), MAX(cons(z1, z2))) SORT(cons(z0, cons(z1, z2))) -> c15(SORT(del(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))), DEL(max(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) SORT(cons(z0, nil)) -> c15(SORT(del(z0, cons(z0, nil))), DEL(max(cons(z0, nil)), cons(z0, nil))) K tuples: SORT(cons(z0, z1)) -> c14(MAX(cons(z0, z1))) Defined Rule Symbols: ge_2, eq_2, del_2, max_1, if1_4, if2_4 Defined Pair Symbols: IF1_4, IF2_4, EQ_2, SORT_1, GE_2, MAX_1, DEL_2 Compound Symbols: c3_1, c4_1, c8_1, c12_1, c14_1, c18_1, c2_2, c2_1, c6_2, c6_1, c15_3, c15_2 ---------------------------------------- (75) 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))) ---------------------------------------- (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, 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) 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)) SORT(cons(z0, z1)) -> c14(MAX(cons(z0, z1))) GE(s(z0), s(z1)) -> c18(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)) SORT(cons(z1, z2)) -> c15(SORT(if2(eq(max(cons(z1, z2)), z1), max(cons(z1, z2)), z1, z2)), DEL(max(cons(z1, z2)), cons(z1, z2)), MAX(cons(z1, z2))) SORT(cons(z0, cons(z1, z2))) -> c15(SORT(del(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))), DEL(max(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) SORT(cons(z0, nil)) -> c15(SORT(del(z0, cons(z0, nil))), DEL(max(cons(z0, nil)), cons(z0, nil))) 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)) -> c18(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)) SORT(cons(z1, z2)) -> c15(SORT(if2(eq(max(cons(z1, z2)), z1), max(cons(z1, z2)), z1, z2)), DEL(max(cons(z1, z2)), cons(z1, z2)), MAX(cons(z1, z2))) SORT(cons(z0, cons(z1, z2))) -> c15(SORT(del(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))), DEL(max(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) SORT(cons(z0, nil)) -> c15(SORT(del(z0, cons(z0, nil))), DEL(max(cons(z0, nil)), cons(z0, nil))) 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: SORT(cons(z0, z1)) -> c14(MAX(cons(z0, z1))) Defined Rule Symbols: ge_2, eq_2, del_2, max_1, if1_4, if2_4 Defined Pair Symbols: IF1_4, IF2_4, EQ_2, SORT_1, GE_2, MAX_1, DEL_2 Compound Symbols: c3_1, c4_1, c8_1, c12_1, c14_1, c18_1, c2_1, c6_2, c6_1, c15_3, c15_2, c2_2 ---------------------------------------- (77) 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))) ---------------------------------------- (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, 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) 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)) SORT(cons(z0, z1)) -> c14(MAX(cons(z0, z1))) GE(s(z0), s(z1)) -> c18(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(z1, z2)) -> c15(SORT(if2(eq(max(cons(z1, z2)), z1), max(cons(z1, z2)), z1, z2)), DEL(max(cons(z1, z2)), cons(z1, z2)), MAX(cons(z1, z2))) SORT(cons(z0, cons(z1, z2))) -> c15(SORT(del(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))), DEL(max(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) SORT(cons(z0, nil)) -> c15(SORT(del(z0, cons(z0, nil))), DEL(max(cons(z0, nil)), cons(z0, nil))) 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)) -> c18(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(z1, z2)) -> c15(SORT(if2(eq(max(cons(z1, z2)), z1), max(cons(z1, z2)), z1, z2)), DEL(max(cons(z1, z2)), cons(z1, z2)), MAX(cons(z1, z2))) SORT(cons(z0, cons(z1, z2))) -> c15(SORT(del(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))), DEL(max(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) SORT(cons(z0, nil)) -> c15(SORT(del(z0, cons(z0, nil))), DEL(max(cons(z0, nil)), cons(z0, nil))) 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: SORT(cons(z0, z1)) -> c14(MAX(cons(z0, z1))) Defined Rule Symbols: ge_2, eq_2, del_2, max_1, if1_4, if2_4 Defined Pair Symbols: IF1_4, IF2_4, EQ_2, SORT_1, GE_2, MAX_1, DEL_2 Compound Symbols: c3_1, c4_1, c8_1, c12_1, c14_1, c18_1, c2_1, c6_1, c15_3, c15_2, c2_2, c6_2 ---------------------------------------- (79) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 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, 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) 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)) SORT(cons(z0, z1)) -> c14(MAX(cons(z0, z1))) GE(s(z0), s(z1)) -> c18(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(z1, z2)) -> c15(SORT(if2(eq(max(cons(z1, z2)), z1), max(cons(z1, z2)), z1, z2)), DEL(max(cons(z1, z2)), cons(z1, z2)), MAX(cons(z1, z2))) SORT(cons(z0, cons(z1, z2))) -> c15(SORT(del(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))), DEL(max(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) SORT(cons(z0, nil)) -> c15(SORT(del(z0, cons(z0, nil))), DEL(max(cons(z0, nil)), cons(z0, nil))) 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)) -> c18(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(z1, z2)) -> c15(SORT(if2(eq(max(cons(z1, z2)), z1), max(cons(z1, z2)), z1, z2)), DEL(max(cons(z1, z2)), cons(z1, z2)), MAX(cons(z1, z2))) SORT(cons(z0, cons(z1, z2))) -> c15(SORT(del(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))), DEL(max(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) SORT(cons(z0, nil)) -> c15(SORT(del(z0, cons(z0, nil))), DEL(max(cons(z0, nil)), cons(z0, nil))) 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: SORT(cons(z0, z1)) -> c14(MAX(cons(z0, z1))) Defined Rule Symbols: ge_2, eq_2, del_2, max_1, if1_4, if2_4 Defined Pair Symbols: IF1_4, IF2_4, EQ_2, SORT_1, GE_2, MAX_1, DEL_2 Compound Symbols: c3_1, c4_1, c8_1, c12_1, c14_1, c18_1, c2_1, c6_1, c15_3, c15_2, c2_2, c6_2 ---------------------------------------- (81) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace SORT(cons(z1, z2)) -> c15(SORT(if2(eq(max(cons(z1, z2)), z1), max(cons(z1, z2)), z1, z2)), DEL(max(cons(z1, z2)), cons(z1, z2)), MAX(cons(z1, z2))) by SORT(cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(max(cons(z0, nil)), cons(z0, nil)), MAX(cons(z0, nil))) SORT(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(max(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(max(cons(z0, nil)), cons(z0, nil)), MAX(cons(z0, nil))) SORT(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(max(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) ---------------------------------------- (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, 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) 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)) SORT(cons(z0, z1)) -> c14(MAX(cons(z0, z1))) GE(s(z0), s(z1)) -> c18(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, cons(z1, z2))) -> c15(SORT(del(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))), DEL(max(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) SORT(cons(z0, nil)) -> c15(SORT(del(z0, cons(z0, nil))), DEL(max(cons(z0, nil)), cons(z0, nil))) 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))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(max(cons(z0, nil)), cons(z0, nil)), MAX(cons(z0, nil))) SORT(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(max(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(max(cons(z0, nil)), cons(z0, nil)), MAX(cons(z0, nil))) SORT(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(max(cons(z0, cons(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)) -> c18(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, cons(z1, z2))) -> c15(SORT(del(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))), DEL(max(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) SORT(cons(z0, nil)) -> c15(SORT(del(z0, cons(z0, nil))), DEL(max(cons(z0, nil)), cons(z0, nil))) 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))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(max(cons(z0, nil)), cons(z0, nil)), MAX(cons(z0, nil))) SORT(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(max(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(max(cons(z0, nil)), cons(z0, nil)), MAX(cons(z0, nil))) SORT(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(max(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) K tuples: SORT(cons(z0, z1)) -> c14(MAX(cons(z0, z1))) Defined Rule Symbols: ge_2, eq_2, del_2, max_1, if1_4, if2_4 Defined Pair Symbols: IF1_4, IF2_4, EQ_2, SORT_1, GE_2, MAX_1, DEL_2 Compound Symbols: c3_1, c4_1, c8_1, c12_1, c14_1, c18_1, c2_1, c6_1, c15_3, c15_2, c2_2, c6_2 ---------------------------------------- (83) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (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, 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) 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)) SORT(cons(z0, z1)) -> c14(MAX(cons(z0, z1))) GE(s(z0), s(z1)) -> c18(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, cons(z1, z2))) -> c15(SORT(del(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))), DEL(max(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) SORT(cons(z0, nil)) -> c15(SORT(del(z0, cons(z0, nil))), DEL(max(cons(z0, nil)), cons(z0, nil))) 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))) SORT(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(max(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) SORT(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(max(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(max(cons(z0, nil)), cons(z0, nil))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(max(cons(z0, nil)), 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)) -> c18(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, cons(z1, z2))) -> c15(SORT(del(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))), DEL(max(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) SORT(cons(z0, nil)) -> c15(SORT(del(z0, cons(z0, nil))), DEL(max(cons(z0, nil)), cons(z0, nil))) 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))) SORT(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(max(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) SORT(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(max(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(max(cons(z0, nil)), cons(z0, nil))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(max(cons(z0, nil)), cons(z0, nil))) K tuples: SORT(cons(z0, z1)) -> c14(MAX(cons(z0, z1))) Defined Rule Symbols: ge_2, eq_2, del_2, max_1, if1_4, if2_4 Defined Pair Symbols: IF1_4, IF2_4, EQ_2, SORT_1, GE_2, MAX_1, DEL_2 Compound Symbols: c3_1, c4_1, c8_1, c12_1, c14_1, c18_1, c2_1, c6_1, c15_3, c15_2, c2_2, c6_2 ---------------------------------------- (85) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace SORT(cons(z0, cons(z1, z2))) -> c15(SORT(del(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))), DEL(max(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) by SORT(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(max(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), MAX(cons(z1, cons(x1, x2)))) SORT(cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2)))), DEL(max(cons(z0, cons(0, x2))), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) SORT(cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2)))), DEL(max(cons(0, cons(s(z0), x2))), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) SORT(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(max(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) SORT(cons(x0, cons(x1, x2))) -> c15(DEL(max(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) ---------------------------------------- (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, 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) 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)) SORT(cons(z0, z1)) -> c14(MAX(cons(z0, z1))) GE(s(z0), s(z1)) -> c18(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, nil)) -> c15(SORT(del(z0, cons(z0, nil))), DEL(max(cons(z0, nil)), cons(z0, nil))) 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))) SORT(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(max(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) SORT(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(max(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(max(cons(z0, nil)), cons(z0, nil))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(max(cons(z0, nil)), cons(z0, nil))) SORT(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(max(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), MAX(cons(z1, cons(x1, x2)))) SORT(cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2)))), DEL(max(cons(z0, cons(0, x2))), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) SORT(cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2)))), DEL(max(cons(0, cons(s(z0), x2))), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) SORT(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(max(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) SORT(cons(x0, cons(x1, x2))) -> c15(DEL(max(cons(x0, cons(x1, x2))), cons(x0, cons(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)) -> c18(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, nil)) -> c15(SORT(del(z0, cons(z0, nil))), DEL(max(cons(z0, nil)), cons(z0, nil))) 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))) SORT(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(max(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) SORT(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(max(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(max(cons(z0, nil)), cons(z0, nil))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(max(cons(z0, nil)), cons(z0, nil))) SORT(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(max(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), MAX(cons(z1, cons(x1, x2)))) SORT(cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2)))), DEL(max(cons(z0, cons(0, x2))), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) SORT(cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2)))), DEL(max(cons(0, cons(s(z0), x2))), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) SORT(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(max(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) SORT(cons(x0, cons(x1, x2))) -> c15(DEL(max(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) K tuples: SORT(cons(z0, z1)) -> c14(MAX(cons(z0, z1))) Defined Rule Symbols: ge_2, eq_2, del_2, max_1, if1_4, if2_4 Defined Pair Symbols: IF1_4, IF2_4, EQ_2, SORT_1, GE_2, MAX_1, DEL_2 Compound Symbols: c3_1, c4_1, c8_1, c12_1, c14_1, c18_1, c2_1, c6_1, c15_2, c2_2, c6_2, c15_3, c15_1 ---------------------------------------- (87) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. SORT(cons(x0, cons(x1, x2))) -> c15(DEL(max(cons(x0, cons(x1, x2))), cons(x0, cons(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)) SORT(cons(z0, z1)) -> c14(MAX(cons(z0, z1))) GE(s(z0), s(z1)) -> c18(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, nil)) -> c15(SORT(del(z0, cons(z0, nil))), DEL(max(cons(z0, nil)), cons(z0, nil))) 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))) SORT(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(max(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) SORT(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(max(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(max(cons(z0, nil)), cons(z0, nil))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(max(cons(z0, nil)), cons(z0, nil))) SORT(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(max(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), MAX(cons(z1, cons(x1, x2)))) SORT(cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2)))), DEL(max(cons(z0, cons(0, x2))), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) SORT(cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2)))), DEL(max(cons(0, cons(s(z0), x2))), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) SORT(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(max(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) SORT(cons(x0, cons(x1, x2))) -> c15(DEL(max(cons(x0, cons(x1, x2))), cons(x0, cons(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(MAX(x_1)) = x_1 POL(SORT(x_1)) = [1] POL(c12(x_1)) = x_1 POL(c14(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(c18(x_1)) = x_1 POL(c2(x_1)) = x_1 POL(c2(x_1, x_2)) = x_1 + x_2 POL(c3(x_1)) = x_1 POL(c4(x_1)) = x_1 POL(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)) = 0 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_3 + x_4 POL(max(x_1)) = 0 POL(nil) = 0 POL(s(x_1)) = 0 POL(true) = 0 ---------------------------------------- (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, 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)) 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)) SORT(cons(z0, z1)) -> c14(MAX(cons(z0, z1))) GE(s(z0), s(z1)) -> c18(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, nil)) -> c15(SORT(del(z0, cons(z0, nil))), DEL(max(cons(z0, nil)), cons(z0, nil))) 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))) SORT(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(max(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) SORT(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(max(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(max(cons(z0, nil)), cons(z0, nil))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(max(cons(z0, nil)), cons(z0, nil))) SORT(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(max(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), MAX(cons(z1, cons(x1, x2)))) SORT(cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2)))), DEL(max(cons(z0, cons(0, x2))), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) SORT(cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2)))), DEL(max(cons(0, cons(s(z0), x2))), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) SORT(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(max(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) SORT(cons(x0, cons(x1, x2))) -> c15(DEL(max(cons(x0, cons(x1, x2))), cons(x0, cons(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)) -> c18(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, nil)) -> c15(SORT(del(z0, cons(z0, nil))), DEL(max(cons(z0, nil)), cons(z0, nil))) 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))) SORT(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(max(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) SORT(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(max(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(max(cons(z0, nil)), cons(z0, nil))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(max(cons(z0, nil)), cons(z0, nil))) SORT(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(max(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), MAX(cons(z1, cons(x1, x2)))) SORT(cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2)))), DEL(max(cons(z0, cons(0, x2))), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) SORT(cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2)))), DEL(max(cons(0, cons(s(z0), x2))), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) SORT(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(max(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) K tuples: SORT(cons(z0, z1)) -> c14(MAX(cons(z0, z1))) SORT(cons(x0, cons(x1, x2))) -> c15(DEL(max(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) Defined Rule Symbols: ge_2, eq_2, del_2, max_1, if1_4, if2_4 Defined Pair Symbols: IF1_4, IF2_4, EQ_2, SORT_1, GE_2, MAX_1, DEL_2 Compound Symbols: c3_1, c4_1, c8_1, c12_1, c14_1, c18_1, c2_1, c6_1, c15_2, c2_2, c6_2, c15_3, c15_1 ---------------------------------------- (89) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace SORT(cons(z0, nil)) -> c15(SORT(del(z0, cons(z0, nil))), DEL(max(cons(z0, nil)), cons(z0, nil))) by SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(max(cons(z0, nil)), cons(z0, nil))) SORT(cons(x0, nil)) -> c15(DEL(max(cons(x0, nil)), cons(x0, nil))) ---------------------------------------- (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, 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)) 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)) SORT(cons(z0, z1)) -> c14(MAX(cons(z0, z1))) GE(s(z0), s(z1)) -> c18(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)) 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))) SORT(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(max(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) SORT(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(max(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(max(cons(z0, nil)), cons(z0, nil))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(max(cons(z0, nil)), cons(z0, nil))) SORT(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(max(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), MAX(cons(z1, cons(x1, x2)))) SORT(cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2)))), DEL(max(cons(z0, cons(0, x2))), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) SORT(cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2)))), DEL(max(cons(0, cons(s(z0), x2))), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) SORT(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(max(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) SORT(cons(x0, cons(x1, x2))) -> c15(DEL(max(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(max(cons(z0, nil)), cons(z0, nil))) SORT(cons(x0, nil)) -> c15(DEL(max(cons(x0, nil)), 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)) -> c18(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)) 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))) SORT(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(max(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) SORT(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(max(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(max(cons(z0, nil)), cons(z0, nil))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(max(cons(z0, nil)), cons(z0, nil))) SORT(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(max(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), MAX(cons(z1, cons(x1, x2)))) SORT(cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2)))), DEL(max(cons(z0, cons(0, x2))), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) SORT(cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2)))), DEL(max(cons(0, cons(s(z0), x2))), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) SORT(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(max(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(max(cons(z0, nil)), cons(z0, nil))) SORT(cons(x0, nil)) -> c15(DEL(max(cons(x0, nil)), cons(x0, nil))) K tuples: SORT(cons(z0, z1)) -> c14(MAX(cons(z0, z1))) SORT(cons(x0, cons(x1, x2))) -> c15(DEL(max(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) Defined Rule Symbols: ge_2, eq_2, del_2, max_1, if1_4, if2_4 Defined Pair Symbols: IF1_4, IF2_4, EQ_2, SORT_1, GE_2, MAX_1, DEL_2 Compound Symbols: c3_1, c4_1, c8_1, c12_1, c14_1, c18_1, c2_1, c6_1, c2_2, c6_2, c15_3, c15_2, c15_1 ---------------------------------------- (91) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. SORT(cons(x0, nil)) -> c15(DEL(max(cons(x0, nil)), cons(x0, nil))) 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)) SORT(cons(z0, z1)) -> c14(MAX(cons(z0, z1))) GE(s(z0), s(z1)) -> c18(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)) 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))) SORT(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(max(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) SORT(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(max(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(max(cons(z0, nil)), cons(z0, nil))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(max(cons(z0, nil)), cons(z0, nil))) SORT(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(max(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), MAX(cons(z1, cons(x1, x2)))) SORT(cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2)))), DEL(max(cons(z0, cons(0, x2))), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) SORT(cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2)))), DEL(max(cons(0, cons(s(z0), x2))), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) SORT(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(max(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) SORT(cons(x0, cons(x1, x2))) -> c15(DEL(max(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(max(cons(z0, nil)), cons(z0, nil))) SORT(cons(x0, nil)) -> c15(DEL(max(cons(x0, nil)), 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(MAX(x_1)) = x_1 POL(SORT(x_1)) = [1] POL(c12(x_1)) = x_1 POL(c14(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(c18(x_1)) = x_1 POL(c2(x_1)) = x_1 POL(c2(x_1, x_2)) = x_1 + x_2 POL(c3(x_1)) = x_1 POL(c4(x_1)) = x_1 POL(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)) = 0 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_3 + x_4 POL(max(x_1)) = 0 POL(nil) = 0 POL(s(x_1)) = 0 POL(true) = 0 ---------------------------------------- (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, 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)) 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)) SORT(cons(z0, z1)) -> c14(MAX(cons(z0, z1))) GE(s(z0), s(z1)) -> c18(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)) 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))) SORT(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(max(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) SORT(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(max(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(max(cons(z0, nil)), cons(z0, nil))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(max(cons(z0, nil)), cons(z0, nil))) SORT(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(max(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), MAX(cons(z1, cons(x1, x2)))) SORT(cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2)))), DEL(max(cons(z0, cons(0, x2))), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) SORT(cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2)))), DEL(max(cons(0, cons(s(z0), x2))), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) SORT(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(max(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) SORT(cons(x0, cons(x1, x2))) -> c15(DEL(max(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(max(cons(z0, nil)), cons(z0, nil))) SORT(cons(x0, nil)) -> c15(DEL(max(cons(x0, nil)), 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)) -> c18(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)) 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))) SORT(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(max(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) SORT(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(max(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(max(cons(z0, nil)), cons(z0, nil))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(max(cons(z0, nil)), cons(z0, nil))) SORT(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(max(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), MAX(cons(z1, cons(x1, x2)))) SORT(cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2)))), DEL(max(cons(z0, cons(0, x2))), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) SORT(cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2)))), DEL(max(cons(0, cons(s(z0), x2))), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) SORT(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(max(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(max(cons(z0, nil)), cons(z0, nil))) K tuples: SORT(cons(z0, z1)) -> c14(MAX(cons(z0, z1))) SORT(cons(x0, cons(x1, x2))) -> c15(DEL(max(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) SORT(cons(x0, nil)) -> c15(DEL(max(cons(x0, nil)), cons(x0, 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, SORT_1, GE_2, MAX_1, DEL_2 Compound Symbols: c3_1, c4_1, c8_1, c12_1, c14_1, c18_1, c2_1, c6_1, c2_2, c6_2, c15_3, c15_2, c15_1 ---------------------------------------- (93) 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(x1)), x2) -> c3(MAX(cons(s(s(x0)), x2))) ---------------------------------------- (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, 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)) 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)) SORT(cons(z0, z1)) -> c14(MAX(cons(z0, z1))) GE(s(z0), s(z1)) -> c18(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)) 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))) SORT(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(max(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) SORT(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(max(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(max(cons(z0, nil)), cons(z0, nil))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(max(cons(z0, nil)), cons(z0, nil))) SORT(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(max(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), MAX(cons(z1, cons(x1, x2)))) SORT(cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2)))), DEL(max(cons(z0, cons(0, x2))), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) SORT(cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2)))), DEL(max(cons(0, cons(s(z0), x2))), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) SORT(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(max(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) SORT(cons(x0, cons(x1, x2))) -> c15(DEL(max(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(max(cons(z0, nil)), cons(z0, nil))) SORT(cons(x0, nil)) -> c15(DEL(max(cons(x0, nil)), cons(x0, 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(x1)), x2) -> c3(MAX(cons(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)) -> c18(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)) 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))) SORT(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(max(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) SORT(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(max(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(max(cons(z0, nil)), cons(z0, nil))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(max(cons(z0, nil)), cons(z0, nil))) SORT(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(max(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), MAX(cons(z1, cons(x1, x2)))) SORT(cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2)))), DEL(max(cons(z0, cons(0, x2))), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) SORT(cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2)))), DEL(max(cons(0, cons(s(z0), x2))), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) SORT(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(max(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(max(cons(z0, nil)), cons(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(x1)), x2) -> c3(MAX(cons(s(s(x0)), x2))) K tuples: SORT(cons(z0, z1)) -> c14(MAX(cons(z0, z1))) SORT(cons(x0, cons(x1, x2))) -> c15(DEL(max(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) SORT(cons(x0, nil)) -> c15(DEL(max(cons(x0, nil)), cons(x0, 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, SORT_1, GE_2, MAX_1, DEL_2 Compound Symbols: c4_1, c8_1, c12_1, c14_1, c18_1, c2_1, c6_1, c2_2, c6_2, c15_3, c15_2, c15_1, c3_1 ---------------------------------------- (95) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace SORT(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(max(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) by SORT(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)))) ---------------------------------------- (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, 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)) 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)) SORT(cons(z0, z1)) -> c14(MAX(cons(z0, z1))) GE(s(z0), s(z1)) -> c18(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)) 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))) SORT(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(max(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(max(cons(z0, nil)), cons(z0, nil))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(max(cons(z0, nil)), cons(z0, nil))) SORT(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(max(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), MAX(cons(z1, cons(x1, x2)))) SORT(cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2)))), DEL(max(cons(z0, cons(0, x2))), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) SORT(cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2)))), DEL(max(cons(0, cons(s(z0), x2))), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) SORT(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(max(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) SORT(cons(x0, cons(x1, x2))) -> c15(DEL(max(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(max(cons(z0, nil)), cons(z0, nil))) SORT(cons(x0, nil)) -> c15(DEL(max(cons(x0, nil)), cons(x0, 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(x1)), x2) -> c3(MAX(cons(s(s(x0)), x2))) SORT(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)))) 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)) -> c18(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)) 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))) SORT(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(max(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(max(cons(z0, nil)), cons(z0, nil))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(max(cons(z0, nil)), cons(z0, nil))) SORT(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(max(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), MAX(cons(z1, cons(x1, x2)))) SORT(cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2)))), DEL(max(cons(z0, cons(0, x2))), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) SORT(cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2)))), DEL(max(cons(0, cons(s(z0), x2))), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) SORT(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(max(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(max(cons(z0, nil)), cons(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(x1)), x2) -> c3(MAX(cons(s(s(x0)), x2))) SORT(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)))) K tuples: SORT(cons(z0, z1)) -> c14(MAX(cons(z0, z1))) SORT(cons(x0, cons(x1, x2))) -> c15(DEL(max(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) SORT(cons(x0, nil)) -> c15(DEL(max(cons(x0, nil)), cons(x0, 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, SORT_1, GE_2, MAX_1, DEL_2 Compound Symbols: c4_1, c8_1, c12_1, c14_1, c18_1, c2_1, c6_1, c2_2, c6_2, c15_3, c15_2, c15_1, c3_1 ---------------------------------------- (97) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace SORT(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(max(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MAX(cons(z0, cons(z1, z2)))) by SORT(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)))) ---------------------------------------- (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, 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)) 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)) SORT(cons(z0, z1)) -> c14(MAX(cons(z0, z1))) GE(s(z0), s(z1)) -> c18(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)) 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))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(max(cons(z0, nil)), cons(z0, nil))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(max(cons(z0, nil)), cons(z0, nil))) SORT(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(max(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), MAX(cons(z1, cons(x1, x2)))) SORT(cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2)))), DEL(max(cons(z0, cons(0, x2))), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) SORT(cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2)))), DEL(max(cons(0, cons(s(z0), x2))), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) SORT(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(max(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) SORT(cons(x0, cons(x1, x2))) -> c15(DEL(max(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(max(cons(z0, nil)), cons(z0, nil))) SORT(cons(x0, nil)) -> c15(DEL(max(cons(x0, nil)), cons(x0, 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(x1)), x2) -> c3(MAX(cons(s(s(x0)), x2))) SORT(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)))) SORT(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(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)) -> c18(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)) 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))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(max(cons(z0, nil)), cons(z0, nil))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(max(cons(z0, nil)), cons(z0, nil))) SORT(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(max(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), MAX(cons(z1, cons(x1, x2)))) SORT(cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2)))), DEL(max(cons(z0, cons(0, x2))), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) SORT(cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2)))), DEL(max(cons(0, cons(s(z0), x2))), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) SORT(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(max(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(max(cons(z0, nil)), cons(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(x1)), x2) -> c3(MAX(cons(s(s(x0)), x2))) SORT(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)))) SORT(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: SORT(cons(z0, z1)) -> c14(MAX(cons(z0, z1))) SORT(cons(x0, cons(x1, x2))) -> c15(DEL(max(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) SORT(cons(x0, nil)) -> c15(DEL(max(cons(x0, nil)), cons(x0, 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, SORT_1, GE_2, MAX_1, DEL_2 Compound Symbols: c4_1, c8_1, c12_1, c14_1, c18_1, c2_1, c6_1, c2_2, c6_2, c15_2, c15_3, c15_1, c3_1 ---------------------------------------- (99) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace SORT(cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(max(cons(z0, nil)), cons(z0, nil))) by SORT(cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, 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, 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)) 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)) SORT(cons(z0, z1)) -> c14(MAX(cons(z0, z1))) GE(s(z0), s(z1)) -> c18(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)) 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))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(max(cons(z0, nil)), cons(z0, nil))) SORT(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(max(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), MAX(cons(z1, cons(x1, x2)))) SORT(cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2)))), DEL(max(cons(z0, cons(0, x2))), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) SORT(cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2)))), DEL(max(cons(0, cons(s(z0), x2))), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) SORT(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(max(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) SORT(cons(x0, cons(x1, x2))) -> c15(DEL(max(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(max(cons(z0, nil)), cons(z0, nil))) SORT(cons(x0, nil)) -> c15(DEL(max(cons(x0, nil)), cons(x0, 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(x1)), x2) -> c3(MAX(cons(s(s(x0)), x2))) SORT(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)))) SORT(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)))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) 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)) -> c18(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)) 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))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(max(cons(z0, nil)), cons(z0, nil))) SORT(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(max(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), MAX(cons(z1, cons(x1, x2)))) SORT(cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2)))), DEL(max(cons(z0, cons(0, x2))), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) SORT(cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2)))), DEL(max(cons(0, cons(s(z0), x2))), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) SORT(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(max(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(max(cons(z0, nil)), cons(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(x1)), x2) -> c3(MAX(cons(s(s(x0)), x2))) SORT(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)))) SORT(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)))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) K tuples: SORT(cons(z0, z1)) -> c14(MAX(cons(z0, z1))) SORT(cons(x0, cons(x1, x2))) -> c15(DEL(max(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) SORT(cons(x0, nil)) -> c15(DEL(max(cons(x0, nil)), cons(x0, 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, SORT_1, GE_2, MAX_1, DEL_2 Compound Symbols: c4_1, c8_1, c12_1, c14_1, c18_1, c2_1, c6_1, c2_2, c6_2, c15_2, c15_3, c15_1, c3_1 ---------------------------------------- (101) 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(x0)), s(s(x1)), x2) -> c4(MAX(cons(s(s(x1)), x2))) ---------------------------------------- (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, 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)) Tuples: IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) SORT(cons(z0, z1)) -> c14(MAX(cons(z0, z1))) GE(s(z0), s(z1)) -> c18(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)) 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))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(max(cons(z0, nil)), cons(z0, nil))) SORT(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(max(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), MAX(cons(z1, cons(x1, x2)))) SORT(cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2)))), DEL(max(cons(z0, cons(0, x2))), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) SORT(cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2)))), DEL(max(cons(0, cons(s(z0), x2))), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) SORT(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(max(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) SORT(cons(x0, cons(x1, x2))) -> c15(DEL(max(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(max(cons(z0, nil)), cons(z0, nil))) SORT(cons(x0, nil)) -> c15(DEL(max(cons(x0, nil)), cons(x0, 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(x1)), x2) -> c3(MAX(cons(s(s(x0)), x2))) SORT(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)))) SORT(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)))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) 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(x0)), s(s(x1)), x2) -> c4(MAX(cons(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)) -> c18(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)) 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))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(max(cons(z0, nil)), cons(z0, nil))) SORT(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(max(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), MAX(cons(z1, cons(x1, x2)))) SORT(cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2)))), DEL(max(cons(z0, cons(0, x2))), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) SORT(cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2)))), DEL(max(cons(0, cons(s(z0), x2))), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) SORT(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(max(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(max(cons(z0, nil)), cons(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(x1)), x2) -> c3(MAX(cons(s(s(x0)), x2))) SORT(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)))) SORT(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)))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) 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(x0)), s(s(x1)), x2) -> c4(MAX(cons(s(s(x1)), x2))) K tuples: SORT(cons(z0, z1)) -> c14(MAX(cons(z0, z1))) SORT(cons(x0, cons(x1, x2))) -> c15(DEL(max(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) SORT(cons(x0, nil)) -> c15(DEL(max(cons(x0, nil)), cons(x0, nil))) Defined Rule Symbols: ge_2, eq_2, del_2, max_1, if1_4, if2_4 Defined Pair Symbols: IF2_4, EQ_2, SORT_1, GE_2, MAX_1, DEL_2, IF1_4 Compound Symbols: c8_1, c12_1, c14_1, c18_1, c2_1, c6_1, c2_2, c6_2, c15_2, c15_3, c15_1, c3_1, c4_1 ---------------------------------------- (103) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(max(cons(z0, nil)), cons(z0, nil))) by SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, 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, 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)) Tuples: IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) SORT(cons(z0, z1)) -> c14(MAX(cons(z0, z1))) GE(s(z0), s(z1)) -> c18(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)) 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))) SORT(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(max(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), MAX(cons(z1, cons(x1, x2)))) SORT(cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2)))), DEL(max(cons(z0, cons(0, x2))), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) SORT(cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2)))), DEL(max(cons(0, cons(s(z0), x2))), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) SORT(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(max(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) SORT(cons(x0, cons(x1, x2))) -> c15(DEL(max(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(max(cons(z0, nil)), cons(z0, nil))) SORT(cons(x0, nil)) -> c15(DEL(max(cons(x0, nil)), cons(x0, 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(x1)), x2) -> c3(MAX(cons(s(s(x0)), x2))) SORT(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)))) SORT(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)))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) 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(x0)), s(s(x1)), x2) -> c4(MAX(cons(s(s(x1)), x2))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) 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)) -> c18(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)) 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))) SORT(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(max(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), MAX(cons(z1, cons(x1, x2)))) SORT(cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2)))), DEL(max(cons(z0, cons(0, x2))), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) SORT(cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2)))), DEL(max(cons(0, cons(s(z0), x2))), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) SORT(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(max(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(max(cons(z0, nil)), cons(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(x1)), x2) -> c3(MAX(cons(s(s(x0)), x2))) SORT(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)))) SORT(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)))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) 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(x0)), s(s(x1)), x2) -> c4(MAX(cons(s(s(x1)), x2))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) K tuples: SORT(cons(z0, z1)) -> c14(MAX(cons(z0, z1))) SORT(cons(x0, cons(x1, x2))) -> c15(DEL(max(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) SORT(cons(x0, nil)) -> c15(DEL(max(cons(x0, nil)), cons(x0, nil))) Defined Rule Symbols: ge_2, eq_2, del_2, max_1, if1_4, if2_4 Defined Pair Symbols: IF2_4, EQ_2, SORT_1, GE_2, MAX_1, DEL_2, IF1_4 Compound Symbols: c8_1, c12_1, c14_1, c18_1, c2_1, c6_1, c2_2, c6_2, c15_3, c15_1, c15_2, c3_1, c4_1 ---------------------------------------- (105) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace SORT(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(max(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), MAX(cons(z1, cons(x1, x2)))) by SORT(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)))) ---------------------------------------- (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, 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)) Tuples: IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) SORT(cons(z0, z1)) -> c14(MAX(cons(z0, z1))) GE(s(z0), s(z1)) -> c18(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)) 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))) SORT(cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2)))), DEL(max(cons(z0, cons(0, x2))), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) SORT(cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2)))), DEL(max(cons(0, cons(s(z0), x2))), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) SORT(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(max(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) SORT(cons(x0, cons(x1, x2))) -> c15(DEL(max(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(max(cons(z0, nil)), cons(z0, nil))) SORT(cons(x0, nil)) -> c15(DEL(max(cons(x0, nil)), cons(x0, 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(x1)), x2) -> c3(MAX(cons(s(s(x0)), x2))) SORT(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)))) SORT(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)))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) 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(x0)), s(s(x1)), x2) -> c4(MAX(cons(s(s(x1)), x2))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) SORT(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)))) 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)) -> c18(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)) 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))) SORT(cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2)))), DEL(max(cons(z0, cons(0, x2))), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) SORT(cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2)))), DEL(max(cons(0, cons(s(z0), x2))), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) SORT(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(max(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(max(cons(z0, nil)), cons(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(x1)), x2) -> c3(MAX(cons(s(s(x0)), x2))) SORT(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)))) SORT(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)))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) 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(x0)), s(s(x1)), x2) -> c4(MAX(cons(s(s(x1)), x2))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) SORT(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)))) K tuples: SORT(cons(z0, z1)) -> c14(MAX(cons(z0, z1))) SORT(cons(x0, cons(x1, x2))) -> c15(DEL(max(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) SORT(cons(x0, nil)) -> c15(DEL(max(cons(x0, nil)), cons(x0, nil))) Defined Rule Symbols: ge_2, eq_2, del_2, max_1, if1_4, if2_4 Defined Pair Symbols: IF2_4, EQ_2, SORT_1, GE_2, MAX_1, DEL_2, IF1_4 Compound Symbols: c8_1, c12_1, c14_1, c18_1, c2_1, c6_1, c2_2, c6_2, c15_3, c15_1, c15_2, c3_1, c4_1 ---------------------------------------- (107) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace SORT(cons(z0, cons(0, x2))) -> c15(SORT(del(if1(true, z0, 0, x2), cons(z0, cons(0, x2)))), DEL(max(cons(z0, cons(0, x2))), cons(z0, cons(0, x2))), MAX(cons(z0, cons(0, x2)))) by SORT(cons(z0, cons(0, z1))) -> c15(SORT(del(if1(true, z0, 0, z1), cons(z0, cons(0, z1)))), DEL(if1(ge(z0, 0), z0, 0, z1), cons(z0, cons(0, z1))), MAX(cons(z0, cons(0, z1)))) ---------------------------------------- (108) 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, 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)) Tuples: IF2(false, z0, z1, z2) -> c8(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) SORT(cons(z0, z1)) -> c14(MAX(cons(z0, z1))) GE(s(z0), s(z1)) -> c18(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)) 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))) SORT(cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2)))), DEL(max(cons(0, cons(s(z0), x2))), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) SORT(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(max(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) SORT(cons(x0, cons(x1, x2))) -> c15(DEL(max(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(max(cons(z0, nil)), cons(z0, nil))) SORT(cons(x0, nil)) -> c15(DEL(max(cons(x0, nil)), cons(x0, 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(x1)), x2) -> c3(MAX(cons(s(s(x0)), x2))) SORT(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)))) SORT(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)))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) 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(x0)), s(s(x1)), x2) -> c4(MAX(cons(s(s(x1)), x2))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) SORT(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)))) SORT(cons(z0, cons(0, z1))) -> c15(SORT(del(if1(true, z0, 0, z1), cons(z0, cons(0, z1)))), DEL(if1(ge(z0, 0), z0, 0, z1), cons(z0, cons(0, z1))), MAX(cons(z0, cons(0, z1)))) 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)) -> c18(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)) 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))) SORT(cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2)))), DEL(max(cons(0, cons(s(z0), x2))), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) SORT(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(max(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(max(cons(z0, nil)), cons(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(x1)), x2) -> c3(MAX(cons(s(s(x0)), x2))) SORT(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)))) SORT(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)))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) 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(x0)), s(s(x1)), x2) -> c4(MAX(cons(s(s(x1)), x2))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) SORT(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)))) SORT(cons(z0, cons(0, z1))) -> c15(SORT(del(if1(true, z0, 0, z1), cons(z0, cons(0, z1)))), DEL(if1(ge(z0, 0), z0, 0, z1), cons(z0, cons(0, z1))), MAX(cons(z0, cons(0, z1)))) K tuples: SORT(cons(z0, z1)) -> c14(MAX(cons(z0, z1))) SORT(cons(x0, cons(x1, x2))) -> c15(DEL(max(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) SORT(cons(x0, nil)) -> c15(DEL(max(cons(x0, nil)), cons(x0, nil))) Defined Rule Symbols: ge_2, eq_2, del_2, max_1, if1_4, if2_4 Defined Pair Symbols: IF2_4, EQ_2, SORT_1, GE_2, MAX_1, DEL_2, IF1_4 Compound Symbols: c8_1, c12_1, c14_1, c18_1, c2_1, c6_1, c2_2, c6_2, c15_3, c15_1, c15_2, c3_1, c4_1 ---------------------------------------- (109) 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(x0)), s(s(x1)), x2) -> c8(DEL(s(s(x0)), x2)) ---------------------------------------- (110) 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, 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)) Tuples: EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) SORT(cons(z0, z1)) -> c14(MAX(cons(z0, z1))) GE(s(z0), s(z1)) -> c18(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)) 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))) SORT(cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2)))), DEL(max(cons(0, cons(s(z0), x2))), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) SORT(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(max(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) SORT(cons(x0, cons(x1, x2))) -> c15(DEL(max(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(max(cons(z0, nil)), cons(z0, nil))) SORT(cons(x0, nil)) -> c15(DEL(max(cons(x0, nil)), cons(x0, 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(x1)), x2) -> c3(MAX(cons(s(s(x0)), x2))) SORT(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)))) SORT(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)))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) 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(x0)), s(s(x1)), x2) -> c4(MAX(cons(s(s(x1)), x2))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) SORT(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)))) SORT(cons(z0, cons(0, z1))) -> c15(SORT(del(if1(true, z0, 0, z1), cons(z0, cons(0, z1)))), DEL(if1(ge(z0, 0), z0, 0, z1), cons(z0, cons(0, z1))), MAX(cons(z0, cons(0, z1)))) 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(x0)), s(s(x1)), x2) -> c8(DEL(s(s(x0)), x2)) S tuples: EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c18(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)) 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))) SORT(cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2)))), DEL(max(cons(0, cons(s(z0), x2))), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) SORT(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(max(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(max(cons(z0, nil)), cons(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(x1)), x2) -> c3(MAX(cons(s(s(x0)), x2))) SORT(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)))) SORT(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)))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) 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(x0)), s(s(x1)), x2) -> c4(MAX(cons(s(s(x1)), x2))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) SORT(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)))) SORT(cons(z0, cons(0, z1))) -> c15(SORT(del(if1(true, z0, 0, z1), cons(z0, cons(0, z1)))), DEL(if1(ge(z0, 0), z0, 0, z1), cons(z0, cons(0, z1))), MAX(cons(z0, cons(0, z1)))) 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(x0)), s(s(x1)), x2) -> c8(DEL(s(s(x0)), x2)) K tuples: SORT(cons(z0, z1)) -> c14(MAX(cons(z0, z1))) SORT(cons(x0, cons(x1, x2))) -> c15(DEL(max(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) SORT(cons(x0, nil)) -> c15(DEL(max(cons(x0, nil)), cons(x0, nil))) Defined Rule Symbols: ge_2, eq_2, del_2, max_1, if1_4, if2_4 Defined Pair Symbols: EQ_2, SORT_1, GE_2, MAX_1, DEL_2, IF1_4, IF2_4 Compound Symbols: c12_1, c14_1, c18_1, c2_1, c6_1, c2_2, c6_2, c15_3, c15_1, c15_2, c3_1, c4_1, c8_1 ---------------------------------------- (111) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace SORT(cons(0, cons(s(z0), x2))) -> c15(SORT(del(if1(false, 0, s(z0), x2), cons(0, cons(s(z0), x2)))), DEL(max(cons(0, cons(s(z0), x2))), cons(0, cons(s(z0), x2))), MAX(cons(0, cons(s(z0), x2)))) by SORT(cons(0, cons(s(z0), z1))) -> c15(SORT(del(if1(false, 0, s(z0), z1), cons(0, cons(s(z0), z1)))), DEL(if1(ge(0, s(z0)), 0, s(z0), z1), cons(0, cons(s(z0), z1))), MAX(cons(0, cons(s(z0), z1)))) ---------------------------------------- (112) 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, 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)) Tuples: EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) SORT(cons(z0, z1)) -> c14(MAX(cons(z0, z1))) GE(s(z0), s(z1)) -> c18(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)) 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))) SORT(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(max(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) SORT(cons(x0, cons(x1, x2))) -> c15(DEL(max(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(max(cons(z0, nil)), cons(z0, nil))) SORT(cons(x0, nil)) -> c15(DEL(max(cons(x0, nil)), cons(x0, 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(x1)), x2) -> c3(MAX(cons(s(s(x0)), x2))) SORT(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)))) SORT(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)))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) 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(x0)), s(s(x1)), x2) -> c4(MAX(cons(s(s(x1)), x2))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) SORT(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)))) SORT(cons(z0, cons(0, z1))) -> c15(SORT(del(if1(true, z0, 0, z1), cons(z0, cons(0, z1)))), DEL(if1(ge(z0, 0), z0, 0, z1), cons(z0, cons(0, z1))), MAX(cons(z0, cons(0, z1)))) 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(x0)), s(s(x1)), x2) -> c8(DEL(s(s(x0)), x2)) SORT(cons(0, cons(s(z0), z1))) -> c15(SORT(del(if1(false, 0, s(z0), z1), cons(0, cons(s(z0), z1)))), DEL(if1(ge(0, s(z0)), 0, s(z0), z1), cons(0, cons(s(z0), z1))), MAX(cons(0, cons(s(z0), z1)))) S tuples: EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c18(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)) 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))) SORT(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(max(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(max(cons(z0, nil)), cons(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(x1)), x2) -> c3(MAX(cons(s(s(x0)), x2))) SORT(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)))) SORT(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)))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) 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(x0)), s(s(x1)), x2) -> c4(MAX(cons(s(s(x1)), x2))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) SORT(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)))) SORT(cons(z0, cons(0, z1))) -> c15(SORT(del(if1(true, z0, 0, z1), cons(z0, cons(0, z1)))), DEL(if1(ge(z0, 0), z0, 0, z1), cons(z0, cons(0, z1))), MAX(cons(z0, cons(0, z1)))) 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(x0)), s(s(x1)), x2) -> c8(DEL(s(s(x0)), x2)) SORT(cons(0, cons(s(z0), z1))) -> c15(SORT(del(if1(false, 0, s(z0), z1), cons(0, cons(s(z0), z1)))), DEL(if1(ge(0, s(z0)), 0, s(z0), z1), cons(0, cons(s(z0), z1))), MAX(cons(0, cons(s(z0), z1)))) K tuples: SORT(cons(z0, z1)) -> c14(MAX(cons(z0, z1))) SORT(cons(x0, cons(x1, x2))) -> c15(DEL(max(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) SORT(cons(x0, nil)) -> c15(DEL(max(cons(x0, nil)), cons(x0, nil))) Defined Rule Symbols: ge_2, eq_2, del_2, max_1, if1_4, if2_4 Defined Pair Symbols: EQ_2, SORT_1, GE_2, MAX_1, DEL_2, IF1_4, IF2_4 Compound Symbols: c12_1, c14_1, c18_1, c2_1, c6_1, c2_2, c6_2, c15_3, c15_1, c15_2, c3_1, c4_1, c8_1 ---------------------------------------- (113) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace SORT(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(max(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MAX(cons(s(z0), cons(s(z1), x2)))) by SORT(cons(s(z0), cons(s(z1), z2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MAX(cons(s(z0), cons(s(z1), z2)))) ---------------------------------------- (114) 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, 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)) Tuples: EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) SORT(cons(z0, z1)) -> c14(MAX(cons(z0, z1))) GE(s(z0), s(z1)) -> c18(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)) 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))) SORT(cons(x0, cons(x1, x2))) -> c15(DEL(max(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(max(cons(z0, nil)), cons(z0, nil))) SORT(cons(x0, nil)) -> c15(DEL(max(cons(x0, nil)), cons(x0, 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(x1)), x2) -> c3(MAX(cons(s(s(x0)), x2))) SORT(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)))) SORT(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)))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) 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(x0)), s(s(x1)), x2) -> c4(MAX(cons(s(s(x1)), x2))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) SORT(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)))) SORT(cons(z0, cons(0, z1))) -> c15(SORT(del(if1(true, z0, 0, z1), cons(z0, cons(0, z1)))), DEL(if1(ge(z0, 0), z0, 0, z1), cons(z0, cons(0, z1))), MAX(cons(z0, cons(0, z1)))) 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(x0)), s(s(x1)), x2) -> c8(DEL(s(s(x0)), x2)) SORT(cons(0, cons(s(z0), z1))) -> c15(SORT(del(if1(false, 0, s(z0), z1), cons(0, cons(s(z0), z1)))), DEL(if1(ge(0, s(z0)), 0, s(z0), z1), cons(0, cons(s(z0), z1))), MAX(cons(0, cons(s(z0), z1)))) SORT(cons(s(z0), cons(s(z1), z2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MAX(cons(s(z0), cons(s(z1), z2)))) S tuples: EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c18(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)) 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))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(max(cons(z0, nil)), cons(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(x1)), x2) -> c3(MAX(cons(s(s(x0)), x2))) SORT(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)))) SORT(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)))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) 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(x0)), s(s(x1)), x2) -> c4(MAX(cons(s(s(x1)), x2))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) SORT(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)))) SORT(cons(z0, cons(0, z1))) -> c15(SORT(del(if1(true, z0, 0, z1), cons(z0, cons(0, z1)))), DEL(if1(ge(z0, 0), z0, 0, z1), cons(z0, cons(0, z1))), MAX(cons(z0, cons(0, z1)))) 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(x0)), s(s(x1)), x2) -> c8(DEL(s(s(x0)), x2)) SORT(cons(0, cons(s(z0), z1))) -> c15(SORT(del(if1(false, 0, s(z0), z1), cons(0, cons(s(z0), z1)))), DEL(if1(ge(0, s(z0)), 0, s(z0), z1), cons(0, cons(s(z0), z1))), MAX(cons(0, cons(s(z0), z1)))) SORT(cons(s(z0), cons(s(z1), z2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MAX(cons(s(z0), cons(s(z1), z2)))) K tuples: SORT(cons(z0, z1)) -> c14(MAX(cons(z0, z1))) SORT(cons(x0, cons(x1, x2))) -> c15(DEL(max(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) SORT(cons(x0, nil)) -> c15(DEL(max(cons(x0, nil)), cons(x0, nil))) Defined Rule Symbols: ge_2, eq_2, del_2, max_1, if1_4, if2_4 Defined Pair Symbols: EQ_2, SORT_1, GE_2, MAX_1, DEL_2, IF1_4, IF2_4 Compound Symbols: c12_1, c14_1, c18_1, c2_1, c6_1, c2_2, c6_2, c15_1, c15_2, c3_1, c15_3, c4_1, c8_1 ---------------------------------------- (115) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace SORT(cons(x0, cons(x1, x2))) -> c15(DEL(max(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) by SORT(cons(z0, cons(z1, z2))) -> c15(DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))) ---------------------------------------- (116) 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, 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)) Tuples: EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) SORT(cons(z0, z1)) -> c14(MAX(cons(z0, z1))) GE(s(z0), s(z1)) -> c18(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)) 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))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(max(cons(z0, nil)), cons(z0, nil))) SORT(cons(x0, nil)) -> c15(DEL(max(cons(x0, nil)), cons(x0, 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(x1)), x2) -> c3(MAX(cons(s(s(x0)), x2))) SORT(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)))) SORT(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)))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) 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(x0)), s(s(x1)), x2) -> c4(MAX(cons(s(s(x1)), x2))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) SORT(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)))) SORT(cons(z0, cons(0, z1))) -> c15(SORT(del(if1(true, z0, 0, z1), cons(z0, cons(0, z1)))), DEL(if1(ge(z0, 0), z0, 0, z1), cons(z0, cons(0, z1))), MAX(cons(z0, cons(0, z1)))) 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(x0)), s(s(x1)), x2) -> c8(DEL(s(s(x0)), x2)) SORT(cons(0, cons(s(z0), z1))) -> c15(SORT(del(if1(false, 0, s(z0), z1), cons(0, cons(s(z0), z1)))), DEL(if1(ge(0, s(z0)), 0, s(z0), z1), cons(0, cons(s(z0), z1))), MAX(cons(0, cons(s(z0), z1)))) SORT(cons(s(z0), cons(s(z1), z2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MAX(cons(s(z0), cons(s(z1), z2)))) SORT(cons(z0, cons(z1, z2))) -> c15(DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))) S tuples: EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c18(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)) 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))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(max(cons(z0, nil)), cons(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(x1)), x2) -> c3(MAX(cons(s(s(x0)), x2))) SORT(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)))) SORT(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)))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) 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(x0)), s(s(x1)), x2) -> c4(MAX(cons(s(s(x1)), x2))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) SORT(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)))) SORT(cons(z0, cons(0, z1))) -> c15(SORT(del(if1(true, z0, 0, z1), cons(z0, cons(0, z1)))), DEL(if1(ge(z0, 0), z0, 0, z1), cons(z0, cons(0, z1))), MAX(cons(z0, cons(0, z1)))) 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(x0)), s(s(x1)), x2) -> c8(DEL(s(s(x0)), x2)) SORT(cons(0, cons(s(z0), z1))) -> c15(SORT(del(if1(false, 0, s(z0), z1), cons(0, cons(s(z0), z1)))), DEL(if1(ge(0, s(z0)), 0, s(z0), z1), cons(0, cons(s(z0), z1))), MAX(cons(0, cons(s(z0), z1)))) SORT(cons(s(z0), cons(s(z1), z2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MAX(cons(s(z0), cons(s(z1), z2)))) K tuples: SORT(cons(z0, z1)) -> c14(MAX(cons(z0, z1))) SORT(cons(x0, cons(x1, x2))) -> c15(DEL(max(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) SORT(cons(x0, nil)) -> c15(DEL(max(cons(x0, nil)), cons(x0, nil))) Defined Rule Symbols: ge_2, eq_2, del_2, max_1, if1_4, if2_4 Defined Pair Symbols: EQ_2, SORT_1, GE_2, MAX_1, DEL_2, IF1_4, IF2_4 Compound Symbols: c12_1, c14_1, c18_1, c2_1, c6_1, c2_2, c6_2, c15_2, c15_1, c3_1, c15_3, c4_1, c8_1 ---------------------------------------- (117) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(max(cons(z0, nil)), cons(z0, nil))) by SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) ---------------------------------------- (118) 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, 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)) Tuples: EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) SORT(cons(z0, z1)) -> c14(MAX(cons(z0, z1))) GE(s(z0), s(z1)) -> c18(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)) 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))) SORT(cons(x0, nil)) -> c15(DEL(max(cons(x0, nil)), cons(x0, 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(x1)), x2) -> c3(MAX(cons(s(s(x0)), x2))) SORT(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)))) SORT(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)))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) 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(x0)), s(s(x1)), x2) -> c4(MAX(cons(s(s(x1)), x2))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) SORT(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)))) SORT(cons(z0, cons(0, z1))) -> c15(SORT(del(if1(true, z0, 0, z1), cons(z0, cons(0, z1)))), DEL(if1(ge(z0, 0), z0, 0, z1), cons(z0, cons(0, z1))), MAX(cons(z0, cons(0, z1)))) 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(x0)), s(s(x1)), x2) -> c8(DEL(s(s(x0)), x2)) SORT(cons(0, cons(s(z0), z1))) -> c15(SORT(del(if1(false, 0, s(z0), z1), cons(0, cons(s(z0), z1)))), DEL(if1(ge(0, s(z0)), 0, s(z0), z1), cons(0, cons(s(z0), z1))), MAX(cons(0, cons(s(z0), z1)))) SORT(cons(s(z0), cons(s(z1), z2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MAX(cons(s(z0), cons(s(z1), z2)))) SORT(cons(z0, cons(z1, z2))) -> c15(DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) S tuples: EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c18(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)) 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))) 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(x1)), x2) -> c3(MAX(cons(s(s(x0)), x2))) SORT(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)))) SORT(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)))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) 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(x0)), s(s(x1)), x2) -> c4(MAX(cons(s(s(x1)), x2))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) SORT(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)))) SORT(cons(z0, cons(0, z1))) -> c15(SORT(del(if1(true, z0, 0, z1), cons(z0, cons(0, z1)))), DEL(if1(ge(z0, 0), z0, 0, z1), cons(z0, cons(0, z1))), MAX(cons(z0, cons(0, z1)))) 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(x0)), s(s(x1)), x2) -> c8(DEL(s(s(x0)), x2)) SORT(cons(0, cons(s(z0), z1))) -> c15(SORT(del(if1(false, 0, s(z0), z1), cons(0, cons(s(z0), z1)))), DEL(if1(ge(0, s(z0)), 0, s(z0), z1), cons(0, cons(s(z0), z1))), MAX(cons(0, cons(s(z0), z1)))) SORT(cons(s(z0), cons(s(z1), z2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MAX(cons(s(z0), cons(s(z1), z2)))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) K tuples: SORT(cons(z0, z1)) -> c14(MAX(cons(z0, z1))) SORT(cons(x0, cons(x1, x2))) -> c15(DEL(max(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) SORT(cons(x0, nil)) -> c15(DEL(max(cons(x0, nil)), cons(x0, nil))) Defined Rule Symbols: ge_2, eq_2, del_2, max_1, if1_4, if2_4 Defined Pair Symbols: EQ_2, SORT_1, GE_2, MAX_1, DEL_2, IF1_4, IF2_4 Compound Symbols: c12_1, c14_1, c18_1, c2_1, c6_1, c2_2, c6_2, c15_1, c3_1, c15_3, c15_2, c4_1, c8_1 ---------------------------------------- (119) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace SORT(cons(x0, nil)) -> c15(DEL(max(cons(x0, nil)), cons(x0, nil))) by SORT(cons(z0, nil)) -> c15(DEL(z0, cons(z0, nil))) ---------------------------------------- (120) 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, 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)) Tuples: EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) SORT(cons(z0, z1)) -> c14(MAX(cons(z0, z1))) GE(s(z0), s(z1)) -> c18(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)) 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))) 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(x1)), x2) -> c3(MAX(cons(s(s(x0)), x2))) SORT(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)))) SORT(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)))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) 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(x0)), s(s(x1)), x2) -> c4(MAX(cons(s(s(x1)), x2))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) SORT(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)))) SORT(cons(z0, cons(0, z1))) -> c15(SORT(del(if1(true, z0, 0, z1), cons(z0, cons(0, z1)))), DEL(if1(ge(z0, 0), z0, 0, z1), cons(z0, cons(0, z1))), MAX(cons(z0, cons(0, z1)))) 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(x0)), s(s(x1)), x2) -> c8(DEL(s(s(x0)), x2)) SORT(cons(0, cons(s(z0), z1))) -> c15(SORT(del(if1(false, 0, s(z0), z1), cons(0, cons(s(z0), z1)))), DEL(if1(ge(0, s(z0)), 0, s(z0), z1), cons(0, cons(s(z0), z1))), MAX(cons(0, cons(s(z0), z1)))) SORT(cons(s(z0), cons(s(z1), z2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MAX(cons(s(z0), cons(s(z1), z2)))) SORT(cons(z0, cons(z1, z2))) -> c15(DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) SORT(cons(z0, nil)) -> c15(DEL(z0, cons(z0, nil))) S tuples: EQ(s(z0), s(z1)) -> c12(EQ(z0, z1)) GE(s(z0), s(z1)) -> c18(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)) 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))) 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(x1)), x2) -> c3(MAX(cons(s(s(x0)), x2))) SORT(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)))) SORT(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)))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) 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(x0)), s(s(x1)), x2) -> c4(MAX(cons(s(s(x1)), x2))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) SORT(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)))) SORT(cons(z0, cons(0, z1))) -> c15(SORT(del(if1(true, z0, 0, z1), cons(z0, cons(0, z1)))), DEL(if1(ge(z0, 0), z0, 0, z1), cons(z0, cons(0, z1))), MAX(cons(z0, cons(0, z1)))) 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(x0)), s(s(x1)), x2) -> c8(DEL(s(s(x0)), x2)) SORT(cons(0, cons(s(z0), z1))) -> c15(SORT(del(if1(false, 0, s(z0), z1), cons(0, cons(s(z0), z1)))), DEL(if1(ge(0, s(z0)), 0, s(z0), z1), cons(0, cons(s(z0), z1))), MAX(cons(0, cons(s(z0), z1)))) SORT(cons(s(z0), cons(s(z1), z2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MAX(cons(s(z0), cons(s(z1), z2)))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) K tuples: SORT(cons(z0, z1)) -> c14(MAX(cons(z0, z1))) SORT(cons(x0, cons(x1, x2))) -> c15(DEL(max(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) SORT(cons(x0, nil)) -> c15(DEL(max(cons(x0, nil)), cons(x0, nil))) Defined Rule Symbols: ge_2, eq_2, del_2, max_1, if1_4, if2_4 Defined Pair Symbols: EQ_2, SORT_1, GE_2, MAX_1, DEL_2, IF1_4, IF2_4 Compound Symbols: c12_1, c14_1, c18_1, c2_1, c6_1, c2_2, c6_2, c3_1, c15_3, c15_2, c4_1, c8_1, c15_1 ---------------------------------------- (121) 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))) ---------------------------------------- (122) 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, 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)) Tuples: SORT(cons(z0, z1)) -> c14(MAX(cons(z0, z1))) GE(s(z0), s(z1)) -> c18(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)) 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))) 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(x1)), x2) -> c3(MAX(cons(s(s(x0)), x2))) SORT(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)))) SORT(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)))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) 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(x0)), s(s(x1)), x2) -> c4(MAX(cons(s(s(x1)), x2))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) SORT(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)))) SORT(cons(z0, cons(0, z1))) -> c15(SORT(del(if1(true, z0, 0, z1), cons(z0, cons(0, z1)))), DEL(if1(ge(z0, 0), z0, 0, z1), cons(z0, cons(0, z1))), MAX(cons(z0, cons(0, z1)))) 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(x0)), s(s(x1)), x2) -> c8(DEL(s(s(x0)), x2)) SORT(cons(0, cons(s(z0), z1))) -> c15(SORT(del(if1(false, 0, s(z0), z1), cons(0, cons(s(z0), z1)))), DEL(if1(ge(0, s(z0)), 0, s(z0), z1), cons(0, cons(s(z0), z1))), MAX(cons(0, cons(s(z0), z1)))) SORT(cons(s(z0), cons(s(z1), z2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MAX(cons(s(z0), cons(s(z1), z2)))) SORT(cons(z0, cons(z1, z2))) -> c15(DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) SORT(cons(z0, nil)) -> c15(DEL(z0, cons(z0, nil))) EQ(s(s(y0)), s(s(y1))) -> c12(EQ(s(y0), s(y1))) S tuples: GE(s(z0), s(z1)) -> c18(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)) 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))) 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(x1)), x2) -> c3(MAX(cons(s(s(x0)), x2))) SORT(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)))) SORT(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)))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) 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(x0)), s(s(x1)), x2) -> c4(MAX(cons(s(s(x1)), x2))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) SORT(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)))) SORT(cons(z0, cons(0, z1))) -> c15(SORT(del(if1(true, z0, 0, z1), cons(z0, cons(0, z1)))), DEL(if1(ge(z0, 0), z0, 0, z1), cons(z0, cons(0, z1))), MAX(cons(z0, cons(0, z1)))) 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(x0)), s(s(x1)), x2) -> c8(DEL(s(s(x0)), x2)) SORT(cons(0, cons(s(z0), z1))) -> c15(SORT(del(if1(false, 0, s(z0), z1), cons(0, cons(s(z0), z1)))), DEL(if1(ge(0, s(z0)), 0, s(z0), z1), cons(0, cons(s(z0), z1))), MAX(cons(0, cons(s(z0), z1)))) SORT(cons(s(z0), cons(s(z1), z2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MAX(cons(s(z0), cons(s(z1), z2)))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) EQ(s(s(y0)), s(s(y1))) -> c12(EQ(s(y0), s(y1))) K tuples: SORT(cons(z0, z1)) -> c14(MAX(cons(z0, z1))) SORT(cons(x0, cons(x1, x2))) -> c15(DEL(max(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) SORT(cons(x0, nil)) -> c15(DEL(max(cons(x0, nil)), cons(x0, nil))) Defined Rule Symbols: ge_2, eq_2, del_2, max_1, if1_4, if2_4 Defined Pair Symbols: SORT_1, GE_2, MAX_1, DEL_2, IF1_4, IF2_4, EQ_2 Compound Symbols: c14_1, c18_1, c2_1, c6_1, c2_2, c6_2, c3_1, c15_3, c15_2, c4_1, c8_1, c15_1, c12_1 ---------------------------------------- (123) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (124) 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, 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)) Tuples: SORT(cons(z0, z1)) -> c14(MAX(cons(z0, z1))) GE(s(z0), s(z1)) -> c18(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)) 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(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))) 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(x1)), x2) -> c3(MAX(cons(s(s(x0)), x2))) SORT(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)))) SORT(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)))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) 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(x0)), s(s(x1)), x2) -> c4(MAX(cons(s(s(x1)), x2))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) SORT(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)))) SORT(cons(z0, cons(0, z1))) -> c15(SORT(del(if1(true, z0, 0, z1), cons(z0, cons(0, z1)))), DEL(if1(ge(z0, 0), z0, 0, z1), cons(z0, cons(0, z1))), MAX(cons(z0, cons(0, z1)))) 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(x0)), s(s(x1)), x2) -> c8(DEL(s(s(x0)), x2)) SORT(cons(0, cons(s(z0), z1))) -> c15(SORT(del(if1(false, 0, s(z0), z1), cons(0, cons(s(z0), z1)))), DEL(if1(ge(0, s(z0)), 0, s(z0), z1), cons(0, cons(s(z0), z1))), MAX(cons(0, cons(s(z0), z1)))) SORT(cons(s(z0), cons(s(z1), z2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MAX(cons(s(z0), cons(s(z1), z2)))) SORT(cons(z0, cons(z1, z2))) -> c15(DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) SORT(cons(z0, nil)) -> c15(DEL(z0, cons(z0, nil))) 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)) -> c18(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)) 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(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))) 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(x1)), x2) -> c3(MAX(cons(s(s(x0)), x2))) SORT(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)))) SORT(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)))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) 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(x0)), s(s(x1)), x2) -> c4(MAX(cons(s(s(x1)), x2))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) SORT(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)))) SORT(cons(z0, cons(0, z1))) -> c15(SORT(del(if1(true, z0, 0, z1), cons(z0, cons(0, z1)))), DEL(if1(ge(z0, 0), z0, 0, z1), cons(z0, cons(0, z1))), MAX(cons(z0, cons(0, z1)))) 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(x0)), s(s(x1)), x2) -> c8(DEL(s(s(x0)), x2)) SORT(cons(0, cons(s(z0), z1))) -> c15(SORT(del(if1(false, 0, s(z0), z1), cons(0, cons(s(z0), z1)))), DEL(if1(ge(0, s(z0)), 0, s(z0), z1), cons(0, cons(s(z0), z1))), MAX(cons(0, cons(s(z0), z1)))) SORT(cons(s(z0), cons(s(z1), z2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MAX(cons(s(z0), cons(s(z1), z2)))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) 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: SORT(cons(z0, z1)) -> c14(MAX(cons(z0, z1))) SORT(cons(x0, cons(x1, x2))) -> c15(DEL(max(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) SORT(cons(x0, nil)) -> c15(DEL(max(cons(x0, nil)), cons(x0, nil))) Defined Rule Symbols: ge_2, eq_2, del_2, max_1, if1_4, if2_4 Defined Pair Symbols: SORT_1, GE_2, MAX_1, DEL_2, IF1_4, IF2_4, EQ_2 Compound Symbols: c14_1, c18_1, c2_1, c6_1, c2_2, c6_2, c3_1, c15_3, c15_2, c4_1, c8_1, c15_1, c12_1, c6 ---------------------------------------- (125) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace SORT(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 SORT(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)))) ---------------------------------------- (126) 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, 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)) Tuples: SORT(cons(z0, z1)) -> c14(MAX(cons(z0, z1))) GE(s(z0), s(z1)) -> c18(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)) 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(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))) 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(x1)), x2) -> c3(MAX(cons(s(s(x0)), x2))) SORT(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)))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) 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(x0)), s(s(x1)), x2) -> c4(MAX(cons(s(s(x1)), x2))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) SORT(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)))) SORT(cons(z0, cons(0, z1))) -> c15(SORT(del(if1(true, z0, 0, z1), cons(z0, cons(0, z1)))), DEL(if1(ge(z0, 0), z0, 0, z1), cons(z0, cons(0, z1))), MAX(cons(z0, cons(0, z1)))) 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(x0)), s(s(x1)), x2) -> c8(DEL(s(s(x0)), x2)) SORT(cons(0, cons(s(z0), z1))) -> c15(SORT(del(if1(false, 0, s(z0), z1), cons(0, cons(s(z0), z1)))), DEL(if1(ge(0, s(z0)), 0, s(z0), z1), cons(0, cons(s(z0), z1))), MAX(cons(0, cons(s(z0), z1)))) SORT(cons(s(z0), cons(s(z1), z2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MAX(cons(s(z0), cons(s(z1), z2)))) SORT(cons(z0, cons(z1, z2))) -> c15(DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) SORT(cons(z0, nil)) -> c15(DEL(z0, cons(z0, nil))) 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)) -> c18(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)) 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(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))) 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(x1)), x2) -> c3(MAX(cons(s(s(x0)), x2))) SORT(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)))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) 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(x0)), s(s(x1)), x2) -> c4(MAX(cons(s(s(x1)), x2))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) SORT(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)))) SORT(cons(z0, cons(0, z1))) -> c15(SORT(del(if1(true, z0, 0, z1), cons(z0, cons(0, z1)))), DEL(if1(ge(z0, 0), z0, 0, z1), cons(z0, cons(0, z1))), MAX(cons(z0, cons(0, z1)))) 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(x0)), s(s(x1)), x2) -> c8(DEL(s(s(x0)), x2)) SORT(cons(0, cons(s(z0), z1))) -> c15(SORT(del(if1(false, 0, s(z0), z1), cons(0, cons(s(z0), z1)))), DEL(if1(ge(0, s(z0)), 0, s(z0), z1), cons(0, cons(s(z0), z1))), MAX(cons(0, cons(s(z0), z1)))) SORT(cons(s(z0), cons(s(z1), z2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MAX(cons(s(z0), cons(s(z1), z2)))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) 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: SORT(cons(z0, z1)) -> c14(MAX(cons(z0, z1))) SORT(cons(x0, cons(x1, x2))) -> c15(DEL(max(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) SORT(cons(x0, nil)) -> c15(DEL(max(cons(x0, nil)), cons(x0, nil))) Defined Rule Symbols: ge_2, eq_2, del_2, max_1, if1_4, if2_4 Defined Pair Symbols: SORT_1, GE_2, MAX_1, DEL_2, IF1_4, IF2_4, EQ_2 Compound Symbols: c14_1, c18_1, c2_1, c6_1, c2_2, c6_2, c3_1, c15_3, c15_2, c4_1, c8_1, c15_1, c12_1, c6 ---------------------------------------- (127) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace SORT(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 SORT(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)))) ---------------------------------------- (128) 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, 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)) Tuples: SORT(cons(z0, z1)) -> c14(MAX(cons(z0, z1))) GE(s(z0), s(z1)) -> c18(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)) 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(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))) 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(x1)), x2) -> c3(MAX(cons(s(s(x0)), x2))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) 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(x0)), s(s(x1)), x2) -> c4(MAX(cons(s(s(x1)), x2))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) SORT(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)))) SORT(cons(z0, cons(0, z1))) -> c15(SORT(del(if1(true, z0, 0, z1), cons(z0, cons(0, z1)))), DEL(if1(ge(z0, 0), z0, 0, z1), cons(z0, cons(0, z1))), MAX(cons(z0, cons(0, z1)))) 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(x0)), s(s(x1)), x2) -> c8(DEL(s(s(x0)), x2)) SORT(cons(0, cons(s(z0), z1))) -> c15(SORT(del(if1(false, 0, s(z0), z1), cons(0, cons(s(z0), z1)))), DEL(if1(ge(0, s(z0)), 0, s(z0), z1), cons(0, cons(s(z0), z1))), MAX(cons(0, cons(s(z0), z1)))) SORT(cons(s(z0), cons(s(z1), z2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MAX(cons(s(z0), cons(s(z1), z2)))) SORT(cons(z0, cons(z1, z2))) -> c15(DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) SORT(cons(z0, nil)) -> c15(DEL(z0, cons(z0, nil))) 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)) -> c18(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)) 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(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))) 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(x1)), x2) -> c3(MAX(cons(s(s(x0)), x2))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) 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(x0)), s(s(x1)), x2) -> c4(MAX(cons(s(s(x1)), x2))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) SORT(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)))) SORT(cons(z0, cons(0, z1))) -> c15(SORT(del(if1(true, z0, 0, z1), cons(z0, cons(0, z1)))), DEL(if1(ge(z0, 0), z0, 0, z1), cons(z0, cons(0, z1))), MAX(cons(z0, cons(0, z1)))) 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(x0)), s(s(x1)), x2) -> c8(DEL(s(s(x0)), x2)) SORT(cons(0, cons(s(z0), z1))) -> c15(SORT(del(if1(false, 0, s(z0), z1), cons(0, cons(s(z0), z1)))), DEL(if1(ge(0, s(z0)), 0, s(z0), z1), cons(0, cons(s(z0), z1))), MAX(cons(0, cons(s(z0), z1)))) SORT(cons(s(z0), cons(s(z1), z2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MAX(cons(s(z0), cons(s(z1), z2)))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) 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: SORT(cons(z0, z1)) -> c14(MAX(cons(z0, z1))) SORT(cons(x0, cons(x1, x2))) -> c15(DEL(max(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) SORT(cons(x0, nil)) -> c15(DEL(max(cons(x0, nil)), cons(x0, nil))) Defined Rule Symbols: ge_2, eq_2, del_2, max_1, if1_4, if2_4 Defined Pair Symbols: SORT_1, GE_2, MAX_1, DEL_2, IF1_4, IF2_4, EQ_2 Compound Symbols: c14_1, c18_1, c2_1, c6_1, c2_2, c6_2, c3_1, c15_2, c4_1, c15_3, c8_1, c15_1, c12_1, c6 ---------------------------------------- (129) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace SORT(cons(z0, nil)) -> c15(SORT(if2(eq(max(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) by SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) ---------------------------------------- (130) 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, 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)) Tuples: SORT(cons(z0, z1)) -> c14(MAX(cons(z0, z1))) GE(s(z0), s(z1)) -> c18(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)) 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(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))) 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(x1)), x2) -> c3(MAX(cons(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(x0)), s(s(x1)), x2) -> c4(MAX(cons(s(s(x1)), x2))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) SORT(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)))) SORT(cons(z0, cons(0, z1))) -> c15(SORT(del(if1(true, z0, 0, z1), cons(z0, cons(0, z1)))), DEL(if1(ge(z0, 0), z0, 0, z1), cons(z0, cons(0, z1))), MAX(cons(z0, cons(0, z1)))) 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(x0)), s(s(x1)), x2) -> c8(DEL(s(s(x0)), x2)) SORT(cons(0, cons(s(z0), z1))) -> c15(SORT(del(if1(false, 0, s(z0), z1), cons(0, cons(s(z0), z1)))), DEL(if1(ge(0, s(z0)), 0, s(z0), z1), cons(0, cons(s(z0), z1))), MAX(cons(0, cons(s(z0), z1)))) SORT(cons(s(z0), cons(s(z1), z2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MAX(cons(s(z0), cons(s(z1), z2)))) SORT(cons(z0, cons(z1, z2))) -> c15(DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) SORT(cons(z0, nil)) -> c15(DEL(z0, cons(z0, nil))) 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)) -> c18(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)) 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(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))) 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(x1)), x2) -> c3(MAX(cons(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(x0)), s(s(x1)), x2) -> c4(MAX(cons(s(s(x1)), x2))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) SORT(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)))) SORT(cons(z0, cons(0, z1))) -> c15(SORT(del(if1(true, z0, 0, z1), cons(z0, cons(0, z1)))), DEL(if1(ge(z0, 0), z0, 0, z1), cons(z0, cons(0, z1))), MAX(cons(z0, cons(0, z1)))) 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(x0)), s(s(x1)), x2) -> c8(DEL(s(s(x0)), x2)) SORT(cons(0, cons(s(z0), z1))) -> c15(SORT(del(if1(false, 0, s(z0), z1), cons(0, cons(s(z0), z1)))), DEL(if1(ge(0, s(z0)), 0, s(z0), z1), cons(0, cons(s(z0), z1))), MAX(cons(0, cons(s(z0), z1)))) SORT(cons(s(z0), cons(s(z1), z2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MAX(cons(s(z0), cons(s(z1), z2)))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) 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: SORT(cons(z0, z1)) -> c14(MAX(cons(z0, z1))) SORT(cons(x0, cons(x1, x2))) -> c15(DEL(max(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) SORT(cons(x0, nil)) -> c15(DEL(max(cons(x0, nil)), cons(x0, nil))) Defined Rule Symbols: ge_2, eq_2, del_2, max_1, if1_4, if2_4 Defined Pair Symbols: SORT_1, GE_2, MAX_1, DEL_2, IF1_4, IF2_4, EQ_2 Compound Symbols: c14_1, c18_1, c2_1, c6_1, c2_2, c6_2, c3_1, c4_1, c15_2, c15_3, c8_1, c15_1, c12_1, c6 ---------------------------------------- (131) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace SORT(cons(z0, z1)) -> c14(MAX(cons(z0, z1))) by SORT(cons(z0, cons(0, y1))) -> c14(MAX(cons(z0, cons(0, y1)))) SORT(cons(0, cons(s(y0), y1))) -> c14(MAX(cons(0, cons(s(y0), y1)))) SORT(cons(s(y0), cons(s(0), y1))) -> c14(MAX(cons(s(y0), cons(s(0), y1)))) SORT(cons(s(0), cons(s(s(y0)), y1))) -> c14(MAX(cons(s(0), cons(s(s(y0)), y1)))) SORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c14(MAX(cons(s(s(y0)), cons(s(s(y1)), y2)))) SORT(cons(s(y0), cons(s(y1), y2))) -> c14(MAX(cons(s(y0), cons(s(y1), y2)))) ---------------------------------------- (132) 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, 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)) Tuples: GE(s(z0), s(z1)) -> c18(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)) 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(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))) 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(x1)), x2) -> c3(MAX(cons(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(x0)), s(s(x1)), x2) -> c4(MAX(cons(s(s(x1)), x2))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) SORT(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)))) SORT(cons(z0, cons(0, z1))) -> c15(SORT(del(if1(true, z0, 0, z1), cons(z0, cons(0, z1)))), DEL(if1(ge(z0, 0), z0, 0, z1), cons(z0, cons(0, z1))), MAX(cons(z0, cons(0, z1)))) 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(x0)), s(s(x1)), x2) -> c8(DEL(s(s(x0)), x2)) SORT(cons(0, cons(s(z0), z1))) -> c15(SORT(del(if1(false, 0, s(z0), z1), cons(0, cons(s(z0), z1)))), DEL(if1(ge(0, s(z0)), 0, s(z0), z1), cons(0, cons(s(z0), z1))), MAX(cons(0, cons(s(z0), z1)))) SORT(cons(s(z0), cons(s(z1), z2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MAX(cons(s(z0), cons(s(z1), z2)))) SORT(cons(z0, cons(z1, z2))) -> c15(DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) SORT(cons(z0, nil)) -> c15(DEL(z0, cons(z0, nil))) 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 SORT(cons(z0, cons(0, y1))) -> c14(MAX(cons(z0, cons(0, y1)))) SORT(cons(0, cons(s(y0), y1))) -> c14(MAX(cons(0, cons(s(y0), y1)))) SORT(cons(s(y0), cons(s(0), y1))) -> c14(MAX(cons(s(y0), cons(s(0), y1)))) SORT(cons(s(0), cons(s(s(y0)), y1))) -> c14(MAX(cons(s(0), cons(s(s(y0)), y1)))) SORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c14(MAX(cons(s(s(y0)), cons(s(s(y1)), y2)))) SORT(cons(s(y0), cons(s(y1), y2))) -> c14(MAX(cons(s(y0), cons(s(y1), y2)))) S tuples: GE(s(z0), s(z1)) -> c18(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)) 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(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))) 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(x1)), x2) -> c3(MAX(cons(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(x0)), s(s(x1)), x2) -> c4(MAX(cons(s(s(x1)), x2))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) SORT(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)))) SORT(cons(z0, cons(0, z1))) -> c15(SORT(del(if1(true, z0, 0, z1), cons(z0, cons(0, z1)))), DEL(if1(ge(z0, 0), z0, 0, z1), cons(z0, cons(0, z1))), MAX(cons(z0, cons(0, z1)))) 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(x0)), s(s(x1)), x2) -> c8(DEL(s(s(x0)), x2)) SORT(cons(0, cons(s(z0), z1))) -> c15(SORT(del(if1(false, 0, s(z0), z1), cons(0, cons(s(z0), z1)))), DEL(if1(ge(0, s(z0)), 0, s(z0), z1), cons(0, cons(s(z0), z1))), MAX(cons(0, cons(s(z0), z1)))) SORT(cons(s(z0), cons(s(z1), z2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MAX(cons(s(z0), cons(s(z1), z2)))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) 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: SORT(cons(x0, cons(x1, x2))) -> c15(DEL(max(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) SORT(cons(x0, nil)) -> c15(DEL(max(cons(x0, nil)), cons(x0, nil))) SORT(cons(z0, cons(0, y1))) -> c14(MAX(cons(z0, cons(0, y1)))) SORT(cons(0, cons(s(y0), y1))) -> c14(MAX(cons(0, cons(s(y0), y1)))) SORT(cons(s(y0), cons(s(0), y1))) -> c14(MAX(cons(s(y0), cons(s(0), y1)))) SORT(cons(s(0), cons(s(s(y0)), y1))) -> c14(MAX(cons(s(0), cons(s(s(y0)), y1)))) SORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c14(MAX(cons(s(s(y0)), cons(s(s(y1)), y2)))) SORT(cons(s(y0), cons(s(y1), y2))) -> c14(MAX(cons(s(y0), cons(s(y1), y2)))) Defined Rule Symbols: ge_2, eq_2, del_2, max_1, if1_4, if2_4 Defined Pair Symbols: GE_2, MAX_1, DEL_2, IF1_4, SORT_1, IF2_4, EQ_2 Compound Symbols: c18_1, c2_1, c6_1, c2_2, c6_2, c3_1, c4_1, c15_2, c15_3, c8_1, c15_1, c12_1, c6, c14_1 ---------------------------------------- (133) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace GE(s(z0), s(z1)) -> c18(GE(z0, z1)) by GE(s(s(y0)), s(s(y1))) -> c18(GE(s(y0), s(y1))) ---------------------------------------- (134) 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, 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)) Tuples: 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)) 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(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))) 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(x1)), x2) -> c3(MAX(cons(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(x0)), s(s(x1)), x2) -> c4(MAX(cons(s(s(x1)), x2))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) SORT(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)))) SORT(cons(z0, cons(0, z1))) -> c15(SORT(del(if1(true, z0, 0, z1), cons(z0, cons(0, z1)))), DEL(if1(ge(z0, 0), z0, 0, z1), cons(z0, cons(0, z1))), MAX(cons(z0, cons(0, z1)))) 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(x0)), s(s(x1)), x2) -> c8(DEL(s(s(x0)), x2)) SORT(cons(0, cons(s(z0), z1))) -> c15(SORT(del(if1(false, 0, s(z0), z1), cons(0, cons(s(z0), z1)))), DEL(if1(ge(0, s(z0)), 0, s(z0), z1), cons(0, cons(s(z0), z1))), MAX(cons(0, cons(s(z0), z1)))) SORT(cons(s(z0), cons(s(z1), z2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MAX(cons(s(z0), cons(s(z1), z2)))) SORT(cons(z0, cons(z1, z2))) -> c15(DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) SORT(cons(z0, nil)) -> c15(DEL(z0, cons(z0, nil))) 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 SORT(cons(z0, cons(0, y1))) -> c14(MAX(cons(z0, cons(0, y1)))) SORT(cons(0, cons(s(y0), y1))) -> c14(MAX(cons(0, cons(s(y0), y1)))) SORT(cons(s(y0), cons(s(0), y1))) -> c14(MAX(cons(s(y0), cons(s(0), y1)))) SORT(cons(s(0), cons(s(s(y0)), y1))) -> c14(MAX(cons(s(0), cons(s(s(y0)), y1)))) SORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c14(MAX(cons(s(s(y0)), cons(s(s(y1)), y2)))) SORT(cons(s(y0), cons(s(y1), y2))) -> c14(MAX(cons(s(y0), cons(s(y1), y2)))) GE(s(s(y0)), s(s(y1))) -> c18(GE(s(y0), s(y1))) S tuples: 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)) 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(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))) 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(x1)), x2) -> c3(MAX(cons(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(x0)), s(s(x1)), x2) -> c4(MAX(cons(s(s(x1)), x2))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) SORT(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)))) SORT(cons(z0, cons(0, z1))) -> c15(SORT(del(if1(true, z0, 0, z1), cons(z0, cons(0, z1)))), DEL(if1(ge(z0, 0), z0, 0, z1), cons(z0, cons(0, z1))), MAX(cons(z0, cons(0, z1)))) 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(x0)), s(s(x1)), x2) -> c8(DEL(s(s(x0)), x2)) SORT(cons(0, cons(s(z0), z1))) -> c15(SORT(del(if1(false, 0, s(z0), z1), cons(0, cons(s(z0), z1)))), DEL(if1(ge(0, s(z0)), 0, s(z0), z1), cons(0, cons(s(z0), z1))), MAX(cons(0, cons(s(z0), z1)))) SORT(cons(s(z0), cons(s(z1), z2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MAX(cons(s(z0), cons(s(z1), z2)))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) 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 GE(s(s(y0)), s(s(y1))) -> c18(GE(s(y0), s(y1))) K tuples: SORT(cons(x0, cons(x1, x2))) -> c15(DEL(max(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) SORT(cons(x0, nil)) -> c15(DEL(max(cons(x0, nil)), cons(x0, nil))) SORT(cons(z0, cons(0, y1))) -> c14(MAX(cons(z0, cons(0, y1)))) SORT(cons(0, cons(s(y0), y1))) -> c14(MAX(cons(0, cons(s(y0), y1)))) SORT(cons(s(y0), cons(s(0), y1))) -> c14(MAX(cons(s(y0), cons(s(0), y1)))) SORT(cons(s(0), cons(s(s(y0)), y1))) -> c14(MAX(cons(s(0), cons(s(s(y0)), y1)))) SORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c14(MAX(cons(s(s(y0)), cons(s(s(y1)), y2)))) SORT(cons(s(y0), cons(s(y1), y2))) -> c14(MAX(cons(s(y0), cons(s(y1), y2)))) Defined Rule Symbols: ge_2, eq_2, del_2, max_1, if1_4, if2_4 Defined Pair Symbols: MAX_1, DEL_2, IF1_4, SORT_1, IF2_4, EQ_2, GE_2 Compound Symbols: c2_1, c6_1, c2_2, c6_2, c3_1, c4_1, c15_2, c15_3, c8_1, c15_1, c12_1, c6, c14_1, c18_1 ---------------------------------------- (135) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (136) 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, 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)) Tuples: 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)) 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(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))) 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(x1)), x2) -> c3(MAX(cons(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(x0)), s(s(x1)), x2) -> c4(MAX(cons(s(s(x1)), x2))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) SORT(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)))) SORT(cons(z0, cons(0, z1))) -> c15(SORT(del(if1(true, z0, 0, z1), cons(z0, cons(0, z1)))), DEL(if1(ge(z0, 0), z0, 0, z1), cons(z0, cons(0, z1))), MAX(cons(z0, cons(0, z1)))) 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(x0)), s(s(x1)), x2) -> c8(DEL(s(s(x0)), x2)) SORT(cons(0, cons(s(z0), z1))) -> c15(SORT(del(if1(false, 0, s(z0), z1), cons(0, cons(s(z0), z1)))), DEL(if1(ge(0, s(z0)), 0, s(z0), z1), cons(0, cons(s(z0), z1))), MAX(cons(0, cons(s(z0), z1)))) SORT(cons(s(z0), cons(s(z1), z2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MAX(cons(s(z0), cons(s(z1), z2)))) SORT(cons(z0, cons(z1, z2))) -> c15(DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) SORT(cons(z0, nil)) -> c15(DEL(z0, cons(z0, nil))) 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 SORT(cons(z0, cons(0, y1))) -> c14(MAX(cons(z0, cons(0, y1)))) SORT(cons(0, cons(s(y0), y1))) -> c14(MAX(cons(0, cons(s(y0), y1)))) SORT(cons(s(y0), cons(s(0), y1))) -> c14(MAX(cons(s(y0), cons(s(0), y1)))) SORT(cons(s(0), cons(s(s(y0)), y1))) -> c14(MAX(cons(s(0), cons(s(s(y0)), y1)))) SORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c14(MAX(cons(s(s(y0)), cons(s(s(y1)), y2)))) SORT(cons(s(y0), cons(s(y1), y2))) -> c14(MAX(cons(s(y0), cons(s(y1), y2)))) GE(s(s(y0)), s(s(y1))) -> c18(GE(s(y0), s(y1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2)) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2)) S tuples: 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)) 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(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))) 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(x1)), x2) -> c3(MAX(cons(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(x0)), s(s(x1)), x2) -> c4(MAX(cons(s(s(x1)), x2))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) SORT(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)))) SORT(cons(z0, cons(0, z1))) -> c15(SORT(del(if1(true, z0, 0, z1), cons(z0, cons(0, z1)))), DEL(if1(ge(z0, 0), z0, 0, z1), cons(z0, cons(0, z1))), MAX(cons(z0, cons(0, z1)))) 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(x0)), s(s(x1)), x2) -> c8(DEL(s(s(x0)), x2)) SORT(cons(0, cons(s(z0), z1))) -> c15(SORT(del(if1(false, 0, s(z0), z1), cons(0, cons(s(z0), z1)))), DEL(if1(ge(0, s(z0)), 0, s(z0), z1), cons(0, cons(s(z0), z1))), MAX(cons(0, cons(s(z0), z1)))) SORT(cons(s(z0), cons(s(z1), z2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MAX(cons(s(z0), cons(s(z1), z2)))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) 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 GE(s(s(y0)), s(s(y1))) -> c18(GE(s(y0), s(y1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2)) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2)) K tuples: SORT(cons(x0, cons(x1, x2))) -> c15(DEL(max(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) SORT(cons(x0, nil)) -> c15(DEL(max(cons(x0, nil)), cons(x0, nil))) SORT(cons(z0, cons(0, y1))) -> c14(MAX(cons(z0, cons(0, y1)))) SORT(cons(0, cons(s(y0), y1))) -> c14(MAX(cons(0, cons(s(y0), y1)))) SORT(cons(s(y0), cons(s(0), y1))) -> c14(MAX(cons(s(y0), cons(s(0), y1)))) SORT(cons(s(0), cons(s(s(y0)), y1))) -> c14(MAX(cons(s(0), cons(s(s(y0)), y1)))) SORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c14(MAX(cons(s(s(y0)), cons(s(s(y1)), y2)))) SORT(cons(s(y0), cons(s(y1), y2))) -> c14(MAX(cons(s(y0), cons(s(y1), y2)))) Defined Rule Symbols: ge_2, eq_2, del_2, max_1, if1_4, if2_4 Defined Pair Symbols: MAX_1, DEL_2, IF1_4, SORT_1, IF2_4, EQ_2, GE_2 Compound Symbols: c2_1, c6_1, c2_2, c6_2, c3_1, c4_1, c15_2, c15_3, c8_1, c15_1, c12_1, c6, c14_1, c18_1 ---------------------------------------- (137) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), max(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) by SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) ---------------------------------------- (138) 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, 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)) Tuples: 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)) 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(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))) 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(x1)), x2) -> c3(MAX(cons(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(x0)), s(s(x1)), x2) -> c4(MAX(cons(s(s(x1)), x2))) SORT(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)))) SORT(cons(z0, cons(0, z1))) -> c15(SORT(del(if1(true, z0, 0, z1), cons(z0, cons(0, z1)))), DEL(if1(ge(z0, 0), z0, 0, z1), cons(z0, cons(0, z1))), MAX(cons(z0, cons(0, z1)))) 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(x0)), s(s(x1)), x2) -> c8(DEL(s(s(x0)), x2)) SORT(cons(0, cons(s(z0), z1))) -> c15(SORT(del(if1(false, 0, s(z0), z1), cons(0, cons(s(z0), z1)))), DEL(if1(ge(0, s(z0)), 0, s(z0), z1), cons(0, cons(s(z0), z1))), MAX(cons(0, cons(s(z0), z1)))) SORT(cons(s(z0), cons(s(z1), z2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MAX(cons(s(z0), cons(s(z1), z2)))) SORT(cons(z0, cons(z1, z2))) -> c15(DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) SORT(cons(z0, nil)) -> c15(DEL(z0, cons(z0, nil))) 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 SORT(cons(z0, cons(0, y1))) -> c14(MAX(cons(z0, cons(0, y1)))) SORT(cons(0, cons(s(y0), y1))) -> c14(MAX(cons(0, cons(s(y0), y1)))) SORT(cons(s(y0), cons(s(0), y1))) -> c14(MAX(cons(s(y0), cons(s(0), y1)))) SORT(cons(s(0), cons(s(s(y0)), y1))) -> c14(MAX(cons(s(0), cons(s(s(y0)), y1)))) SORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c14(MAX(cons(s(s(y0)), cons(s(s(y1)), y2)))) SORT(cons(s(y0), cons(s(y1), y2))) -> c14(MAX(cons(s(y0), cons(s(y1), y2)))) GE(s(s(y0)), s(s(y1))) -> c18(GE(s(y0), s(y1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2)) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2)) S tuples: 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)) 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(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))) 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(x1)), x2) -> c3(MAX(cons(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(x0)), s(s(x1)), x2) -> c4(MAX(cons(s(s(x1)), x2))) SORT(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)))) SORT(cons(z0, cons(0, z1))) -> c15(SORT(del(if1(true, z0, 0, z1), cons(z0, cons(0, z1)))), DEL(if1(ge(z0, 0), z0, 0, z1), cons(z0, cons(0, z1))), MAX(cons(z0, cons(0, z1)))) 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(x0)), s(s(x1)), x2) -> c8(DEL(s(s(x0)), x2)) SORT(cons(0, cons(s(z0), z1))) -> c15(SORT(del(if1(false, 0, s(z0), z1), cons(0, cons(s(z0), z1)))), DEL(if1(ge(0, s(z0)), 0, s(z0), z1), cons(0, cons(s(z0), z1))), MAX(cons(0, cons(s(z0), z1)))) SORT(cons(s(z0), cons(s(z1), z2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MAX(cons(s(z0), cons(s(z1), z2)))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) 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 GE(s(s(y0)), s(s(y1))) -> c18(GE(s(y0), s(y1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2)) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2)) K tuples: SORT(cons(x0, cons(x1, x2))) -> c15(DEL(max(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) SORT(cons(x0, nil)) -> c15(DEL(max(cons(x0, nil)), cons(x0, nil))) SORT(cons(z0, cons(0, y1))) -> c14(MAX(cons(z0, cons(0, y1)))) SORT(cons(0, cons(s(y0), y1))) -> c14(MAX(cons(0, cons(s(y0), y1)))) SORT(cons(s(y0), cons(s(0), y1))) -> c14(MAX(cons(s(y0), cons(s(0), y1)))) SORT(cons(s(0), cons(s(s(y0)), y1))) -> c14(MAX(cons(s(0), cons(s(s(y0)), y1)))) SORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c14(MAX(cons(s(s(y0)), cons(s(s(y1)), y2)))) SORT(cons(s(y0), cons(s(y1), y2))) -> c14(MAX(cons(s(y0), cons(s(y1), y2)))) Defined Rule Symbols: ge_2, eq_2, del_2, max_1, if1_4, if2_4 Defined Pair Symbols: MAX_1, DEL_2, IF1_4, SORT_1, IF2_4, EQ_2, GE_2 Compound Symbols: c2_1, c6_1, c2_2, c6_2, c3_1, c4_1, c15_3, c8_1, c15_1, c15_2, c12_1, c6, c14_1, c18_1 ---------------------------------------- (139) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace MAX(cons(s(x0), cons(s(x1), x2))) -> c2(GE(s(x0), s(x1))) by MAX(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c2(GE(s(s(y0)), s(s(y1)))) ---------------------------------------- (140) 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, 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)) Tuples: 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)) 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)))) 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))) 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(x1)), x2) -> c3(MAX(cons(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(x0)), s(s(x1)), x2) -> c4(MAX(cons(s(s(x1)), x2))) SORT(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)))) SORT(cons(z0, cons(0, z1))) -> c15(SORT(del(if1(true, z0, 0, z1), cons(z0, cons(0, z1)))), DEL(if1(ge(z0, 0), z0, 0, z1), cons(z0, cons(0, z1))), MAX(cons(z0, cons(0, z1)))) 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(x0)), s(s(x1)), x2) -> c8(DEL(s(s(x0)), x2)) SORT(cons(0, cons(s(z0), z1))) -> c15(SORT(del(if1(false, 0, s(z0), z1), cons(0, cons(s(z0), z1)))), DEL(if1(ge(0, s(z0)), 0, s(z0), z1), cons(0, cons(s(z0), z1))), MAX(cons(0, cons(s(z0), z1)))) SORT(cons(s(z0), cons(s(z1), z2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MAX(cons(s(z0), cons(s(z1), z2)))) SORT(cons(z0, cons(z1, z2))) -> c15(DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) SORT(cons(z0, nil)) -> c15(DEL(z0, cons(z0, nil))) 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 SORT(cons(z0, cons(0, y1))) -> c14(MAX(cons(z0, cons(0, y1)))) SORT(cons(0, cons(s(y0), y1))) -> c14(MAX(cons(0, cons(s(y0), y1)))) SORT(cons(s(y0), cons(s(0), y1))) -> c14(MAX(cons(s(y0), cons(s(0), y1)))) SORT(cons(s(0), cons(s(s(y0)), y1))) -> c14(MAX(cons(s(0), cons(s(s(y0)), y1)))) SORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c14(MAX(cons(s(s(y0)), cons(s(s(y1)), y2)))) SORT(cons(s(y0), cons(s(y1), y2))) -> c14(MAX(cons(s(y0), cons(s(y1), y2)))) GE(s(s(y0)), s(s(y1))) -> c18(GE(s(y0), s(y1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2)) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2)) MAX(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c2(GE(s(s(y0)), s(s(y1)))) S tuples: 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)) 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)))) 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))) 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(x1)), x2) -> c3(MAX(cons(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(x0)), s(s(x1)), x2) -> c4(MAX(cons(s(s(x1)), x2))) SORT(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)))) SORT(cons(z0, cons(0, z1))) -> c15(SORT(del(if1(true, z0, 0, z1), cons(z0, cons(0, z1)))), DEL(if1(ge(z0, 0), z0, 0, z1), cons(z0, cons(0, z1))), MAX(cons(z0, cons(0, z1)))) 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(x0)), s(s(x1)), x2) -> c8(DEL(s(s(x0)), x2)) SORT(cons(0, cons(s(z0), z1))) -> c15(SORT(del(if1(false, 0, s(z0), z1), cons(0, cons(s(z0), z1)))), DEL(if1(ge(0, s(z0)), 0, s(z0), z1), cons(0, cons(s(z0), z1))), MAX(cons(0, cons(s(z0), z1)))) SORT(cons(s(z0), cons(s(z1), z2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MAX(cons(s(z0), cons(s(z1), z2)))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) 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 GE(s(s(y0)), s(s(y1))) -> c18(GE(s(y0), s(y1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2)) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2)) MAX(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c2(GE(s(s(y0)), s(s(y1)))) K tuples: SORT(cons(x0, cons(x1, x2))) -> c15(DEL(max(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) SORT(cons(x0, nil)) -> c15(DEL(max(cons(x0, nil)), cons(x0, nil))) SORT(cons(z0, cons(0, y1))) -> c14(MAX(cons(z0, cons(0, y1)))) SORT(cons(0, cons(s(y0), y1))) -> c14(MAX(cons(0, cons(s(y0), y1)))) SORT(cons(s(y0), cons(s(0), y1))) -> c14(MAX(cons(s(y0), cons(s(0), y1)))) SORT(cons(s(0), cons(s(s(y0)), y1))) -> c14(MAX(cons(s(0), cons(s(s(y0)), y1)))) SORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c14(MAX(cons(s(s(y0)), cons(s(s(y1)), y2)))) SORT(cons(s(y0), cons(s(y1), y2))) -> c14(MAX(cons(s(y0), cons(s(y1), y2)))) Defined Rule Symbols: ge_2, eq_2, del_2, max_1, if1_4, if2_4 Defined Pair Symbols: MAX_1, DEL_2, IF1_4, SORT_1, IF2_4, EQ_2, GE_2 Compound Symbols: c2_1, c6_1, c2_2, c6_2, c3_1, c4_1, c15_3, c8_1, c15_1, c15_2, c12_1, c6, c14_1, c18_1 ---------------------------------------- (141) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace SORT(cons(z0, cons(0, z1))) -> c15(SORT(del(if1(true, z0, 0, z1), cons(z0, cons(0, z1)))), DEL(if1(ge(z0, 0), z0, 0, z1), cons(z0, cons(0, z1))), MAX(cons(z0, cons(0, z1)))) by SORT(cons(z0, cons(0, z1))) -> c15(SORT(del(if1(true, z0, 0, z1), cons(z0, cons(0, z1)))), DEL(if1(true, z0, 0, z1), cons(z0, cons(0, z1))), MAX(cons(z0, cons(0, z1)))) ---------------------------------------- (142) 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, 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)) Tuples: 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)) 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)))) 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))) 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(x1)), x2) -> c3(MAX(cons(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(x0)), s(s(x1)), x2) -> c4(MAX(cons(s(s(x1)), x2))) SORT(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)))) 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(x0)), s(s(x1)), x2) -> c8(DEL(s(s(x0)), x2)) SORT(cons(0, cons(s(z0), z1))) -> c15(SORT(del(if1(false, 0, s(z0), z1), cons(0, cons(s(z0), z1)))), DEL(if1(ge(0, s(z0)), 0, s(z0), z1), cons(0, cons(s(z0), z1))), MAX(cons(0, cons(s(z0), z1)))) SORT(cons(s(z0), cons(s(z1), z2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MAX(cons(s(z0), cons(s(z1), z2)))) SORT(cons(z0, cons(z1, z2))) -> c15(DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) SORT(cons(z0, nil)) -> c15(DEL(z0, cons(z0, nil))) 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 SORT(cons(z0, cons(0, y1))) -> c14(MAX(cons(z0, cons(0, y1)))) SORT(cons(0, cons(s(y0), y1))) -> c14(MAX(cons(0, cons(s(y0), y1)))) SORT(cons(s(y0), cons(s(0), y1))) -> c14(MAX(cons(s(y0), cons(s(0), y1)))) SORT(cons(s(0), cons(s(s(y0)), y1))) -> c14(MAX(cons(s(0), cons(s(s(y0)), y1)))) SORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c14(MAX(cons(s(s(y0)), cons(s(s(y1)), y2)))) SORT(cons(s(y0), cons(s(y1), y2))) -> c14(MAX(cons(s(y0), cons(s(y1), y2)))) GE(s(s(y0)), s(s(y1))) -> c18(GE(s(y0), s(y1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2)) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2)) MAX(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c2(GE(s(s(y0)), s(s(y1)))) SORT(cons(z0, cons(0, z1))) -> c15(SORT(del(if1(true, z0, 0, z1), cons(z0, cons(0, z1)))), DEL(if1(true, z0, 0, z1), cons(z0, cons(0, z1))), MAX(cons(z0, cons(0, z1)))) S tuples: 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)) 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)))) 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))) 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(x1)), x2) -> c3(MAX(cons(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(x0)), s(s(x1)), x2) -> c4(MAX(cons(s(s(x1)), x2))) SORT(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)))) 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(x0)), s(s(x1)), x2) -> c8(DEL(s(s(x0)), x2)) SORT(cons(0, cons(s(z0), z1))) -> c15(SORT(del(if1(false, 0, s(z0), z1), cons(0, cons(s(z0), z1)))), DEL(if1(ge(0, s(z0)), 0, s(z0), z1), cons(0, cons(s(z0), z1))), MAX(cons(0, cons(s(z0), z1)))) SORT(cons(s(z0), cons(s(z1), z2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MAX(cons(s(z0), cons(s(z1), z2)))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) 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 GE(s(s(y0)), s(s(y1))) -> c18(GE(s(y0), s(y1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2)) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2)) MAX(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c2(GE(s(s(y0)), s(s(y1)))) SORT(cons(z0, cons(0, z1))) -> c15(SORT(del(if1(true, z0, 0, z1), cons(z0, cons(0, z1)))), DEL(if1(true, z0, 0, z1), cons(z0, cons(0, z1))), MAX(cons(z0, cons(0, z1)))) K tuples: SORT(cons(x0, cons(x1, x2))) -> c15(DEL(max(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) SORT(cons(x0, nil)) -> c15(DEL(max(cons(x0, nil)), cons(x0, nil))) SORT(cons(z0, cons(0, y1))) -> c14(MAX(cons(z0, cons(0, y1)))) SORT(cons(0, cons(s(y0), y1))) -> c14(MAX(cons(0, cons(s(y0), y1)))) SORT(cons(s(y0), cons(s(0), y1))) -> c14(MAX(cons(s(y0), cons(s(0), y1)))) SORT(cons(s(0), cons(s(s(y0)), y1))) -> c14(MAX(cons(s(0), cons(s(s(y0)), y1)))) SORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c14(MAX(cons(s(s(y0)), cons(s(s(y1)), y2)))) SORT(cons(s(y0), cons(s(y1), y2))) -> c14(MAX(cons(s(y0), cons(s(y1), y2)))) Defined Rule Symbols: ge_2, eq_2, del_2, max_1, if1_4, if2_4 Defined Pair Symbols: MAX_1, DEL_2, IF1_4, SORT_1, IF2_4, EQ_2, GE_2 Compound Symbols: c2_1, c6_1, c2_2, c6_2, c3_1, c4_1, c15_3, c8_1, c15_1, c15_2, c12_1, c6, c14_1, c18_1 ---------------------------------------- (143) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace SORT(cons(0, cons(s(z0), z1))) -> c15(SORT(del(if1(false, 0, s(z0), z1), cons(0, cons(s(z0), z1)))), DEL(if1(ge(0, s(z0)), 0, s(z0), z1), cons(0, cons(s(z0), z1))), MAX(cons(0, cons(s(z0), z1)))) by SORT(cons(0, cons(s(z0), z1))) -> c15(SORT(del(if1(false, 0, s(z0), z1), cons(0, cons(s(z0), z1)))), DEL(if1(false, 0, s(z0), z1), cons(0, cons(s(z0), z1))), MAX(cons(0, cons(s(z0), z1)))) ---------------------------------------- (144) 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, 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)) Tuples: 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)) 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)))) 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))) 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(x1)), x2) -> c3(MAX(cons(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(x0)), s(s(x1)), x2) -> c4(MAX(cons(s(s(x1)), x2))) SORT(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)))) 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(x0)), s(s(x1)), x2) -> c8(DEL(s(s(x0)), x2)) SORT(cons(s(z0), cons(s(z1), z2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MAX(cons(s(z0), cons(s(z1), z2)))) SORT(cons(z0, cons(z1, z2))) -> c15(DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) SORT(cons(z0, nil)) -> c15(DEL(z0, cons(z0, nil))) 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 SORT(cons(z0, cons(0, y1))) -> c14(MAX(cons(z0, cons(0, y1)))) SORT(cons(0, cons(s(y0), y1))) -> c14(MAX(cons(0, cons(s(y0), y1)))) SORT(cons(s(y0), cons(s(0), y1))) -> c14(MAX(cons(s(y0), cons(s(0), y1)))) SORT(cons(s(0), cons(s(s(y0)), y1))) -> c14(MAX(cons(s(0), cons(s(s(y0)), y1)))) SORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c14(MAX(cons(s(s(y0)), cons(s(s(y1)), y2)))) SORT(cons(s(y0), cons(s(y1), y2))) -> c14(MAX(cons(s(y0), cons(s(y1), y2)))) GE(s(s(y0)), s(s(y1))) -> c18(GE(s(y0), s(y1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2)) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2)) MAX(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c2(GE(s(s(y0)), s(s(y1)))) SORT(cons(z0, cons(0, z1))) -> c15(SORT(del(if1(true, z0, 0, z1), cons(z0, cons(0, z1)))), DEL(if1(true, z0, 0, z1), cons(z0, cons(0, z1))), MAX(cons(z0, cons(0, z1)))) SORT(cons(0, cons(s(z0), z1))) -> c15(SORT(del(if1(false, 0, s(z0), z1), cons(0, cons(s(z0), z1)))), DEL(if1(false, 0, s(z0), z1), cons(0, cons(s(z0), z1))), MAX(cons(0, cons(s(z0), z1)))) S tuples: 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)) 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)))) 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))) 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(x1)), x2) -> c3(MAX(cons(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(x0)), s(s(x1)), x2) -> c4(MAX(cons(s(s(x1)), x2))) SORT(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)))) 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(x0)), s(s(x1)), x2) -> c8(DEL(s(s(x0)), x2)) SORT(cons(s(z0), cons(s(z1), z2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MAX(cons(s(z0), cons(s(z1), z2)))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) 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 GE(s(s(y0)), s(s(y1))) -> c18(GE(s(y0), s(y1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2)) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2)) MAX(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c2(GE(s(s(y0)), s(s(y1)))) SORT(cons(z0, cons(0, z1))) -> c15(SORT(del(if1(true, z0, 0, z1), cons(z0, cons(0, z1)))), DEL(if1(true, z0, 0, z1), cons(z0, cons(0, z1))), MAX(cons(z0, cons(0, z1)))) SORT(cons(0, cons(s(z0), z1))) -> c15(SORT(del(if1(false, 0, s(z0), z1), cons(0, cons(s(z0), z1)))), DEL(if1(false, 0, s(z0), z1), cons(0, cons(s(z0), z1))), MAX(cons(0, cons(s(z0), z1)))) K tuples: SORT(cons(x0, cons(x1, x2))) -> c15(DEL(max(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) SORT(cons(x0, nil)) -> c15(DEL(max(cons(x0, nil)), cons(x0, nil))) SORT(cons(z0, cons(0, y1))) -> c14(MAX(cons(z0, cons(0, y1)))) SORT(cons(0, cons(s(y0), y1))) -> c14(MAX(cons(0, cons(s(y0), y1)))) SORT(cons(s(y0), cons(s(0), y1))) -> c14(MAX(cons(s(y0), cons(s(0), y1)))) SORT(cons(s(0), cons(s(s(y0)), y1))) -> c14(MAX(cons(s(0), cons(s(s(y0)), y1)))) SORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c14(MAX(cons(s(s(y0)), cons(s(s(y1)), y2)))) SORT(cons(s(y0), cons(s(y1), y2))) -> c14(MAX(cons(s(y0), cons(s(y1), y2)))) Defined Rule Symbols: ge_2, eq_2, del_2, max_1, if1_4, if2_4 Defined Pair Symbols: MAX_1, DEL_2, IF1_4, SORT_1, IF2_4, EQ_2, GE_2 Compound Symbols: c2_1, c6_1, c2_2, c6_2, c3_1, c4_1, c15_3, c8_1, c15_1, c15_2, c12_1, c6, c14_1, c18_1 ---------------------------------------- (145) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace SORT(cons(s(z0), cons(s(z1), z2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(ge(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MAX(cons(s(z0), cons(s(z1), z2)))) by SORT(cons(s(z0), cons(s(z1), z2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(ge(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MAX(cons(s(z0), cons(s(z1), z2)))) ---------------------------------------- (146) 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, 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)) Tuples: 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)) 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)))) 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))) 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(x1)), x2) -> c3(MAX(cons(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(x0)), s(s(x1)), x2) -> c4(MAX(cons(s(s(x1)), x2))) SORT(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)))) 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(x0)), s(s(x1)), x2) -> c8(DEL(s(s(x0)), x2)) SORT(cons(z0, cons(z1, z2))) -> c15(DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) SORT(cons(z0, nil)) -> c15(DEL(z0, cons(z0, nil))) 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 SORT(cons(z0, cons(0, y1))) -> c14(MAX(cons(z0, cons(0, y1)))) SORT(cons(0, cons(s(y0), y1))) -> c14(MAX(cons(0, cons(s(y0), y1)))) SORT(cons(s(y0), cons(s(0), y1))) -> c14(MAX(cons(s(y0), cons(s(0), y1)))) SORT(cons(s(0), cons(s(s(y0)), y1))) -> c14(MAX(cons(s(0), cons(s(s(y0)), y1)))) SORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c14(MAX(cons(s(s(y0)), cons(s(s(y1)), y2)))) SORT(cons(s(y0), cons(s(y1), y2))) -> c14(MAX(cons(s(y0), cons(s(y1), y2)))) GE(s(s(y0)), s(s(y1))) -> c18(GE(s(y0), s(y1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2)) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2)) MAX(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c2(GE(s(s(y0)), s(s(y1)))) SORT(cons(z0, cons(0, z1))) -> c15(SORT(del(if1(true, z0, 0, z1), cons(z0, cons(0, z1)))), DEL(if1(true, z0, 0, z1), cons(z0, cons(0, z1))), MAX(cons(z0, cons(0, z1)))) SORT(cons(0, cons(s(z0), z1))) -> c15(SORT(del(if1(false, 0, s(z0), z1), cons(0, cons(s(z0), z1)))), DEL(if1(false, 0, s(z0), z1), cons(0, cons(s(z0), z1))), MAX(cons(0, cons(s(z0), z1)))) SORT(cons(s(z0), cons(s(z1), z2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(ge(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MAX(cons(s(z0), cons(s(z1), z2)))) S tuples: 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)) 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)))) 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))) 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(x1)), x2) -> c3(MAX(cons(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(x0)), s(s(x1)), x2) -> c4(MAX(cons(s(s(x1)), x2))) SORT(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)))) 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(x0)), s(s(x1)), x2) -> c8(DEL(s(s(x0)), x2)) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) 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 GE(s(s(y0)), s(s(y1))) -> c18(GE(s(y0), s(y1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2)) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2)) MAX(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c2(GE(s(s(y0)), s(s(y1)))) SORT(cons(z0, cons(0, z1))) -> c15(SORT(del(if1(true, z0, 0, z1), cons(z0, cons(0, z1)))), DEL(if1(true, z0, 0, z1), cons(z0, cons(0, z1))), MAX(cons(z0, cons(0, z1)))) SORT(cons(0, cons(s(z0), z1))) -> c15(SORT(del(if1(false, 0, s(z0), z1), cons(0, cons(s(z0), z1)))), DEL(if1(false, 0, s(z0), z1), cons(0, cons(s(z0), z1))), MAX(cons(0, cons(s(z0), z1)))) SORT(cons(s(z0), cons(s(z1), z2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(ge(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MAX(cons(s(z0), cons(s(z1), z2)))) K tuples: SORT(cons(x0, cons(x1, x2))) -> c15(DEL(max(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) SORT(cons(x0, nil)) -> c15(DEL(max(cons(x0, nil)), cons(x0, nil))) SORT(cons(z0, cons(0, y1))) -> c14(MAX(cons(z0, cons(0, y1)))) SORT(cons(0, cons(s(y0), y1))) -> c14(MAX(cons(0, cons(s(y0), y1)))) SORT(cons(s(y0), cons(s(0), y1))) -> c14(MAX(cons(s(y0), cons(s(0), y1)))) SORT(cons(s(0), cons(s(s(y0)), y1))) -> c14(MAX(cons(s(0), cons(s(s(y0)), y1)))) SORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c14(MAX(cons(s(s(y0)), cons(s(s(y1)), y2)))) SORT(cons(s(y0), cons(s(y1), y2))) -> c14(MAX(cons(s(y0), cons(s(y1), y2)))) Defined Rule Symbols: ge_2, eq_2, del_2, max_1, if1_4, if2_4 Defined Pair Symbols: MAX_1, DEL_2, IF1_4, SORT_1, IF2_4, EQ_2, GE_2 Compound Symbols: c2_1, c6_1, c2_2, c6_2, c3_1, c4_1, c15_3, c8_1, c15_1, c15_2, c12_1, c6, c14_1, c18_1 ---------------------------------------- (147) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace DEL(s(x0), cons(s(x1), x2)) -> c6(EQ(s(x0), s(x1))) by DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c6(EQ(s(s(y0)), s(s(y1)))) ---------------------------------------- (148) 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, 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)) Tuples: 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)) 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)))) 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)))) 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(x1)), x2) -> c3(MAX(cons(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(x0)), s(s(x1)), x2) -> c4(MAX(cons(s(s(x1)), x2))) SORT(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)))) 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(x0)), s(s(x1)), x2) -> c8(DEL(s(s(x0)), x2)) SORT(cons(z0, cons(z1, z2))) -> c15(DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) SORT(cons(z0, nil)) -> c15(DEL(z0, cons(z0, nil))) 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 SORT(cons(z0, cons(0, y1))) -> c14(MAX(cons(z0, cons(0, y1)))) SORT(cons(0, cons(s(y0), y1))) -> c14(MAX(cons(0, cons(s(y0), y1)))) SORT(cons(s(y0), cons(s(0), y1))) -> c14(MAX(cons(s(y0), cons(s(0), y1)))) SORT(cons(s(0), cons(s(s(y0)), y1))) -> c14(MAX(cons(s(0), cons(s(s(y0)), y1)))) SORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c14(MAX(cons(s(s(y0)), cons(s(s(y1)), y2)))) SORT(cons(s(y0), cons(s(y1), y2))) -> c14(MAX(cons(s(y0), cons(s(y1), y2)))) GE(s(s(y0)), s(s(y1))) -> c18(GE(s(y0), s(y1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2)) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2)) MAX(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c2(GE(s(s(y0)), s(s(y1)))) SORT(cons(z0, cons(0, z1))) -> c15(SORT(del(if1(true, z0, 0, z1), cons(z0, cons(0, z1)))), DEL(if1(true, z0, 0, z1), cons(z0, cons(0, z1))), MAX(cons(z0, cons(0, z1)))) SORT(cons(0, cons(s(z0), z1))) -> c15(SORT(del(if1(false, 0, s(z0), z1), cons(0, cons(s(z0), z1)))), DEL(if1(false, 0, s(z0), z1), cons(0, cons(s(z0), z1))), MAX(cons(0, cons(s(z0), z1)))) SORT(cons(s(z0), cons(s(z1), z2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(ge(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MAX(cons(s(z0), cons(s(z1), z2)))) DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c6(EQ(s(s(y0)), s(s(y1)))) S tuples: 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)) 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)))) 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)))) 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(x1)), x2) -> c3(MAX(cons(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(x0)), s(s(x1)), x2) -> c4(MAX(cons(s(s(x1)), x2))) SORT(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)))) 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(x0)), s(s(x1)), x2) -> c8(DEL(s(s(x0)), x2)) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) 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 GE(s(s(y0)), s(s(y1))) -> c18(GE(s(y0), s(y1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2)) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2)) MAX(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c2(GE(s(s(y0)), s(s(y1)))) SORT(cons(z0, cons(0, z1))) -> c15(SORT(del(if1(true, z0, 0, z1), cons(z0, cons(0, z1)))), DEL(if1(true, z0, 0, z1), cons(z0, cons(0, z1))), MAX(cons(z0, cons(0, z1)))) SORT(cons(0, cons(s(z0), z1))) -> c15(SORT(del(if1(false, 0, s(z0), z1), cons(0, cons(s(z0), z1)))), DEL(if1(false, 0, s(z0), z1), cons(0, cons(s(z0), z1))), MAX(cons(0, cons(s(z0), z1)))) SORT(cons(s(z0), cons(s(z1), z2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(ge(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MAX(cons(s(z0), cons(s(z1), z2)))) DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c6(EQ(s(s(y0)), s(s(y1)))) K tuples: SORT(cons(x0, cons(x1, x2))) -> c15(DEL(max(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) SORT(cons(x0, nil)) -> c15(DEL(max(cons(x0, nil)), cons(x0, nil))) SORT(cons(z0, cons(0, y1))) -> c14(MAX(cons(z0, cons(0, y1)))) SORT(cons(0, cons(s(y0), y1))) -> c14(MAX(cons(0, cons(s(y0), y1)))) SORT(cons(s(y0), cons(s(0), y1))) -> c14(MAX(cons(s(y0), cons(s(0), y1)))) SORT(cons(s(0), cons(s(s(y0)), y1))) -> c14(MAX(cons(s(0), cons(s(s(y0)), y1)))) SORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c14(MAX(cons(s(s(y0)), cons(s(s(y1)), y2)))) SORT(cons(s(y0), cons(s(y1), y2))) -> c14(MAX(cons(s(y0), cons(s(y1), y2)))) Defined Rule Symbols: ge_2, eq_2, del_2, max_1, if1_4, if2_4 Defined Pair Symbols: MAX_1, DEL_2, IF1_4, SORT_1, IF2_4, EQ_2, GE_2 Compound Symbols: c2_1, c6_1, c2_2, c6_2, c3_1, c4_1, c15_3, c8_1, c15_1, c15_2, c12_1, c6, c14_1, c18_1 ---------------------------------------- (149) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace IF1(true, x0, 0, x1) -> c3(MAX(cons(x0, x1))) by IF1(true, z0, 0, cons(0, y1)) -> c3(MAX(cons(z0, cons(0, y1)))) IF1(true, 0, 0, cons(s(y0), y1)) -> c3(MAX(cons(0, cons(s(y0), y1)))) IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c3(MAX(cons(s(s(y0)), cons(s(s(y1)), y2)))) IF1(true, s(y0), 0, cons(s(0), y1)) -> c3(MAX(cons(s(y0), cons(s(0), y1)))) IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c3(MAX(cons(s(0), cons(s(s(y0)), y1)))) ---------------------------------------- (150) 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, 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)) Tuples: 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)) 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)))) 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)))) IF1(true, s(x0), s(0), x1) -> c3(MAX(cons(s(x0), x1))) IF1(true, s(s(x0)), s(s(x1)), x2) -> c3(MAX(cons(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(x0)), s(s(x1)), x2) -> c4(MAX(cons(s(s(x1)), x2))) SORT(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)))) 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(x0)), s(s(x1)), x2) -> c8(DEL(s(s(x0)), x2)) SORT(cons(z0, cons(z1, z2))) -> c15(DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) SORT(cons(z0, nil)) -> c15(DEL(z0, cons(z0, nil))) 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 SORT(cons(z0, cons(0, y1))) -> c14(MAX(cons(z0, cons(0, y1)))) SORT(cons(0, cons(s(y0), y1))) -> c14(MAX(cons(0, cons(s(y0), y1)))) SORT(cons(s(y0), cons(s(0), y1))) -> c14(MAX(cons(s(y0), cons(s(0), y1)))) SORT(cons(s(0), cons(s(s(y0)), y1))) -> c14(MAX(cons(s(0), cons(s(s(y0)), y1)))) SORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c14(MAX(cons(s(s(y0)), cons(s(s(y1)), y2)))) SORT(cons(s(y0), cons(s(y1), y2))) -> c14(MAX(cons(s(y0), cons(s(y1), y2)))) GE(s(s(y0)), s(s(y1))) -> c18(GE(s(y0), s(y1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2)) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2)) MAX(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c2(GE(s(s(y0)), s(s(y1)))) SORT(cons(z0, cons(0, z1))) -> c15(SORT(del(if1(true, z0, 0, z1), cons(z0, cons(0, z1)))), DEL(if1(true, z0, 0, z1), cons(z0, cons(0, z1))), MAX(cons(z0, cons(0, z1)))) SORT(cons(0, cons(s(z0), z1))) -> c15(SORT(del(if1(false, 0, s(z0), z1), cons(0, cons(s(z0), z1)))), DEL(if1(false, 0, s(z0), z1), cons(0, cons(s(z0), z1))), MAX(cons(0, cons(s(z0), z1)))) SORT(cons(s(z0), cons(s(z1), z2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(ge(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MAX(cons(s(z0), cons(s(z1), z2)))) DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c6(EQ(s(s(y0)), s(s(y1)))) IF1(true, z0, 0, cons(0, y1)) -> c3(MAX(cons(z0, cons(0, y1)))) IF1(true, 0, 0, cons(s(y0), y1)) -> c3(MAX(cons(0, cons(s(y0), y1)))) IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c3(MAX(cons(s(s(y0)), cons(s(s(y1)), y2)))) IF1(true, s(y0), 0, cons(s(0), y1)) -> c3(MAX(cons(s(y0), cons(s(0), y1)))) IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c3(MAX(cons(s(0), cons(s(s(y0)), y1)))) S tuples: 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)) 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)))) 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)))) IF1(true, s(x0), s(0), x1) -> c3(MAX(cons(s(x0), x1))) IF1(true, s(s(x0)), s(s(x1)), x2) -> c3(MAX(cons(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(x0)), s(s(x1)), x2) -> c4(MAX(cons(s(s(x1)), x2))) SORT(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)))) 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(x0)), s(s(x1)), x2) -> c8(DEL(s(s(x0)), x2)) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) 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 GE(s(s(y0)), s(s(y1))) -> c18(GE(s(y0), s(y1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2)) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2)) MAX(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c2(GE(s(s(y0)), s(s(y1)))) SORT(cons(z0, cons(0, z1))) -> c15(SORT(del(if1(true, z0, 0, z1), cons(z0, cons(0, z1)))), DEL(if1(true, z0, 0, z1), cons(z0, cons(0, z1))), MAX(cons(z0, cons(0, z1)))) SORT(cons(0, cons(s(z0), z1))) -> c15(SORT(del(if1(false, 0, s(z0), z1), cons(0, cons(s(z0), z1)))), DEL(if1(false, 0, s(z0), z1), cons(0, cons(s(z0), z1))), MAX(cons(0, cons(s(z0), z1)))) SORT(cons(s(z0), cons(s(z1), z2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(ge(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MAX(cons(s(z0), cons(s(z1), z2)))) DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c6(EQ(s(s(y0)), s(s(y1)))) IF1(true, z0, 0, cons(0, y1)) -> c3(MAX(cons(z0, cons(0, y1)))) IF1(true, 0, 0, cons(s(y0), y1)) -> c3(MAX(cons(0, cons(s(y0), y1)))) IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c3(MAX(cons(s(s(y0)), cons(s(s(y1)), y2)))) IF1(true, s(y0), 0, cons(s(0), y1)) -> c3(MAX(cons(s(y0), cons(s(0), y1)))) IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c3(MAX(cons(s(0), cons(s(s(y0)), y1)))) K tuples: SORT(cons(x0, cons(x1, x2))) -> c15(DEL(max(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) SORT(cons(x0, nil)) -> c15(DEL(max(cons(x0, nil)), cons(x0, nil))) SORT(cons(z0, cons(0, y1))) -> c14(MAX(cons(z0, cons(0, y1)))) SORT(cons(0, cons(s(y0), y1))) -> c14(MAX(cons(0, cons(s(y0), y1)))) SORT(cons(s(y0), cons(s(0), y1))) -> c14(MAX(cons(s(y0), cons(s(0), y1)))) SORT(cons(s(0), cons(s(s(y0)), y1))) -> c14(MAX(cons(s(0), cons(s(s(y0)), y1)))) SORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c14(MAX(cons(s(s(y0)), cons(s(s(y1)), y2)))) SORT(cons(s(y0), cons(s(y1), y2))) -> c14(MAX(cons(s(y0), cons(s(y1), y2)))) Defined Rule Symbols: ge_2, eq_2, del_2, max_1, if1_4, if2_4 Defined Pair Symbols: MAX_1, DEL_2, IF1_4, SORT_1, IF2_4, EQ_2, GE_2 Compound Symbols: c2_1, c6_1, c2_2, c6_2, c3_1, c4_1, c15_3, c8_1, c15_1, c15_2, c12_1, c6, c14_1, c18_1 ---------------------------------------- (151) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace MAX(cons(z0, cons(0, x2))) -> c2(IF1(true, z0, 0, x2)) by MAX(cons(z0, cons(0, cons(0, y1)))) -> c2(IF1(true, z0, 0, cons(0, y1))) MAX(cons(0, cons(0, cons(s(y0), y1)))) -> c2(IF1(true, 0, 0, cons(s(y0), y1))) MAX(cons(s(s(y0)), cons(0, cons(s(s(y1)), y2)))) -> c2(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2))) MAX(cons(s(y0), cons(0, cons(s(0), y1)))) -> c2(IF1(true, s(y0), 0, cons(s(0), y1))) MAX(cons(s(0), cons(0, cons(s(s(y0)), y1)))) -> c2(IF1(true, s(0), 0, cons(s(s(y0)), y1))) ---------------------------------------- (152) 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, 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)) Tuples: 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)) 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)))) 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)))) IF1(true, s(x0), s(0), x1) -> c3(MAX(cons(s(x0), x1))) IF1(true, s(s(x0)), s(s(x1)), x2) -> c3(MAX(cons(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(x0)), s(s(x1)), x2) -> c4(MAX(cons(s(s(x1)), x2))) SORT(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)))) 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(x0)), s(s(x1)), x2) -> c8(DEL(s(s(x0)), x2)) SORT(cons(z0, cons(z1, z2))) -> c15(DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) SORT(cons(z0, nil)) -> c15(DEL(z0, cons(z0, nil))) 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 SORT(cons(z0, cons(0, y1))) -> c14(MAX(cons(z0, cons(0, y1)))) SORT(cons(0, cons(s(y0), y1))) -> c14(MAX(cons(0, cons(s(y0), y1)))) SORT(cons(s(y0), cons(s(0), y1))) -> c14(MAX(cons(s(y0), cons(s(0), y1)))) SORT(cons(s(0), cons(s(s(y0)), y1))) -> c14(MAX(cons(s(0), cons(s(s(y0)), y1)))) SORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c14(MAX(cons(s(s(y0)), cons(s(s(y1)), y2)))) SORT(cons(s(y0), cons(s(y1), y2))) -> c14(MAX(cons(s(y0), cons(s(y1), y2)))) GE(s(s(y0)), s(s(y1))) -> c18(GE(s(y0), s(y1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2)) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2)) MAX(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c2(GE(s(s(y0)), s(s(y1)))) SORT(cons(z0, cons(0, z1))) -> c15(SORT(del(if1(true, z0, 0, z1), cons(z0, cons(0, z1)))), DEL(if1(true, z0, 0, z1), cons(z0, cons(0, z1))), MAX(cons(z0, cons(0, z1)))) SORT(cons(0, cons(s(z0), z1))) -> c15(SORT(del(if1(false, 0, s(z0), z1), cons(0, cons(s(z0), z1)))), DEL(if1(false, 0, s(z0), z1), cons(0, cons(s(z0), z1))), MAX(cons(0, cons(s(z0), z1)))) SORT(cons(s(z0), cons(s(z1), z2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(ge(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MAX(cons(s(z0), cons(s(z1), z2)))) DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c6(EQ(s(s(y0)), s(s(y1)))) IF1(true, z0, 0, cons(0, y1)) -> c3(MAX(cons(z0, cons(0, y1)))) IF1(true, 0, 0, cons(s(y0), y1)) -> c3(MAX(cons(0, cons(s(y0), y1)))) IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c3(MAX(cons(s(s(y0)), cons(s(s(y1)), y2)))) IF1(true, s(y0), 0, cons(s(0), y1)) -> c3(MAX(cons(s(y0), cons(s(0), y1)))) IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c3(MAX(cons(s(0), cons(s(s(y0)), y1)))) MAX(cons(z0, cons(0, cons(0, y1)))) -> c2(IF1(true, z0, 0, cons(0, y1))) MAX(cons(0, cons(0, cons(s(y0), y1)))) -> c2(IF1(true, 0, 0, cons(s(y0), y1))) MAX(cons(s(s(y0)), cons(0, cons(s(s(y1)), y2)))) -> c2(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2))) MAX(cons(s(y0), cons(0, cons(s(0), y1)))) -> c2(IF1(true, s(y0), 0, cons(s(0), y1))) MAX(cons(s(0), cons(0, cons(s(s(y0)), y1)))) -> c2(IF1(true, s(0), 0, cons(s(s(y0)), y1))) S tuples: 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)) 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)))) 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)))) IF1(true, s(x0), s(0), x1) -> c3(MAX(cons(s(x0), x1))) IF1(true, s(s(x0)), s(s(x1)), x2) -> c3(MAX(cons(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(x0)), s(s(x1)), x2) -> c4(MAX(cons(s(s(x1)), x2))) SORT(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)))) 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(x0)), s(s(x1)), x2) -> c8(DEL(s(s(x0)), x2)) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) 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 GE(s(s(y0)), s(s(y1))) -> c18(GE(s(y0), s(y1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2)) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2)) MAX(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c2(GE(s(s(y0)), s(s(y1)))) SORT(cons(z0, cons(0, z1))) -> c15(SORT(del(if1(true, z0, 0, z1), cons(z0, cons(0, z1)))), DEL(if1(true, z0, 0, z1), cons(z0, cons(0, z1))), MAX(cons(z0, cons(0, z1)))) SORT(cons(0, cons(s(z0), z1))) -> c15(SORT(del(if1(false, 0, s(z0), z1), cons(0, cons(s(z0), z1)))), DEL(if1(false, 0, s(z0), z1), cons(0, cons(s(z0), z1))), MAX(cons(0, cons(s(z0), z1)))) SORT(cons(s(z0), cons(s(z1), z2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(ge(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MAX(cons(s(z0), cons(s(z1), z2)))) DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c6(EQ(s(s(y0)), s(s(y1)))) IF1(true, z0, 0, cons(0, y1)) -> c3(MAX(cons(z0, cons(0, y1)))) IF1(true, 0, 0, cons(s(y0), y1)) -> c3(MAX(cons(0, cons(s(y0), y1)))) IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c3(MAX(cons(s(s(y0)), cons(s(s(y1)), y2)))) IF1(true, s(y0), 0, cons(s(0), y1)) -> c3(MAX(cons(s(y0), cons(s(0), y1)))) IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c3(MAX(cons(s(0), cons(s(s(y0)), y1)))) MAX(cons(z0, cons(0, cons(0, y1)))) -> c2(IF1(true, z0, 0, cons(0, y1))) MAX(cons(0, cons(0, cons(s(y0), y1)))) -> c2(IF1(true, 0, 0, cons(s(y0), y1))) MAX(cons(s(s(y0)), cons(0, cons(s(s(y1)), y2)))) -> c2(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2))) MAX(cons(s(y0), cons(0, cons(s(0), y1)))) -> c2(IF1(true, s(y0), 0, cons(s(0), y1))) MAX(cons(s(0), cons(0, cons(s(s(y0)), y1)))) -> c2(IF1(true, s(0), 0, cons(s(s(y0)), y1))) K tuples: SORT(cons(x0, cons(x1, x2))) -> c15(DEL(max(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) SORT(cons(x0, nil)) -> c15(DEL(max(cons(x0, nil)), cons(x0, nil))) SORT(cons(z0, cons(0, y1))) -> c14(MAX(cons(z0, cons(0, y1)))) SORT(cons(0, cons(s(y0), y1))) -> c14(MAX(cons(0, cons(s(y0), y1)))) SORT(cons(s(y0), cons(s(0), y1))) -> c14(MAX(cons(s(y0), cons(s(0), y1)))) SORT(cons(s(0), cons(s(s(y0)), y1))) -> c14(MAX(cons(s(0), cons(s(s(y0)), y1)))) SORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c14(MAX(cons(s(s(y0)), cons(s(s(y1)), y2)))) SORT(cons(s(y0), cons(s(y1), y2))) -> c14(MAX(cons(s(y0), cons(s(y1), y2)))) Defined Rule Symbols: ge_2, eq_2, del_2, max_1, if1_4, if2_4 Defined Pair Symbols: MAX_1, DEL_2, IF1_4, SORT_1, IF2_4, EQ_2, GE_2 Compound Symbols: c2_1, c6_1, c2_2, c6_2, c3_1, c4_1, c15_3, c8_1, c15_1, c15_2, c12_1, c6, c14_1, c18_1 ---------------------------------------- (153) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace IF1(true, s(x0), s(0), x1) -> c3(MAX(cons(s(x0), x1))) by IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)) -> c3(MAX(cons(s(s(y0)), cons(s(s(y1)), y2)))) IF1(true, s(z0), s(0), cons(s(0), y1)) -> c3(MAX(cons(s(z0), cons(s(0), y1)))) IF1(true, s(0), s(0), cons(s(s(y0)), y1)) -> c3(MAX(cons(s(0), cons(s(s(y0)), y1)))) IF1(true, s(z0), s(0), cons(0, cons(0, y1))) -> c3(MAX(cons(s(z0), cons(0, cons(0, y1))))) IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c3(MAX(cons(s(s(y0)), cons(0, cons(s(s(y1)), y2))))) IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))) -> c3(MAX(cons(s(z0), cons(0, cons(s(0), y1))))) IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))) -> c3(MAX(cons(s(0), cons(0, cons(s(s(y0)), y1))))) ---------------------------------------- (154) 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, 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)) Tuples: 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)) 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)))) 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)))) IF1(true, s(s(x0)), s(s(x1)), x2) -> c3(MAX(cons(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(x0)), s(s(x1)), x2) -> c4(MAX(cons(s(s(x1)), x2))) SORT(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)))) 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(x0)), s(s(x1)), x2) -> c8(DEL(s(s(x0)), x2)) SORT(cons(z0, cons(z1, z2))) -> c15(DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) SORT(cons(z0, nil)) -> c15(DEL(z0, cons(z0, nil))) 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 SORT(cons(z0, cons(0, y1))) -> c14(MAX(cons(z0, cons(0, y1)))) SORT(cons(0, cons(s(y0), y1))) -> c14(MAX(cons(0, cons(s(y0), y1)))) SORT(cons(s(y0), cons(s(0), y1))) -> c14(MAX(cons(s(y0), cons(s(0), y1)))) SORT(cons(s(0), cons(s(s(y0)), y1))) -> c14(MAX(cons(s(0), cons(s(s(y0)), y1)))) SORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c14(MAX(cons(s(s(y0)), cons(s(s(y1)), y2)))) SORT(cons(s(y0), cons(s(y1), y2))) -> c14(MAX(cons(s(y0), cons(s(y1), y2)))) GE(s(s(y0)), s(s(y1))) -> c18(GE(s(y0), s(y1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2)) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2)) MAX(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c2(GE(s(s(y0)), s(s(y1)))) SORT(cons(z0, cons(0, z1))) -> c15(SORT(del(if1(true, z0, 0, z1), cons(z0, cons(0, z1)))), DEL(if1(true, z0, 0, z1), cons(z0, cons(0, z1))), MAX(cons(z0, cons(0, z1)))) SORT(cons(0, cons(s(z0), z1))) -> c15(SORT(del(if1(false, 0, s(z0), z1), cons(0, cons(s(z0), z1)))), DEL(if1(false, 0, s(z0), z1), cons(0, cons(s(z0), z1))), MAX(cons(0, cons(s(z0), z1)))) SORT(cons(s(z0), cons(s(z1), z2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(ge(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MAX(cons(s(z0), cons(s(z1), z2)))) DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c6(EQ(s(s(y0)), s(s(y1)))) IF1(true, z0, 0, cons(0, y1)) -> c3(MAX(cons(z0, cons(0, y1)))) IF1(true, 0, 0, cons(s(y0), y1)) -> c3(MAX(cons(0, cons(s(y0), y1)))) IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c3(MAX(cons(s(s(y0)), cons(s(s(y1)), y2)))) IF1(true, s(y0), 0, cons(s(0), y1)) -> c3(MAX(cons(s(y0), cons(s(0), y1)))) IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c3(MAX(cons(s(0), cons(s(s(y0)), y1)))) MAX(cons(z0, cons(0, cons(0, y1)))) -> c2(IF1(true, z0, 0, cons(0, y1))) MAX(cons(0, cons(0, cons(s(y0), y1)))) -> c2(IF1(true, 0, 0, cons(s(y0), y1))) MAX(cons(s(s(y0)), cons(0, cons(s(s(y1)), y2)))) -> c2(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2))) MAX(cons(s(y0), cons(0, cons(s(0), y1)))) -> c2(IF1(true, s(y0), 0, cons(s(0), y1))) MAX(cons(s(0), cons(0, cons(s(s(y0)), y1)))) -> c2(IF1(true, s(0), 0, cons(s(s(y0)), y1))) IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)) -> c3(MAX(cons(s(s(y0)), cons(s(s(y1)), y2)))) IF1(true, s(z0), s(0), cons(s(0), y1)) -> c3(MAX(cons(s(z0), cons(s(0), y1)))) IF1(true, s(0), s(0), cons(s(s(y0)), y1)) -> c3(MAX(cons(s(0), cons(s(s(y0)), y1)))) IF1(true, s(z0), s(0), cons(0, cons(0, y1))) -> c3(MAX(cons(s(z0), cons(0, cons(0, y1))))) IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c3(MAX(cons(s(s(y0)), cons(0, cons(s(s(y1)), y2))))) IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))) -> c3(MAX(cons(s(z0), cons(0, cons(s(0), y1))))) IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))) -> c3(MAX(cons(s(0), cons(0, cons(s(s(y0)), y1))))) S tuples: 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)) 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)))) 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)))) IF1(true, s(s(x0)), s(s(x1)), x2) -> c3(MAX(cons(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(x0)), s(s(x1)), x2) -> c4(MAX(cons(s(s(x1)), x2))) SORT(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)))) 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(x0)), s(s(x1)), x2) -> c8(DEL(s(s(x0)), x2)) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) 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 GE(s(s(y0)), s(s(y1))) -> c18(GE(s(y0), s(y1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2)) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2)) MAX(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c2(GE(s(s(y0)), s(s(y1)))) SORT(cons(z0, cons(0, z1))) -> c15(SORT(del(if1(true, z0, 0, z1), cons(z0, cons(0, z1)))), DEL(if1(true, z0, 0, z1), cons(z0, cons(0, z1))), MAX(cons(z0, cons(0, z1)))) SORT(cons(0, cons(s(z0), z1))) -> c15(SORT(del(if1(false, 0, s(z0), z1), cons(0, cons(s(z0), z1)))), DEL(if1(false, 0, s(z0), z1), cons(0, cons(s(z0), z1))), MAX(cons(0, cons(s(z0), z1)))) SORT(cons(s(z0), cons(s(z1), z2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(ge(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MAX(cons(s(z0), cons(s(z1), z2)))) DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c6(EQ(s(s(y0)), s(s(y1)))) IF1(true, z0, 0, cons(0, y1)) -> c3(MAX(cons(z0, cons(0, y1)))) IF1(true, 0, 0, cons(s(y0), y1)) -> c3(MAX(cons(0, cons(s(y0), y1)))) IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c3(MAX(cons(s(s(y0)), cons(s(s(y1)), y2)))) IF1(true, s(y0), 0, cons(s(0), y1)) -> c3(MAX(cons(s(y0), cons(s(0), y1)))) IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c3(MAX(cons(s(0), cons(s(s(y0)), y1)))) MAX(cons(z0, cons(0, cons(0, y1)))) -> c2(IF1(true, z0, 0, cons(0, y1))) MAX(cons(0, cons(0, cons(s(y0), y1)))) -> c2(IF1(true, 0, 0, cons(s(y0), y1))) MAX(cons(s(s(y0)), cons(0, cons(s(s(y1)), y2)))) -> c2(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2))) MAX(cons(s(y0), cons(0, cons(s(0), y1)))) -> c2(IF1(true, s(y0), 0, cons(s(0), y1))) MAX(cons(s(0), cons(0, cons(s(s(y0)), y1)))) -> c2(IF1(true, s(0), 0, cons(s(s(y0)), y1))) IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)) -> c3(MAX(cons(s(s(y0)), cons(s(s(y1)), y2)))) IF1(true, s(z0), s(0), cons(s(0), y1)) -> c3(MAX(cons(s(z0), cons(s(0), y1)))) IF1(true, s(0), s(0), cons(s(s(y0)), y1)) -> c3(MAX(cons(s(0), cons(s(s(y0)), y1)))) IF1(true, s(z0), s(0), cons(0, cons(0, y1))) -> c3(MAX(cons(s(z0), cons(0, cons(0, y1))))) IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c3(MAX(cons(s(s(y0)), cons(0, cons(s(s(y1)), y2))))) IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))) -> c3(MAX(cons(s(z0), cons(0, cons(s(0), y1))))) IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))) -> c3(MAX(cons(s(0), cons(0, cons(s(s(y0)), y1))))) K tuples: SORT(cons(x0, cons(x1, x2))) -> c15(DEL(max(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) SORT(cons(x0, nil)) -> c15(DEL(max(cons(x0, nil)), cons(x0, nil))) SORT(cons(z0, cons(0, y1))) -> c14(MAX(cons(z0, cons(0, y1)))) SORT(cons(0, cons(s(y0), y1))) -> c14(MAX(cons(0, cons(s(y0), y1)))) SORT(cons(s(y0), cons(s(0), y1))) -> c14(MAX(cons(s(y0), cons(s(0), y1)))) SORT(cons(s(0), cons(s(s(y0)), y1))) -> c14(MAX(cons(s(0), cons(s(s(y0)), y1)))) SORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c14(MAX(cons(s(s(y0)), cons(s(s(y1)), y2)))) SORT(cons(s(y0), cons(s(y1), y2))) -> c14(MAX(cons(s(y0), cons(s(y1), y2)))) Defined Rule Symbols: ge_2, eq_2, del_2, max_1, if1_4, if2_4 Defined Pair Symbols: MAX_1, DEL_2, IF1_4, SORT_1, IF2_4, EQ_2, GE_2 Compound Symbols: c2_1, c6_1, c2_2, c6_2, c3_1, c4_1, c15_3, c8_1, c15_1, c15_2, c12_1, c6, c14_1, c18_1 ---------------------------------------- (155) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace IF1(true, s(s(x0)), s(s(x1)), x2) -> c3(MAX(cons(s(s(x0)), x2))) by IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c3(MAX(cons(s(s(z0)), cons(s(s(y1)), y2)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c3(MAX(cons(s(s(z0)), cons(s(0), y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c3(MAX(cons(s(s(z0)), cons(0, cons(0, y1))))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c3(MAX(cons(s(s(z0)), cons(0, cons(s(s(y1)), y2))))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c3(MAX(cons(s(s(z0)), cons(0, cons(s(0), y1))))) ---------------------------------------- (156) 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, 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)) Tuples: 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)) 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)))) 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)))) 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(x0)), s(s(x1)), x2) -> c4(MAX(cons(s(s(x1)), x2))) SORT(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)))) 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(x0)), s(s(x1)), x2) -> c8(DEL(s(s(x0)), x2)) SORT(cons(z0, cons(z1, z2))) -> c15(DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) SORT(cons(z0, nil)) -> c15(DEL(z0, cons(z0, nil))) 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 SORT(cons(z0, cons(0, y1))) -> c14(MAX(cons(z0, cons(0, y1)))) SORT(cons(0, cons(s(y0), y1))) -> c14(MAX(cons(0, cons(s(y0), y1)))) SORT(cons(s(y0), cons(s(0), y1))) -> c14(MAX(cons(s(y0), cons(s(0), y1)))) SORT(cons(s(0), cons(s(s(y0)), y1))) -> c14(MAX(cons(s(0), cons(s(s(y0)), y1)))) SORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c14(MAX(cons(s(s(y0)), cons(s(s(y1)), y2)))) SORT(cons(s(y0), cons(s(y1), y2))) -> c14(MAX(cons(s(y0), cons(s(y1), y2)))) GE(s(s(y0)), s(s(y1))) -> c18(GE(s(y0), s(y1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2)) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2)) MAX(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c2(GE(s(s(y0)), s(s(y1)))) SORT(cons(z0, cons(0, z1))) -> c15(SORT(del(if1(true, z0, 0, z1), cons(z0, cons(0, z1)))), DEL(if1(true, z0, 0, z1), cons(z0, cons(0, z1))), MAX(cons(z0, cons(0, z1)))) SORT(cons(0, cons(s(z0), z1))) -> c15(SORT(del(if1(false, 0, s(z0), z1), cons(0, cons(s(z0), z1)))), DEL(if1(false, 0, s(z0), z1), cons(0, cons(s(z0), z1))), MAX(cons(0, cons(s(z0), z1)))) SORT(cons(s(z0), cons(s(z1), z2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(ge(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MAX(cons(s(z0), cons(s(z1), z2)))) DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c6(EQ(s(s(y0)), s(s(y1)))) IF1(true, z0, 0, cons(0, y1)) -> c3(MAX(cons(z0, cons(0, y1)))) IF1(true, 0, 0, cons(s(y0), y1)) -> c3(MAX(cons(0, cons(s(y0), y1)))) IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c3(MAX(cons(s(s(y0)), cons(s(s(y1)), y2)))) IF1(true, s(y0), 0, cons(s(0), y1)) -> c3(MAX(cons(s(y0), cons(s(0), y1)))) IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c3(MAX(cons(s(0), cons(s(s(y0)), y1)))) MAX(cons(z0, cons(0, cons(0, y1)))) -> c2(IF1(true, z0, 0, cons(0, y1))) MAX(cons(0, cons(0, cons(s(y0), y1)))) -> c2(IF1(true, 0, 0, cons(s(y0), y1))) MAX(cons(s(s(y0)), cons(0, cons(s(s(y1)), y2)))) -> c2(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2))) MAX(cons(s(y0), cons(0, cons(s(0), y1)))) -> c2(IF1(true, s(y0), 0, cons(s(0), y1))) MAX(cons(s(0), cons(0, cons(s(s(y0)), y1)))) -> c2(IF1(true, s(0), 0, cons(s(s(y0)), y1))) IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)) -> c3(MAX(cons(s(s(y0)), cons(s(s(y1)), y2)))) IF1(true, s(z0), s(0), cons(s(0), y1)) -> c3(MAX(cons(s(z0), cons(s(0), y1)))) IF1(true, s(0), s(0), cons(s(s(y0)), y1)) -> c3(MAX(cons(s(0), cons(s(s(y0)), y1)))) IF1(true, s(z0), s(0), cons(0, cons(0, y1))) -> c3(MAX(cons(s(z0), cons(0, cons(0, y1))))) IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c3(MAX(cons(s(s(y0)), cons(0, cons(s(s(y1)), y2))))) IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))) -> c3(MAX(cons(s(z0), cons(0, cons(s(0), y1))))) IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))) -> c3(MAX(cons(s(0), cons(0, cons(s(s(y0)), y1))))) IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c3(MAX(cons(s(s(z0)), cons(s(s(y1)), y2)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c3(MAX(cons(s(s(z0)), cons(s(0), y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c3(MAX(cons(s(s(z0)), cons(0, cons(0, y1))))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c3(MAX(cons(s(s(z0)), cons(0, cons(s(s(y1)), y2))))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c3(MAX(cons(s(s(z0)), cons(0, cons(s(0), y1))))) S tuples: 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)) 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)))) 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)))) 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(x0)), s(s(x1)), x2) -> c4(MAX(cons(s(s(x1)), x2))) SORT(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)))) 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(x0)), s(s(x1)), x2) -> c8(DEL(s(s(x0)), x2)) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) 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 GE(s(s(y0)), s(s(y1))) -> c18(GE(s(y0), s(y1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2)) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2)) MAX(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c2(GE(s(s(y0)), s(s(y1)))) SORT(cons(z0, cons(0, z1))) -> c15(SORT(del(if1(true, z0, 0, z1), cons(z0, cons(0, z1)))), DEL(if1(true, z0, 0, z1), cons(z0, cons(0, z1))), MAX(cons(z0, cons(0, z1)))) SORT(cons(0, cons(s(z0), z1))) -> c15(SORT(del(if1(false, 0, s(z0), z1), cons(0, cons(s(z0), z1)))), DEL(if1(false, 0, s(z0), z1), cons(0, cons(s(z0), z1))), MAX(cons(0, cons(s(z0), z1)))) SORT(cons(s(z0), cons(s(z1), z2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(ge(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MAX(cons(s(z0), cons(s(z1), z2)))) DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c6(EQ(s(s(y0)), s(s(y1)))) IF1(true, z0, 0, cons(0, y1)) -> c3(MAX(cons(z0, cons(0, y1)))) IF1(true, 0, 0, cons(s(y0), y1)) -> c3(MAX(cons(0, cons(s(y0), y1)))) IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c3(MAX(cons(s(s(y0)), cons(s(s(y1)), y2)))) IF1(true, s(y0), 0, cons(s(0), y1)) -> c3(MAX(cons(s(y0), cons(s(0), y1)))) IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c3(MAX(cons(s(0), cons(s(s(y0)), y1)))) MAX(cons(z0, cons(0, cons(0, y1)))) -> c2(IF1(true, z0, 0, cons(0, y1))) MAX(cons(0, cons(0, cons(s(y0), y1)))) -> c2(IF1(true, 0, 0, cons(s(y0), y1))) MAX(cons(s(s(y0)), cons(0, cons(s(s(y1)), y2)))) -> c2(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2))) MAX(cons(s(y0), cons(0, cons(s(0), y1)))) -> c2(IF1(true, s(y0), 0, cons(s(0), y1))) MAX(cons(s(0), cons(0, cons(s(s(y0)), y1)))) -> c2(IF1(true, s(0), 0, cons(s(s(y0)), y1))) IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)) -> c3(MAX(cons(s(s(y0)), cons(s(s(y1)), y2)))) IF1(true, s(z0), s(0), cons(s(0), y1)) -> c3(MAX(cons(s(z0), cons(s(0), y1)))) IF1(true, s(0), s(0), cons(s(s(y0)), y1)) -> c3(MAX(cons(s(0), cons(s(s(y0)), y1)))) IF1(true, s(z0), s(0), cons(0, cons(0, y1))) -> c3(MAX(cons(s(z0), cons(0, cons(0, y1))))) IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c3(MAX(cons(s(s(y0)), cons(0, cons(s(s(y1)), y2))))) IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))) -> c3(MAX(cons(s(z0), cons(0, cons(s(0), y1))))) IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))) -> c3(MAX(cons(s(0), cons(0, cons(s(s(y0)), y1))))) IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c3(MAX(cons(s(s(z0)), cons(s(s(y1)), y2)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c3(MAX(cons(s(s(z0)), cons(s(0), y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c3(MAX(cons(s(s(z0)), cons(0, cons(0, y1))))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c3(MAX(cons(s(s(z0)), cons(0, cons(s(s(y1)), y2))))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c3(MAX(cons(s(s(z0)), cons(0, cons(s(0), y1))))) K tuples: SORT(cons(x0, cons(x1, x2))) -> c15(DEL(max(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) SORT(cons(x0, nil)) -> c15(DEL(max(cons(x0, nil)), cons(x0, nil))) SORT(cons(z0, cons(0, y1))) -> c14(MAX(cons(z0, cons(0, y1)))) SORT(cons(0, cons(s(y0), y1))) -> c14(MAX(cons(0, cons(s(y0), y1)))) SORT(cons(s(y0), cons(s(0), y1))) -> c14(MAX(cons(s(y0), cons(s(0), y1)))) SORT(cons(s(0), cons(s(s(y0)), y1))) -> c14(MAX(cons(s(0), cons(s(s(y0)), y1)))) SORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c14(MAX(cons(s(s(y0)), cons(s(s(y1)), y2)))) SORT(cons(s(y0), cons(s(y1), y2))) -> c14(MAX(cons(s(y0), cons(s(y1), y2)))) Defined Rule Symbols: ge_2, eq_2, del_2, max_1, if1_4, if2_4 Defined Pair Symbols: MAX_1, DEL_2, IF1_4, SORT_1, IF2_4, EQ_2, GE_2 Compound Symbols: c2_1, c6_1, c2_2, c6_2, c4_1, c15_3, c8_1, c15_1, c15_2, c12_1, c6, c14_1, c18_1, c3_1 ---------------------------------------- (157) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace IF1(false, 0, s(x0), x1) -> c4(MAX(cons(s(x0), x1))) by IF1(false, 0, s(s(y0)), cons(s(s(y1)), y2)) -> c4(MAX(cons(s(s(y0)), cons(s(s(y1)), y2)))) IF1(false, 0, s(z0), cons(s(0), y1)) -> c4(MAX(cons(s(z0), cons(s(0), y1)))) IF1(false, 0, s(0), cons(s(s(y0)), y1)) -> c4(MAX(cons(s(0), cons(s(s(y0)), y1)))) IF1(false, 0, s(z0), cons(0, cons(0, y1))) -> c4(MAX(cons(s(z0), cons(0, cons(0, y1))))) IF1(false, 0, s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(MAX(cons(s(s(y0)), cons(0, cons(s(s(y1)), y2))))) IF1(false, 0, s(z0), cons(0, cons(s(0), y1))) -> c4(MAX(cons(s(z0), cons(0, cons(s(0), y1))))) IF1(false, 0, s(0), cons(0, cons(s(s(y0)), y1))) -> c4(MAX(cons(s(0), cons(0, cons(s(s(y0)), y1))))) ---------------------------------------- (158) 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, 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)) Tuples: 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)) 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)))) 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)))) IF1(false, s(0), s(s(x0)), x1) -> c4(MAX(cons(s(s(x0)), x1))) IF1(false, s(s(x0)), s(s(x1)), x2) -> c4(MAX(cons(s(s(x1)), x2))) SORT(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)))) 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(x0)), s(s(x1)), x2) -> c8(DEL(s(s(x0)), x2)) SORT(cons(z0, cons(z1, z2))) -> c15(DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) SORT(cons(z0, nil)) -> c15(DEL(z0, cons(z0, nil))) 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 SORT(cons(z0, cons(0, y1))) -> c14(MAX(cons(z0, cons(0, y1)))) SORT(cons(0, cons(s(y0), y1))) -> c14(MAX(cons(0, cons(s(y0), y1)))) SORT(cons(s(y0), cons(s(0), y1))) -> c14(MAX(cons(s(y0), cons(s(0), y1)))) SORT(cons(s(0), cons(s(s(y0)), y1))) -> c14(MAX(cons(s(0), cons(s(s(y0)), y1)))) SORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c14(MAX(cons(s(s(y0)), cons(s(s(y1)), y2)))) SORT(cons(s(y0), cons(s(y1), y2))) -> c14(MAX(cons(s(y0), cons(s(y1), y2)))) GE(s(s(y0)), s(s(y1))) -> c18(GE(s(y0), s(y1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2)) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2)) MAX(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c2(GE(s(s(y0)), s(s(y1)))) SORT(cons(z0, cons(0, z1))) -> c15(SORT(del(if1(true, z0, 0, z1), cons(z0, cons(0, z1)))), DEL(if1(true, z0, 0, z1), cons(z0, cons(0, z1))), MAX(cons(z0, cons(0, z1)))) SORT(cons(0, cons(s(z0), z1))) -> c15(SORT(del(if1(false, 0, s(z0), z1), cons(0, cons(s(z0), z1)))), DEL(if1(false, 0, s(z0), z1), cons(0, cons(s(z0), z1))), MAX(cons(0, cons(s(z0), z1)))) SORT(cons(s(z0), cons(s(z1), z2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(ge(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MAX(cons(s(z0), cons(s(z1), z2)))) DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c6(EQ(s(s(y0)), s(s(y1)))) IF1(true, z0, 0, cons(0, y1)) -> c3(MAX(cons(z0, cons(0, y1)))) IF1(true, 0, 0, cons(s(y0), y1)) -> c3(MAX(cons(0, cons(s(y0), y1)))) IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c3(MAX(cons(s(s(y0)), cons(s(s(y1)), y2)))) IF1(true, s(y0), 0, cons(s(0), y1)) -> c3(MAX(cons(s(y0), cons(s(0), y1)))) IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c3(MAX(cons(s(0), cons(s(s(y0)), y1)))) MAX(cons(z0, cons(0, cons(0, y1)))) -> c2(IF1(true, z0, 0, cons(0, y1))) MAX(cons(0, cons(0, cons(s(y0), y1)))) -> c2(IF1(true, 0, 0, cons(s(y0), y1))) MAX(cons(s(s(y0)), cons(0, cons(s(s(y1)), y2)))) -> c2(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2))) MAX(cons(s(y0), cons(0, cons(s(0), y1)))) -> c2(IF1(true, s(y0), 0, cons(s(0), y1))) MAX(cons(s(0), cons(0, cons(s(s(y0)), y1)))) -> c2(IF1(true, s(0), 0, cons(s(s(y0)), y1))) IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)) -> c3(MAX(cons(s(s(y0)), cons(s(s(y1)), y2)))) IF1(true, s(z0), s(0), cons(s(0), y1)) -> c3(MAX(cons(s(z0), cons(s(0), y1)))) IF1(true, s(0), s(0), cons(s(s(y0)), y1)) -> c3(MAX(cons(s(0), cons(s(s(y0)), y1)))) IF1(true, s(z0), s(0), cons(0, cons(0, y1))) -> c3(MAX(cons(s(z0), cons(0, cons(0, y1))))) IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c3(MAX(cons(s(s(y0)), cons(0, cons(s(s(y1)), y2))))) IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))) -> c3(MAX(cons(s(z0), cons(0, cons(s(0), y1))))) IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))) -> c3(MAX(cons(s(0), cons(0, cons(s(s(y0)), y1))))) IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c3(MAX(cons(s(s(z0)), cons(s(s(y1)), y2)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c3(MAX(cons(s(s(z0)), cons(s(0), y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c3(MAX(cons(s(s(z0)), cons(0, cons(0, y1))))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c3(MAX(cons(s(s(z0)), cons(0, cons(s(s(y1)), y2))))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c3(MAX(cons(s(s(z0)), cons(0, cons(s(0), y1))))) IF1(false, 0, s(s(y0)), cons(s(s(y1)), y2)) -> c4(MAX(cons(s(s(y0)), cons(s(s(y1)), y2)))) IF1(false, 0, s(z0), cons(s(0), y1)) -> c4(MAX(cons(s(z0), cons(s(0), y1)))) IF1(false, 0, s(0), cons(s(s(y0)), y1)) -> c4(MAX(cons(s(0), cons(s(s(y0)), y1)))) IF1(false, 0, s(z0), cons(0, cons(0, y1))) -> c4(MAX(cons(s(z0), cons(0, cons(0, y1))))) IF1(false, 0, s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(MAX(cons(s(s(y0)), cons(0, cons(s(s(y1)), y2))))) IF1(false, 0, s(z0), cons(0, cons(s(0), y1))) -> c4(MAX(cons(s(z0), cons(0, cons(s(0), y1))))) IF1(false, 0, s(0), cons(0, cons(s(s(y0)), y1))) -> c4(MAX(cons(s(0), cons(0, cons(s(s(y0)), y1))))) S tuples: 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)) 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)))) 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)))) IF1(false, s(0), s(s(x0)), x1) -> c4(MAX(cons(s(s(x0)), x1))) IF1(false, s(s(x0)), s(s(x1)), x2) -> c4(MAX(cons(s(s(x1)), x2))) SORT(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)))) 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(x0)), s(s(x1)), x2) -> c8(DEL(s(s(x0)), x2)) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) 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 GE(s(s(y0)), s(s(y1))) -> c18(GE(s(y0), s(y1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2)) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2)) MAX(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c2(GE(s(s(y0)), s(s(y1)))) SORT(cons(z0, cons(0, z1))) -> c15(SORT(del(if1(true, z0, 0, z1), cons(z0, cons(0, z1)))), DEL(if1(true, z0, 0, z1), cons(z0, cons(0, z1))), MAX(cons(z0, cons(0, z1)))) SORT(cons(0, cons(s(z0), z1))) -> c15(SORT(del(if1(false, 0, s(z0), z1), cons(0, cons(s(z0), z1)))), DEL(if1(false, 0, s(z0), z1), cons(0, cons(s(z0), z1))), MAX(cons(0, cons(s(z0), z1)))) SORT(cons(s(z0), cons(s(z1), z2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(ge(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MAX(cons(s(z0), cons(s(z1), z2)))) DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c6(EQ(s(s(y0)), s(s(y1)))) IF1(true, z0, 0, cons(0, y1)) -> c3(MAX(cons(z0, cons(0, y1)))) IF1(true, 0, 0, cons(s(y0), y1)) -> c3(MAX(cons(0, cons(s(y0), y1)))) IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c3(MAX(cons(s(s(y0)), cons(s(s(y1)), y2)))) IF1(true, s(y0), 0, cons(s(0), y1)) -> c3(MAX(cons(s(y0), cons(s(0), y1)))) IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c3(MAX(cons(s(0), cons(s(s(y0)), y1)))) MAX(cons(z0, cons(0, cons(0, y1)))) -> c2(IF1(true, z0, 0, cons(0, y1))) MAX(cons(0, cons(0, cons(s(y0), y1)))) -> c2(IF1(true, 0, 0, cons(s(y0), y1))) MAX(cons(s(s(y0)), cons(0, cons(s(s(y1)), y2)))) -> c2(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2))) MAX(cons(s(y0), cons(0, cons(s(0), y1)))) -> c2(IF1(true, s(y0), 0, cons(s(0), y1))) MAX(cons(s(0), cons(0, cons(s(s(y0)), y1)))) -> c2(IF1(true, s(0), 0, cons(s(s(y0)), y1))) IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)) -> c3(MAX(cons(s(s(y0)), cons(s(s(y1)), y2)))) IF1(true, s(z0), s(0), cons(s(0), y1)) -> c3(MAX(cons(s(z0), cons(s(0), y1)))) IF1(true, s(0), s(0), cons(s(s(y0)), y1)) -> c3(MAX(cons(s(0), cons(s(s(y0)), y1)))) IF1(true, s(z0), s(0), cons(0, cons(0, y1))) -> c3(MAX(cons(s(z0), cons(0, cons(0, y1))))) IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c3(MAX(cons(s(s(y0)), cons(0, cons(s(s(y1)), y2))))) IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))) -> c3(MAX(cons(s(z0), cons(0, cons(s(0), y1))))) IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))) -> c3(MAX(cons(s(0), cons(0, cons(s(s(y0)), y1))))) IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c3(MAX(cons(s(s(z0)), cons(s(s(y1)), y2)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c3(MAX(cons(s(s(z0)), cons(s(0), y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c3(MAX(cons(s(s(z0)), cons(0, cons(0, y1))))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c3(MAX(cons(s(s(z0)), cons(0, cons(s(s(y1)), y2))))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c3(MAX(cons(s(s(z0)), cons(0, cons(s(0), y1))))) IF1(false, 0, s(s(y0)), cons(s(s(y1)), y2)) -> c4(MAX(cons(s(s(y0)), cons(s(s(y1)), y2)))) IF1(false, 0, s(z0), cons(s(0), y1)) -> c4(MAX(cons(s(z0), cons(s(0), y1)))) IF1(false, 0, s(0), cons(s(s(y0)), y1)) -> c4(MAX(cons(s(0), cons(s(s(y0)), y1)))) IF1(false, 0, s(z0), cons(0, cons(0, y1))) -> c4(MAX(cons(s(z0), cons(0, cons(0, y1))))) IF1(false, 0, s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(MAX(cons(s(s(y0)), cons(0, cons(s(s(y1)), y2))))) IF1(false, 0, s(z0), cons(0, cons(s(0), y1))) -> c4(MAX(cons(s(z0), cons(0, cons(s(0), y1))))) IF1(false, 0, s(0), cons(0, cons(s(s(y0)), y1))) -> c4(MAX(cons(s(0), cons(0, cons(s(s(y0)), y1))))) K tuples: SORT(cons(x0, cons(x1, x2))) -> c15(DEL(max(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) SORT(cons(x0, nil)) -> c15(DEL(max(cons(x0, nil)), cons(x0, nil))) SORT(cons(z0, cons(0, y1))) -> c14(MAX(cons(z0, cons(0, y1)))) SORT(cons(0, cons(s(y0), y1))) -> c14(MAX(cons(0, cons(s(y0), y1)))) SORT(cons(s(y0), cons(s(0), y1))) -> c14(MAX(cons(s(y0), cons(s(0), y1)))) SORT(cons(s(0), cons(s(s(y0)), y1))) -> c14(MAX(cons(s(0), cons(s(s(y0)), y1)))) SORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c14(MAX(cons(s(s(y0)), cons(s(s(y1)), y2)))) SORT(cons(s(y0), cons(s(y1), y2))) -> c14(MAX(cons(s(y0), cons(s(y1), y2)))) Defined Rule Symbols: ge_2, eq_2, del_2, max_1, if1_4, if2_4 Defined Pair Symbols: MAX_1, DEL_2, IF1_4, SORT_1, IF2_4, EQ_2, GE_2 Compound Symbols: c2_1, c6_1, c2_2, c6_2, c4_1, c15_3, c8_1, c15_1, c15_2, c12_1, c6, c14_1, c18_1, c3_1 ---------------------------------------- (159) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace MAX(cons(0, cons(s(z0), x2))) -> c2(IF1(false, 0, s(z0), x2)) by MAX(cons(0, cons(s(s(y0)), cons(s(s(y1)), y2)))) -> c2(IF1(false, 0, s(s(y0)), cons(s(s(y1)), y2))) MAX(cons(0, cons(s(z0), cons(s(0), y1)))) -> c2(IF1(false, 0, s(z0), cons(s(0), y1))) MAX(cons(0, cons(s(0), cons(s(s(y0)), y1)))) -> c2(IF1(false, 0, s(0), cons(s(s(y0)), y1))) MAX(cons(0, cons(s(z0), cons(0, cons(0, y1))))) -> c2(IF1(false, 0, s(z0), cons(0, cons(0, y1)))) MAX(cons(0, cons(s(s(y0)), cons(0, cons(s(s(y1)), y2))))) -> c2(IF1(false, 0, s(s(y0)), cons(0, cons(s(s(y1)), y2)))) MAX(cons(0, cons(s(z0), cons(0, cons(s(0), y1))))) -> c2(IF1(false, 0, s(z0), cons(0, cons(s(0), y1)))) MAX(cons(0, cons(s(0), cons(0, cons(s(s(y0)), y1))))) -> c2(IF1(false, 0, s(0), cons(0, cons(s(s(y0)), y1)))) ---------------------------------------- (160) 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, 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)) Tuples: 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)) 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)))) 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)))) IF1(false, s(0), s(s(x0)), x1) -> c4(MAX(cons(s(s(x0)), x1))) IF1(false, s(s(x0)), s(s(x1)), x2) -> c4(MAX(cons(s(s(x1)), x2))) SORT(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)))) 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(x0)), s(s(x1)), x2) -> c8(DEL(s(s(x0)), x2)) SORT(cons(z0, cons(z1, z2))) -> c15(DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) SORT(cons(z0, nil)) -> c15(DEL(z0, cons(z0, nil))) 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 SORT(cons(z0, cons(0, y1))) -> c14(MAX(cons(z0, cons(0, y1)))) SORT(cons(0, cons(s(y0), y1))) -> c14(MAX(cons(0, cons(s(y0), y1)))) SORT(cons(s(y0), cons(s(0), y1))) -> c14(MAX(cons(s(y0), cons(s(0), y1)))) SORT(cons(s(0), cons(s(s(y0)), y1))) -> c14(MAX(cons(s(0), cons(s(s(y0)), y1)))) SORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c14(MAX(cons(s(s(y0)), cons(s(s(y1)), y2)))) SORT(cons(s(y0), cons(s(y1), y2))) -> c14(MAX(cons(s(y0), cons(s(y1), y2)))) GE(s(s(y0)), s(s(y1))) -> c18(GE(s(y0), s(y1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2)) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2)) MAX(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c2(GE(s(s(y0)), s(s(y1)))) SORT(cons(z0, cons(0, z1))) -> c15(SORT(del(if1(true, z0, 0, z1), cons(z0, cons(0, z1)))), DEL(if1(true, z0, 0, z1), cons(z0, cons(0, z1))), MAX(cons(z0, cons(0, z1)))) SORT(cons(0, cons(s(z0), z1))) -> c15(SORT(del(if1(false, 0, s(z0), z1), cons(0, cons(s(z0), z1)))), DEL(if1(false, 0, s(z0), z1), cons(0, cons(s(z0), z1))), MAX(cons(0, cons(s(z0), z1)))) SORT(cons(s(z0), cons(s(z1), z2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(ge(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MAX(cons(s(z0), cons(s(z1), z2)))) DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c6(EQ(s(s(y0)), s(s(y1)))) IF1(true, z0, 0, cons(0, y1)) -> c3(MAX(cons(z0, cons(0, y1)))) IF1(true, 0, 0, cons(s(y0), y1)) -> c3(MAX(cons(0, cons(s(y0), y1)))) IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c3(MAX(cons(s(s(y0)), cons(s(s(y1)), y2)))) IF1(true, s(y0), 0, cons(s(0), y1)) -> c3(MAX(cons(s(y0), cons(s(0), y1)))) IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c3(MAX(cons(s(0), cons(s(s(y0)), y1)))) MAX(cons(z0, cons(0, cons(0, y1)))) -> c2(IF1(true, z0, 0, cons(0, y1))) MAX(cons(0, cons(0, cons(s(y0), y1)))) -> c2(IF1(true, 0, 0, cons(s(y0), y1))) MAX(cons(s(s(y0)), cons(0, cons(s(s(y1)), y2)))) -> c2(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2))) MAX(cons(s(y0), cons(0, cons(s(0), y1)))) -> c2(IF1(true, s(y0), 0, cons(s(0), y1))) MAX(cons(s(0), cons(0, cons(s(s(y0)), y1)))) -> c2(IF1(true, s(0), 0, cons(s(s(y0)), y1))) IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)) -> c3(MAX(cons(s(s(y0)), cons(s(s(y1)), y2)))) IF1(true, s(z0), s(0), cons(s(0), y1)) -> c3(MAX(cons(s(z0), cons(s(0), y1)))) IF1(true, s(0), s(0), cons(s(s(y0)), y1)) -> c3(MAX(cons(s(0), cons(s(s(y0)), y1)))) IF1(true, s(z0), s(0), cons(0, cons(0, y1))) -> c3(MAX(cons(s(z0), cons(0, cons(0, y1))))) IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c3(MAX(cons(s(s(y0)), cons(0, cons(s(s(y1)), y2))))) IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))) -> c3(MAX(cons(s(z0), cons(0, cons(s(0), y1))))) IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))) -> c3(MAX(cons(s(0), cons(0, cons(s(s(y0)), y1))))) IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c3(MAX(cons(s(s(z0)), cons(s(s(y1)), y2)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c3(MAX(cons(s(s(z0)), cons(s(0), y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c3(MAX(cons(s(s(z0)), cons(0, cons(0, y1))))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c3(MAX(cons(s(s(z0)), cons(0, cons(s(s(y1)), y2))))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c3(MAX(cons(s(s(z0)), cons(0, cons(s(0), y1))))) IF1(false, 0, s(s(y0)), cons(s(s(y1)), y2)) -> c4(MAX(cons(s(s(y0)), cons(s(s(y1)), y2)))) IF1(false, 0, s(z0), cons(s(0), y1)) -> c4(MAX(cons(s(z0), cons(s(0), y1)))) IF1(false, 0, s(0), cons(s(s(y0)), y1)) -> c4(MAX(cons(s(0), cons(s(s(y0)), y1)))) IF1(false, 0, s(z0), cons(0, cons(0, y1))) -> c4(MAX(cons(s(z0), cons(0, cons(0, y1))))) IF1(false, 0, s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(MAX(cons(s(s(y0)), cons(0, cons(s(s(y1)), y2))))) IF1(false, 0, s(z0), cons(0, cons(s(0), y1))) -> c4(MAX(cons(s(z0), cons(0, cons(s(0), y1))))) IF1(false, 0, s(0), cons(0, cons(s(s(y0)), y1))) -> c4(MAX(cons(s(0), cons(0, cons(s(s(y0)), y1))))) MAX(cons(0, cons(s(s(y0)), cons(s(s(y1)), y2)))) -> c2(IF1(false, 0, s(s(y0)), cons(s(s(y1)), y2))) MAX(cons(0, cons(s(z0), cons(s(0), y1)))) -> c2(IF1(false, 0, s(z0), cons(s(0), y1))) MAX(cons(0, cons(s(0), cons(s(s(y0)), y1)))) -> c2(IF1(false, 0, s(0), cons(s(s(y0)), y1))) MAX(cons(0, cons(s(z0), cons(0, cons(0, y1))))) -> c2(IF1(false, 0, s(z0), cons(0, cons(0, y1)))) MAX(cons(0, cons(s(s(y0)), cons(0, cons(s(s(y1)), y2))))) -> c2(IF1(false, 0, s(s(y0)), cons(0, cons(s(s(y1)), y2)))) MAX(cons(0, cons(s(z0), cons(0, cons(s(0), y1))))) -> c2(IF1(false, 0, s(z0), cons(0, cons(s(0), y1)))) MAX(cons(0, cons(s(0), cons(0, cons(s(s(y0)), y1))))) -> c2(IF1(false, 0, s(0), cons(0, cons(s(s(y0)), y1)))) S tuples: 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)) 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)))) 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)))) IF1(false, s(0), s(s(x0)), x1) -> c4(MAX(cons(s(s(x0)), x1))) IF1(false, s(s(x0)), s(s(x1)), x2) -> c4(MAX(cons(s(s(x1)), x2))) SORT(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)))) 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(x0)), s(s(x1)), x2) -> c8(DEL(s(s(x0)), x2)) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) 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 GE(s(s(y0)), s(s(y1))) -> c18(GE(s(y0), s(y1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2)) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2)) MAX(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c2(GE(s(s(y0)), s(s(y1)))) SORT(cons(z0, cons(0, z1))) -> c15(SORT(del(if1(true, z0, 0, z1), cons(z0, cons(0, z1)))), DEL(if1(true, z0, 0, z1), cons(z0, cons(0, z1))), MAX(cons(z0, cons(0, z1)))) SORT(cons(0, cons(s(z0), z1))) -> c15(SORT(del(if1(false, 0, s(z0), z1), cons(0, cons(s(z0), z1)))), DEL(if1(false, 0, s(z0), z1), cons(0, cons(s(z0), z1))), MAX(cons(0, cons(s(z0), z1)))) SORT(cons(s(z0), cons(s(z1), z2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(ge(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MAX(cons(s(z0), cons(s(z1), z2)))) DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c6(EQ(s(s(y0)), s(s(y1)))) IF1(true, z0, 0, cons(0, y1)) -> c3(MAX(cons(z0, cons(0, y1)))) IF1(true, 0, 0, cons(s(y0), y1)) -> c3(MAX(cons(0, cons(s(y0), y1)))) IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c3(MAX(cons(s(s(y0)), cons(s(s(y1)), y2)))) IF1(true, s(y0), 0, cons(s(0), y1)) -> c3(MAX(cons(s(y0), cons(s(0), y1)))) IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c3(MAX(cons(s(0), cons(s(s(y0)), y1)))) MAX(cons(z0, cons(0, cons(0, y1)))) -> c2(IF1(true, z0, 0, cons(0, y1))) MAX(cons(0, cons(0, cons(s(y0), y1)))) -> c2(IF1(true, 0, 0, cons(s(y0), y1))) MAX(cons(s(s(y0)), cons(0, cons(s(s(y1)), y2)))) -> c2(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2))) MAX(cons(s(y0), cons(0, cons(s(0), y1)))) -> c2(IF1(true, s(y0), 0, cons(s(0), y1))) MAX(cons(s(0), cons(0, cons(s(s(y0)), y1)))) -> c2(IF1(true, s(0), 0, cons(s(s(y0)), y1))) IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)) -> c3(MAX(cons(s(s(y0)), cons(s(s(y1)), y2)))) IF1(true, s(z0), s(0), cons(s(0), y1)) -> c3(MAX(cons(s(z0), cons(s(0), y1)))) IF1(true, s(0), s(0), cons(s(s(y0)), y1)) -> c3(MAX(cons(s(0), cons(s(s(y0)), y1)))) IF1(true, s(z0), s(0), cons(0, cons(0, y1))) -> c3(MAX(cons(s(z0), cons(0, cons(0, y1))))) IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c3(MAX(cons(s(s(y0)), cons(0, cons(s(s(y1)), y2))))) IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))) -> c3(MAX(cons(s(z0), cons(0, cons(s(0), y1))))) IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))) -> c3(MAX(cons(s(0), cons(0, cons(s(s(y0)), y1))))) IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c3(MAX(cons(s(s(z0)), cons(s(s(y1)), y2)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c3(MAX(cons(s(s(z0)), cons(s(0), y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c3(MAX(cons(s(s(z0)), cons(0, cons(0, y1))))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c3(MAX(cons(s(s(z0)), cons(0, cons(s(s(y1)), y2))))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c3(MAX(cons(s(s(z0)), cons(0, cons(s(0), y1))))) IF1(false, 0, s(s(y0)), cons(s(s(y1)), y2)) -> c4(MAX(cons(s(s(y0)), cons(s(s(y1)), y2)))) IF1(false, 0, s(z0), cons(s(0), y1)) -> c4(MAX(cons(s(z0), cons(s(0), y1)))) IF1(false, 0, s(0), cons(s(s(y0)), y1)) -> c4(MAX(cons(s(0), cons(s(s(y0)), y1)))) IF1(false, 0, s(z0), cons(0, cons(0, y1))) -> c4(MAX(cons(s(z0), cons(0, cons(0, y1))))) IF1(false, 0, s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(MAX(cons(s(s(y0)), cons(0, cons(s(s(y1)), y2))))) IF1(false, 0, s(z0), cons(0, cons(s(0), y1))) -> c4(MAX(cons(s(z0), cons(0, cons(s(0), y1))))) IF1(false, 0, s(0), cons(0, cons(s(s(y0)), y1))) -> c4(MAX(cons(s(0), cons(0, cons(s(s(y0)), y1))))) MAX(cons(0, cons(s(s(y0)), cons(s(s(y1)), y2)))) -> c2(IF1(false, 0, s(s(y0)), cons(s(s(y1)), y2))) MAX(cons(0, cons(s(z0), cons(s(0), y1)))) -> c2(IF1(false, 0, s(z0), cons(s(0), y1))) MAX(cons(0, cons(s(0), cons(s(s(y0)), y1)))) -> c2(IF1(false, 0, s(0), cons(s(s(y0)), y1))) MAX(cons(0, cons(s(z0), cons(0, cons(0, y1))))) -> c2(IF1(false, 0, s(z0), cons(0, cons(0, y1)))) MAX(cons(0, cons(s(s(y0)), cons(0, cons(s(s(y1)), y2))))) -> c2(IF1(false, 0, s(s(y0)), cons(0, cons(s(s(y1)), y2)))) MAX(cons(0, cons(s(z0), cons(0, cons(s(0), y1))))) -> c2(IF1(false, 0, s(z0), cons(0, cons(s(0), y1)))) MAX(cons(0, cons(s(0), cons(0, cons(s(s(y0)), y1))))) -> c2(IF1(false, 0, s(0), cons(0, cons(s(s(y0)), y1)))) K tuples: SORT(cons(x0, cons(x1, x2))) -> c15(DEL(max(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) SORT(cons(x0, nil)) -> c15(DEL(max(cons(x0, nil)), cons(x0, nil))) SORT(cons(z0, cons(0, y1))) -> c14(MAX(cons(z0, cons(0, y1)))) SORT(cons(0, cons(s(y0), y1))) -> c14(MAX(cons(0, cons(s(y0), y1)))) SORT(cons(s(y0), cons(s(0), y1))) -> c14(MAX(cons(s(y0), cons(s(0), y1)))) SORT(cons(s(0), cons(s(s(y0)), y1))) -> c14(MAX(cons(s(0), cons(s(s(y0)), y1)))) SORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c14(MAX(cons(s(s(y0)), cons(s(s(y1)), y2)))) SORT(cons(s(y0), cons(s(y1), y2))) -> c14(MAX(cons(s(y0), cons(s(y1), y2)))) Defined Rule Symbols: ge_2, eq_2, del_2, max_1, if1_4, if2_4 Defined Pair Symbols: DEL_2, MAX_1, IF1_4, SORT_1, IF2_4, EQ_2, GE_2 Compound Symbols: c6_1, c2_2, c6_2, c4_1, c15_3, c8_1, c15_1, c15_2, c12_1, c6, c14_1, c18_1, c2_1, c3_1 ---------------------------------------- (161) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace IF1(false, s(0), s(s(x0)), x1) -> c4(MAX(cons(s(s(x0)), x1))) by IF1(false, s(0), s(s(z0)), cons(s(s(y1)), y2)) -> c4(MAX(cons(s(s(z0)), cons(s(s(y1)), y2)))) IF1(false, s(0), s(s(z0)), cons(s(0), y1)) -> c4(MAX(cons(s(s(z0)), cons(s(0), y1)))) IF1(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c4(MAX(cons(s(s(z0)), cons(0, cons(0, y1))))) IF1(false, s(0), s(s(z0)), cons(0, cons(s(s(y1)), y2))) -> c4(MAX(cons(s(s(z0)), cons(0, cons(s(s(y1)), y2))))) IF1(false, s(0), s(s(z0)), cons(0, cons(s(0), y1))) -> c4(MAX(cons(s(s(z0)), cons(0, cons(s(0), y1))))) ---------------------------------------- (162) 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, 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)) Tuples: 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)) 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)))) 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)))) IF1(false, s(s(x0)), s(s(x1)), x2) -> c4(MAX(cons(s(s(x1)), x2))) SORT(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)))) 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(x0)), s(s(x1)), x2) -> c8(DEL(s(s(x0)), x2)) SORT(cons(z0, cons(z1, z2))) -> c15(DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) SORT(cons(z0, nil)) -> c15(DEL(z0, cons(z0, nil))) 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 SORT(cons(z0, cons(0, y1))) -> c14(MAX(cons(z0, cons(0, y1)))) SORT(cons(0, cons(s(y0), y1))) -> c14(MAX(cons(0, cons(s(y0), y1)))) SORT(cons(s(y0), cons(s(0), y1))) -> c14(MAX(cons(s(y0), cons(s(0), y1)))) SORT(cons(s(0), cons(s(s(y0)), y1))) -> c14(MAX(cons(s(0), cons(s(s(y0)), y1)))) SORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c14(MAX(cons(s(s(y0)), cons(s(s(y1)), y2)))) SORT(cons(s(y0), cons(s(y1), y2))) -> c14(MAX(cons(s(y0), cons(s(y1), y2)))) GE(s(s(y0)), s(s(y1))) -> c18(GE(s(y0), s(y1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2)) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2)) MAX(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c2(GE(s(s(y0)), s(s(y1)))) SORT(cons(z0, cons(0, z1))) -> c15(SORT(del(if1(true, z0, 0, z1), cons(z0, cons(0, z1)))), DEL(if1(true, z0, 0, z1), cons(z0, cons(0, z1))), MAX(cons(z0, cons(0, z1)))) SORT(cons(0, cons(s(z0), z1))) -> c15(SORT(del(if1(false, 0, s(z0), z1), cons(0, cons(s(z0), z1)))), DEL(if1(false, 0, s(z0), z1), cons(0, cons(s(z0), z1))), MAX(cons(0, cons(s(z0), z1)))) SORT(cons(s(z0), cons(s(z1), z2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(ge(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MAX(cons(s(z0), cons(s(z1), z2)))) DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c6(EQ(s(s(y0)), s(s(y1)))) IF1(true, z0, 0, cons(0, y1)) -> c3(MAX(cons(z0, cons(0, y1)))) IF1(true, 0, 0, cons(s(y0), y1)) -> c3(MAX(cons(0, cons(s(y0), y1)))) IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c3(MAX(cons(s(s(y0)), cons(s(s(y1)), y2)))) IF1(true, s(y0), 0, cons(s(0), y1)) -> c3(MAX(cons(s(y0), cons(s(0), y1)))) IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c3(MAX(cons(s(0), cons(s(s(y0)), y1)))) MAX(cons(z0, cons(0, cons(0, y1)))) -> c2(IF1(true, z0, 0, cons(0, y1))) MAX(cons(0, cons(0, cons(s(y0), y1)))) -> c2(IF1(true, 0, 0, cons(s(y0), y1))) MAX(cons(s(s(y0)), cons(0, cons(s(s(y1)), y2)))) -> c2(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2))) MAX(cons(s(y0), cons(0, cons(s(0), y1)))) -> c2(IF1(true, s(y0), 0, cons(s(0), y1))) MAX(cons(s(0), cons(0, cons(s(s(y0)), y1)))) -> c2(IF1(true, s(0), 0, cons(s(s(y0)), y1))) IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)) -> c3(MAX(cons(s(s(y0)), cons(s(s(y1)), y2)))) IF1(true, s(z0), s(0), cons(s(0), y1)) -> c3(MAX(cons(s(z0), cons(s(0), y1)))) IF1(true, s(0), s(0), cons(s(s(y0)), y1)) -> c3(MAX(cons(s(0), cons(s(s(y0)), y1)))) IF1(true, s(z0), s(0), cons(0, cons(0, y1))) -> c3(MAX(cons(s(z0), cons(0, cons(0, y1))))) IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c3(MAX(cons(s(s(y0)), cons(0, cons(s(s(y1)), y2))))) IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))) -> c3(MAX(cons(s(z0), cons(0, cons(s(0), y1))))) IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))) -> c3(MAX(cons(s(0), cons(0, cons(s(s(y0)), y1))))) IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c3(MAX(cons(s(s(z0)), cons(s(s(y1)), y2)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c3(MAX(cons(s(s(z0)), cons(s(0), y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c3(MAX(cons(s(s(z0)), cons(0, cons(0, y1))))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c3(MAX(cons(s(s(z0)), cons(0, cons(s(s(y1)), y2))))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c3(MAX(cons(s(s(z0)), cons(0, cons(s(0), y1))))) IF1(false, 0, s(s(y0)), cons(s(s(y1)), y2)) -> c4(MAX(cons(s(s(y0)), cons(s(s(y1)), y2)))) IF1(false, 0, s(z0), cons(s(0), y1)) -> c4(MAX(cons(s(z0), cons(s(0), y1)))) IF1(false, 0, s(0), cons(s(s(y0)), y1)) -> c4(MAX(cons(s(0), cons(s(s(y0)), y1)))) IF1(false, 0, s(z0), cons(0, cons(0, y1))) -> c4(MAX(cons(s(z0), cons(0, cons(0, y1))))) IF1(false, 0, s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(MAX(cons(s(s(y0)), cons(0, cons(s(s(y1)), y2))))) IF1(false, 0, s(z0), cons(0, cons(s(0), y1))) -> c4(MAX(cons(s(z0), cons(0, cons(s(0), y1))))) IF1(false, 0, s(0), cons(0, cons(s(s(y0)), y1))) -> c4(MAX(cons(s(0), cons(0, cons(s(s(y0)), y1))))) MAX(cons(0, cons(s(s(y0)), cons(s(s(y1)), y2)))) -> c2(IF1(false, 0, s(s(y0)), cons(s(s(y1)), y2))) MAX(cons(0, cons(s(z0), cons(s(0), y1)))) -> c2(IF1(false, 0, s(z0), cons(s(0), y1))) MAX(cons(0, cons(s(0), cons(s(s(y0)), y1)))) -> c2(IF1(false, 0, s(0), cons(s(s(y0)), y1))) MAX(cons(0, cons(s(z0), cons(0, cons(0, y1))))) -> c2(IF1(false, 0, s(z0), cons(0, cons(0, y1)))) MAX(cons(0, cons(s(s(y0)), cons(0, cons(s(s(y1)), y2))))) -> c2(IF1(false, 0, s(s(y0)), cons(0, cons(s(s(y1)), y2)))) MAX(cons(0, cons(s(z0), cons(0, cons(s(0), y1))))) -> c2(IF1(false, 0, s(z0), cons(0, cons(s(0), y1)))) MAX(cons(0, cons(s(0), cons(0, cons(s(s(y0)), y1))))) -> c2(IF1(false, 0, s(0), cons(0, cons(s(s(y0)), y1)))) IF1(false, s(0), s(s(z0)), cons(s(s(y1)), y2)) -> c4(MAX(cons(s(s(z0)), cons(s(s(y1)), y2)))) IF1(false, s(0), s(s(z0)), cons(s(0), y1)) -> c4(MAX(cons(s(s(z0)), cons(s(0), y1)))) IF1(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c4(MAX(cons(s(s(z0)), cons(0, cons(0, y1))))) IF1(false, s(0), s(s(z0)), cons(0, cons(s(s(y1)), y2))) -> c4(MAX(cons(s(s(z0)), cons(0, cons(s(s(y1)), y2))))) IF1(false, s(0), s(s(z0)), cons(0, cons(s(0), y1))) -> c4(MAX(cons(s(s(z0)), cons(0, cons(s(0), y1))))) S tuples: 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)) 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)))) 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)))) IF1(false, s(s(x0)), s(s(x1)), x2) -> c4(MAX(cons(s(s(x1)), x2))) SORT(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)))) 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(x0)), s(s(x1)), x2) -> c8(DEL(s(s(x0)), x2)) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) 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 GE(s(s(y0)), s(s(y1))) -> c18(GE(s(y0), s(y1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2)) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2)) MAX(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c2(GE(s(s(y0)), s(s(y1)))) SORT(cons(z0, cons(0, z1))) -> c15(SORT(del(if1(true, z0, 0, z1), cons(z0, cons(0, z1)))), DEL(if1(true, z0, 0, z1), cons(z0, cons(0, z1))), MAX(cons(z0, cons(0, z1)))) SORT(cons(0, cons(s(z0), z1))) -> c15(SORT(del(if1(false, 0, s(z0), z1), cons(0, cons(s(z0), z1)))), DEL(if1(false, 0, s(z0), z1), cons(0, cons(s(z0), z1))), MAX(cons(0, cons(s(z0), z1)))) SORT(cons(s(z0), cons(s(z1), z2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(ge(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MAX(cons(s(z0), cons(s(z1), z2)))) DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c6(EQ(s(s(y0)), s(s(y1)))) IF1(true, z0, 0, cons(0, y1)) -> c3(MAX(cons(z0, cons(0, y1)))) IF1(true, 0, 0, cons(s(y0), y1)) -> c3(MAX(cons(0, cons(s(y0), y1)))) IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c3(MAX(cons(s(s(y0)), cons(s(s(y1)), y2)))) IF1(true, s(y0), 0, cons(s(0), y1)) -> c3(MAX(cons(s(y0), cons(s(0), y1)))) IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c3(MAX(cons(s(0), cons(s(s(y0)), y1)))) MAX(cons(z0, cons(0, cons(0, y1)))) -> c2(IF1(true, z0, 0, cons(0, y1))) MAX(cons(0, cons(0, cons(s(y0), y1)))) -> c2(IF1(true, 0, 0, cons(s(y0), y1))) MAX(cons(s(s(y0)), cons(0, cons(s(s(y1)), y2)))) -> c2(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2))) MAX(cons(s(y0), cons(0, cons(s(0), y1)))) -> c2(IF1(true, s(y0), 0, cons(s(0), y1))) MAX(cons(s(0), cons(0, cons(s(s(y0)), y1)))) -> c2(IF1(true, s(0), 0, cons(s(s(y0)), y1))) IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)) -> c3(MAX(cons(s(s(y0)), cons(s(s(y1)), y2)))) IF1(true, s(z0), s(0), cons(s(0), y1)) -> c3(MAX(cons(s(z0), cons(s(0), y1)))) IF1(true, s(0), s(0), cons(s(s(y0)), y1)) -> c3(MAX(cons(s(0), cons(s(s(y0)), y1)))) IF1(true, s(z0), s(0), cons(0, cons(0, y1))) -> c3(MAX(cons(s(z0), cons(0, cons(0, y1))))) IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c3(MAX(cons(s(s(y0)), cons(0, cons(s(s(y1)), y2))))) IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))) -> c3(MAX(cons(s(z0), cons(0, cons(s(0), y1))))) IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))) -> c3(MAX(cons(s(0), cons(0, cons(s(s(y0)), y1))))) IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c3(MAX(cons(s(s(z0)), cons(s(s(y1)), y2)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c3(MAX(cons(s(s(z0)), cons(s(0), y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c3(MAX(cons(s(s(z0)), cons(0, cons(0, y1))))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c3(MAX(cons(s(s(z0)), cons(0, cons(s(s(y1)), y2))))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c3(MAX(cons(s(s(z0)), cons(0, cons(s(0), y1))))) IF1(false, 0, s(s(y0)), cons(s(s(y1)), y2)) -> c4(MAX(cons(s(s(y0)), cons(s(s(y1)), y2)))) IF1(false, 0, s(z0), cons(s(0), y1)) -> c4(MAX(cons(s(z0), cons(s(0), y1)))) IF1(false, 0, s(0), cons(s(s(y0)), y1)) -> c4(MAX(cons(s(0), cons(s(s(y0)), y1)))) IF1(false, 0, s(z0), cons(0, cons(0, y1))) -> c4(MAX(cons(s(z0), cons(0, cons(0, y1))))) IF1(false, 0, s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(MAX(cons(s(s(y0)), cons(0, cons(s(s(y1)), y2))))) IF1(false, 0, s(z0), cons(0, cons(s(0), y1))) -> c4(MAX(cons(s(z0), cons(0, cons(s(0), y1))))) IF1(false, 0, s(0), cons(0, cons(s(s(y0)), y1))) -> c4(MAX(cons(s(0), cons(0, cons(s(s(y0)), y1))))) MAX(cons(0, cons(s(s(y0)), cons(s(s(y1)), y2)))) -> c2(IF1(false, 0, s(s(y0)), cons(s(s(y1)), y2))) MAX(cons(0, cons(s(z0), cons(s(0), y1)))) -> c2(IF1(false, 0, s(z0), cons(s(0), y1))) MAX(cons(0, cons(s(0), cons(s(s(y0)), y1)))) -> c2(IF1(false, 0, s(0), cons(s(s(y0)), y1))) MAX(cons(0, cons(s(z0), cons(0, cons(0, y1))))) -> c2(IF1(false, 0, s(z0), cons(0, cons(0, y1)))) MAX(cons(0, cons(s(s(y0)), cons(0, cons(s(s(y1)), y2))))) -> c2(IF1(false, 0, s(s(y0)), cons(0, cons(s(s(y1)), y2)))) MAX(cons(0, cons(s(z0), cons(0, cons(s(0), y1))))) -> c2(IF1(false, 0, s(z0), cons(0, cons(s(0), y1)))) MAX(cons(0, cons(s(0), cons(0, cons(s(s(y0)), y1))))) -> c2(IF1(false, 0, s(0), cons(0, cons(s(s(y0)), y1)))) IF1(false, s(0), s(s(z0)), cons(s(s(y1)), y2)) -> c4(MAX(cons(s(s(z0)), cons(s(s(y1)), y2)))) IF1(false, s(0), s(s(z0)), cons(s(0), y1)) -> c4(MAX(cons(s(s(z0)), cons(s(0), y1)))) IF1(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c4(MAX(cons(s(s(z0)), cons(0, cons(0, y1))))) IF1(false, s(0), s(s(z0)), cons(0, cons(s(s(y1)), y2))) -> c4(MAX(cons(s(s(z0)), cons(0, cons(s(s(y1)), y2))))) IF1(false, s(0), s(s(z0)), cons(0, cons(s(0), y1))) -> c4(MAX(cons(s(s(z0)), cons(0, cons(s(0), y1))))) K tuples: SORT(cons(x0, cons(x1, x2))) -> c15(DEL(max(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) SORT(cons(x0, nil)) -> c15(DEL(max(cons(x0, nil)), cons(x0, nil))) SORT(cons(z0, cons(0, y1))) -> c14(MAX(cons(z0, cons(0, y1)))) SORT(cons(0, cons(s(y0), y1))) -> c14(MAX(cons(0, cons(s(y0), y1)))) SORT(cons(s(y0), cons(s(0), y1))) -> c14(MAX(cons(s(y0), cons(s(0), y1)))) SORT(cons(s(0), cons(s(s(y0)), y1))) -> c14(MAX(cons(s(0), cons(s(s(y0)), y1)))) SORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c14(MAX(cons(s(s(y0)), cons(s(s(y1)), y2)))) SORT(cons(s(y0), cons(s(y1), y2))) -> c14(MAX(cons(s(y0), cons(s(y1), y2)))) Defined Rule Symbols: ge_2, eq_2, del_2, max_1, if1_4, if2_4 Defined Pair Symbols: DEL_2, MAX_1, IF1_4, SORT_1, IF2_4, EQ_2, GE_2 Compound Symbols: c6_1, c2_2, c6_2, c4_1, c15_3, c8_1, c15_1, c15_2, c12_1, c6, c14_1, c18_1, c2_1, c3_1 ---------------------------------------- (163) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace IF1(false, s(s(x0)), s(s(x1)), x2) -> c4(MAX(cons(s(s(x1)), x2))) by IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c4(MAX(cons(s(s(z1)), cons(s(s(y1)), y2)))) IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c4(MAX(cons(s(s(z1)), cons(s(0), y1)))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c4(MAX(cons(s(s(z1)), cons(0, cons(0, y1))))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c4(MAX(cons(s(s(z1)), cons(0, cons(s(s(y1)), y2))))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c4(MAX(cons(s(s(z1)), cons(0, cons(s(0), y1))))) ---------------------------------------- (164) 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, 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)) Tuples: 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)) 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)))) 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)))) SORT(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)))) 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(x0)), s(s(x1)), x2) -> c8(DEL(s(s(x0)), x2)) SORT(cons(z0, cons(z1, z2))) -> c15(DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) SORT(cons(z0, nil)) -> c15(DEL(z0, cons(z0, nil))) 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 SORT(cons(z0, cons(0, y1))) -> c14(MAX(cons(z0, cons(0, y1)))) SORT(cons(0, cons(s(y0), y1))) -> c14(MAX(cons(0, cons(s(y0), y1)))) SORT(cons(s(y0), cons(s(0), y1))) -> c14(MAX(cons(s(y0), cons(s(0), y1)))) SORT(cons(s(0), cons(s(s(y0)), y1))) -> c14(MAX(cons(s(0), cons(s(s(y0)), y1)))) SORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c14(MAX(cons(s(s(y0)), cons(s(s(y1)), y2)))) SORT(cons(s(y0), cons(s(y1), y2))) -> c14(MAX(cons(s(y0), cons(s(y1), y2)))) GE(s(s(y0)), s(s(y1))) -> c18(GE(s(y0), s(y1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2)) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2)) MAX(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c2(GE(s(s(y0)), s(s(y1)))) SORT(cons(z0, cons(0, z1))) -> c15(SORT(del(if1(true, z0, 0, z1), cons(z0, cons(0, z1)))), DEL(if1(true, z0, 0, z1), cons(z0, cons(0, z1))), MAX(cons(z0, cons(0, z1)))) SORT(cons(0, cons(s(z0), z1))) -> c15(SORT(del(if1(false, 0, s(z0), z1), cons(0, cons(s(z0), z1)))), DEL(if1(false, 0, s(z0), z1), cons(0, cons(s(z0), z1))), MAX(cons(0, cons(s(z0), z1)))) SORT(cons(s(z0), cons(s(z1), z2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(ge(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MAX(cons(s(z0), cons(s(z1), z2)))) DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c6(EQ(s(s(y0)), s(s(y1)))) IF1(true, z0, 0, cons(0, y1)) -> c3(MAX(cons(z0, cons(0, y1)))) IF1(true, 0, 0, cons(s(y0), y1)) -> c3(MAX(cons(0, cons(s(y0), y1)))) IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c3(MAX(cons(s(s(y0)), cons(s(s(y1)), y2)))) IF1(true, s(y0), 0, cons(s(0), y1)) -> c3(MAX(cons(s(y0), cons(s(0), y1)))) IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c3(MAX(cons(s(0), cons(s(s(y0)), y1)))) MAX(cons(z0, cons(0, cons(0, y1)))) -> c2(IF1(true, z0, 0, cons(0, y1))) MAX(cons(0, cons(0, cons(s(y0), y1)))) -> c2(IF1(true, 0, 0, cons(s(y0), y1))) MAX(cons(s(s(y0)), cons(0, cons(s(s(y1)), y2)))) -> c2(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2))) MAX(cons(s(y0), cons(0, cons(s(0), y1)))) -> c2(IF1(true, s(y0), 0, cons(s(0), y1))) MAX(cons(s(0), cons(0, cons(s(s(y0)), y1)))) -> c2(IF1(true, s(0), 0, cons(s(s(y0)), y1))) IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)) -> c3(MAX(cons(s(s(y0)), cons(s(s(y1)), y2)))) IF1(true, s(z0), s(0), cons(s(0), y1)) -> c3(MAX(cons(s(z0), cons(s(0), y1)))) IF1(true, s(0), s(0), cons(s(s(y0)), y1)) -> c3(MAX(cons(s(0), cons(s(s(y0)), y1)))) IF1(true, s(z0), s(0), cons(0, cons(0, y1))) -> c3(MAX(cons(s(z0), cons(0, cons(0, y1))))) IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c3(MAX(cons(s(s(y0)), cons(0, cons(s(s(y1)), y2))))) IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))) -> c3(MAX(cons(s(z0), cons(0, cons(s(0), y1))))) IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))) -> c3(MAX(cons(s(0), cons(0, cons(s(s(y0)), y1))))) IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c3(MAX(cons(s(s(z0)), cons(s(s(y1)), y2)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c3(MAX(cons(s(s(z0)), cons(s(0), y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c3(MAX(cons(s(s(z0)), cons(0, cons(0, y1))))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c3(MAX(cons(s(s(z0)), cons(0, cons(s(s(y1)), y2))))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c3(MAX(cons(s(s(z0)), cons(0, cons(s(0), y1))))) IF1(false, 0, s(s(y0)), cons(s(s(y1)), y2)) -> c4(MAX(cons(s(s(y0)), cons(s(s(y1)), y2)))) IF1(false, 0, s(z0), cons(s(0), y1)) -> c4(MAX(cons(s(z0), cons(s(0), y1)))) IF1(false, 0, s(0), cons(s(s(y0)), y1)) -> c4(MAX(cons(s(0), cons(s(s(y0)), y1)))) IF1(false, 0, s(z0), cons(0, cons(0, y1))) -> c4(MAX(cons(s(z0), cons(0, cons(0, y1))))) IF1(false, 0, s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(MAX(cons(s(s(y0)), cons(0, cons(s(s(y1)), y2))))) IF1(false, 0, s(z0), cons(0, cons(s(0), y1))) -> c4(MAX(cons(s(z0), cons(0, cons(s(0), y1))))) IF1(false, 0, s(0), cons(0, cons(s(s(y0)), y1))) -> c4(MAX(cons(s(0), cons(0, cons(s(s(y0)), y1))))) MAX(cons(0, cons(s(s(y0)), cons(s(s(y1)), y2)))) -> c2(IF1(false, 0, s(s(y0)), cons(s(s(y1)), y2))) MAX(cons(0, cons(s(z0), cons(s(0), y1)))) -> c2(IF1(false, 0, s(z0), cons(s(0), y1))) MAX(cons(0, cons(s(0), cons(s(s(y0)), y1)))) -> c2(IF1(false, 0, s(0), cons(s(s(y0)), y1))) MAX(cons(0, cons(s(z0), cons(0, cons(0, y1))))) -> c2(IF1(false, 0, s(z0), cons(0, cons(0, y1)))) MAX(cons(0, cons(s(s(y0)), cons(0, cons(s(s(y1)), y2))))) -> c2(IF1(false, 0, s(s(y0)), cons(0, cons(s(s(y1)), y2)))) MAX(cons(0, cons(s(z0), cons(0, cons(s(0), y1))))) -> c2(IF1(false, 0, s(z0), cons(0, cons(s(0), y1)))) MAX(cons(0, cons(s(0), cons(0, cons(s(s(y0)), y1))))) -> c2(IF1(false, 0, s(0), cons(0, cons(s(s(y0)), y1)))) IF1(false, s(0), s(s(z0)), cons(s(s(y1)), y2)) -> c4(MAX(cons(s(s(z0)), cons(s(s(y1)), y2)))) IF1(false, s(0), s(s(z0)), cons(s(0), y1)) -> c4(MAX(cons(s(s(z0)), cons(s(0), y1)))) IF1(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c4(MAX(cons(s(s(z0)), cons(0, cons(0, y1))))) IF1(false, s(0), s(s(z0)), cons(0, cons(s(s(y1)), y2))) -> c4(MAX(cons(s(s(z0)), cons(0, cons(s(s(y1)), y2))))) IF1(false, s(0), s(s(z0)), cons(0, cons(s(0), y1))) -> c4(MAX(cons(s(s(z0)), cons(0, cons(s(0), y1))))) IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c4(MAX(cons(s(s(z1)), cons(s(s(y1)), y2)))) IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c4(MAX(cons(s(s(z1)), cons(s(0), y1)))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c4(MAX(cons(s(s(z1)), cons(0, cons(0, y1))))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c4(MAX(cons(s(s(z1)), cons(0, cons(s(s(y1)), y2))))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c4(MAX(cons(s(s(z1)), cons(0, cons(s(0), y1))))) S tuples: 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)) 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)))) 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)))) SORT(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)))) 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(x0)), s(s(x1)), x2) -> c8(DEL(s(s(x0)), x2)) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) 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 GE(s(s(y0)), s(s(y1))) -> c18(GE(s(y0), s(y1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2)) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2)) MAX(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c2(GE(s(s(y0)), s(s(y1)))) SORT(cons(z0, cons(0, z1))) -> c15(SORT(del(if1(true, z0, 0, z1), cons(z0, cons(0, z1)))), DEL(if1(true, z0, 0, z1), cons(z0, cons(0, z1))), MAX(cons(z0, cons(0, z1)))) SORT(cons(0, cons(s(z0), z1))) -> c15(SORT(del(if1(false, 0, s(z0), z1), cons(0, cons(s(z0), z1)))), DEL(if1(false, 0, s(z0), z1), cons(0, cons(s(z0), z1))), MAX(cons(0, cons(s(z0), z1)))) SORT(cons(s(z0), cons(s(z1), z2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(ge(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MAX(cons(s(z0), cons(s(z1), z2)))) DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c6(EQ(s(s(y0)), s(s(y1)))) IF1(true, z0, 0, cons(0, y1)) -> c3(MAX(cons(z0, cons(0, y1)))) IF1(true, 0, 0, cons(s(y0), y1)) -> c3(MAX(cons(0, cons(s(y0), y1)))) IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c3(MAX(cons(s(s(y0)), cons(s(s(y1)), y2)))) IF1(true, s(y0), 0, cons(s(0), y1)) -> c3(MAX(cons(s(y0), cons(s(0), y1)))) IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c3(MAX(cons(s(0), cons(s(s(y0)), y1)))) MAX(cons(z0, cons(0, cons(0, y1)))) -> c2(IF1(true, z0, 0, cons(0, y1))) MAX(cons(0, cons(0, cons(s(y0), y1)))) -> c2(IF1(true, 0, 0, cons(s(y0), y1))) MAX(cons(s(s(y0)), cons(0, cons(s(s(y1)), y2)))) -> c2(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2))) MAX(cons(s(y0), cons(0, cons(s(0), y1)))) -> c2(IF1(true, s(y0), 0, cons(s(0), y1))) MAX(cons(s(0), cons(0, cons(s(s(y0)), y1)))) -> c2(IF1(true, s(0), 0, cons(s(s(y0)), y1))) IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)) -> c3(MAX(cons(s(s(y0)), cons(s(s(y1)), y2)))) IF1(true, s(z0), s(0), cons(s(0), y1)) -> c3(MAX(cons(s(z0), cons(s(0), y1)))) IF1(true, s(0), s(0), cons(s(s(y0)), y1)) -> c3(MAX(cons(s(0), cons(s(s(y0)), y1)))) IF1(true, s(z0), s(0), cons(0, cons(0, y1))) -> c3(MAX(cons(s(z0), cons(0, cons(0, y1))))) IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c3(MAX(cons(s(s(y0)), cons(0, cons(s(s(y1)), y2))))) IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))) -> c3(MAX(cons(s(z0), cons(0, cons(s(0), y1))))) IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))) -> c3(MAX(cons(s(0), cons(0, cons(s(s(y0)), y1))))) IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c3(MAX(cons(s(s(z0)), cons(s(s(y1)), y2)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c3(MAX(cons(s(s(z0)), cons(s(0), y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c3(MAX(cons(s(s(z0)), cons(0, cons(0, y1))))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c3(MAX(cons(s(s(z0)), cons(0, cons(s(s(y1)), y2))))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c3(MAX(cons(s(s(z0)), cons(0, cons(s(0), y1))))) IF1(false, 0, s(s(y0)), cons(s(s(y1)), y2)) -> c4(MAX(cons(s(s(y0)), cons(s(s(y1)), y2)))) IF1(false, 0, s(z0), cons(s(0), y1)) -> c4(MAX(cons(s(z0), cons(s(0), y1)))) IF1(false, 0, s(0), cons(s(s(y0)), y1)) -> c4(MAX(cons(s(0), cons(s(s(y0)), y1)))) IF1(false, 0, s(z0), cons(0, cons(0, y1))) -> c4(MAX(cons(s(z0), cons(0, cons(0, y1))))) IF1(false, 0, s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(MAX(cons(s(s(y0)), cons(0, cons(s(s(y1)), y2))))) IF1(false, 0, s(z0), cons(0, cons(s(0), y1))) -> c4(MAX(cons(s(z0), cons(0, cons(s(0), y1))))) IF1(false, 0, s(0), cons(0, cons(s(s(y0)), y1))) -> c4(MAX(cons(s(0), cons(0, cons(s(s(y0)), y1))))) MAX(cons(0, cons(s(s(y0)), cons(s(s(y1)), y2)))) -> c2(IF1(false, 0, s(s(y0)), cons(s(s(y1)), y2))) MAX(cons(0, cons(s(z0), cons(s(0), y1)))) -> c2(IF1(false, 0, s(z0), cons(s(0), y1))) MAX(cons(0, cons(s(0), cons(s(s(y0)), y1)))) -> c2(IF1(false, 0, s(0), cons(s(s(y0)), y1))) MAX(cons(0, cons(s(z0), cons(0, cons(0, y1))))) -> c2(IF1(false, 0, s(z0), cons(0, cons(0, y1)))) MAX(cons(0, cons(s(s(y0)), cons(0, cons(s(s(y1)), y2))))) -> c2(IF1(false, 0, s(s(y0)), cons(0, cons(s(s(y1)), y2)))) MAX(cons(0, cons(s(z0), cons(0, cons(s(0), y1))))) -> c2(IF1(false, 0, s(z0), cons(0, cons(s(0), y1)))) MAX(cons(0, cons(s(0), cons(0, cons(s(s(y0)), y1))))) -> c2(IF1(false, 0, s(0), cons(0, cons(s(s(y0)), y1)))) IF1(false, s(0), s(s(z0)), cons(s(s(y1)), y2)) -> c4(MAX(cons(s(s(z0)), cons(s(s(y1)), y2)))) IF1(false, s(0), s(s(z0)), cons(s(0), y1)) -> c4(MAX(cons(s(s(z0)), cons(s(0), y1)))) IF1(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c4(MAX(cons(s(s(z0)), cons(0, cons(0, y1))))) IF1(false, s(0), s(s(z0)), cons(0, cons(s(s(y1)), y2))) -> c4(MAX(cons(s(s(z0)), cons(0, cons(s(s(y1)), y2))))) IF1(false, s(0), s(s(z0)), cons(0, cons(s(0), y1))) -> c4(MAX(cons(s(s(z0)), cons(0, cons(s(0), y1))))) IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c4(MAX(cons(s(s(z1)), cons(s(s(y1)), y2)))) IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c4(MAX(cons(s(s(z1)), cons(s(0), y1)))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c4(MAX(cons(s(s(z1)), cons(0, cons(0, y1))))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c4(MAX(cons(s(s(z1)), cons(0, cons(s(s(y1)), y2))))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c4(MAX(cons(s(s(z1)), cons(0, cons(s(0), y1))))) K tuples: SORT(cons(x0, cons(x1, x2))) -> c15(DEL(max(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) SORT(cons(x0, nil)) -> c15(DEL(max(cons(x0, nil)), cons(x0, nil))) SORT(cons(z0, cons(0, y1))) -> c14(MAX(cons(z0, cons(0, y1)))) SORT(cons(0, cons(s(y0), y1))) -> c14(MAX(cons(0, cons(s(y0), y1)))) SORT(cons(s(y0), cons(s(0), y1))) -> c14(MAX(cons(s(y0), cons(s(0), y1)))) SORT(cons(s(0), cons(s(s(y0)), y1))) -> c14(MAX(cons(s(0), cons(s(s(y0)), y1)))) SORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c14(MAX(cons(s(s(y0)), cons(s(s(y1)), y2)))) SORT(cons(s(y0), cons(s(y1), y2))) -> c14(MAX(cons(s(y0), cons(s(y1), y2)))) Defined Rule Symbols: ge_2, eq_2, del_2, max_1, if1_4, if2_4 Defined Pair Symbols: DEL_2, MAX_1, SORT_1, IF2_4, EQ_2, GE_2, IF1_4 Compound Symbols: c6_1, c2_2, c6_2, c15_3, c8_1, c15_1, c15_2, c12_1, c6, c14_1, c18_1, c2_1, c3_1, c4_1 ---------------------------------------- (165) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace IF2(false, 0, s(x0), x1) -> c8(DEL(0, x1)) by IF2(false, 0, s(z0), cons(s(y0), y1)) -> c8(DEL(0, cons(s(y0), y1))) ---------------------------------------- (166) 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, 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)) Tuples: 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)) 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)))) 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)))) SORT(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)))) 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(x0)), s(s(x1)), x2) -> c8(DEL(s(s(x0)), x2)) SORT(cons(z0, cons(z1, z2))) -> c15(DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) SORT(cons(z0, nil)) -> c15(DEL(z0, cons(z0, nil))) 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 SORT(cons(z0, cons(0, y1))) -> c14(MAX(cons(z0, cons(0, y1)))) SORT(cons(0, cons(s(y0), y1))) -> c14(MAX(cons(0, cons(s(y0), y1)))) SORT(cons(s(y0), cons(s(0), y1))) -> c14(MAX(cons(s(y0), cons(s(0), y1)))) SORT(cons(s(0), cons(s(s(y0)), y1))) -> c14(MAX(cons(s(0), cons(s(s(y0)), y1)))) SORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c14(MAX(cons(s(s(y0)), cons(s(s(y1)), y2)))) SORT(cons(s(y0), cons(s(y1), y2))) -> c14(MAX(cons(s(y0), cons(s(y1), y2)))) GE(s(s(y0)), s(s(y1))) -> c18(GE(s(y0), s(y1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2)) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2)) MAX(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c2(GE(s(s(y0)), s(s(y1)))) SORT(cons(z0, cons(0, z1))) -> c15(SORT(del(if1(true, z0, 0, z1), cons(z0, cons(0, z1)))), DEL(if1(true, z0, 0, z1), cons(z0, cons(0, z1))), MAX(cons(z0, cons(0, z1)))) SORT(cons(0, cons(s(z0), z1))) -> c15(SORT(del(if1(false, 0, s(z0), z1), cons(0, cons(s(z0), z1)))), DEL(if1(false, 0, s(z0), z1), cons(0, cons(s(z0), z1))), MAX(cons(0, cons(s(z0), z1)))) SORT(cons(s(z0), cons(s(z1), z2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(ge(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MAX(cons(s(z0), cons(s(z1), z2)))) DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c6(EQ(s(s(y0)), s(s(y1)))) IF1(true, z0, 0, cons(0, y1)) -> c3(MAX(cons(z0, cons(0, y1)))) IF1(true, 0, 0, cons(s(y0), y1)) -> c3(MAX(cons(0, cons(s(y0), y1)))) IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c3(MAX(cons(s(s(y0)), cons(s(s(y1)), y2)))) IF1(true, s(y0), 0, cons(s(0), y1)) -> c3(MAX(cons(s(y0), cons(s(0), y1)))) IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c3(MAX(cons(s(0), cons(s(s(y0)), y1)))) MAX(cons(z0, cons(0, cons(0, y1)))) -> c2(IF1(true, z0, 0, cons(0, y1))) MAX(cons(0, cons(0, cons(s(y0), y1)))) -> c2(IF1(true, 0, 0, cons(s(y0), y1))) MAX(cons(s(s(y0)), cons(0, cons(s(s(y1)), y2)))) -> c2(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2))) MAX(cons(s(y0), cons(0, cons(s(0), y1)))) -> c2(IF1(true, s(y0), 0, cons(s(0), y1))) MAX(cons(s(0), cons(0, cons(s(s(y0)), y1)))) -> c2(IF1(true, s(0), 0, cons(s(s(y0)), y1))) IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)) -> c3(MAX(cons(s(s(y0)), cons(s(s(y1)), y2)))) IF1(true, s(z0), s(0), cons(s(0), y1)) -> c3(MAX(cons(s(z0), cons(s(0), y1)))) IF1(true, s(0), s(0), cons(s(s(y0)), y1)) -> c3(MAX(cons(s(0), cons(s(s(y0)), y1)))) IF1(true, s(z0), s(0), cons(0, cons(0, y1))) -> c3(MAX(cons(s(z0), cons(0, cons(0, y1))))) IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c3(MAX(cons(s(s(y0)), cons(0, cons(s(s(y1)), y2))))) IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))) -> c3(MAX(cons(s(z0), cons(0, cons(s(0), y1))))) IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))) -> c3(MAX(cons(s(0), cons(0, cons(s(s(y0)), y1))))) IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c3(MAX(cons(s(s(z0)), cons(s(s(y1)), y2)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c3(MAX(cons(s(s(z0)), cons(s(0), y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c3(MAX(cons(s(s(z0)), cons(0, cons(0, y1))))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c3(MAX(cons(s(s(z0)), cons(0, cons(s(s(y1)), y2))))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c3(MAX(cons(s(s(z0)), cons(0, cons(s(0), y1))))) IF1(false, 0, s(s(y0)), cons(s(s(y1)), y2)) -> c4(MAX(cons(s(s(y0)), cons(s(s(y1)), y2)))) IF1(false, 0, s(z0), cons(s(0), y1)) -> c4(MAX(cons(s(z0), cons(s(0), y1)))) IF1(false, 0, s(0), cons(s(s(y0)), y1)) -> c4(MAX(cons(s(0), cons(s(s(y0)), y1)))) IF1(false, 0, s(z0), cons(0, cons(0, y1))) -> c4(MAX(cons(s(z0), cons(0, cons(0, y1))))) IF1(false, 0, s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(MAX(cons(s(s(y0)), cons(0, cons(s(s(y1)), y2))))) IF1(false, 0, s(z0), cons(0, cons(s(0), y1))) -> c4(MAX(cons(s(z0), cons(0, cons(s(0), y1))))) IF1(false, 0, s(0), cons(0, cons(s(s(y0)), y1))) -> c4(MAX(cons(s(0), cons(0, cons(s(s(y0)), y1))))) MAX(cons(0, cons(s(s(y0)), cons(s(s(y1)), y2)))) -> c2(IF1(false, 0, s(s(y0)), cons(s(s(y1)), y2))) MAX(cons(0, cons(s(z0), cons(s(0), y1)))) -> c2(IF1(false, 0, s(z0), cons(s(0), y1))) MAX(cons(0, cons(s(0), cons(s(s(y0)), y1)))) -> c2(IF1(false, 0, s(0), cons(s(s(y0)), y1))) MAX(cons(0, cons(s(z0), cons(0, cons(0, y1))))) -> c2(IF1(false, 0, s(z0), cons(0, cons(0, y1)))) MAX(cons(0, cons(s(s(y0)), cons(0, cons(s(s(y1)), y2))))) -> c2(IF1(false, 0, s(s(y0)), cons(0, cons(s(s(y1)), y2)))) MAX(cons(0, cons(s(z0), cons(0, cons(s(0), y1))))) -> c2(IF1(false, 0, s(z0), cons(0, cons(s(0), y1)))) MAX(cons(0, cons(s(0), cons(0, cons(s(s(y0)), y1))))) -> c2(IF1(false, 0, s(0), cons(0, cons(s(s(y0)), y1)))) IF1(false, s(0), s(s(z0)), cons(s(s(y1)), y2)) -> c4(MAX(cons(s(s(z0)), cons(s(s(y1)), y2)))) IF1(false, s(0), s(s(z0)), cons(s(0), y1)) -> c4(MAX(cons(s(s(z0)), cons(s(0), y1)))) IF1(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c4(MAX(cons(s(s(z0)), cons(0, cons(0, y1))))) IF1(false, s(0), s(s(z0)), cons(0, cons(s(s(y1)), y2))) -> c4(MAX(cons(s(s(z0)), cons(0, cons(s(s(y1)), y2))))) IF1(false, s(0), s(s(z0)), cons(0, cons(s(0), y1))) -> c4(MAX(cons(s(s(z0)), cons(0, cons(s(0), y1))))) IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c4(MAX(cons(s(s(z1)), cons(s(s(y1)), y2)))) IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c4(MAX(cons(s(s(z1)), cons(s(0), y1)))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c4(MAX(cons(s(s(z1)), cons(0, cons(0, y1))))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c4(MAX(cons(s(s(z1)), cons(0, cons(s(s(y1)), y2))))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c4(MAX(cons(s(s(z1)), cons(0, cons(s(0), y1))))) IF2(false, 0, s(z0), cons(s(y0), y1)) -> c8(DEL(0, cons(s(y0), y1))) S tuples: 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)) 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)))) 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)))) SORT(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)))) 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(x0)), s(s(x1)), x2) -> c8(DEL(s(s(x0)), x2)) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) 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 GE(s(s(y0)), s(s(y1))) -> c18(GE(s(y0), s(y1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2)) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2)) MAX(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c2(GE(s(s(y0)), s(s(y1)))) SORT(cons(z0, cons(0, z1))) -> c15(SORT(del(if1(true, z0, 0, z1), cons(z0, cons(0, z1)))), DEL(if1(true, z0, 0, z1), cons(z0, cons(0, z1))), MAX(cons(z0, cons(0, z1)))) SORT(cons(0, cons(s(z0), z1))) -> c15(SORT(del(if1(false, 0, s(z0), z1), cons(0, cons(s(z0), z1)))), DEL(if1(false, 0, s(z0), z1), cons(0, cons(s(z0), z1))), MAX(cons(0, cons(s(z0), z1)))) SORT(cons(s(z0), cons(s(z1), z2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(ge(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MAX(cons(s(z0), cons(s(z1), z2)))) DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c6(EQ(s(s(y0)), s(s(y1)))) IF1(true, z0, 0, cons(0, y1)) -> c3(MAX(cons(z0, cons(0, y1)))) IF1(true, 0, 0, cons(s(y0), y1)) -> c3(MAX(cons(0, cons(s(y0), y1)))) IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c3(MAX(cons(s(s(y0)), cons(s(s(y1)), y2)))) IF1(true, s(y0), 0, cons(s(0), y1)) -> c3(MAX(cons(s(y0), cons(s(0), y1)))) IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c3(MAX(cons(s(0), cons(s(s(y0)), y1)))) MAX(cons(z0, cons(0, cons(0, y1)))) -> c2(IF1(true, z0, 0, cons(0, y1))) MAX(cons(0, cons(0, cons(s(y0), y1)))) -> c2(IF1(true, 0, 0, cons(s(y0), y1))) MAX(cons(s(s(y0)), cons(0, cons(s(s(y1)), y2)))) -> c2(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2))) MAX(cons(s(y0), cons(0, cons(s(0), y1)))) -> c2(IF1(true, s(y0), 0, cons(s(0), y1))) MAX(cons(s(0), cons(0, cons(s(s(y0)), y1)))) -> c2(IF1(true, s(0), 0, cons(s(s(y0)), y1))) IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)) -> c3(MAX(cons(s(s(y0)), cons(s(s(y1)), y2)))) IF1(true, s(z0), s(0), cons(s(0), y1)) -> c3(MAX(cons(s(z0), cons(s(0), y1)))) IF1(true, s(0), s(0), cons(s(s(y0)), y1)) -> c3(MAX(cons(s(0), cons(s(s(y0)), y1)))) IF1(true, s(z0), s(0), cons(0, cons(0, y1))) -> c3(MAX(cons(s(z0), cons(0, cons(0, y1))))) IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c3(MAX(cons(s(s(y0)), cons(0, cons(s(s(y1)), y2))))) IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))) -> c3(MAX(cons(s(z0), cons(0, cons(s(0), y1))))) IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))) -> c3(MAX(cons(s(0), cons(0, cons(s(s(y0)), y1))))) IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c3(MAX(cons(s(s(z0)), cons(s(s(y1)), y2)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c3(MAX(cons(s(s(z0)), cons(s(0), y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c3(MAX(cons(s(s(z0)), cons(0, cons(0, y1))))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c3(MAX(cons(s(s(z0)), cons(0, cons(s(s(y1)), y2))))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c3(MAX(cons(s(s(z0)), cons(0, cons(s(0), y1))))) IF1(false, 0, s(s(y0)), cons(s(s(y1)), y2)) -> c4(MAX(cons(s(s(y0)), cons(s(s(y1)), y2)))) IF1(false, 0, s(z0), cons(s(0), y1)) -> c4(MAX(cons(s(z0), cons(s(0), y1)))) IF1(false, 0, s(0), cons(s(s(y0)), y1)) -> c4(MAX(cons(s(0), cons(s(s(y0)), y1)))) IF1(false, 0, s(z0), cons(0, cons(0, y1))) -> c4(MAX(cons(s(z0), cons(0, cons(0, y1))))) IF1(false, 0, s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(MAX(cons(s(s(y0)), cons(0, cons(s(s(y1)), y2))))) IF1(false, 0, s(z0), cons(0, cons(s(0), y1))) -> c4(MAX(cons(s(z0), cons(0, cons(s(0), y1))))) IF1(false, 0, s(0), cons(0, cons(s(s(y0)), y1))) -> c4(MAX(cons(s(0), cons(0, cons(s(s(y0)), y1))))) MAX(cons(0, cons(s(s(y0)), cons(s(s(y1)), y2)))) -> c2(IF1(false, 0, s(s(y0)), cons(s(s(y1)), y2))) MAX(cons(0, cons(s(z0), cons(s(0), y1)))) -> c2(IF1(false, 0, s(z0), cons(s(0), y1))) MAX(cons(0, cons(s(0), cons(s(s(y0)), y1)))) -> c2(IF1(false, 0, s(0), cons(s(s(y0)), y1))) MAX(cons(0, cons(s(z0), cons(0, cons(0, y1))))) -> c2(IF1(false, 0, s(z0), cons(0, cons(0, y1)))) MAX(cons(0, cons(s(s(y0)), cons(0, cons(s(s(y1)), y2))))) -> c2(IF1(false, 0, s(s(y0)), cons(0, cons(s(s(y1)), y2)))) MAX(cons(0, cons(s(z0), cons(0, cons(s(0), y1))))) -> c2(IF1(false, 0, s(z0), cons(0, cons(s(0), y1)))) MAX(cons(0, cons(s(0), cons(0, cons(s(s(y0)), y1))))) -> c2(IF1(false, 0, s(0), cons(0, cons(s(s(y0)), y1)))) IF1(false, s(0), s(s(z0)), cons(s(s(y1)), y2)) -> c4(MAX(cons(s(s(z0)), cons(s(s(y1)), y2)))) IF1(false, s(0), s(s(z0)), cons(s(0), y1)) -> c4(MAX(cons(s(s(z0)), cons(s(0), y1)))) IF1(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c4(MAX(cons(s(s(z0)), cons(0, cons(0, y1))))) IF1(false, s(0), s(s(z0)), cons(0, cons(s(s(y1)), y2))) -> c4(MAX(cons(s(s(z0)), cons(0, cons(s(s(y1)), y2))))) IF1(false, s(0), s(s(z0)), cons(0, cons(s(0), y1))) -> c4(MAX(cons(s(s(z0)), cons(0, cons(s(0), y1))))) IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c4(MAX(cons(s(s(z1)), cons(s(s(y1)), y2)))) IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c4(MAX(cons(s(s(z1)), cons(s(0), y1)))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c4(MAX(cons(s(s(z1)), cons(0, cons(0, y1))))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c4(MAX(cons(s(s(z1)), cons(0, cons(s(s(y1)), y2))))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c4(MAX(cons(s(s(z1)), cons(0, cons(s(0), y1))))) IF2(false, 0, s(z0), cons(s(y0), y1)) -> c8(DEL(0, cons(s(y0), y1))) K tuples: SORT(cons(x0, cons(x1, x2))) -> c15(DEL(max(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) SORT(cons(x0, nil)) -> c15(DEL(max(cons(x0, nil)), cons(x0, nil))) SORT(cons(z0, cons(0, y1))) -> c14(MAX(cons(z0, cons(0, y1)))) SORT(cons(0, cons(s(y0), y1))) -> c14(MAX(cons(0, cons(s(y0), y1)))) SORT(cons(s(y0), cons(s(0), y1))) -> c14(MAX(cons(s(y0), cons(s(0), y1)))) SORT(cons(s(0), cons(s(s(y0)), y1))) -> c14(MAX(cons(s(0), cons(s(s(y0)), y1)))) SORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c14(MAX(cons(s(s(y0)), cons(s(s(y1)), y2)))) SORT(cons(s(y0), cons(s(y1), y2))) -> c14(MAX(cons(s(y0), cons(s(y1), y2)))) Defined Rule Symbols: ge_2, eq_2, del_2, max_1, if1_4, if2_4 Defined Pair Symbols: DEL_2, MAX_1, SORT_1, IF2_4, EQ_2, GE_2, IF1_4 Compound Symbols: c6_1, c2_2, c6_2, c15_3, c8_1, c15_1, c15_2, c12_1, c6, c14_1, c18_1, c2_1, c3_1, c4_1 ---------------------------------------- (167) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace DEL(0, cons(s(z0), x2)) -> c6(IF2(false, 0, s(z0), x2)) by DEL(0, cons(s(z0), cons(s(y1), y2))) -> c6(IF2(false, 0, s(z0), cons(s(y1), y2))) ---------------------------------------- (168) 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, 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)) Tuples: DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) 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)))) 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)))) SORT(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)))) 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(x0)), s(s(x1)), x2) -> c8(DEL(s(s(x0)), x2)) SORT(cons(z0, cons(z1, z2))) -> c15(DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) SORT(cons(z0, nil)) -> c15(DEL(z0, cons(z0, nil))) 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 SORT(cons(z0, cons(0, y1))) -> c14(MAX(cons(z0, cons(0, y1)))) SORT(cons(0, cons(s(y0), y1))) -> c14(MAX(cons(0, cons(s(y0), y1)))) SORT(cons(s(y0), cons(s(0), y1))) -> c14(MAX(cons(s(y0), cons(s(0), y1)))) SORT(cons(s(0), cons(s(s(y0)), y1))) -> c14(MAX(cons(s(0), cons(s(s(y0)), y1)))) SORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c14(MAX(cons(s(s(y0)), cons(s(s(y1)), y2)))) SORT(cons(s(y0), cons(s(y1), y2))) -> c14(MAX(cons(s(y0), cons(s(y1), y2)))) GE(s(s(y0)), s(s(y1))) -> c18(GE(s(y0), s(y1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2)) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2)) MAX(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c2(GE(s(s(y0)), s(s(y1)))) SORT(cons(z0, cons(0, z1))) -> c15(SORT(del(if1(true, z0, 0, z1), cons(z0, cons(0, z1)))), DEL(if1(true, z0, 0, z1), cons(z0, cons(0, z1))), MAX(cons(z0, cons(0, z1)))) SORT(cons(0, cons(s(z0), z1))) -> c15(SORT(del(if1(false, 0, s(z0), z1), cons(0, cons(s(z0), z1)))), DEL(if1(false, 0, s(z0), z1), cons(0, cons(s(z0), z1))), MAX(cons(0, cons(s(z0), z1)))) SORT(cons(s(z0), cons(s(z1), z2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(ge(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MAX(cons(s(z0), cons(s(z1), z2)))) DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c6(EQ(s(s(y0)), s(s(y1)))) IF1(true, z0, 0, cons(0, y1)) -> c3(MAX(cons(z0, cons(0, y1)))) IF1(true, 0, 0, cons(s(y0), y1)) -> c3(MAX(cons(0, cons(s(y0), y1)))) IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c3(MAX(cons(s(s(y0)), cons(s(s(y1)), y2)))) IF1(true, s(y0), 0, cons(s(0), y1)) -> c3(MAX(cons(s(y0), cons(s(0), y1)))) IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c3(MAX(cons(s(0), cons(s(s(y0)), y1)))) MAX(cons(z0, cons(0, cons(0, y1)))) -> c2(IF1(true, z0, 0, cons(0, y1))) MAX(cons(0, cons(0, cons(s(y0), y1)))) -> c2(IF1(true, 0, 0, cons(s(y0), y1))) MAX(cons(s(s(y0)), cons(0, cons(s(s(y1)), y2)))) -> c2(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2))) MAX(cons(s(y0), cons(0, cons(s(0), y1)))) -> c2(IF1(true, s(y0), 0, cons(s(0), y1))) MAX(cons(s(0), cons(0, cons(s(s(y0)), y1)))) -> c2(IF1(true, s(0), 0, cons(s(s(y0)), y1))) IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)) -> c3(MAX(cons(s(s(y0)), cons(s(s(y1)), y2)))) IF1(true, s(z0), s(0), cons(s(0), y1)) -> c3(MAX(cons(s(z0), cons(s(0), y1)))) IF1(true, s(0), s(0), cons(s(s(y0)), y1)) -> c3(MAX(cons(s(0), cons(s(s(y0)), y1)))) IF1(true, s(z0), s(0), cons(0, cons(0, y1))) -> c3(MAX(cons(s(z0), cons(0, cons(0, y1))))) IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c3(MAX(cons(s(s(y0)), cons(0, cons(s(s(y1)), y2))))) IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))) -> c3(MAX(cons(s(z0), cons(0, cons(s(0), y1))))) IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))) -> c3(MAX(cons(s(0), cons(0, cons(s(s(y0)), y1))))) IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c3(MAX(cons(s(s(z0)), cons(s(s(y1)), y2)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c3(MAX(cons(s(s(z0)), cons(s(0), y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c3(MAX(cons(s(s(z0)), cons(0, cons(0, y1))))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c3(MAX(cons(s(s(z0)), cons(0, cons(s(s(y1)), y2))))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c3(MAX(cons(s(s(z0)), cons(0, cons(s(0), y1))))) IF1(false, 0, s(s(y0)), cons(s(s(y1)), y2)) -> c4(MAX(cons(s(s(y0)), cons(s(s(y1)), y2)))) IF1(false, 0, s(z0), cons(s(0), y1)) -> c4(MAX(cons(s(z0), cons(s(0), y1)))) IF1(false, 0, s(0), cons(s(s(y0)), y1)) -> c4(MAX(cons(s(0), cons(s(s(y0)), y1)))) IF1(false, 0, s(z0), cons(0, cons(0, y1))) -> c4(MAX(cons(s(z0), cons(0, cons(0, y1))))) IF1(false, 0, s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(MAX(cons(s(s(y0)), cons(0, cons(s(s(y1)), y2))))) IF1(false, 0, s(z0), cons(0, cons(s(0), y1))) -> c4(MAX(cons(s(z0), cons(0, cons(s(0), y1))))) IF1(false, 0, s(0), cons(0, cons(s(s(y0)), y1))) -> c4(MAX(cons(s(0), cons(0, cons(s(s(y0)), y1))))) MAX(cons(0, cons(s(s(y0)), cons(s(s(y1)), y2)))) -> c2(IF1(false, 0, s(s(y0)), cons(s(s(y1)), y2))) MAX(cons(0, cons(s(z0), cons(s(0), y1)))) -> c2(IF1(false, 0, s(z0), cons(s(0), y1))) MAX(cons(0, cons(s(0), cons(s(s(y0)), y1)))) -> c2(IF1(false, 0, s(0), cons(s(s(y0)), y1))) MAX(cons(0, cons(s(z0), cons(0, cons(0, y1))))) -> c2(IF1(false, 0, s(z0), cons(0, cons(0, y1)))) MAX(cons(0, cons(s(s(y0)), cons(0, cons(s(s(y1)), y2))))) -> c2(IF1(false, 0, s(s(y0)), cons(0, cons(s(s(y1)), y2)))) MAX(cons(0, cons(s(z0), cons(0, cons(s(0), y1))))) -> c2(IF1(false, 0, s(z0), cons(0, cons(s(0), y1)))) MAX(cons(0, cons(s(0), cons(0, cons(s(s(y0)), y1))))) -> c2(IF1(false, 0, s(0), cons(0, cons(s(s(y0)), y1)))) IF1(false, s(0), s(s(z0)), cons(s(s(y1)), y2)) -> c4(MAX(cons(s(s(z0)), cons(s(s(y1)), y2)))) IF1(false, s(0), s(s(z0)), cons(s(0), y1)) -> c4(MAX(cons(s(s(z0)), cons(s(0), y1)))) IF1(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c4(MAX(cons(s(s(z0)), cons(0, cons(0, y1))))) IF1(false, s(0), s(s(z0)), cons(0, cons(s(s(y1)), y2))) -> c4(MAX(cons(s(s(z0)), cons(0, cons(s(s(y1)), y2))))) IF1(false, s(0), s(s(z0)), cons(0, cons(s(0), y1))) -> c4(MAX(cons(s(s(z0)), cons(0, cons(s(0), y1))))) IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c4(MAX(cons(s(s(z1)), cons(s(s(y1)), y2)))) IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c4(MAX(cons(s(s(z1)), cons(s(0), y1)))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c4(MAX(cons(s(s(z1)), cons(0, cons(0, y1))))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c4(MAX(cons(s(s(z1)), cons(0, cons(s(s(y1)), y2))))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c4(MAX(cons(s(s(z1)), cons(0, cons(s(0), y1))))) IF2(false, 0, s(z0), cons(s(y0), y1)) -> c8(DEL(0, cons(s(y0), y1))) DEL(0, cons(s(z0), cons(s(y1), y2))) -> c6(IF2(false, 0, s(z0), cons(s(y1), y2))) S tuples: DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) 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)))) 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)))) SORT(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)))) 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(x0)), s(s(x1)), x2) -> c8(DEL(s(s(x0)), x2)) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) 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 GE(s(s(y0)), s(s(y1))) -> c18(GE(s(y0), s(y1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2)) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2)) MAX(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c2(GE(s(s(y0)), s(s(y1)))) SORT(cons(z0, cons(0, z1))) -> c15(SORT(del(if1(true, z0, 0, z1), cons(z0, cons(0, z1)))), DEL(if1(true, z0, 0, z1), cons(z0, cons(0, z1))), MAX(cons(z0, cons(0, z1)))) SORT(cons(0, cons(s(z0), z1))) -> c15(SORT(del(if1(false, 0, s(z0), z1), cons(0, cons(s(z0), z1)))), DEL(if1(false, 0, s(z0), z1), cons(0, cons(s(z0), z1))), MAX(cons(0, cons(s(z0), z1)))) SORT(cons(s(z0), cons(s(z1), z2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(ge(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MAX(cons(s(z0), cons(s(z1), z2)))) DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c6(EQ(s(s(y0)), s(s(y1)))) IF1(true, z0, 0, cons(0, y1)) -> c3(MAX(cons(z0, cons(0, y1)))) IF1(true, 0, 0, cons(s(y0), y1)) -> c3(MAX(cons(0, cons(s(y0), y1)))) IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c3(MAX(cons(s(s(y0)), cons(s(s(y1)), y2)))) IF1(true, s(y0), 0, cons(s(0), y1)) -> c3(MAX(cons(s(y0), cons(s(0), y1)))) IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c3(MAX(cons(s(0), cons(s(s(y0)), y1)))) MAX(cons(z0, cons(0, cons(0, y1)))) -> c2(IF1(true, z0, 0, cons(0, y1))) MAX(cons(0, cons(0, cons(s(y0), y1)))) -> c2(IF1(true, 0, 0, cons(s(y0), y1))) MAX(cons(s(s(y0)), cons(0, cons(s(s(y1)), y2)))) -> c2(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2))) MAX(cons(s(y0), cons(0, cons(s(0), y1)))) -> c2(IF1(true, s(y0), 0, cons(s(0), y1))) MAX(cons(s(0), cons(0, cons(s(s(y0)), y1)))) -> c2(IF1(true, s(0), 0, cons(s(s(y0)), y1))) IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)) -> c3(MAX(cons(s(s(y0)), cons(s(s(y1)), y2)))) IF1(true, s(z0), s(0), cons(s(0), y1)) -> c3(MAX(cons(s(z0), cons(s(0), y1)))) IF1(true, s(0), s(0), cons(s(s(y0)), y1)) -> c3(MAX(cons(s(0), cons(s(s(y0)), y1)))) IF1(true, s(z0), s(0), cons(0, cons(0, y1))) -> c3(MAX(cons(s(z0), cons(0, cons(0, y1))))) IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c3(MAX(cons(s(s(y0)), cons(0, cons(s(s(y1)), y2))))) IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))) -> c3(MAX(cons(s(z0), cons(0, cons(s(0), y1))))) IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))) -> c3(MAX(cons(s(0), cons(0, cons(s(s(y0)), y1))))) IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c3(MAX(cons(s(s(z0)), cons(s(s(y1)), y2)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c3(MAX(cons(s(s(z0)), cons(s(0), y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c3(MAX(cons(s(s(z0)), cons(0, cons(0, y1))))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c3(MAX(cons(s(s(z0)), cons(0, cons(s(s(y1)), y2))))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c3(MAX(cons(s(s(z0)), cons(0, cons(s(0), y1))))) IF1(false, 0, s(s(y0)), cons(s(s(y1)), y2)) -> c4(MAX(cons(s(s(y0)), cons(s(s(y1)), y2)))) IF1(false, 0, s(z0), cons(s(0), y1)) -> c4(MAX(cons(s(z0), cons(s(0), y1)))) IF1(false, 0, s(0), cons(s(s(y0)), y1)) -> c4(MAX(cons(s(0), cons(s(s(y0)), y1)))) IF1(false, 0, s(z0), cons(0, cons(0, y1))) -> c4(MAX(cons(s(z0), cons(0, cons(0, y1))))) IF1(false, 0, s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(MAX(cons(s(s(y0)), cons(0, cons(s(s(y1)), y2))))) IF1(false, 0, s(z0), cons(0, cons(s(0), y1))) -> c4(MAX(cons(s(z0), cons(0, cons(s(0), y1))))) IF1(false, 0, s(0), cons(0, cons(s(s(y0)), y1))) -> c4(MAX(cons(s(0), cons(0, cons(s(s(y0)), y1))))) MAX(cons(0, cons(s(s(y0)), cons(s(s(y1)), y2)))) -> c2(IF1(false, 0, s(s(y0)), cons(s(s(y1)), y2))) MAX(cons(0, cons(s(z0), cons(s(0), y1)))) -> c2(IF1(false, 0, s(z0), cons(s(0), y1))) MAX(cons(0, cons(s(0), cons(s(s(y0)), y1)))) -> c2(IF1(false, 0, s(0), cons(s(s(y0)), y1))) MAX(cons(0, cons(s(z0), cons(0, cons(0, y1))))) -> c2(IF1(false, 0, s(z0), cons(0, cons(0, y1)))) MAX(cons(0, cons(s(s(y0)), cons(0, cons(s(s(y1)), y2))))) -> c2(IF1(false, 0, s(s(y0)), cons(0, cons(s(s(y1)), y2)))) MAX(cons(0, cons(s(z0), cons(0, cons(s(0), y1))))) -> c2(IF1(false, 0, s(z0), cons(0, cons(s(0), y1)))) MAX(cons(0, cons(s(0), cons(0, cons(s(s(y0)), y1))))) -> c2(IF1(false, 0, s(0), cons(0, cons(s(s(y0)), y1)))) IF1(false, s(0), s(s(z0)), cons(s(s(y1)), y2)) -> c4(MAX(cons(s(s(z0)), cons(s(s(y1)), y2)))) IF1(false, s(0), s(s(z0)), cons(s(0), y1)) -> c4(MAX(cons(s(s(z0)), cons(s(0), y1)))) IF1(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c4(MAX(cons(s(s(z0)), cons(0, cons(0, y1))))) IF1(false, s(0), s(s(z0)), cons(0, cons(s(s(y1)), y2))) -> c4(MAX(cons(s(s(z0)), cons(0, cons(s(s(y1)), y2))))) IF1(false, s(0), s(s(z0)), cons(0, cons(s(0), y1))) -> c4(MAX(cons(s(s(z0)), cons(0, cons(s(0), y1))))) IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c4(MAX(cons(s(s(z1)), cons(s(s(y1)), y2)))) IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c4(MAX(cons(s(s(z1)), cons(s(0), y1)))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c4(MAX(cons(s(s(z1)), cons(0, cons(0, y1))))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c4(MAX(cons(s(s(z1)), cons(0, cons(s(s(y1)), y2))))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c4(MAX(cons(s(s(z1)), cons(0, cons(s(0), y1))))) IF2(false, 0, s(z0), cons(s(y0), y1)) -> c8(DEL(0, cons(s(y0), y1))) DEL(0, cons(s(z0), cons(s(y1), y2))) -> c6(IF2(false, 0, s(z0), cons(s(y1), y2))) K tuples: SORT(cons(x0, cons(x1, x2))) -> c15(DEL(max(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) SORT(cons(x0, nil)) -> c15(DEL(max(cons(x0, nil)), cons(x0, nil))) SORT(cons(z0, cons(0, y1))) -> c14(MAX(cons(z0, cons(0, y1)))) SORT(cons(0, cons(s(y0), y1))) -> c14(MAX(cons(0, cons(s(y0), y1)))) SORT(cons(s(y0), cons(s(0), y1))) -> c14(MAX(cons(s(y0), cons(s(0), y1)))) SORT(cons(s(0), cons(s(s(y0)), y1))) -> c14(MAX(cons(s(0), cons(s(s(y0)), y1)))) SORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c14(MAX(cons(s(s(y0)), cons(s(s(y1)), y2)))) SORT(cons(s(y0), cons(s(y1), y2))) -> c14(MAX(cons(s(y0), cons(s(y1), y2)))) Defined Rule Symbols: ge_2, eq_2, del_2, max_1, if1_4, if2_4 Defined Pair Symbols: DEL_2, MAX_1, SORT_1, IF2_4, EQ_2, GE_2, IF1_4 Compound Symbols: c6_1, c2_2, c6_2, c15_3, c8_1, c15_1, c15_2, c12_1, c6, c14_1, c18_1, c2_1, c3_1, c4_1 ---------------------------------------- (169) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace IF2(false, s(x0), 0, x1) -> c8(DEL(s(x0), x1)) by IF2(false, s(z0), 0, cons(0, y1)) -> c8(DEL(s(z0), cons(0, y1))) IF2(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c8(DEL(s(s(y0)), cons(s(s(y1)), y2))) IF2(false, s(0), 0, cons(s(s(y0)), y1)) -> c8(DEL(s(0), cons(s(s(y0)), y1))) IF2(false, s(s(y0)), 0, cons(s(0), y1)) -> c8(DEL(s(s(y0)), cons(s(0), y1))) IF2(false, s(0), 0, cons(s(0), y0)) -> c8(DEL(s(0), cons(s(0), y0))) ---------------------------------------- (170) 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, 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)) Tuples: DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) 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)))) 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)))) SORT(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)))) 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(x0)), s(s(x1)), x2) -> c8(DEL(s(s(x0)), x2)) SORT(cons(z0, cons(z1, z2))) -> c15(DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) SORT(cons(z0, nil)) -> c15(DEL(z0, cons(z0, nil))) 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 SORT(cons(z0, cons(0, y1))) -> c14(MAX(cons(z0, cons(0, y1)))) SORT(cons(0, cons(s(y0), y1))) -> c14(MAX(cons(0, cons(s(y0), y1)))) SORT(cons(s(y0), cons(s(0), y1))) -> c14(MAX(cons(s(y0), cons(s(0), y1)))) SORT(cons(s(0), cons(s(s(y0)), y1))) -> c14(MAX(cons(s(0), cons(s(s(y0)), y1)))) SORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c14(MAX(cons(s(s(y0)), cons(s(s(y1)), y2)))) SORT(cons(s(y0), cons(s(y1), y2))) -> c14(MAX(cons(s(y0), cons(s(y1), y2)))) GE(s(s(y0)), s(s(y1))) -> c18(GE(s(y0), s(y1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2)) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2)) MAX(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c2(GE(s(s(y0)), s(s(y1)))) SORT(cons(z0, cons(0, z1))) -> c15(SORT(del(if1(true, z0, 0, z1), cons(z0, cons(0, z1)))), DEL(if1(true, z0, 0, z1), cons(z0, cons(0, z1))), MAX(cons(z0, cons(0, z1)))) SORT(cons(0, cons(s(z0), z1))) -> c15(SORT(del(if1(false, 0, s(z0), z1), cons(0, cons(s(z0), z1)))), DEL(if1(false, 0, s(z0), z1), cons(0, cons(s(z0), z1))), MAX(cons(0, cons(s(z0), z1)))) SORT(cons(s(z0), cons(s(z1), z2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(ge(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MAX(cons(s(z0), cons(s(z1), z2)))) DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c6(EQ(s(s(y0)), s(s(y1)))) IF1(true, z0, 0, cons(0, y1)) -> c3(MAX(cons(z0, cons(0, y1)))) IF1(true, 0, 0, cons(s(y0), y1)) -> c3(MAX(cons(0, cons(s(y0), y1)))) IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c3(MAX(cons(s(s(y0)), cons(s(s(y1)), y2)))) IF1(true, s(y0), 0, cons(s(0), y1)) -> c3(MAX(cons(s(y0), cons(s(0), y1)))) IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c3(MAX(cons(s(0), cons(s(s(y0)), y1)))) MAX(cons(z0, cons(0, cons(0, y1)))) -> c2(IF1(true, z0, 0, cons(0, y1))) MAX(cons(0, cons(0, cons(s(y0), y1)))) -> c2(IF1(true, 0, 0, cons(s(y0), y1))) MAX(cons(s(s(y0)), cons(0, cons(s(s(y1)), y2)))) -> c2(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2))) MAX(cons(s(y0), cons(0, cons(s(0), y1)))) -> c2(IF1(true, s(y0), 0, cons(s(0), y1))) MAX(cons(s(0), cons(0, cons(s(s(y0)), y1)))) -> c2(IF1(true, s(0), 0, cons(s(s(y0)), y1))) IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)) -> c3(MAX(cons(s(s(y0)), cons(s(s(y1)), y2)))) IF1(true, s(z0), s(0), cons(s(0), y1)) -> c3(MAX(cons(s(z0), cons(s(0), y1)))) IF1(true, s(0), s(0), cons(s(s(y0)), y1)) -> c3(MAX(cons(s(0), cons(s(s(y0)), y1)))) IF1(true, s(z0), s(0), cons(0, cons(0, y1))) -> c3(MAX(cons(s(z0), cons(0, cons(0, y1))))) IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c3(MAX(cons(s(s(y0)), cons(0, cons(s(s(y1)), y2))))) IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))) -> c3(MAX(cons(s(z0), cons(0, cons(s(0), y1))))) IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))) -> c3(MAX(cons(s(0), cons(0, cons(s(s(y0)), y1))))) IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c3(MAX(cons(s(s(z0)), cons(s(s(y1)), y2)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c3(MAX(cons(s(s(z0)), cons(s(0), y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c3(MAX(cons(s(s(z0)), cons(0, cons(0, y1))))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c3(MAX(cons(s(s(z0)), cons(0, cons(s(s(y1)), y2))))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c3(MAX(cons(s(s(z0)), cons(0, cons(s(0), y1))))) IF1(false, 0, s(s(y0)), cons(s(s(y1)), y2)) -> c4(MAX(cons(s(s(y0)), cons(s(s(y1)), y2)))) IF1(false, 0, s(z0), cons(s(0), y1)) -> c4(MAX(cons(s(z0), cons(s(0), y1)))) IF1(false, 0, s(0), cons(s(s(y0)), y1)) -> c4(MAX(cons(s(0), cons(s(s(y0)), y1)))) IF1(false, 0, s(z0), cons(0, cons(0, y1))) -> c4(MAX(cons(s(z0), cons(0, cons(0, y1))))) IF1(false, 0, s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(MAX(cons(s(s(y0)), cons(0, cons(s(s(y1)), y2))))) IF1(false, 0, s(z0), cons(0, cons(s(0), y1))) -> c4(MAX(cons(s(z0), cons(0, cons(s(0), y1))))) IF1(false, 0, s(0), cons(0, cons(s(s(y0)), y1))) -> c4(MAX(cons(s(0), cons(0, cons(s(s(y0)), y1))))) MAX(cons(0, cons(s(s(y0)), cons(s(s(y1)), y2)))) -> c2(IF1(false, 0, s(s(y0)), cons(s(s(y1)), y2))) MAX(cons(0, cons(s(z0), cons(s(0), y1)))) -> c2(IF1(false, 0, s(z0), cons(s(0), y1))) MAX(cons(0, cons(s(0), cons(s(s(y0)), y1)))) -> c2(IF1(false, 0, s(0), cons(s(s(y0)), y1))) MAX(cons(0, cons(s(z0), cons(0, cons(0, y1))))) -> c2(IF1(false, 0, s(z0), cons(0, cons(0, y1)))) MAX(cons(0, cons(s(s(y0)), cons(0, cons(s(s(y1)), y2))))) -> c2(IF1(false, 0, s(s(y0)), cons(0, cons(s(s(y1)), y2)))) MAX(cons(0, cons(s(z0), cons(0, cons(s(0), y1))))) -> c2(IF1(false, 0, s(z0), cons(0, cons(s(0), y1)))) MAX(cons(0, cons(s(0), cons(0, cons(s(s(y0)), y1))))) -> c2(IF1(false, 0, s(0), cons(0, cons(s(s(y0)), y1)))) IF1(false, s(0), s(s(z0)), cons(s(s(y1)), y2)) -> c4(MAX(cons(s(s(z0)), cons(s(s(y1)), y2)))) IF1(false, s(0), s(s(z0)), cons(s(0), y1)) -> c4(MAX(cons(s(s(z0)), cons(s(0), y1)))) IF1(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c4(MAX(cons(s(s(z0)), cons(0, cons(0, y1))))) IF1(false, s(0), s(s(z0)), cons(0, cons(s(s(y1)), y2))) -> c4(MAX(cons(s(s(z0)), cons(0, cons(s(s(y1)), y2))))) IF1(false, s(0), s(s(z0)), cons(0, cons(s(0), y1))) -> c4(MAX(cons(s(s(z0)), cons(0, cons(s(0), y1))))) IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c4(MAX(cons(s(s(z1)), cons(s(s(y1)), y2)))) IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c4(MAX(cons(s(s(z1)), cons(s(0), y1)))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c4(MAX(cons(s(s(z1)), cons(0, cons(0, y1))))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c4(MAX(cons(s(s(z1)), cons(0, cons(s(s(y1)), y2))))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c4(MAX(cons(s(s(z1)), cons(0, cons(s(0), y1))))) IF2(false, 0, s(z0), cons(s(y0), y1)) -> c8(DEL(0, cons(s(y0), y1))) DEL(0, cons(s(z0), cons(s(y1), y2))) -> c6(IF2(false, 0, s(z0), cons(s(y1), y2))) IF2(false, s(z0), 0, cons(0, y1)) -> c8(DEL(s(z0), cons(0, y1))) IF2(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c8(DEL(s(s(y0)), cons(s(s(y1)), y2))) IF2(false, s(0), 0, cons(s(s(y0)), y1)) -> c8(DEL(s(0), cons(s(s(y0)), y1))) IF2(false, s(s(y0)), 0, cons(s(0), y1)) -> c8(DEL(s(s(y0)), cons(s(0), y1))) IF2(false, s(0), 0, cons(s(0), y0)) -> c8(DEL(s(0), cons(s(0), y0))) S tuples: DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) 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)))) 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)))) SORT(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)))) 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(x0)), s(s(x1)), x2) -> c8(DEL(s(s(x0)), x2)) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) 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 GE(s(s(y0)), s(s(y1))) -> c18(GE(s(y0), s(y1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2)) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2)) MAX(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c2(GE(s(s(y0)), s(s(y1)))) SORT(cons(z0, cons(0, z1))) -> c15(SORT(del(if1(true, z0, 0, z1), cons(z0, cons(0, z1)))), DEL(if1(true, z0, 0, z1), cons(z0, cons(0, z1))), MAX(cons(z0, cons(0, z1)))) SORT(cons(0, cons(s(z0), z1))) -> c15(SORT(del(if1(false, 0, s(z0), z1), cons(0, cons(s(z0), z1)))), DEL(if1(false, 0, s(z0), z1), cons(0, cons(s(z0), z1))), MAX(cons(0, cons(s(z0), z1)))) SORT(cons(s(z0), cons(s(z1), z2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(ge(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MAX(cons(s(z0), cons(s(z1), z2)))) DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c6(EQ(s(s(y0)), s(s(y1)))) IF1(true, z0, 0, cons(0, y1)) -> c3(MAX(cons(z0, cons(0, y1)))) IF1(true, 0, 0, cons(s(y0), y1)) -> c3(MAX(cons(0, cons(s(y0), y1)))) IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c3(MAX(cons(s(s(y0)), cons(s(s(y1)), y2)))) IF1(true, s(y0), 0, cons(s(0), y1)) -> c3(MAX(cons(s(y0), cons(s(0), y1)))) IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c3(MAX(cons(s(0), cons(s(s(y0)), y1)))) MAX(cons(z0, cons(0, cons(0, y1)))) -> c2(IF1(true, z0, 0, cons(0, y1))) MAX(cons(0, cons(0, cons(s(y0), y1)))) -> c2(IF1(true, 0, 0, cons(s(y0), y1))) MAX(cons(s(s(y0)), cons(0, cons(s(s(y1)), y2)))) -> c2(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2))) MAX(cons(s(y0), cons(0, cons(s(0), y1)))) -> c2(IF1(true, s(y0), 0, cons(s(0), y1))) MAX(cons(s(0), cons(0, cons(s(s(y0)), y1)))) -> c2(IF1(true, s(0), 0, cons(s(s(y0)), y1))) IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)) -> c3(MAX(cons(s(s(y0)), cons(s(s(y1)), y2)))) IF1(true, s(z0), s(0), cons(s(0), y1)) -> c3(MAX(cons(s(z0), cons(s(0), y1)))) IF1(true, s(0), s(0), cons(s(s(y0)), y1)) -> c3(MAX(cons(s(0), cons(s(s(y0)), y1)))) IF1(true, s(z0), s(0), cons(0, cons(0, y1))) -> c3(MAX(cons(s(z0), cons(0, cons(0, y1))))) IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c3(MAX(cons(s(s(y0)), cons(0, cons(s(s(y1)), y2))))) IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))) -> c3(MAX(cons(s(z0), cons(0, cons(s(0), y1))))) IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))) -> c3(MAX(cons(s(0), cons(0, cons(s(s(y0)), y1))))) IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c3(MAX(cons(s(s(z0)), cons(s(s(y1)), y2)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c3(MAX(cons(s(s(z0)), cons(s(0), y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c3(MAX(cons(s(s(z0)), cons(0, cons(0, y1))))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c3(MAX(cons(s(s(z0)), cons(0, cons(s(s(y1)), y2))))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c3(MAX(cons(s(s(z0)), cons(0, cons(s(0), y1))))) IF1(false, 0, s(s(y0)), cons(s(s(y1)), y2)) -> c4(MAX(cons(s(s(y0)), cons(s(s(y1)), y2)))) IF1(false, 0, s(z0), cons(s(0), y1)) -> c4(MAX(cons(s(z0), cons(s(0), y1)))) IF1(false, 0, s(0), cons(s(s(y0)), y1)) -> c4(MAX(cons(s(0), cons(s(s(y0)), y1)))) IF1(false, 0, s(z0), cons(0, cons(0, y1))) -> c4(MAX(cons(s(z0), cons(0, cons(0, y1))))) IF1(false, 0, s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(MAX(cons(s(s(y0)), cons(0, cons(s(s(y1)), y2))))) IF1(false, 0, s(z0), cons(0, cons(s(0), y1))) -> c4(MAX(cons(s(z0), cons(0, cons(s(0), y1))))) IF1(false, 0, s(0), cons(0, cons(s(s(y0)), y1))) -> c4(MAX(cons(s(0), cons(0, cons(s(s(y0)), y1))))) MAX(cons(0, cons(s(s(y0)), cons(s(s(y1)), y2)))) -> c2(IF1(false, 0, s(s(y0)), cons(s(s(y1)), y2))) MAX(cons(0, cons(s(z0), cons(s(0), y1)))) -> c2(IF1(false, 0, s(z0), cons(s(0), y1))) MAX(cons(0, cons(s(0), cons(s(s(y0)), y1)))) -> c2(IF1(false, 0, s(0), cons(s(s(y0)), y1))) MAX(cons(0, cons(s(z0), cons(0, cons(0, y1))))) -> c2(IF1(false, 0, s(z0), cons(0, cons(0, y1)))) MAX(cons(0, cons(s(s(y0)), cons(0, cons(s(s(y1)), y2))))) -> c2(IF1(false, 0, s(s(y0)), cons(0, cons(s(s(y1)), y2)))) MAX(cons(0, cons(s(z0), cons(0, cons(s(0), y1))))) -> c2(IF1(false, 0, s(z0), cons(0, cons(s(0), y1)))) MAX(cons(0, cons(s(0), cons(0, cons(s(s(y0)), y1))))) -> c2(IF1(false, 0, s(0), cons(0, cons(s(s(y0)), y1)))) IF1(false, s(0), s(s(z0)), cons(s(s(y1)), y2)) -> c4(MAX(cons(s(s(z0)), cons(s(s(y1)), y2)))) IF1(false, s(0), s(s(z0)), cons(s(0), y1)) -> c4(MAX(cons(s(s(z0)), cons(s(0), y1)))) IF1(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c4(MAX(cons(s(s(z0)), cons(0, cons(0, y1))))) IF1(false, s(0), s(s(z0)), cons(0, cons(s(s(y1)), y2))) -> c4(MAX(cons(s(s(z0)), cons(0, cons(s(s(y1)), y2))))) IF1(false, s(0), s(s(z0)), cons(0, cons(s(0), y1))) -> c4(MAX(cons(s(s(z0)), cons(0, cons(s(0), y1))))) IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c4(MAX(cons(s(s(z1)), cons(s(s(y1)), y2)))) IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c4(MAX(cons(s(s(z1)), cons(s(0), y1)))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c4(MAX(cons(s(s(z1)), cons(0, cons(0, y1))))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c4(MAX(cons(s(s(z1)), cons(0, cons(s(s(y1)), y2))))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c4(MAX(cons(s(s(z1)), cons(0, cons(s(0), y1))))) IF2(false, 0, s(z0), cons(s(y0), y1)) -> c8(DEL(0, cons(s(y0), y1))) DEL(0, cons(s(z0), cons(s(y1), y2))) -> c6(IF2(false, 0, s(z0), cons(s(y1), y2))) IF2(false, s(z0), 0, cons(0, y1)) -> c8(DEL(s(z0), cons(0, y1))) IF2(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c8(DEL(s(s(y0)), cons(s(s(y1)), y2))) IF2(false, s(0), 0, cons(s(s(y0)), y1)) -> c8(DEL(s(0), cons(s(s(y0)), y1))) IF2(false, s(s(y0)), 0, cons(s(0), y1)) -> c8(DEL(s(s(y0)), cons(s(0), y1))) IF2(false, s(0), 0, cons(s(0), y0)) -> c8(DEL(s(0), cons(s(0), y0))) K tuples: SORT(cons(x0, cons(x1, x2))) -> c15(DEL(max(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) SORT(cons(x0, nil)) -> c15(DEL(max(cons(x0, nil)), cons(x0, nil))) SORT(cons(z0, cons(0, y1))) -> c14(MAX(cons(z0, cons(0, y1)))) SORT(cons(0, cons(s(y0), y1))) -> c14(MAX(cons(0, cons(s(y0), y1)))) SORT(cons(s(y0), cons(s(0), y1))) -> c14(MAX(cons(s(y0), cons(s(0), y1)))) SORT(cons(s(0), cons(s(s(y0)), y1))) -> c14(MAX(cons(s(0), cons(s(s(y0)), y1)))) SORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c14(MAX(cons(s(s(y0)), cons(s(s(y1)), y2)))) SORT(cons(s(y0), cons(s(y1), y2))) -> c14(MAX(cons(s(y0), cons(s(y1), y2)))) Defined Rule Symbols: ge_2, eq_2, del_2, max_1, if1_4, if2_4 Defined Pair Symbols: DEL_2, MAX_1, SORT_1, IF2_4, EQ_2, GE_2, IF1_4 Compound Symbols: c6_1, c2_2, c6_2, c15_3, c8_1, c15_1, c15_2, c12_1, c6, c14_1, c18_1, c2_1, c3_1, c4_1 ---------------------------------------- (171) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace DEL(s(z0), cons(0, x2)) -> c6(IF2(false, s(z0), 0, x2)) by DEL(s(z0), cons(0, cons(0, y1))) -> c6(IF2(false, s(z0), 0, cons(0, y1))) DEL(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c6(IF2(false, s(s(y0)), 0, cons(s(s(y1)), y2))) DEL(s(0), cons(0, cons(s(s(y0)), y1))) -> c6(IF2(false, s(0), 0, cons(s(s(y0)), y1))) DEL(s(s(y0)), cons(0, cons(s(0), y1))) -> c6(IF2(false, s(s(y0)), 0, cons(s(0), y1))) DEL(s(0), cons(0, cons(s(0), y0))) -> c6(IF2(false, s(0), 0, cons(s(0), y0))) ---------------------------------------- (172) 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, 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)) Tuples: 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)))) 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)))) SORT(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)))) 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(x0)), s(s(x1)), x2) -> c8(DEL(s(s(x0)), x2)) SORT(cons(z0, cons(z1, z2))) -> c15(DEL(if1(ge(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) SORT(cons(z0, nil)) -> c15(DEL(z0, cons(z0, nil))) 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 SORT(cons(z0, cons(0, y1))) -> c14(MAX(cons(z0, cons(0, y1)))) SORT(cons(0, cons(s(y0), y1))) -> c14(MAX(cons(0, cons(s(y0), y1)))) SORT(cons(s(y0), cons(s(0), y1))) -> c14(MAX(cons(s(y0), cons(s(0), y1)))) SORT(cons(s(0), cons(s(s(y0)), y1))) -> c14(MAX(cons(s(0), cons(s(s(y0)), y1)))) SORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c14(MAX(cons(s(s(y0)), cons(s(s(y1)), y2)))) SORT(cons(s(y0), cons(s(y1), y2))) -> c14(MAX(cons(s(y0), cons(s(y1), y2)))) GE(s(s(y0)), s(s(y1))) -> c18(GE(s(y0), s(y1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2)) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2)) MAX(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c2(GE(s(s(y0)), s(s(y1)))) SORT(cons(z0, cons(0, z1))) -> c15(SORT(del(if1(true, z0, 0, z1), cons(z0, cons(0, z1)))), DEL(if1(true, z0, 0, z1), cons(z0, cons(0, z1))), MAX(cons(z0, cons(0, z1)))) SORT(cons(0, cons(s(z0), z1))) -> c15(SORT(del(if1(false, 0, s(z0), z1), cons(0, cons(s(z0), z1)))), DEL(if1(false, 0, s(z0), z1), cons(0, cons(s(z0), z1))), MAX(cons(0, cons(s(z0), z1)))) SORT(cons(s(z0), cons(s(z1), z2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(ge(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MAX(cons(s(z0), cons(s(z1), z2)))) DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c6(EQ(s(s(y0)), s(s(y1)))) IF1(true, z0, 0, cons(0, y1)) -> c3(MAX(cons(z0, cons(0, y1)))) IF1(true, 0, 0, cons(s(y0), y1)) -> c3(MAX(cons(0, cons(s(y0), y1)))) IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c3(MAX(cons(s(s(y0)), cons(s(s(y1)), y2)))) IF1(true, s(y0), 0, cons(s(0), y1)) -> c3(MAX(cons(s(y0), cons(s(0), y1)))) IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c3(MAX(cons(s(0), cons(s(s(y0)), y1)))) MAX(cons(z0, cons(0, cons(0, y1)))) -> c2(IF1(true, z0, 0, cons(0, y1))) MAX(cons(0, cons(0, cons(s(y0), y1)))) -> c2(IF1(true, 0, 0, cons(s(y0), y1))) MAX(cons(s(s(y0)), cons(0, cons(s(s(y1)), y2)))) -> c2(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2))) MAX(cons(s(y0), cons(0, cons(s(0), y1)))) -> c2(IF1(true, s(y0), 0, cons(s(0), y1))) MAX(cons(s(0), cons(0, cons(s(s(y0)), y1)))) -> c2(IF1(true, s(0), 0, cons(s(s(y0)), y1))) IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)) -> c3(MAX(cons(s(s(y0)), cons(s(s(y1)), y2)))) IF1(true, s(z0), s(0), cons(s(0), y1)) -> c3(MAX(cons(s(z0), cons(s(0), y1)))) IF1(true, s(0), s(0), cons(s(s(y0)), y1)) -> c3(MAX(cons(s(0), cons(s(s(y0)), y1)))) IF1(true, s(z0), s(0), cons(0, cons(0, y1))) -> c3(MAX(cons(s(z0), cons(0, cons(0, y1))))) IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c3(MAX(cons(s(s(y0)), cons(0, cons(s(s(y1)), y2))))) IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))) -> c3(MAX(cons(s(z0), cons(0, cons(s(0), y1))))) IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))) -> c3(MAX(cons(s(0), cons(0, cons(s(s(y0)), y1))))) IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c3(MAX(cons(s(s(z0)), cons(s(s(y1)), y2)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c3(MAX(cons(s(s(z0)), cons(s(0), y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c3(MAX(cons(s(s(z0)), cons(0, cons(0, y1))))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c3(MAX(cons(s(s(z0)), cons(0, cons(s(s(y1)), y2))))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c3(MAX(cons(s(s(z0)), cons(0, cons(s(0), y1))))) IF1(false, 0, s(s(y0)), cons(s(s(y1)), y2)) -> c4(MAX(cons(s(s(y0)), cons(s(s(y1)), y2)))) IF1(false, 0, s(z0), cons(s(0), y1)) -> c4(MAX(cons(s(z0), cons(s(0), y1)))) IF1(false, 0, s(0), cons(s(s(y0)), y1)) -> c4(MAX(cons(s(0), cons(s(s(y0)), y1)))) IF1(false, 0, s(z0), cons(0, cons(0, y1))) -> c4(MAX(cons(s(z0), cons(0, cons(0, y1))))) IF1(false, 0, s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(MAX(cons(s(s(y0)), cons(0, cons(s(s(y1)), y2))))) IF1(false, 0, s(z0), cons(0, cons(s(0), y1))) -> c4(MAX(cons(s(z0), cons(0, cons(s(0), y1))))) IF1(false, 0, s(0), cons(0, cons(s(s(y0)), y1))) -> c4(MAX(cons(s(0), cons(0, cons(s(s(y0)), y1))))) MAX(cons(0, cons(s(s(y0)), cons(s(s(y1)), y2)))) -> c2(IF1(false, 0, s(s(y0)), cons(s(s(y1)), y2))) MAX(cons(0, cons(s(z0), cons(s(0), y1)))) -> c2(IF1(false, 0, s(z0), cons(s(0), y1))) MAX(cons(0, cons(s(0), cons(s(s(y0)), y1)))) -> c2(IF1(false, 0, s(0), cons(s(s(y0)), y1))) MAX(cons(0, cons(s(z0), cons(0, cons(0, y1))))) -> c2(IF1(false, 0, s(z0), cons(0, cons(0, y1)))) MAX(cons(0, cons(s(s(y0)), cons(0, cons(s(s(y1)), y2))))) -> c2(IF1(false, 0, s(s(y0)), cons(0, cons(s(s(y1)), y2)))) MAX(cons(0, cons(s(z0), cons(0, cons(s(0), y1))))) -> c2(IF1(false, 0, s(z0), cons(0, cons(s(0), y1)))) MAX(cons(0, cons(s(0), cons(0, cons(s(s(y0)), y1))))) -> c2(IF1(false, 0, s(0), cons(0, cons(s(s(y0)), y1)))) IF1(false, s(0), s(s(z0)), cons(s(s(y1)), y2)) -> c4(MAX(cons(s(s(z0)), cons(s(s(y1)), y2)))) IF1(false, s(0), s(s(z0)), cons(s(0), y1)) -> c4(MAX(cons(s(s(z0)), cons(s(0), y1)))) IF1(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c4(MAX(cons(s(s(z0)), cons(0, cons(0, y1))))) IF1(false, s(0), s(s(z0)), cons(0, cons(s(s(y1)), y2))) -> c4(MAX(cons(s(s(z0)), cons(0, cons(s(s(y1)), y2))))) IF1(false, s(0), s(s(z0)), cons(0, cons(s(0), y1))) -> c4(MAX(cons(s(s(z0)), cons(0, cons(s(0), y1))))) IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c4(MAX(cons(s(s(z1)), cons(s(s(y1)), y2)))) IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c4(MAX(cons(s(s(z1)), cons(s(0), y1)))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c4(MAX(cons(s(s(z1)), cons(0, cons(0, y1))))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c4(MAX(cons(s(s(z1)), cons(0, cons(s(s(y1)), y2))))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c4(MAX(cons(s(s(z1)), cons(0, cons(s(0), y1))))) IF2(false, 0, s(z0), cons(s(y0), y1)) -> c8(DEL(0, cons(s(y0), y1))) DEL(0, cons(s(z0), cons(s(y1), y2))) -> c6(IF2(false, 0, s(z0), cons(s(y1), y2))) IF2(false, s(z0), 0, cons(0, y1)) -> c8(DEL(s(z0), cons(0, y1))) IF2(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c8(DEL(s(s(y0)), cons(s(s(y1)), y2))) IF2(false, s(0), 0, cons(s(s(y0)), y1)) -> c8(DEL(s(0), cons(s(s(y0)), y1))) IF2(false, s(s(y0)), 0, cons(s(0), y1)) -> c8(DEL(s(s(y0)), cons(s(0), y1))) IF2(false, s(0), 0, cons(s(0), y0)) -> c8(DEL(s(0), cons(s(0), y0))) DEL(s(z0), cons(0, cons(0, y1))) -> c6(IF2(false, s(z0), 0, cons(0, y1))) DEL(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c6(IF2(false, s(s(y0)), 0, cons(s(s(y1)), y2))) DEL(s(0), cons(0, cons(s(s(y0)), y1))) -> c6(IF2(false, s(0), 0, cons(s(s(y0)), y1))) DEL(s(s(y0)), cons(0, cons(s(0), y1))) -> c6(IF2(false, s(s(y0)), 0, cons(s(0), y1))) DEL(s(0), cons(0, cons(s(0), y0))) -> c6(IF2(false, s(0), 0, cons(s(0), y0))) S tuples: 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)))) 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)))) SORT(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)))) 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(x0)), s(s(x1)), x2) -> c8(DEL(s(s(x0)), x2)) SORT(cons(z0, nil)) -> c15(SORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) 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 GE(s(s(y0)), s(s(y1))) -> c18(GE(s(y0), s(y1))) MAX(cons(s(z0), cons(s(0), x2))) -> c2(IF1(true, s(z0), s(0), x2)) MAX(cons(s(0), cons(s(s(z0)), x2))) -> c2(IF1(false, s(0), s(s(z0)), x2)) MAX(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c2(GE(s(s(y0)), s(s(y1)))) SORT(cons(z0, cons(0, z1))) -> c15(SORT(del(if1(true, z0, 0, z1), cons(z0, cons(0, z1)))), DEL(if1(true, z0, 0, z1), cons(z0, cons(0, z1))), MAX(cons(z0, cons(0, z1)))) SORT(cons(0, cons(s(z0), z1))) -> c15(SORT(del(if1(false, 0, s(z0), z1), cons(0, cons(s(z0), z1)))), DEL(if1(false, 0, s(z0), z1), cons(0, cons(s(z0), z1))), MAX(cons(0, cons(s(z0), z1)))) SORT(cons(s(z0), cons(s(z1), z2))) -> c15(SORT(del(if1(ge(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(ge(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MAX(cons(s(z0), cons(s(z1), z2)))) DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c6(EQ(s(s(y0)), s(s(y1)))) IF1(true, z0, 0, cons(0, y1)) -> c3(MAX(cons(z0, cons(0, y1)))) IF1(true, 0, 0, cons(s(y0), y1)) -> c3(MAX(cons(0, cons(s(y0), y1)))) IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c3(MAX(cons(s(s(y0)), cons(s(s(y1)), y2)))) IF1(true, s(y0), 0, cons(s(0), y1)) -> c3(MAX(cons(s(y0), cons(s(0), y1)))) IF1(true, s(0), 0, cons(s(s(y0)), y1)) -> c3(MAX(cons(s(0), cons(s(s(y0)), y1)))) MAX(cons(z0, cons(0, cons(0, y1)))) -> c2(IF1(true, z0, 0, cons(0, y1))) MAX(cons(0, cons(0, cons(s(y0), y1)))) -> c2(IF1(true, 0, 0, cons(s(y0), y1))) MAX(cons(s(s(y0)), cons(0, cons(s(s(y1)), y2)))) -> c2(IF1(true, s(s(y0)), 0, cons(s(s(y1)), y2))) MAX(cons(s(y0), cons(0, cons(s(0), y1)))) -> c2(IF1(true, s(y0), 0, cons(s(0), y1))) MAX(cons(s(0), cons(0, cons(s(s(y0)), y1)))) -> c2(IF1(true, s(0), 0, cons(s(s(y0)), y1))) IF1(true, s(s(y0)), s(0), cons(s(s(y1)), y2)) -> c3(MAX(cons(s(s(y0)), cons(s(s(y1)), y2)))) IF1(true, s(z0), s(0), cons(s(0), y1)) -> c3(MAX(cons(s(z0), cons(s(0), y1)))) IF1(true, s(0), s(0), cons(s(s(y0)), y1)) -> c3(MAX(cons(s(0), cons(s(s(y0)), y1)))) IF1(true, s(z0), s(0), cons(0, cons(0, y1))) -> c3(MAX(cons(s(z0), cons(0, cons(0, y1))))) IF1(true, s(s(y0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c3(MAX(cons(s(s(y0)), cons(0, cons(s(s(y1)), y2))))) IF1(true, s(z0), s(0), cons(0, cons(s(0), y1))) -> c3(MAX(cons(s(z0), cons(0, cons(s(0), y1))))) IF1(true, s(0), s(0), cons(0, cons(s(s(y0)), y1))) -> c3(MAX(cons(s(0), cons(0, cons(s(s(y0)), y1))))) IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c3(MAX(cons(s(s(z0)), cons(s(s(y1)), y2)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c3(MAX(cons(s(s(z0)), cons(s(0), y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c3(MAX(cons(s(s(z0)), cons(0, cons(0, y1))))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c3(MAX(cons(s(s(z0)), cons(0, cons(s(s(y1)), y2))))) IF1(true, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c3(MAX(cons(s(s(z0)), cons(0, cons(s(0), y1))))) IF1(false, 0, s(s(y0)), cons(s(s(y1)), y2)) -> c4(MAX(cons(s(s(y0)), cons(s(s(y1)), y2)))) IF1(false, 0, s(z0), cons(s(0), y1)) -> c4(MAX(cons(s(z0), cons(s(0), y1)))) IF1(false, 0, s(0), cons(s(s(y0)), y1)) -> c4(MAX(cons(s(0), cons(s(s(y0)), y1)))) IF1(false, 0, s(z0), cons(0, cons(0, y1))) -> c4(MAX(cons(s(z0), cons(0, cons(0, y1))))) IF1(false, 0, s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(MAX(cons(s(s(y0)), cons(0, cons(s(s(y1)), y2))))) IF1(false, 0, s(z0), cons(0, cons(s(0), y1))) -> c4(MAX(cons(s(z0), cons(0, cons(s(0), y1))))) IF1(false, 0, s(0), cons(0, cons(s(s(y0)), y1))) -> c4(MAX(cons(s(0), cons(0, cons(s(s(y0)), y1))))) MAX(cons(0, cons(s(s(y0)), cons(s(s(y1)), y2)))) -> c2(IF1(false, 0, s(s(y0)), cons(s(s(y1)), y2))) MAX(cons(0, cons(s(z0), cons(s(0), y1)))) -> c2(IF1(false, 0, s(z0), cons(s(0), y1))) MAX(cons(0, cons(s(0), cons(s(s(y0)), y1)))) -> c2(IF1(false, 0, s(0), cons(s(s(y0)), y1))) MAX(cons(0, cons(s(z0), cons(0, cons(0, y1))))) -> c2(IF1(false, 0, s(z0), cons(0, cons(0, y1)))) MAX(cons(0, cons(s(s(y0)), cons(0, cons(s(s(y1)), y2))))) -> c2(IF1(false, 0, s(s(y0)), cons(0, cons(s(s(y1)), y2)))) MAX(cons(0, cons(s(z0), cons(0, cons(s(0), y1))))) -> c2(IF1(false, 0, s(z0), cons(0, cons(s(0), y1)))) MAX(cons(0, cons(s(0), cons(0, cons(s(s(y0)), y1))))) -> c2(IF1(false, 0, s(0), cons(0, cons(s(s(y0)), y1)))) IF1(false, s(0), s(s(z0)), cons(s(s(y1)), y2)) -> c4(MAX(cons(s(s(z0)), cons(s(s(y1)), y2)))) IF1(false, s(0), s(s(z0)), cons(s(0), y1)) -> c4(MAX(cons(s(s(z0)), cons(s(0), y1)))) IF1(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c4(MAX(cons(s(s(z0)), cons(0, cons(0, y1))))) IF1(false, s(0), s(s(z0)), cons(0, cons(s(s(y1)), y2))) -> c4(MAX(cons(s(s(z0)), cons(0, cons(s(s(y1)), y2))))) IF1(false, s(0), s(s(z0)), cons(0, cons(s(0), y1))) -> c4(MAX(cons(s(s(z0)), cons(0, cons(s(0), y1))))) IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c4(MAX(cons(s(s(z1)), cons(s(s(y1)), y2)))) IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c4(MAX(cons(s(s(z1)), cons(s(0), y1)))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c4(MAX(cons(s(s(z1)), cons(0, cons(0, y1))))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c4(MAX(cons(s(s(z1)), cons(0, cons(s(s(y1)), y2))))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c4(MAX(cons(s(s(z1)), cons(0, cons(s(0), y1))))) IF2(false, 0, s(z0), cons(s(y0), y1)) -> c8(DEL(0, cons(s(y0), y1))) DEL(0, cons(s(z0), cons(s(y1), y2))) -> c6(IF2(false, 0, s(z0), cons(s(y1), y2))) IF2(false, s(z0), 0, cons(0, y1)) -> c8(DEL(s(z0), cons(0, y1))) IF2(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c8(DEL(s(s(y0)), cons(s(s(y1)), y2))) IF2(false, s(0), 0, cons(s(s(y0)), y1)) -> c8(DEL(s(0), cons(s(s(y0)), y1))) IF2(false, s(s(y0)), 0, cons(s(0), y1)) -> c8(DEL(s(s(y0)), cons(s(0), y1))) IF2(false, s(0), 0, cons(s(0), y0)) -> c8(DEL(s(0), cons(s(0), y0))) DEL(s(z0), cons(0, cons(0, y1))) -> c6(IF2(false, s(z0), 0, cons(0, y1))) DEL(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c6(IF2(false, s(s(y0)), 0, cons(s(s(y1)), y2))) DEL(s(0), cons(0, cons(s(s(y0)), y1))) -> c6(IF2(false, s(0), 0, cons(s(s(y0)), y1))) DEL(s(s(y0)), cons(0, cons(s(0), y1))) -> c6(IF2(false, s(s(y0)), 0, cons(s(0), y1))) DEL(s(0), cons(0, cons(s(0), y0))) -> c6(IF2(false, s(0), 0, cons(s(0), y0))) K tuples: SORT(cons(x0, cons(x1, x2))) -> c15(DEL(max(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) SORT(cons(x0, nil)) -> c15(DEL(max(cons(x0, nil)), cons(x0, nil))) SORT(cons(z0, cons(0, y1))) -> c14(MAX(cons(z0, cons(0, y1)))) SORT(cons(0, cons(s(y0), y1))) -> c14(MAX(cons(0, cons(s(y0), y1)))) SORT(cons(s(y0), cons(s(0), y1))) -> c14(MAX(cons(s(y0), cons(s(0), y1)))) SORT(cons(s(0), cons(s(s(y0)), y1))) -> c14(MAX(cons(s(0), cons(s(s(y0)), y1)))) SORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c14(MAX(cons(s(s(y0)), cons(s(s(y1)), y2)))) SORT(cons(s(y0), cons(s(y1), y2))) -> c14(MAX(cons(s(y0), cons(s(y1), y2)))) Defined Rule Symbols: ge_2, eq_2, del_2, max_1, if1_4, if2_4 Defined Pair Symbols: MAX_1, DEL_2, SORT_1, IF2_4, EQ_2, GE_2, IF1_4 Compound Symbols: c2_2, c6_2, c15_3, c8_1, c15_1, c15_2, c12_1, c6_1, c6, c14_1, c18_1, c2_1, c3_1, c4_1