KILLED proof of input_j7MptV71Tn.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). (0) CpxTRS (1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxTRS (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (4) CpxTRS (5) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxWeightedTrs (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (8) CpxTypedWeightedTrs (9) CompletionProof [UPPER BOUND(ID), 0 ms] (10) CpxTypedWeightedCompleteTrs (11) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (12) CpxRNTS (13) CompletionProof [UPPER BOUND(ID), 0 ms] (14) CpxTypedWeightedCompleteTrs (15) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (16) CpxTypedWeightedCompleteTrs (17) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (18) CpxRNTS (19) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (20) CpxRNTS (21) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms] (22) CpxRNTS (23) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (24) CpxRNTS (25) IntTrsBoundProof [UPPER BOUND(ID), 333 ms] (26) CpxRNTS (27) IntTrsBoundProof [UPPER BOUND(ID), 132 ms] (28) CpxRNTS (29) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (30) CpxRNTS (31) IntTrsBoundProof [UPPER BOUND(ID), 388 ms] (32) CpxRNTS (33) IntTrsBoundProof [UPPER BOUND(ID), 149 ms] (34) CpxRNTS (35) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (36) CpxRNTS (37) IntTrsBoundProof [UPPER BOUND(ID), 1173 ms] (38) CpxRNTS (39) IntTrsBoundProof [UPPER BOUND(ID), 659 ms] (40) CpxRNTS (41) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (42) CpxRNTS (43) IntTrsBoundProof [UPPER BOUND(ID), 1050 ms] (44) CpxRNTS (45) IntTrsBoundProof [UPPER BOUND(ID), 457 ms] (46) CpxRNTS (47) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (48) CpxRNTS (49) IntTrsBoundProof [UPPER BOUND(ID), 1158 ms] (50) CpxRNTS (51) IntTrsBoundProof [UPPER BOUND(ID), 277 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^1)), 169 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) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (68) CdtProblem (69) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (70) CdtProblem (71) CdtLeafRemovalProof [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) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (78) CdtProblem (79) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (80) CdtProblem (81) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 79 ms] (82) CdtProblem (83) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (84) CdtProblem (85) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 69 ms] (86) CdtProblem (87) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (88) CdtProblem (89) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (90) CdtProblem (91) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (92) CdtProblem (93) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 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) CdtGraphSplitRhsProof [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) CdtRewritingProof [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) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (122) CdtProblem (123) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (124) CdtProblem (125) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (126) CdtProblem (127) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (128) CdtProblem (129) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (130) CdtProblem (131) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (132) CdtProblem (133) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (134) CdtProblem (135) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (136) CdtProblem (137) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (138) CdtProblem (139) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (140) CdtProblem (141) CdtForwardInstantiationProof [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), 0 ms] (156) CdtProblem (157) CdtRewritingProof [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 (173) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (174) CdtProblem (175) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (176) 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: le(0, y) -> true le(s(x), 0) -> false le(s(x), s(y)) -> le(x, y) eq(0, 0) -> true eq(0, s(y)) -> false eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) if1(true, x, y, xs) -> min(x, xs) if1(false, x, y, xs) -> min(y, xs) if2(true, x, y, xs) -> xs if2(false, x, y, xs) -> cons(y, del(x, xs)) minsort(nil) -> nil minsort(cons(x, y)) -> cons(min(x, y), minsort(del(min(x, y), cons(x, y)))) min(x, nil) -> x min(x, cons(y, z)) -> if1(le(x, y), x, y, z) del(x, nil) -> nil del(x, cons(y, z)) -> if2(eq(x, y), x, y, z) S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (1) RenamingProof (BOTH BOUNDS(ID, ID)) Renamed function symbols to avoid clashes with predefined symbol. ---------------------------------------- (2) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: le(0', y) -> true le(s(x), 0') -> false le(s(x), s(y)) -> le(x, y) eq(0', 0') -> true eq(0', s(y)) -> false eq(s(x), 0') -> false eq(s(x), s(y)) -> eq(x, y) if1(true, x, y, xs) -> min(x, xs) if1(false, x, y, xs) -> min(y, xs) if2(true, x, y, xs) -> xs if2(false, x, y, xs) -> cons(y, del(x, xs)) minsort(nil) -> nil minsort(cons(x, y)) -> cons(min(x, y), minsort(del(min(x, y), cons(x, y)))) min(x, nil) -> x min(x, cons(y, z)) -> if1(le(x, y), x, y, z) del(x, nil) -> nil del(x, cons(y, z)) -> if2(eq(x, y), x, y, z) S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (3) RelTrsToTrsProof (UPPER BOUND(ID)) transformed relative TRS to TRS ---------------------------------------- (4) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: le(0, y) -> true le(s(x), 0) -> false le(s(x), s(y)) -> le(x, y) eq(0, 0) -> true eq(0, s(y)) -> false eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) if1(true, x, y, xs) -> min(x, xs) if1(false, x, y, xs) -> min(y, xs) if2(true, x, y, xs) -> xs if2(false, x, y, xs) -> cons(y, del(x, xs)) minsort(nil) -> nil minsort(cons(x, y)) -> cons(min(x, y), minsort(del(min(x, y), cons(x, y)))) min(x, nil) -> x min(x, cons(y, z)) -> if1(le(x, y), x, y, z) del(x, nil) -> nil del(x, cons(y, z)) -> if2(eq(x, y), x, y, z) S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (5) RelTrsToWeightedTrsProof (UPPER BOUND(ID)) Transformed relative TRS to weighted TRS ---------------------------------------- (6) Obligation: The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: le(0, y) -> true [1] le(s(x), 0) -> false [1] le(s(x), s(y)) -> le(x, y) [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] if1(true, x, y, xs) -> min(x, xs) [1] if1(false, x, y, xs) -> min(y, xs) [1] if2(true, x, y, xs) -> xs [1] if2(false, x, y, xs) -> cons(y, del(x, xs)) [1] minsort(nil) -> nil [1] minsort(cons(x, y)) -> cons(min(x, y), minsort(del(min(x, y), cons(x, y)))) [1] min(x, nil) -> x [1] min(x, cons(y, z)) -> if1(le(x, y), x, y, z) [1] del(x, nil) -> nil [1] del(x, cons(y, z)) -> if2(eq(x, y), x, y, z) [1] Rewrite Strategy: INNERMOST ---------------------------------------- (7) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (8) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: le(0, y) -> true [1] le(s(x), 0) -> false [1] le(s(x), s(y)) -> le(x, y) [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] if1(true, x, y, xs) -> min(x, xs) [1] if1(false, x, y, xs) -> min(y, xs) [1] if2(true, x, y, xs) -> xs [1] if2(false, x, y, xs) -> cons(y, del(x, xs)) [1] minsort(nil) -> nil [1] minsort(cons(x, y)) -> cons(min(x, y), minsort(del(min(x, y), cons(x, y)))) [1] min(x, nil) -> x [1] min(x, cons(y, z)) -> if1(le(x, y), x, y, z) [1] del(x, nil) -> nil [1] del(x, cons(y, z)) -> if2(eq(x, y), x, y, z) [1] The TRS has the following type information: le :: 0:s -> 0:s -> true:false 0 :: 0:s true :: true:false s :: 0:s -> 0:s false :: true:false eq :: 0:s -> 0:s -> true:false if1 :: true:false -> 0:s -> 0:s -> cons:nil -> 0:s min :: 0:s -> cons:nil -> 0:s if2 :: true:false -> 0:s -> 0:s -> cons:nil -> cons:nil cons :: 0:s -> cons:nil -> cons:nil del :: 0:s -> cons:nil -> cons:nil minsort :: cons:nil -> cons:nil nil :: cons:nil 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: le(0, y) -> true [1] le(s(x), 0) -> false [1] le(s(x), s(y)) -> le(x, y) [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] if1(true, x, y, xs) -> min(x, xs) [1] if1(false, x, y, xs) -> min(y, xs) [1] if2(true, x, y, xs) -> xs [1] if2(false, x, y, xs) -> cons(y, del(x, xs)) [1] minsort(nil) -> nil [1] minsort(cons(x, y)) -> cons(min(x, y), minsort(del(min(x, y), cons(x, y)))) [1] min(x, nil) -> x [1] min(x, cons(y, z)) -> if1(le(x, y), x, y, z) [1] del(x, nil) -> nil [1] del(x, cons(y, z)) -> if2(eq(x, y), x, y, z) [1] The TRS has the following type information: le :: 0:s -> 0:s -> true:false 0 :: 0:s true :: true:false s :: 0:s -> 0:s false :: true:false eq :: 0:s -> 0:s -> true:false if1 :: true:false -> 0:s -> 0:s -> cons:nil -> 0:s min :: 0:s -> cons:nil -> 0:s if2 :: true:false -> 0:s -> 0:s -> cons:nil -> cons:nil cons :: 0:s -> cons:nil -> cons:nil del :: 0:s -> cons:nil -> cons:nil minsort :: cons:nil -> cons:nil nil :: cons:nil 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: 0 => 0 true => 1 false => 0 nil => 0 ---------------------------------------- (12) Obligation: Complexity RNTS consisting of the following rules: del(z', z'') -{ 1 }-> if2(eq(x, y), x, y, z) :|: z >= 0, z' = x, x >= 0, y >= 0, z'' = 1 + y + z del(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' = x, x >= 0 eq(z', z'') -{ 1 }-> eq(x, y) :|: z' = 1 + x, x >= 0, y >= 0, z'' = 1 + y eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' = 0 eq(z', z'') -{ 1 }-> 0 :|: y >= 0, z'' = 1 + y, z' = 0 eq(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' = 1 + x, x >= 0 if1(z', z'', z1, z2) -{ 1 }-> min(x, xs) :|: xs >= 0, z1 = y, x >= 0, y >= 0, z'' = x, z' = 1, z2 = xs if1(z', z'', z1, z2) -{ 1 }-> min(y, xs) :|: xs >= 0, z1 = y, x >= 0, y >= 0, z'' = x, z2 = xs, z' = 0 if2(z', z'', z1, z2) -{ 1 }-> xs :|: xs >= 0, z1 = y, x >= 0, y >= 0, z'' = x, z' = 1, z2 = xs if2(z', z'', z1, z2) -{ 1 }-> 1 + y + del(x, xs) :|: xs >= 0, z1 = y, x >= 0, y >= 0, z'' = x, z2 = xs, z' = 0 le(z', z'') -{ 1 }-> le(x, y) :|: z' = 1 + x, x >= 0, y >= 0, z'' = 1 + y le(z', z'') -{ 1 }-> 1 :|: z'' = y, y >= 0, z' = 0 le(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' = 1 + x, x >= 0 min(z', z'') -{ 1 }-> x :|: z'' = 0, z' = x, x >= 0 min(z', z'') -{ 1 }-> if1(le(x, y), x, y, z) :|: z >= 0, z' = x, x >= 0, y >= 0, z'' = 1 + y + z minsort(z') -{ 1 }-> 0 :|: z' = 0 minsort(z') -{ 1 }-> 1 + min(x, y) + minsort(del(min(x, y), 1 + x + y)) :|: z' = 1 + x + y, x >= 0, y >= 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: minsort_1 (c) The following functions are completely defined: del_2 min_2 eq_2 le_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: le(0, y) -> true [1] le(s(x), 0) -> false [1] le(s(x), s(y)) -> le(x, y) [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] if1(true, x, y, xs) -> min(x, xs) [1] if1(false, x, y, xs) -> min(y, xs) [1] if2(true, x, y, xs) -> xs [1] if2(false, x, y, xs) -> cons(y, del(x, xs)) [1] minsort(nil) -> nil [1] minsort(cons(x, y)) -> cons(min(x, y), minsort(del(min(x, y), cons(x, y)))) [1] min(x, nil) -> x [1] min(x, cons(y, z)) -> if1(le(x, y), x, y, z) [1] del(x, nil) -> nil [1] del(x, cons(y, z)) -> if2(eq(x, y), x, y, z) [1] The TRS has the following type information: le :: 0:s -> 0:s -> true:false 0 :: 0:s true :: true:false s :: 0:s -> 0:s false :: true:false eq :: 0:s -> 0:s -> true:false if1 :: true:false -> 0:s -> 0:s -> cons:nil -> 0:s min :: 0:s -> cons:nil -> 0:s if2 :: true:false -> 0:s -> 0:s -> cons:nil -> cons:nil cons :: 0:s -> cons:nil -> cons:nil del :: 0:s -> cons:nil -> cons:nil minsort :: cons:nil -> cons:nil nil :: cons:nil 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: le(0, y) -> true [1] le(s(x), 0) -> false [1] le(s(x), s(y)) -> le(x, y) [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] if1(true, x, y, xs) -> min(x, xs) [1] if1(false, x, y, xs) -> min(y, xs) [1] if2(true, x, y, xs) -> xs [1] if2(false, x, y, xs) -> cons(y, del(x, xs)) [1] minsort(nil) -> nil [1] minsort(cons(x, nil)) -> cons(min(x, nil), minsort(del(x, cons(x, nil)))) [2] minsort(cons(x, cons(y', z'))) -> cons(min(x, cons(y', z')), minsort(del(if1(le(x, y'), x, y', z'), cons(x, cons(y', z'))))) [2] min(x, nil) -> x [1] min(0, cons(y, z)) -> if1(true, 0, y, z) [2] min(s(x'), cons(0, z)) -> if1(false, s(x'), 0, z) [2] min(s(x''), cons(s(y''), z)) -> if1(le(x'', y''), s(x''), s(y''), z) [2] del(x, nil) -> nil [1] del(0, cons(0, z)) -> if2(true, 0, 0, z) [2] del(0, cons(s(y1), z)) -> if2(false, 0, s(y1), z) [2] del(s(x1), cons(0, z)) -> if2(false, s(x1), 0, z) [2] del(s(x2), cons(s(y2), z)) -> if2(eq(x2, y2), s(x2), s(y2), z) [2] The TRS has the following type information: le :: 0:s -> 0:s -> true:false 0 :: 0:s true :: true:false s :: 0:s -> 0:s false :: true:false eq :: 0:s -> 0:s -> true:false if1 :: true:false -> 0:s -> 0:s -> cons:nil -> 0:s min :: 0:s -> cons:nil -> 0:s if2 :: true:false -> 0:s -> 0:s -> cons:nil -> cons:nil cons :: 0:s -> cons:nil -> cons:nil del :: 0:s -> cons:nil -> cons:nil minsort :: cons:nil -> cons:nil nil :: cons:nil 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: 0 => 0 true => 1 false => 0 nil => 0 ---------------------------------------- (18) Obligation: Complexity RNTS consisting of the following rules: del(z'', z1) -{ 2 }-> if2(eq(x2, y2), 1 + x2, 1 + y2, z) :|: z >= 0, z1 = 1 + (1 + y2) + z, z'' = 1 + x2, y2 >= 0, x2 >= 0 del(z'', z1) -{ 2 }-> if2(1, 0, 0, z) :|: z'' = 0, z >= 0, z1 = 1 + 0 + z del(z'', z1) -{ 2 }-> if2(0, 0, 1 + y1, z) :|: z'' = 0, y1 >= 0, z >= 0, z1 = 1 + (1 + y1) + z del(z'', z1) -{ 2 }-> if2(0, 1 + x1, 0, z) :|: x1 >= 0, z >= 0, z'' = 1 + x1, z1 = 1 + 0 + z del(z'', z1) -{ 1 }-> 0 :|: z1 = 0, x >= 0, z'' = x eq(z'', z1) -{ 1 }-> eq(x, y) :|: x >= 0, y >= 0, z1 = 1 + y, z'' = 1 + x eq(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 = 0 eq(z'', z1) -{ 1 }-> 0 :|: z'' = 0, y >= 0, z1 = 1 + y eq(z'', z1) -{ 1 }-> 0 :|: z1 = 0, x >= 0, z'' = 1 + x if1(z'', z1, z2, z3) -{ 1 }-> min(x, xs) :|: xs >= 0, z2 = y, x >= 0, y >= 0, z3 = xs, z'' = 1, z1 = x if1(z'', z1, z2, z3) -{ 1 }-> min(y, xs) :|: z'' = 0, xs >= 0, z2 = y, x >= 0, y >= 0, z3 = xs, z1 = x if2(z'', z1, z2, z3) -{ 1 }-> xs :|: xs >= 0, z2 = y, x >= 0, y >= 0, z3 = xs, z'' = 1, z1 = x if2(z'', z1, z2, z3) -{ 1 }-> 1 + y + del(x, xs) :|: z'' = 0, xs >= 0, z2 = y, x >= 0, y >= 0, z3 = xs, z1 = x le(z'', z1) -{ 1 }-> le(x, y) :|: x >= 0, y >= 0, z1 = 1 + y, z'' = 1 + x le(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 = y, y >= 0 le(z'', z1) -{ 1 }-> 0 :|: z1 = 0, x >= 0, z'' = 1 + x min(z'', z1) -{ 1 }-> x :|: z1 = 0, x >= 0, z'' = x min(z'', z1) -{ 2 }-> if1(le(x'', y''), 1 + x'', 1 + y'', z) :|: z >= 0, z1 = 1 + (1 + y'') + z, y'' >= 0, z'' = 1 + x'', x'' >= 0 min(z'', z1) -{ 2 }-> if1(1, 0, y, z) :|: z'' = 0, z >= 0, z1 = 1 + y + z, y >= 0 min(z'', z1) -{ 2 }-> if1(0, 1 + x', 0, z) :|: z >= 0, z'' = 1 + x', x' >= 0, z1 = 1 + 0 + z minsort(z'') -{ 1 }-> 0 :|: z'' = 0 minsort(z'') -{ 2 }-> 1 + min(x, 0) + minsort(del(x, 1 + x + 0)) :|: z'' = 1 + x + 0, x >= 0 minsort(z'') -{ 2 }-> 1 + min(x, 1 + y' + z') + minsort(del(if1(le(x, y'), x, y', z'), 1 + x + (1 + y' + z'))) :|: z'' = 1 + x + (1 + y' + z'), x >= 0, z' >= 0, y' >= 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'', z1) -{ 2 }-> if2(eq(z'' - 1, y2), 1 + (z'' - 1), 1 + y2, z) :|: z >= 0, z1 = 1 + (1 + y2) + z, y2 >= 0, z'' - 1 >= 0 del(z'', z1) -{ 2 }-> if2(1, 0, 0, z1 - 1) :|: z'' = 0, z1 - 1 >= 0 del(z'', z1) -{ 2 }-> if2(0, 0, 1 + y1, z) :|: z'' = 0, y1 >= 0, z >= 0, z1 = 1 + (1 + y1) + z del(z'', z1) -{ 2 }-> if2(0, 1 + (z'' - 1), 0, z1 - 1) :|: z'' - 1 >= 0, z1 - 1 >= 0 del(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' >= 0 eq(z'', z1) -{ 1 }-> eq(z'' - 1, z1 - 1) :|: z'' - 1 >= 0, z1 - 1 >= 0 eq(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 = 0 eq(z'', z1) -{ 1 }-> 0 :|: z'' = 0, z1 - 1 >= 0 eq(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' - 1 >= 0 if1(z'', z1, z2, z3) -{ 1 }-> min(z1, z3) :|: z3 >= 0, z1 >= 0, z2 >= 0, z'' = 1 if1(z'', z1, z2, z3) -{ 1 }-> min(z2, z3) :|: z'' = 0, z3 >= 0, z1 >= 0, z2 >= 0 if2(z'', z1, z2, z3) -{ 1 }-> z3 :|: z3 >= 0, z1 >= 0, z2 >= 0, z'' = 1 if2(z'', z1, z2, z3) -{ 1 }-> 1 + z2 + del(z1, z3) :|: z'' = 0, z3 >= 0, z1 >= 0, z2 >= 0 le(z'', z1) -{ 1 }-> le(z'' - 1, z1 - 1) :|: z'' - 1 >= 0, z1 - 1 >= 0 le(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 >= 0 le(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' - 1 >= 0 min(z'', z1) -{ 1 }-> z'' :|: z1 = 0, z'' >= 0 min(z'', z1) -{ 2 }-> if1(le(z'' - 1, y''), 1 + (z'' - 1), 1 + y'', z) :|: z >= 0, z1 = 1 + (1 + y'') + z, y'' >= 0, z'' - 1 >= 0 min(z'', z1) -{ 2 }-> if1(1, 0, y, z) :|: z'' = 0, z >= 0, z1 = 1 + y + z, y >= 0 min(z'', z1) -{ 2 }-> if1(0, 1 + (z'' - 1), 0, z1 - 1) :|: z1 - 1 >= 0, z'' - 1 >= 0 minsort(z'') -{ 1 }-> 0 :|: z'' = 0 minsort(z'') -{ 2 }-> 1 + min(x, 1 + y' + z') + minsort(del(if1(le(x, y'), x, y', z'), 1 + x + (1 + y' + z'))) :|: z'' = 1 + x + (1 + y' + z'), x >= 0, z' >= 0, y' >= 0 minsort(z'') -{ 2 }-> 1 + min(z'' - 1, 0) + minsort(del(z'' - 1, 1 + (z'' - 1) + 0)) :|: z'' - 1 >= 0 ---------------------------------------- (21) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { le } { eq } { min, if1 } { del, if2 } { minsort } ---------------------------------------- (22) Obligation: Complexity RNTS consisting of the following rules: del(z'', z1) -{ 2 }-> if2(eq(z'' - 1, y2), 1 + (z'' - 1), 1 + y2, z) :|: z >= 0, z1 = 1 + (1 + y2) + z, y2 >= 0, z'' - 1 >= 0 del(z'', z1) -{ 2 }-> if2(1, 0, 0, z1 - 1) :|: z'' = 0, z1 - 1 >= 0 del(z'', z1) -{ 2 }-> if2(0, 0, 1 + y1, z) :|: z'' = 0, y1 >= 0, z >= 0, z1 = 1 + (1 + y1) + z del(z'', z1) -{ 2 }-> if2(0, 1 + (z'' - 1), 0, z1 - 1) :|: z'' - 1 >= 0, z1 - 1 >= 0 del(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' >= 0 eq(z'', z1) -{ 1 }-> eq(z'' - 1, z1 - 1) :|: z'' - 1 >= 0, z1 - 1 >= 0 eq(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 = 0 eq(z'', z1) -{ 1 }-> 0 :|: z'' = 0, z1 - 1 >= 0 eq(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' - 1 >= 0 if1(z'', z1, z2, z3) -{ 1 }-> min(z1, z3) :|: z3 >= 0, z1 >= 0, z2 >= 0, z'' = 1 if1(z'', z1, z2, z3) -{ 1 }-> min(z2, z3) :|: z'' = 0, z3 >= 0, z1 >= 0, z2 >= 0 if2(z'', z1, z2, z3) -{ 1 }-> z3 :|: z3 >= 0, z1 >= 0, z2 >= 0, z'' = 1 if2(z'', z1, z2, z3) -{ 1 }-> 1 + z2 + del(z1, z3) :|: z'' = 0, z3 >= 0, z1 >= 0, z2 >= 0 le(z'', z1) -{ 1 }-> le(z'' - 1, z1 - 1) :|: z'' - 1 >= 0, z1 - 1 >= 0 le(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 >= 0 le(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' - 1 >= 0 min(z'', z1) -{ 1 }-> z'' :|: z1 = 0, z'' >= 0 min(z'', z1) -{ 2 }-> if1(le(z'' - 1, y''), 1 + (z'' - 1), 1 + y'', z) :|: z >= 0, z1 = 1 + (1 + y'') + z, y'' >= 0, z'' - 1 >= 0 min(z'', z1) -{ 2 }-> if1(1, 0, y, z) :|: z'' = 0, z >= 0, z1 = 1 + y + z, y >= 0 min(z'', z1) -{ 2 }-> if1(0, 1 + (z'' - 1), 0, z1 - 1) :|: z1 - 1 >= 0, z'' - 1 >= 0 minsort(z'') -{ 1 }-> 0 :|: z'' = 0 minsort(z'') -{ 2 }-> 1 + min(x, 1 + y' + z') + minsort(del(if1(le(x, y'), x, y', z'), 1 + x + (1 + y' + z'))) :|: z'' = 1 + x + (1 + y' + z'), x >= 0, z' >= 0, y' >= 0 minsort(z'') -{ 2 }-> 1 + min(z'' - 1, 0) + minsort(del(z'' - 1, 1 + (z'' - 1) + 0)) :|: z'' - 1 >= 0 Function symbols to be analyzed: {le}, {eq}, {min,if1}, {del,if2}, {minsort} ---------------------------------------- (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'', z1) -{ 2 }-> if2(eq(z'' - 1, y2), 1 + (z'' - 1), 1 + y2, z) :|: z >= 0, z1 = 1 + (1 + y2) + z, y2 >= 0, z'' - 1 >= 0 del(z'', z1) -{ 2 }-> if2(1, 0, 0, z1 - 1) :|: z'' = 0, z1 - 1 >= 0 del(z'', z1) -{ 2 }-> if2(0, 0, 1 + y1, z) :|: z'' = 0, y1 >= 0, z >= 0, z1 = 1 + (1 + y1) + z del(z'', z1) -{ 2 }-> if2(0, 1 + (z'' - 1), 0, z1 - 1) :|: z'' - 1 >= 0, z1 - 1 >= 0 del(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' >= 0 eq(z'', z1) -{ 1 }-> eq(z'' - 1, z1 - 1) :|: z'' - 1 >= 0, z1 - 1 >= 0 eq(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 = 0 eq(z'', z1) -{ 1 }-> 0 :|: z'' = 0, z1 - 1 >= 0 eq(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' - 1 >= 0 if1(z'', z1, z2, z3) -{ 1 }-> min(z1, z3) :|: z3 >= 0, z1 >= 0, z2 >= 0, z'' = 1 if1(z'', z1, z2, z3) -{ 1 }-> min(z2, z3) :|: z'' = 0, z3 >= 0, z1 >= 0, z2 >= 0 if2(z'', z1, z2, z3) -{ 1 }-> z3 :|: z3 >= 0, z1 >= 0, z2 >= 0, z'' = 1 if2(z'', z1, z2, z3) -{ 1 }-> 1 + z2 + del(z1, z3) :|: z'' = 0, z3 >= 0, z1 >= 0, z2 >= 0 le(z'', z1) -{ 1 }-> le(z'' - 1, z1 - 1) :|: z'' - 1 >= 0, z1 - 1 >= 0 le(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 >= 0 le(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' - 1 >= 0 min(z'', z1) -{ 1 }-> z'' :|: z1 = 0, z'' >= 0 min(z'', z1) -{ 2 }-> if1(le(z'' - 1, y''), 1 + (z'' - 1), 1 + y'', z) :|: z >= 0, z1 = 1 + (1 + y'') + z, y'' >= 0, z'' - 1 >= 0 min(z'', z1) -{ 2 }-> if1(1, 0, y, z) :|: z'' = 0, z >= 0, z1 = 1 + y + z, y >= 0 min(z'', z1) -{ 2 }-> if1(0, 1 + (z'' - 1), 0, z1 - 1) :|: z1 - 1 >= 0, z'' - 1 >= 0 minsort(z'') -{ 1 }-> 0 :|: z'' = 0 minsort(z'') -{ 2 }-> 1 + min(x, 1 + y' + z') + minsort(del(if1(le(x, y'), x, y', z'), 1 + x + (1 + y' + z'))) :|: z'' = 1 + x + (1 + y' + z'), x >= 0, z' >= 0, y' >= 0 minsort(z'') -{ 2 }-> 1 + min(z'' - 1, 0) + minsort(del(z'' - 1, 1 + (z'' - 1) + 0)) :|: z'' - 1 >= 0 Function symbols to be analyzed: {le}, {eq}, {min,if1}, {del,if2}, {minsort} ---------------------------------------- (25) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: le after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (26) Obligation: Complexity RNTS consisting of the following rules: del(z'', z1) -{ 2 }-> if2(eq(z'' - 1, y2), 1 + (z'' - 1), 1 + y2, z) :|: z >= 0, z1 = 1 + (1 + y2) + z, y2 >= 0, z'' - 1 >= 0 del(z'', z1) -{ 2 }-> if2(1, 0, 0, z1 - 1) :|: z'' = 0, z1 - 1 >= 0 del(z'', z1) -{ 2 }-> if2(0, 0, 1 + y1, z) :|: z'' = 0, y1 >= 0, z >= 0, z1 = 1 + (1 + y1) + z del(z'', z1) -{ 2 }-> if2(0, 1 + (z'' - 1), 0, z1 - 1) :|: z'' - 1 >= 0, z1 - 1 >= 0 del(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' >= 0 eq(z'', z1) -{ 1 }-> eq(z'' - 1, z1 - 1) :|: z'' - 1 >= 0, z1 - 1 >= 0 eq(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 = 0 eq(z'', z1) -{ 1 }-> 0 :|: z'' = 0, z1 - 1 >= 0 eq(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' - 1 >= 0 if1(z'', z1, z2, z3) -{ 1 }-> min(z1, z3) :|: z3 >= 0, z1 >= 0, z2 >= 0, z'' = 1 if1(z'', z1, z2, z3) -{ 1 }-> min(z2, z3) :|: z'' = 0, z3 >= 0, z1 >= 0, z2 >= 0 if2(z'', z1, z2, z3) -{ 1 }-> z3 :|: z3 >= 0, z1 >= 0, z2 >= 0, z'' = 1 if2(z'', z1, z2, z3) -{ 1 }-> 1 + z2 + del(z1, z3) :|: z'' = 0, z3 >= 0, z1 >= 0, z2 >= 0 le(z'', z1) -{ 1 }-> le(z'' - 1, z1 - 1) :|: z'' - 1 >= 0, z1 - 1 >= 0 le(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 >= 0 le(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' - 1 >= 0 min(z'', z1) -{ 1 }-> z'' :|: z1 = 0, z'' >= 0 min(z'', z1) -{ 2 }-> if1(le(z'' - 1, y''), 1 + (z'' - 1), 1 + y'', z) :|: z >= 0, z1 = 1 + (1 + y'') + z, y'' >= 0, z'' - 1 >= 0 min(z'', z1) -{ 2 }-> if1(1, 0, y, z) :|: z'' = 0, z >= 0, z1 = 1 + y + z, y >= 0 min(z'', z1) -{ 2 }-> if1(0, 1 + (z'' - 1), 0, z1 - 1) :|: z1 - 1 >= 0, z'' - 1 >= 0 minsort(z'') -{ 1 }-> 0 :|: z'' = 0 minsort(z'') -{ 2 }-> 1 + min(x, 1 + y' + z') + minsort(del(if1(le(x, y'), x, y', z'), 1 + x + (1 + y' + z'))) :|: z'' = 1 + x + (1 + y' + z'), x >= 0, z' >= 0, y' >= 0 minsort(z'') -{ 2 }-> 1 + min(z'' - 1, 0) + minsort(del(z'' - 1, 1 + (z'' - 1) + 0)) :|: z'' - 1 >= 0 Function symbols to be analyzed: {le}, {eq}, {min,if1}, {del,if2}, {minsort} Previous analysis results are: le: runtime: ?, size: O(1) [1] ---------------------------------------- (27) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: le after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2 + z1 ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: del(z'', z1) -{ 2 }-> if2(eq(z'' - 1, y2), 1 + (z'' - 1), 1 + y2, z) :|: z >= 0, z1 = 1 + (1 + y2) + z, y2 >= 0, z'' - 1 >= 0 del(z'', z1) -{ 2 }-> if2(1, 0, 0, z1 - 1) :|: z'' = 0, z1 - 1 >= 0 del(z'', z1) -{ 2 }-> if2(0, 0, 1 + y1, z) :|: z'' = 0, y1 >= 0, z >= 0, z1 = 1 + (1 + y1) + z del(z'', z1) -{ 2 }-> if2(0, 1 + (z'' - 1), 0, z1 - 1) :|: z'' - 1 >= 0, z1 - 1 >= 0 del(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' >= 0 eq(z'', z1) -{ 1 }-> eq(z'' - 1, z1 - 1) :|: z'' - 1 >= 0, z1 - 1 >= 0 eq(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 = 0 eq(z'', z1) -{ 1 }-> 0 :|: z'' = 0, z1 - 1 >= 0 eq(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' - 1 >= 0 if1(z'', z1, z2, z3) -{ 1 }-> min(z1, z3) :|: z3 >= 0, z1 >= 0, z2 >= 0, z'' = 1 if1(z'', z1, z2, z3) -{ 1 }-> min(z2, z3) :|: z'' = 0, z3 >= 0, z1 >= 0, z2 >= 0 if2(z'', z1, z2, z3) -{ 1 }-> z3 :|: z3 >= 0, z1 >= 0, z2 >= 0, z'' = 1 if2(z'', z1, z2, z3) -{ 1 }-> 1 + z2 + del(z1, z3) :|: z'' = 0, z3 >= 0, z1 >= 0, z2 >= 0 le(z'', z1) -{ 1 }-> le(z'' - 1, z1 - 1) :|: z'' - 1 >= 0, z1 - 1 >= 0 le(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 >= 0 le(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' - 1 >= 0 min(z'', z1) -{ 1 }-> z'' :|: z1 = 0, z'' >= 0 min(z'', z1) -{ 2 }-> if1(le(z'' - 1, y''), 1 + (z'' - 1), 1 + y'', z) :|: z >= 0, z1 = 1 + (1 + y'') + z, y'' >= 0, z'' - 1 >= 0 min(z'', z1) -{ 2 }-> if1(1, 0, y, z) :|: z'' = 0, z >= 0, z1 = 1 + y + z, y >= 0 min(z'', z1) -{ 2 }-> if1(0, 1 + (z'' - 1), 0, z1 - 1) :|: z1 - 1 >= 0, z'' - 1 >= 0 minsort(z'') -{ 1 }-> 0 :|: z'' = 0 minsort(z'') -{ 2 }-> 1 + min(x, 1 + y' + z') + minsort(del(if1(le(x, y'), x, y', z'), 1 + x + (1 + y' + z'))) :|: z'' = 1 + x + (1 + y' + z'), x >= 0, z' >= 0, y' >= 0 minsort(z'') -{ 2 }-> 1 + min(z'' - 1, 0) + minsort(del(z'' - 1, 1 + (z'' - 1) + 0)) :|: z'' - 1 >= 0 Function symbols to be analyzed: {eq}, {min,if1}, {del,if2}, {minsort} Previous analysis results are: le: runtime: O(n^1) [2 + z1], 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'', z1) -{ 2 }-> if2(eq(z'' - 1, y2), 1 + (z'' - 1), 1 + y2, z) :|: z >= 0, z1 = 1 + (1 + y2) + z, y2 >= 0, z'' - 1 >= 0 del(z'', z1) -{ 2 }-> if2(1, 0, 0, z1 - 1) :|: z'' = 0, z1 - 1 >= 0 del(z'', z1) -{ 2 }-> if2(0, 0, 1 + y1, z) :|: z'' = 0, y1 >= 0, z >= 0, z1 = 1 + (1 + y1) + z del(z'', z1) -{ 2 }-> if2(0, 1 + (z'' - 1), 0, z1 - 1) :|: z'' - 1 >= 0, z1 - 1 >= 0 del(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' >= 0 eq(z'', z1) -{ 1 }-> eq(z'' - 1, z1 - 1) :|: z'' - 1 >= 0, z1 - 1 >= 0 eq(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 = 0 eq(z'', z1) -{ 1 }-> 0 :|: z'' = 0, z1 - 1 >= 0 eq(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' - 1 >= 0 if1(z'', z1, z2, z3) -{ 1 }-> min(z1, z3) :|: z3 >= 0, z1 >= 0, z2 >= 0, z'' = 1 if1(z'', z1, z2, z3) -{ 1 }-> min(z2, z3) :|: z'' = 0, z3 >= 0, z1 >= 0, z2 >= 0 if2(z'', z1, z2, z3) -{ 1 }-> z3 :|: z3 >= 0, z1 >= 0, z2 >= 0, z'' = 1 if2(z'', z1, z2, z3) -{ 1 }-> 1 + z2 + del(z1, z3) :|: z'' = 0, z3 >= 0, z1 >= 0, z2 >= 0 le(z'', z1) -{ 2 + z1 }-> s :|: s >= 0, s <= 1, z'' - 1 >= 0, z1 - 1 >= 0 le(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 >= 0 le(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' - 1 >= 0 min(z'', z1) -{ 1 }-> z'' :|: z1 = 0, z'' >= 0 min(z'', z1) -{ 4 + y'' }-> if1(s'', 1 + (z'' - 1), 1 + y'', z) :|: s'' >= 0, s'' <= 1, z >= 0, z1 = 1 + (1 + y'') + z, y'' >= 0, z'' - 1 >= 0 min(z'', z1) -{ 2 }-> if1(1, 0, y, z) :|: z'' = 0, z >= 0, z1 = 1 + y + z, y >= 0 min(z'', z1) -{ 2 }-> if1(0, 1 + (z'' - 1), 0, z1 - 1) :|: z1 - 1 >= 0, z'' - 1 >= 0 minsort(z'') -{ 1 }-> 0 :|: z'' = 0 minsort(z'') -{ 4 + y' }-> 1 + min(x, 1 + y' + z') + minsort(del(if1(s', x, y', z'), 1 + x + (1 + y' + z'))) :|: s' >= 0, s' <= 1, z'' = 1 + x + (1 + y' + z'), x >= 0, z' >= 0, y' >= 0 minsort(z'') -{ 2 }-> 1 + min(z'' - 1, 0) + minsort(del(z'' - 1, 1 + (z'' - 1) + 0)) :|: z'' - 1 >= 0 Function symbols to be analyzed: {eq}, {min,if1}, {del,if2}, {minsort} Previous analysis results are: le: runtime: O(n^1) [2 + z1], 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'', z1) -{ 2 }-> if2(eq(z'' - 1, y2), 1 + (z'' - 1), 1 + y2, z) :|: z >= 0, z1 = 1 + (1 + y2) + z, y2 >= 0, z'' - 1 >= 0 del(z'', z1) -{ 2 }-> if2(1, 0, 0, z1 - 1) :|: z'' = 0, z1 - 1 >= 0 del(z'', z1) -{ 2 }-> if2(0, 0, 1 + y1, z) :|: z'' = 0, y1 >= 0, z >= 0, z1 = 1 + (1 + y1) + z del(z'', z1) -{ 2 }-> if2(0, 1 + (z'' - 1), 0, z1 - 1) :|: z'' - 1 >= 0, z1 - 1 >= 0 del(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' >= 0 eq(z'', z1) -{ 1 }-> eq(z'' - 1, z1 - 1) :|: z'' - 1 >= 0, z1 - 1 >= 0 eq(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 = 0 eq(z'', z1) -{ 1 }-> 0 :|: z'' = 0, z1 - 1 >= 0 eq(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' - 1 >= 0 if1(z'', z1, z2, z3) -{ 1 }-> min(z1, z3) :|: z3 >= 0, z1 >= 0, z2 >= 0, z'' = 1 if1(z'', z1, z2, z3) -{ 1 }-> min(z2, z3) :|: z'' = 0, z3 >= 0, z1 >= 0, z2 >= 0 if2(z'', z1, z2, z3) -{ 1 }-> z3 :|: z3 >= 0, z1 >= 0, z2 >= 0, z'' = 1 if2(z'', z1, z2, z3) -{ 1 }-> 1 + z2 + del(z1, z3) :|: z'' = 0, z3 >= 0, z1 >= 0, z2 >= 0 le(z'', z1) -{ 2 + z1 }-> s :|: s >= 0, s <= 1, z'' - 1 >= 0, z1 - 1 >= 0 le(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 >= 0 le(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' - 1 >= 0 min(z'', z1) -{ 1 }-> z'' :|: z1 = 0, z'' >= 0 min(z'', z1) -{ 4 + y'' }-> if1(s'', 1 + (z'' - 1), 1 + y'', z) :|: s'' >= 0, s'' <= 1, z >= 0, z1 = 1 + (1 + y'') + z, y'' >= 0, z'' - 1 >= 0 min(z'', z1) -{ 2 }-> if1(1, 0, y, z) :|: z'' = 0, z >= 0, z1 = 1 + y + z, y >= 0 min(z'', z1) -{ 2 }-> if1(0, 1 + (z'' - 1), 0, z1 - 1) :|: z1 - 1 >= 0, z'' - 1 >= 0 minsort(z'') -{ 1 }-> 0 :|: z'' = 0 minsort(z'') -{ 4 + y' }-> 1 + min(x, 1 + y' + z') + minsort(del(if1(s', x, y', z'), 1 + x + (1 + y' + z'))) :|: s' >= 0, s' <= 1, z'' = 1 + x + (1 + y' + z'), x >= 0, z' >= 0, y' >= 0 minsort(z'') -{ 2 }-> 1 + min(z'' - 1, 0) + minsort(del(z'' - 1, 1 + (z'' - 1) + 0)) :|: z'' - 1 >= 0 Function symbols to be analyzed: {eq}, {min,if1}, {del,if2}, {minsort} Previous analysis results are: le: runtime: O(n^1) [2 + z1], 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 + z1 ---------------------------------------- (34) Obligation: Complexity RNTS consisting of the following rules: del(z'', z1) -{ 2 }-> if2(eq(z'' - 1, y2), 1 + (z'' - 1), 1 + y2, z) :|: z >= 0, z1 = 1 + (1 + y2) + z, y2 >= 0, z'' - 1 >= 0 del(z'', z1) -{ 2 }-> if2(1, 0, 0, z1 - 1) :|: z'' = 0, z1 - 1 >= 0 del(z'', z1) -{ 2 }-> if2(0, 0, 1 + y1, z) :|: z'' = 0, y1 >= 0, z >= 0, z1 = 1 + (1 + y1) + z del(z'', z1) -{ 2 }-> if2(0, 1 + (z'' - 1), 0, z1 - 1) :|: z'' - 1 >= 0, z1 - 1 >= 0 del(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' >= 0 eq(z'', z1) -{ 1 }-> eq(z'' - 1, z1 - 1) :|: z'' - 1 >= 0, z1 - 1 >= 0 eq(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 = 0 eq(z'', z1) -{ 1 }-> 0 :|: z'' = 0, z1 - 1 >= 0 eq(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' - 1 >= 0 if1(z'', z1, z2, z3) -{ 1 }-> min(z1, z3) :|: z3 >= 0, z1 >= 0, z2 >= 0, z'' = 1 if1(z'', z1, z2, z3) -{ 1 }-> min(z2, z3) :|: z'' = 0, z3 >= 0, z1 >= 0, z2 >= 0 if2(z'', z1, z2, z3) -{ 1 }-> z3 :|: z3 >= 0, z1 >= 0, z2 >= 0, z'' = 1 if2(z'', z1, z2, z3) -{ 1 }-> 1 + z2 + del(z1, z3) :|: z'' = 0, z3 >= 0, z1 >= 0, z2 >= 0 le(z'', z1) -{ 2 + z1 }-> s :|: s >= 0, s <= 1, z'' - 1 >= 0, z1 - 1 >= 0 le(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 >= 0 le(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' - 1 >= 0 min(z'', z1) -{ 1 }-> z'' :|: z1 = 0, z'' >= 0 min(z'', z1) -{ 4 + y'' }-> if1(s'', 1 + (z'' - 1), 1 + y'', z) :|: s'' >= 0, s'' <= 1, z >= 0, z1 = 1 + (1 + y'') + z, y'' >= 0, z'' - 1 >= 0 min(z'', z1) -{ 2 }-> if1(1, 0, y, z) :|: z'' = 0, z >= 0, z1 = 1 + y + z, y >= 0 min(z'', z1) -{ 2 }-> if1(0, 1 + (z'' - 1), 0, z1 - 1) :|: z1 - 1 >= 0, z'' - 1 >= 0 minsort(z'') -{ 1 }-> 0 :|: z'' = 0 minsort(z'') -{ 4 + y' }-> 1 + min(x, 1 + y' + z') + minsort(del(if1(s', x, y', z'), 1 + x + (1 + y' + z'))) :|: s' >= 0, s' <= 1, z'' = 1 + x + (1 + y' + z'), x >= 0, z' >= 0, y' >= 0 minsort(z'') -{ 2 }-> 1 + min(z'' - 1, 0) + minsort(del(z'' - 1, 1 + (z'' - 1) + 0)) :|: z'' - 1 >= 0 Function symbols to be analyzed: {min,if1}, {del,if2}, {minsort} Previous analysis results are: le: runtime: O(n^1) [2 + z1], size: O(1) [1] eq: runtime: O(n^1) [3 + z1], 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'', z1) -{ 5 + y2 }-> if2(s2, 1 + (z'' - 1), 1 + y2, z) :|: s2 >= 0, s2 <= 1, z >= 0, z1 = 1 + (1 + y2) + z, y2 >= 0, z'' - 1 >= 0 del(z'', z1) -{ 2 }-> if2(1, 0, 0, z1 - 1) :|: z'' = 0, z1 - 1 >= 0 del(z'', z1) -{ 2 }-> if2(0, 0, 1 + y1, z) :|: z'' = 0, y1 >= 0, z >= 0, z1 = 1 + (1 + y1) + z del(z'', z1) -{ 2 }-> if2(0, 1 + (z'' - 1), 0, z1 - 1) :|: z'' - 1 >= 0, z1 - 1 >= 0 del(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' >= 0 eq(z'', z1) -{ 3 + z1 }-> s1 :|: s1 >= 0, s1 <= 1, z'' - 1 >= 0, z1 - 1 >= 0 eq(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 = 0 eq(z'', z1) -{ 1 }-> 0 :|: z'' = 0, z1 - 1 >= 0 eq(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' - 1 >= 0 if1(z'', z1, z2, z3) -{ 1 }-> min(z1, z3) :|: z3 >= 0, z1 >= 0, z2 >= 0, z'' = 1 if1(z'', z1, z2, z3) -{ 1 }-> min(z2, z3) :|: z'' = 0, z3 >= 0, z1 >= 0, z2 >= 0 if2(z'', z1, z2, z3) -{ 1 }-> z3 :|: z3 >= 0, z1 >= 0, z2 >= 0, z'' = 1 if2(z'', z1, z2, z3) -{ 1 }-> 1 + z2 + del(z1, z3) :|: z'' = 0, z3 >= 0, z1 >= 0, z2 >= 0 le(z'', z1) -{ 2 + z1 }-> s :|: s >= 0, s <= 1, z'' - 1 >= 0, z1 - 1 >= 0 le(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 >= 0 le(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' - 1 >= 0 min(z'', z1) -{ 1 }-> z'' :|: z1 = 0, z'' >= 0 min(z'', z1) -{ 4 + y'' }-> if1(s'', 1 + (z'' - 1), 1 + y'', z) :|: s'' >= 0, s'' <= 1, z >= 0, z1 = 1 + (1 + y'') + z, y'' >= 0, z'' - 1 >= 0 min(z'', z1) -{ 2 }-> if1(1, 0, y, z) :|: z'' = 0, z >= 0, z1 = 1 + y + z, y >= 0 min(z'', z1) -{ 2 }-> if1(0, 1 + (z'' - 1), 0, z1 - 1) :|: z1 - 1 >= 0, z'' - 1 >= 0 minsort(z'') -{ 1 }-> 0 :|: z'' = 0 minsort(z'') -{ 4 + y' }-> 1 + min(x, 1 + y' + z') + minsort(del(if1(s', x, y', z'), 1 + x + (1 + y' + z'))) :|: s' >= 0, s' <= 1, z'' = 1 + x + (1 + y' + z'), x >= 0, z' >= 0, y' >= 0 minsort(z'') -{ 2 }-> 1 + min(z'' - 1, 0) + minsort(del(z'' - 1, 1 + (z'' - 1) + 0)) :|: z'' - 1 >= 0 Function symbols to be analyzed: {min,if1}, {del,if2}, {minsort} Previous analysis results are: le: runtime: O(n^1) [2 + z1], size: O(1) [1] eq: runtime: O(n^1) [3 + z1], size: O(1) [1] ---------------------------------------- (37) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: min after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z'' + z1 Computed SIZE bound using CoFloCo for: if1 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z1 + z2 + z3 ---------------------------------------- (38) Obligation: Complexity RNTS consisting of the following rules: del(z'', z1) -{ 5 + y2 }-> if2(s2, 1 + (z'' - 1), 1 + y2, z) :|: s2 >= 0, s2 <= 1, z >= 0, z1 = 1 + (1 + y2) + z, y2 >= 0, z'' - 1 >= 0 del(z'', z1) -{ 2 }-> if2(1, 0, 0, z1 - 1) :|: z'' = 0, z1 - 1 >= 0 del(z'', z1) -{ 2 }-> if2(0, 0, 1 + y1, z) :|: z'' = 0, y1 >= 0, z >= 0, z1 = 1 + (1 + y1) + z del(z'', z1) -{ 2 }-> if2(0, 1 + (z'' - 1), 0, z1 - 1) :|: z'' - 1 >= 0, z1 - 1 >= 0 del(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' >= 0 eq(z'', z1) -{ 3 + z1 }-> s1 :|: s1 >= 0, s1 <= 1, z'' - 1 >= 0, z1 - 1 >= 0 eq(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 = 0 eq(z'', z1) -{ 1 }-> 0 :|: z'' = 0, z1 - 1 >= 0 eq(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' - 1 >= 0 if1(z'', z1, z2, z3) -{ 1 }-> min(z1, z3) :|: z3 >= 0, z1 >= 0, z2 >= 0, z'' = 1 if1(z'', z1, z2, z3) -{ 1 }-> min(z2, z3) :|: z'' = 0, z3 >= 0, z1 >= 0, z2 >= 0 if2(z'', z1, z2, z3) -{ 1 }-> z3 :|: z3 >= 0, z1 >= 0, z2 >= 0, z'' = 1 if2(z'', z1, z2, z3) -{ 1 }-> 1 + z2 + del(z1, z3) :|: z'' = 0, z3 >= 0, z1 >= 0, z2 >= 0 le(z'', z1) -{ 2 + z1 }-> s :|: s >= 0, s <= 1, z'' - 1 >= 0, z1 - 1 >= 0 le(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 >= 0 le(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' - 1 >= 0 min(z'', z1) -{ 1 }-> z'' :|: z1 = 0, z'' >= 0 min(z'', z1) -{ 4 + y'' }-> if1(s'', 1 + (z'' - 1), 1 + y'', z) :|: s'' >= 0, s'' <= 1, z >= 0, z1 = 1 + (1 + y'') + z, y'' >= 0, z'' - 1 >= 0 min(z'', z1) -{ 2 }-> if1(1, 0, y, z) :|: z'' = 0, z >= 0, z1 = 1 + y + z, y >= 0 min(z'', z1) -{ 2 }-> if1(0, 1 + (z'' - 1), 0, z1 - 1) :|: z1 - 1 >= 0, z'' - 1 >= 0 minsort(z'') -{ 1 }-> 0 :|: z'' = 0 minsort(z'') -{ 4 + y' }-> 1 + min(x, 1 + y' + z') + minsort(del(if1(s', x, y', z'), 1 + x + (1 + y' + z'))) :|: s' >= 0, s' <= 1, z'' = 1 + x + (1 + y' + z'), x >= 0, z' >= 0, y' >= 0 minsort(z'') -{ 2 }-> 1 + min(z'' - 1, 0) + minsort(del(z'' - 1, 1 + (z'' - 1) + 0)) :|: z'' - 1 >= 0 Function symbols to be analyzed: {min,if1}, {del,if2}, {minsort} Previous analysis results are: le: runtime: O(n^1) [2 + z1], size: O(1) [1] eq: runtime: O(n^1) [3 + z1], size: O(1) [1] min: runtime: ?, size: O(n^1) [z'' + z1] if1: runtime: ?, size: O(n^1) [z1 + z2 + z3] ---------------------------------------- (39) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: min after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 20 + z'' + 7*z1 Computed RUNTIME bound using CoFloCo for: if1 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 21 + z1 + z2 + 7*z3 ---------------------------------------- (40) Obligation: Complexity RNTS consisting of the following rules: del(z'', z1) -{ 5 + y2 }-> if2(s2, 1 + (z'' - 1), 1 + y2, z) :|: s2 >= 0, s2 <= 1, z >= 0, z1 = 1 + (1 + y2) + z, y2 >= 0, z'' - 1 >= 0 del(z'', z1) -{ 2 }-> if2(1, 0, 0, z1 - 1) :|: z'' = 0, z1 - 1 >= 0 del(z'', z1) -{ 2 }-> if2(0, 0, 1 + y1, z) :|: z'' = 0, y1 >= 0, z >= 0, z1 = 1 + (1 + y1) + z del(z'', z1) -{ 2 }-> if2(0, 1 + (z'' - 1), 0, z1 - 1) :|: z'' - 1 >= 0, z1 - 1 >= 0 del(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' >= 0 eq(z'', z1) -{ 3 + z1 }-> s1 :|: s1 >= 0, s1 <= 1, z'' - 1 >= 0, z1 - 1 >= 0 eq(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 = 0 eq(z'', z1) -{ 1 }-> 0 :|: z'' = 0, z1 - 1 >= 0 eq(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' - 1 >= 0 if1(z'', z1, z2, z3) -{ 1 }-> min(z1, z3) :|: z3 >= 0, z1 >= 0, z2 >= 0, z'' = 1 if1(z'', z1, z2, z3) -{ 1 }-> min(z2, z3) :|: z'' = 0, z3 >= 0, z1 >= 0, z2 >= 0 if2(z'', z1, z2, z3) -{ 1 }-> z3 :|: z3 >= 0, z1 >= 0, z2 >= 0, z'' = 1 if2(z'', z1, z2, z3) -{ 1 }-> 1 + z2 + del(z1, z3) :|: z'' = 0, z3 >= 0, z1 >= 0, z2 >= 0 le(z'', z1) -{ 2 + z1 }-> s :|: s >= 0, s <= 1, z'' - 1 >= 0, z1 - 1 >= 0 le(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 >= 0 le(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' - 1 >= 0 min(z'', z1) -{ 1 }-> z'' :|: z1 = 0, z'' >= 0 min(z'', z1) -{ 4 + y'' }-> if1(s'', 1 + (z'' - 1), 1 + y'', z) :|: s'' >= 0, s'' <= 1, z >= 0, z1 = 1 + (1 + y'') + z, y'' >= 0, z'' - 1 >= 0 min(z'', z1) -{ 2 }-> if1(1, 0, y, z) :|: z'' = 0, z >= 0, z1 = 1 + y + z, y >= 0 min(z'', z1) -{ 2 }-> if1(0, 1 + (z'' - 1), 0, z1 - 1) :|: z1 - 1 >= 0, z'' - 1 >= 0 minsort(z'') -{ 1 }-> 0 :|: z'' = 0 minsort(z'') -{ 4 + y' }-> 1 + min(x, 1 + y' + z') + minsort(del(if1(s', x, y', z'), 1 + x + (1 + y' + z'))) :|: s' >= 0, s' <= 1, z'' = 1 + x + (1 + y' + z'), x >= 0, z' >= 0, y' >= 0 minsort(z'') -{ 2 }-> 1 + min(z'' - 1, 0) + minsort(del(z'' - 1, 1 + (z'' - 1) + 0)) :|: z'' - 1 >= 0 Function symbols to be analyzed: {del,if2}, {minsort} Previous analysis results are: le: runtime: O(n^1) [2 + z1], size: O(1) [1] eq: runtime: O(n^1) [3 + z1], size: O(1) [1] min: runtime: O(n^1) [20 + z'' + 7*z1], size: O(n^1) [z'' + z1] if1: runtime: O(n^1) [21 + z1 + z2 + 7*z3], size: O(n^1) [z1 + z2 + z3] ---------------------------------------- (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'', z1) -{ 5 + y2 }-> if2(s2, 1 + (z'' - 1), 1 + y2, z) :|: s2 >= 0, s2 <= 1, z >= 0, z1 = 1 + (1 + y2) + z, y2 >= 0, z'' - 1 >= 0 del(z'', z1) -{ 2 }-> if2(1, 0, 0, z1 - 1) :|: z'' = 0, z1 - 1 >= 0 del(z'', z1) -{ 2 }-> if2(0, 0, 1 + y1, z) :|: z'' = 0, y1 >= 0, z >= 0, z1 = 1 + (1 + y1) + z del(z'', z1) -{ 2 }-> if2(0, 1 + (z'' - 1), 0, z1 - 1) :|: z'' - 1 >= 0, z1 - 1 >= 0 del(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' >= 0 eq(z'', z1) -{ 3 + z1 }-> s1 :|: s1 >= 0, s1 <= 1, z'' - 1 >= 0, z1 - 1 >= 0 eq(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 = 0 eq(z'', z1) -{ 1 }-> 0 :|: z'' = 0, z1 - 1 >= 0 eq(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' - 1 >= 0 if1(z'', z1, z2, z3) -{ 21 + z1 + 7*z3 }-> s3 :|: s3 >= 0, s3 <= z1 + z3, z3 >= 0, z1 >= 0, z2 >= 0, z'' = 1 if1(z'', z1, z2, z3) -{ 21 + z2 + 7*z3 }-> s4 :|: s4 >= 0, s4 <= z2 + z3, z'' = 0, z3 >= 0, z1 >= 0, z2 >= 0 if2(z'', z1, z2, z3) -{ 1 }-> z3 :|: z3 >= 0, z1 >= 0, z2 >= 0, z'' = 1 if2(z'', z1, z2, z3) -{ 1 }-> 1 + z2 + del(z1, z3) :|: z'' = 0, z3 >= 0, z1 >= 0, z2 >= 0 le(z'', z1) -{ 2 + z1 }-> s :|: s >= 0, s <= 1, z'' - 1 >= 0, z1 - 1 >= 0 le(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 >= 0 le(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' - 1 >= 0 min(z'', z1) -{ 26 + 2*y'' + 7*z + z'' }-> s10 :|: s10 >= 0, s10 <= 1 + y'' + z + (1 + (z'' - 1)), s'' >= 0, s'' <= 1, z >= 0, z1 = 1 + (1 + y'') + z, y'' >= 0, z'' - 1 >= 0 min(z'', z1) -{ 23 + y + 7*z }-> s8 :|: s8 >= 0, s8 <= y + z + 0, z'' = 0, z >= 0, z1 = 1 + y + z, y >= 0 min(z'', z1) -{ 16 + z'' + 7*z1 }-> s9 :|: s9 >= 0, s9 <= 0 + (z1 - 1) + (1 + (z'' - 1)), z1 - 1 >= 0, z'' - 1 >= 0 min(z'', z1) -{ 1 }-> z'' :|: z1 = 0, z'' >= 0 minsort(z'') -{ 1 }-> 0 :|: z'' = 0 minsort(z'') -{ 21 + z'' }-> 1 + s5 + minsort(del(z'' - 1, 1 + (z'' - 1) + 0)) :|: s5 >= 0, s5 <= z'' - 1 + 0, z'' - 1 >= 0 minsort(z'') -{ 52 + 2*x + 9*y' + 14*z' }-> 1 + s6 + minsort(del(s7, 1 + x + (1 + y' + z'))) :|: s6 >= 0, s6 <= x + (1 + y' + z'), s7 >= 0, s7 <= y' + z' + x, s' >= 0, s' <= 1, z'' = 1 + x + (1 + y' + z'), x >= 0, z' >= 0, y' >= 0 Function symbols to be analyzed: {del,if2}, {minsort} Previous analysis results are: le: runtime: O(n^1) [2 + z1], size: O(1) [1] eq: runtime: O(n^1) [3 + z1], size: O(1) [1] min: runtime: O(n^1) [20 + z'' + 7*z1], size: O(n^1) [z'' + z1] if1: runtime: O(n^1) [21 + z1 + z2 + 7*z3], size: O(n^1) [z1 + z2 + z3] ---------------------------------------- (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: z1 Computed SIZE bound using CoFloCo for: if2 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z2 + z3 ---------------------------------------- (44) Obligation: Complexity RNTS consisting of the following rules: del(z'', z1) -{ 5 + y2 }-> if2(s2, 1 + (z'' - 1), 1 + y2, z) :|: s2 >= 0, s2 <= 1, z >= 0, z1 = 1 + (1 + y2) + z, y2 >= 0, z'' - 1 >= 0 del(z'', z1) -{ 2 }-> if2(1, 0, 0, z1 - 1) :|: z'' = 0, z1 - 1 >= 0 del(z'', z1) -{ 2 }-> if2(0, 0, 1 + y1, z) :|: z'' = 0, y1 >= 0, z >= 0, z1 = 1 + (1 + y1) + z del(z'', z1) -{ 2 }-> if2(0, 1 + (z'' - 1), 0, z1 - 1) :|: z'' - 1 >= 0, z1 - 1 >= 0 del(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' >= 0 eq(z'', z1) -{ 3 + z1 }-> s1 :|: s1 >= 0, s1 <= 1, z'' - 1 >= 0, z1 - 1 >= 0 eq(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 = 0 eq(z'', z1) -{ 1 }-> 0 :|: z'' = 0, z1 - 1 >= 0 eq(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' - 1 >= 0 if1(z'', z1, z2, z3) -{ 21 + z1 + 7*z3 }-> s3 :|: s3 >= 0, s3 <= z1 + z3, z3 >= 0, z1 >= 0, z2 >= 0, z'' = 1 if1(z'', z1, z2, z3) -{ 21 + z2 + 7*z3 }-> s4 :|: s4 >= 0, s4 <= z2 + z3, z'' = 0, z3 >= 0, z1 >= 0, z2 >= 0 if2(z'', z1, z2, z3) -{ 1 }-> z3 :|: z3 >= 0, z1 >= 0, z2 >= 0, z'' = 1 if2(z'', z1, z2, z3) -{ 1 }-> 1 + z2 + del(z1, z3) :|: z'' = 0, z3 >= 0, z1 >= 0, z2 >= 0 le(z'', z1) -{ 2 + z1 }-> s :|: s >= 0, s <= 1, z'' - 1 >= 0, z1 - 1 >= 0 le(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 >= 0 le(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' - 1 >= 0 min(z'', z1) -{ 26 + 2*y'' + 7*z + z'' }-> s10 :|: s10 >= 0, s10 <= 1 + y'' + z + (1 + (z'' - 1)), s'' >= 0, s'' <= 1, z >= 0, z1 = 1 + (1 + y'') + z, y'' >= 0, z'' - 1 >= 0 min(z'', z1) -{ 23 + y + 7*z }-> s8 :|: s8 >= 0, s8 <= y + z + 0, z'' = 0, z >= 0, z1 = 1 + y + z, y >= 0 min(z'', z1) -{ 16 + z'' + 7*z1 }-> s9 :|: s9 >= 0, s9 <= 0 + (z1 - 1) + (1 + (z'' - 1)), z1 - 1 >= 0, z'' - 1 >= 0 min(z'', z1) -{ 1 }-> z'' :|: z1 = 0, z'' >= 0 minsort(z'') -{ 1 }-> 0 :|: z'' = 0 minsort(z'') -{ 21 + z'' }-> 1 + s5 + minsort(del(z'' - 1, 1 + (z'' - 1) + 0)) :|: s5 >= 0, s5 <= z'' - 1 + 0, z'' - 1 >= 0 minsort(z'') -{ 52 + 2*x + 9*y' + 14*z' }-> 1 + s6 + minsort(del(s7, 1 + x + (1 + y' + z'))) :|: s6 >= 0, s6 <= x + (1 + y' + z'), s7 >= 0, s7 <= y' + z' + x, s' >= 0, s' <= 1, z'' = 1 + x + (1 + y' + z'), x >= 0, z' >= 0, y' >= 0 Function symbols to be analyzed: {del,if2}, {minsort} Previous analysis results are: le: runtime: O(n^1) [2 + z1], size: O(1) [1] eq: runtime: O(n^1) [3 + z1], size: O(1) [1] min: runtime: O(n^1) [20 + z'' + 7*z1], size: O(n^1) [z'' + z1] if1: runtime: O(n^1) [21 + z1 + z2 + 7*z3], size: O(n^1) [z1 + z2 + z3] del: runtime: ?, size: O(n^1) [z1] if2: runtime: ?, size: O(n^1) [1 + z2 + z3] ---------------------------------------- (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*z1 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*z3 ---------------------------------------- (46) Obligation: Complexity RNTS consisting of the following rules: del(z'', z1) -{ 5 + y2 }-> if2(s2, 1 + (z'' - 1), 1 + y2, z) :|: s2 >= 0, s2 <= 1, z >= 0, z1 = 1 + (1 + y2) + z, y2 >= 0, z'' - 1 >= 0 del(z'', z1) -{ 2 }-> if2(1, 0, 0, z1 - 1) :|: z'' = 0, z1 - 1 >= 0 del(z'', z1) -{ 2 }-> if2(0, 0, 1 + y1, z) :|: z'' = 0, y1 >= 0, z >= 0, z1 = 1 + (1 + y1) + z del(z'', z1) -{ 2 }-> if2(0, 1 + (z'' - 1), 0, z1 - 1) :|: z'' - 1 >= 0, z1 - 1 >= 0 del(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' >= 0 eq(z'', z1) -{ 3 + z1 }-> s1 :|: s1 >= 0, s1 <= 1, z'' - 1 >= 0, z1 - 1 >= 0 eq(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 = 0 eq(z'', z1) -{ 1 }-> 0 :|: z'' = 0, z1 - 1 >= 0 eq(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' - 1 >= 0 if1(z'', z1, z2, z3) -{ 21 + z1 + 7*z3 }-> s3 :|: s3 >= 0, s3 <= z1 + z3, z3 >= 0, z1 >= 0, z2 >= 0, z'' = 1 if1(z'', z1, z2, z3) -{ 21 + z2 + 7*z3 }-> s4 :|: s4 >= 0, s4 <= z2 + z3, z'' = 0, z3 >= 0, z1 >= 0, z2 >= 0 if2(z'', z1, z2, z3) -{ 1 }-> z3 :|: z3 >= 0, z1 >= 0, z2 >= 0, z'' = 1 if2(z'', z1, z2, z3) -{ 1 }-> 1 + z2 + del(z1, z3) :|: z'' = 0, z3 >= 0, z1 >= 0, z2 >= 0 le(z'', z1) -{ 2 + z1 }-> s :|: s >= 0, s <= 1, z'' - 1 >= 0, z1 - 1 >= 0 le(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 >= 0 le(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' - 1 >= 0 min(z'', z1) -{ 26 + 2*y'' + 7*z + z'' }-> s10 :|: s10 >= 0, s10 <= 1 + y'' + z + (1 + (z'' - 1)), s'' >= 0, s'' <= 1, z >= 0, z1 = 1 + (1 + y'') + z, y'' >= 0, z'' - 1 >= 0 min(z'', z1) -{ 23 + y + 7*z }-> s8 :|: s8 >= 0, s8 <= y + z + 0, z'' = 0, z >= 0, z1 = 1 + y + z, y >= 0 min(z'', z1) -{ 16 + z'' + 7*z1 }-> s9 :|: s9 >= 0, s9 <= 0 + (z1 - 1) + (1 + (z'' - 1)), z1 - 1 >= 0, z'' - 1 >= 0 min(z'', z1) -{ 1 }-> z'' :|: z1 = 0, z'' >= 0 minsort(z'') -{ 1 }-> 0 :|: z'' = 0 minsort(z'') -{ 21 + z'' }-> 1 + s5 + minsort(del(z'' - 1, 1 + (z'' - 1) + 0)) :|: s5 >= 0, s5 <= z'' - 1 + 0, z'' - 1 >= 0 minsort(z'') -{ 52 + 2*x + 9*y' + 14*z' }-> 1 + s6 + minsort(del(s7, 1 + x + (1 + y' + z'))) :|: s6 >= 0, s6 <= x + (1 + y' + z'), s7 >= 0, s7 <= y' + z' + x, s' >= 0, s' <= 1, z'' = 1 + x + (1 + y' + z'), x >= 0, z' >= 0, y' >= 0 Function symbols to be analyzed: {minsort} Previous analysis results are: le: runtime: O(n^1) [2 + z1], size: O(1) [1] eq: runtime: O(n^1) [3 + z1], size: O(1) [1] min: runtime: O(n^1) [20 + z'' + 7*z1], size: O(n^1) [z'' + z1] if1: runtime: O(n^1) [21 + z1 + z2 + 7*z3], size: O(n^1) [z1 + z2 + z3] del: runtime: O(n^1) [49 + 17*z1], size: O(n^1) [z1] if2: runtime: O(n^1) [50 + 17*z3], size: O(n^1) [1 + z2 + z3] ---------------------------------------- (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'', z1) -{ 35 + 17*z1 }-> s14 :|: s14 >= 0, s14 <= 0 + (z1 - 1) + 1, z'' = 0, z1 - 1 >= 0 del(z'', z1) -{ 52 + 17*z }-> s15 :|: s15 >= 0, s15 <= 1 + y1 + z + 1, z'' = 0, y1 >= 0, z >= 0, z1 = 1 + (1 + y1) + z del(z'', z1) -{ 35 + 17*z1 }-> s16 :|: s16 >= 0, s16 <= 0 + (z1 - 1) + 1, z'' - 1 >= 0, z1 - 1 >= 0 del(z'', z1) -{ 55 + y2 + 17*z }-> s17 :|: s17 >= 0, s17 <= 1 + y2 + z + 1, s2 >= 0, s2 <= 1, z >= 0, z1 = 1 + (1 + y2) + z, y2 >= 0, z'' - 1 >= 0 del(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' >= 0 eq(z'', z1) -{ 3 + z1 }-> s1 :|: s1 >= 0, s1 <= 1, z'' - 1 >= 0, z1 - 1 >= 0 eq(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 = 0 eq(z'', z1) -{ 1 }-> 0 :|: z'' = 0, z1 - 1 >= 0 eq(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' - 1 >= 0 if1(z'', z1, z2, z3) -{ 21 + z1 + 7*z3 }-> s3 :|: s3 >= 0, s3 <= z1 + z3, z3 >= 0, z1 >= 0, z2 >= 0, z'' = 1 if1(z'', z1, z2, z3) -{ 21 + z2 + 7*z3 }-> s4 :|: s4 >= 0, s4 <= z2 + z3, z'' = 0, z3 >= 0, z1 >= 0, z2 >= 0 if2(z'', z1, z2, z3) -{ 1 }-> z3 :|: z3 >= 0, z1 >= 0, z2 >= 0, z'' = 1 if2(z'', z1, z2, z3) -{ 50 + 17*z3 }-> 1 + z2 + s11 :|: s11 >= 0, s11 <= z3, z'' = 0, z3 >= 0, z1 >= 0, z2 >= 0 le(z'', z1) -{ 2 + z1 }-> s :|: s >= 0, s <= 1, z'' - 1 >= 0, z1 - 1 >= 0 le(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 >= 0 le(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' - 1 >= 0 min(z'', z1) -{ 26 + 2*y'' + 7*z + z'' }-> s10 :|: s10 >= 0, s10 <= 1 + y'' + z + (1 + (z'' - 1)), s'' >= 0, s'' <= 1, z >= 0, z1 = 1 + (1 + y'') + z, y'' >= 0, z'' - 1 >= 0 min(z'', z1) -{ 23 + y + 7*z }-> s8 :|: s8 >= 0, s8 <= y + z + 0, z'' = 0, z >= 0, z1 = 1 + y + z, y >= 0 min(z'', z1) -{ 16 + z'' + 7*z1 }-> s9 :|: s9 >= 0, s9 <= 0 + (z1 - 1) + (1 + (z'' - 1)), z1 - 1 >= 0, z'' - 1 >= 0 min(z'', z1) -{ 1 }-> z'' :|: z1 = 0, z'' >= 0 minsort(z'') -{ 1 }-> 0 :|: z'' = 0 minsort(z'') -{ 70 + 18*z'' }-> 1 + s5 + minsort(s12) :|: s12 >= 0, s12 <= 1 + (z'' - 1) + 0, s5 >= 0, s5 <= z'' - 1 + 0, z'' - 1 >= 0 minsort(z'') -{ 135 + 19*x + 26*y' + 31*z' }-> 1 + s6 + minsort(s13) :|: s13 >= 0, s13 <= 1 + x + (1 + y' + z'), s6 >= 0, s6 <= x + (1 + y' + z'), s7 >= 0, s7 <= y' + z' + x, s' >= 0, s' <= 1, z'' = 1 + x + (1 + y' + z'), x >= 0, z' >= 0, y' >= 0 Function symbols to be analyzed: {minsort} Previous analysis results are: le: runtime: O(n^1) [2 + z1], size: O(1) [1] eq: runtime: O(n^1) [3 + z1], size: O(1) [1] min: runtime: O(n^1) [20 + z'' + 7*z1], size: O(n^1) [z'' + z1] if1: runtime: O(n^1) [21 + z1 + z2 + 7*z3], size: O(n^1) [z1 + z2 + z3] del: runtime: O(n^1) [49 + 17*z1], size: O(n^1) [z1] if2: runtime: O(n^1) [50 + 17*z3], size: O(n^1) [1 + z2 + z3] ---------------------------------------- (49) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: minsort 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'', z1) -{ 35 + 17*z1 }-> s14 :|: s14 >= 0, s14 <= 0 + (z1 - 1) + 1, z'' = 0, z1 - 1 >= 0 del(z'', z1) -{ 52 + 17*z }-> s15 :|: s15 >= 0, s15 <= 1 + y1 + z + 1, z'' = 0, y1 >= 0, z >= 0, z1 = 1 + (1 + y1) + z del(z'', z1) -{ 35 + 17*z1 }-> s16 :|: s16 >= 0, s16 <= 0 + (z1 - 1) + 1, z'' - 1 >= 0, z1 - 1 >= 0 del(z'', z1) -{ 55 + y2 + 17*z }-> s17 :|: s17 >= 0, s17 <= 1 + y2 + z + 1, s2 >= 0, s2 <= 1, z >= 0, z1 = 1 + (1 + y2) + z, y2 >= 0, z'' - 1 >= 0 del(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' >= 0 eq(z'', z1) -{ 3 + z1 }-> s1 :|: s1 >= 0, s1 <= 1, z'' - 1 >= 0, z1 - 1 >= 0 eq(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 = 0 eq(z'', z1) -{ 1 }-> 0 :|: z'' = 0, z1 - 1 >= 0 eq(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' - 1 >= 0 if1(z'', z1, z2, z3) -{ 21 + z1 + 7*z3 }-> s3 :|: s3 >= 0, s3 <= z1 + z3, z3 >= 0, z1 >= 0, z2 >= 0, z'' = 1 if1(z'', z1, z2, z3) -{ 21 + z2 + 7*z3 }-> s4 :|: s4 >= 0, s4 <= z2 + z3, z'' = 0, z3 >= 0, z1 >= 0, z2 >= 0 if2(z'', z1, z2, z3) -{ 1 }-> z3 :|: z3 >= 0, z1 >= 0, z2 >= 0, z'' = 1 if2(z'', z1, z2, z3) -{ 50 + 17*z3 }-> 1 + z2 + s11 :|: s11 >= 0, s11 <= z3, z'' = 0, z3 >= 0, z1 >= 0, z2 >= 0 le(z'', z1) -{ 2 + z1 }-> s :|: s >= 0, s <= 1, z'' - 1 >= 0, z1 - 1 >= 0 le(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 >= 0 le(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' - 1 >= 0 min(z'', z1) -{ 26 + 2*y'' + 7*z + z'' }-> s10 :|: s10 >= 0, s10 <= 1 + y'' + z + (1 + (z'' - 1)), s'' >= 0, s'' <= 1, z >= 0, z1 = 1 + (1 + y'') + z, y'' >= 0, z'' - 1 >= 0 min(z'', z1) -{ 23 + y + 7*z }-> s8 :|: s8 >= 0, s8 <= y + z + 0, z'' = 0, z >= 0, z1 = 1 + y + z, y >= 0 min(z'', z1) -{ 16 + z'' + 7*z1 }-> s9 :|: s9 >= 0, s9 <= 0 + (z1 - 1) + (1 + (z'' - 1)), z1 - 1 >= 0, z'' - 1 >= 0 min(z'', z1) -{ 1 }-> z'' :|: z1 = 0, z'' >= 0 minsort(z'') -{ 1 }-> 0 :|: z'' = 0 minsort(z'') -{ 70 + 18*z'' }-> 1 + s5 + minsort(s12) :|: s12 >= 0, s12 <= 1 + (z'' - 1) + 0, s5 >= 0, s5 <= z'' - 1 + 0, z'' - 1 >= 0 minsort(z'') -{ 135 + 19*x + 26*y' + 31*z' }-> 1 + s6 + minsort(s13) :|: s13 >= 0, s13 <= 1 + x + (1 + y' + z'), s6 >= 0, s6 <= x + (1 + y' + z'), s7 >= 0, s7 <= y' + z' + x, s' >= 0, s' <= 1, z'' = 1 + x + (1 + y' + z'), x >= 0, z' >= 0, y' >= 0 Function symbols to be analyzed: {minsort} Previous analysis results are: le: runtime: O(n^1) [2 + z1], size: O(1) [1] eq: runtime: O(n^1) [3 + z1], size: O(1) [1] min: runtime: O(n^1) [20 + z'' + 7*z1], size: O(n^1) [z'' + z1] if1: runtime: O(n^1) [21 + z1 + z2 + 7*z3], size: O(n^1) [z1 + z2 + z3] del: runtime: O(n^1) [49 + 17*z1], size: O(n^1) [z1] if2: runtime: O(n^1) [50 + 17*z3], size: O(n^1) [1 + z2 + z3] minsort: runtime: ?, size: INF ---------------------------------------- (51) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: minsort 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'', z1) -{ 35 + 17*z1 }-> s14 :|: s14 >= 0, s14 <= 0 + (z1 - 1) + 1, z'' = 0, z1 - 1 >= 0 del(z'', z1) -{ 52 + 17*z }-> s15 :|: s15 >= 0, s15 <= 1 + y1 + z + 1, z'' = 0, y1 >= 0, z >= 0, z1 = 1 + (1 + y1) + z del(z'', z1) -{ 35 + 17*z1 }-> s16 :|: s16 >= 0, s16 <= 0 + (z1 - 1) + 1, z'' - 1 >= 0, z1 - 1 >= 0 del(z'', z1) -{ 55 + y2 + 17*z }-> s17 :|: s17 >= 0, s17 <= 1 + y2 + z + 1, s2 >= 0, s2 <= 1, z >= 0, z1 = 1 + (1 + y2) + z, y2 >= 0, z'' - 1 >= 0 del(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' >= 0 eq(z'', z1) -{ 3 + z1 }-> s1 :|: s1 >= 0, s1 <= 1, z'' - 1 >= 0, z1 - 1 >= 0 eq(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 = 0 eq(z'', z1) -{ 1 }-> 0 :|: z'' = 0, z1 - 1 >= 0 eq(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' - 1 >= 0 if1(z'', z1, z2, z3) -{ 21 + z1 + 7*z3 }-> s3 :|: s3 >= 0, s3 <= z1 + z3, z3 >= 0, z1 >= 0, z2 >= 0, z'' = 1 if1(z'', z1, z2, z3) -{ 21 + z2 + 7*z3 }-> s4 :|: s4 >= 0, s4 <= z2 + z3, z'' = 0, z3 >= 0, z1 >= 0, z2 >= 0 if2(z'', z1, z2, z3) -{ 1 }-> z3 :|: z3 >= 0, z1 >= 0, z2 >= 0, z'' = 1 if2(z'', z1, z2, z3) -{ 50 + 17*z3 }-> 1 + z2 + s11 :|: s11 >= 0, s11 <= z3, z'' = 0, z3 >= 0, z1 >= 0, z2 >= 0 le(z'', z1) -{ 2 + z1 }-> s :|: s >= 0, s <= 1, z'' - 1 >= 0, z1 - 1 >= 0 le(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 >= 0 le(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' - 1 >= 0 min(z'', z1) -{ 26 + 2*y'' + 7*z + z'' }-> s10 :|: s10 >= 0, s10 <= 1 + y'' + z + (1 + (z'' - 1)), s'' >= 0, s'' <= 1, z >= 0, z1 = 1 + (1 + y'') + z, y'' >= 0, z'' - 1 >= 0 min(z'', z1) -{ 23 + y + 7*z }-> s8 :|: s8 >= 0, s8 <= y + z + 0, z'' = 0, z >= 0, z1 = 1 + y + z, y >= 0 min(z'', z1) -{ 16 + z'' + 7*z1 }-> s9 :|: s9 >= 0, s9 <= 0 + (z1 - 1) + (1 + (z'' - 1)), z1 - 1 >= 0, z'' - 1 >= 0 min(z'', z1) -{ 1 }-> z'' :|: z1 = 0, z'' >= 0 minsort(z'') -{ 1 }-> 0 :|: z'' = 0 minsort(z'') -{ 70 + 18*z'' }-> 1 + s5 + minsort(s12) :|: s12 >= 0, s12 <= 1 + (z'' - 1) + 0, s5 >= 0, s5 <= z'' - 1 + 0, z'' - 1 >= 0 minsort(z'') -{ 135 + 19*x + 26*y' + 31*z' }-> 1 + s6 + minsort(s13) :|: s13 >= 0, s13 <= 1 + x + (1 + y' + z'), s6 >= 0, s6 <= x + (1 + y' + z'), s7 >= 0, s7 <= y' + z' + x, s' >= 0, s' <= 1, z'' = 1 + x + (1 + y' + z'), x >= 0, z' >= 0, y' >= 0 Function symbols to be analyzed: {minsort} Previous analysis results are: le: runtime: O(n^1) [2 + z1], size: O(1) [1] eq: runtime: O(n^1) [3 + z1], size: O(1) [1] min: runtime: O(n^1) [20 + z'' + 7*z1], size: O(n^1) [z'' + z1] if1: runtime: O(n^1) [21 + z1 + z2 + 7*z3], size: O(n^1) [z1 + z2 + z3] del: runtime: O(n^1) [49 + 17*z1], size: O(n^1) [z1] if2: runtime: O(n^1) [50 + 17*z3], size: O(n^1) [1 + z2 + z3] minsort: 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: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) if1(true, z0, z1, z2) -> min(z0, z2) if1(false, z0, z1, z2) -> min(z1, z2) if2(true, z0, z1, z2) -> z2 if2(false, z0, z1, z2) -> cons(z1, del(z0, z2)) minsort(nil) -> nil minsort(cons(z0, z1)) -> cons(min(z0, z1), minsort(del(min(z0, z1), cons(z0, z1)))) min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2) del(z0, nil) -> nil del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) Tuples: LE(0, z0) -> c LE(s(z0), 0) -> c1 LE(s(z0), s(z1)) -> c2(LE(z0, z1)) EQ(0, 0) -> c3 EQ(0, s(z0)) -> c4 EQ(s(z0), 0) -> c5 EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) IF1(true, z0, z1, z2) -> c7(MIN(z0, z2)) IF1(false, z0, z1, z2) -> c8(MIN(z1, z2)) IF2(true, z0, z1, z2) -> c9 IF2(false, z0, z1, z2) -> c10(DEL(z0, z2)) MINSORT(nil) -> c11 MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1)) MINSORT(cons(z0, z1)) -> c13(MINSORT(del(min(z0, z1), cons(z0, z1))), DEL(min(z0, z1), cons(z0, z1)), MIN(z0, z1)) MIN(z0, nil) -> c14 MIN(z0, cons(z1, z2)) -> c15(IF1(le(z0, z1), z0, z1, z2), LE(z0, z1)) DEL(z0, nil) -> c16 DEL(z0, cons(z1, z2)) -> c17(IF2(eq(z0, z1), z0, z1, z2), EQ(z0, z1)) S tuples: LE(0, z0) -> c LE(s(z0), 0) -> c1 LE(s(z0), s(z1)) -> c2(LE(z0, z1)) EQ(0, 0) -> c3 EQ(0, s(z0)) -> c4 EQ(s(z0), 0) -> c5 EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) IF1(true, z0, z1, z2) -> c7(MIN(z0, z2)) IF1(false, z0, z1, z2) -> c8(MIN(z1, z2)) IF2(true, z0, z1, z2) -> c9 IF2(false, z0, z1, z2) -> c10(DEL(z0, z2)) MINSORT(nil) -> c11 MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1)) MINSORT(cons(z0, z1)) -> c13(MINSORT(del(min(z0, z1), cons(z0, z1))), DEL(min(z0, z1), cons(z0, z1)), MIN(z0, z1)) MIN(z0, nil) -> c14 MIN(z0, cons(z1, z2)) -> c15(IF1(le(z0, z1), z0, z1, z2), LE(z0, z1)) DEL(z0, nil) -> c16 DEL(z0, cons(z1, z2)) -> c17(IF2(eq(z0, z1), z0, z1, z2), EQ(z0, z1)) K tuples:none Defined Rule Symbols: le_2, eq_2, if1_4, if2_4, minsort_1, min_2, del_2 Defined Pair Symbols: LE_2, EQ_2, IF1_4, IF2_4, MINSORT_1, MIN_2, DEL_2 Compound Symbols: c, c1, c2_1, c3, c4, c5, c6_1, c7_1, c8_1, c9, c10_1, c11, c12_1, c13_3, c14, c15_2, c16, c17_2 ---------------------------------------- (55) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 9 trailing nodes: MINSORT(nil) -> c11 EQ(s(z0), 0) -> c5 EQ(0, 0) -> c3 LE(s(z0), 0) -> c1 EQ(0, s(z0)) -> c4 DEL(z0, nil) -> c16 IF2(true, z0, z1, z2) -> c9 MIN(z0, nil) -> c14 LE(0, z0) -> c ---------------------------------------- (56) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) if1(true, z0, z1, z2) -> min(z0, z2) if1(false, z0, z1, z2) -> min(z1, z2) if2(true, z0, z1, z2) -> z2 if2(false, z0, z1, z2) -> cons(z1, del(z0, z2)) minsort(nil) -> nil minsort(cons(z0, z1)) -> cons(min(z0, z1), minsort(del(min(z0, z1), cons(z0, z1)))) min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2) del(z0, nil) -> nil del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) IF1(true, z0, z1, z2) -> c7(MIN(z0, z2)) IF1(false, z0, z1, z2) -> c8(MIN(z1, z2)) IF2(false, z0, z1, z2) -> c10(DEL(z0, z2)) MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1)) MINSORT(cons(z0, z1)) -> c13(MINSORT(del(min(z0, z1), cons(z0, z1))), DEL(min(z0, z1), cons(z0, z1)), MIN(z0, z1)) MIN(z0, cons(z1, z2)) -> c15(IF1(le(z0, z1), z0, z1, z2), LE(z0, z1)) DEL(z0, cons(z1, z2)) -> c17(IF2(eq(z0, z1), z0, z1, z2), EQ(z0, z1)) S tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) IF1(true, z0, z1, z2) -> c7(MIN(z0, z2)) IF1(false, z0, z1, z2) -> c8(MIN(z1, z2)) IF2(false, z0, z1, z2) -> c10(DEL(z0, z2)) MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1)) MINSORT(cons(z0, z1)) -> c13(MINSORT(del(min(z0, z1), cons(z0, z1))), DEL(min(z0, z1), cons(z0, z1)), MIN(z0, z1)) MIN(z0, cons(z1, z2)) -> c15(IF1(le(z0, z1), z0, z1, z2), LE(z0, z1)) DEL(z0, cons(z1, z2)) -> c17(IF2(eq(z0, z1), z0, z1, z2), EQ(z0, z1)) K tuples:none Defined Rule Symbols: le_2, eq_2, if1_4, if2_4, minsort_1, min_2, del_2 Defined Pair Symbols: LE_2, EQ_2, IF1_4, IF2_4, MINSORT_1, MIN_2, DEL_2 Compound Symbols: c2_1, c6_1, c7_1, c8_1, c10_1, c12_1, c13_3, c15_2, c17_2 ---------------------------------------- (57) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: minsort(nil) -> nil minsort(cons(z0, z1)) -> cons(min(z0, z1), minsort(del(min(z0, z1), cons(z0, z1)))) ---------------------------------------- (58) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(z0, z2) if1(false, z0, z1, z2) -> min(z1, z2) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) 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) Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) IF1(true, z0, z1, z2) -> c7(MIN(z0, z2)) IF1(false, z0, z1, z2) -> c8(MIN(z1, z2)) IF2(false, z0, z1, z2) -> c10(DEL(z0, z2)) MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1)) MINSORT(cons(z0, z1)) -> c13(MINSORT(del(min(z0, z1), cons(z0, z1))), DEL(min(z0, z1), cons(z0, z1)), MIN(z0, z1)) MIN(z0, cons(z1, z2)) -> c15(IF1(le(z0, z1), z0, z1, z2), LE(z0, z1)) DEL(z0, cons(z1, z2)) -> c17(IF2(eq(z0, z1), z0, z1, z2), EQ(z0, z1)) S tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) IF1(true, z0, z1, z2) -> c7(MIN(z0, z2)) IF1(false, z0, z1, z2) -> c8(MIN(z1, z2)) IF2(false, z0, z1, z2) -> c10(DEL(z0, z2)) MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1)) MINSORT(cons(z0, z1)) -> c13(MINSORT(del(min(z0, z1), cons(z0, z1))), DEL(min(z0, z1), cons(z0, z1)), MIN(z0, z1)) MIN(z0, cons(z1, z2)) -> c15(IF1(le(z0, z1), z0, z1, z2), LE(z0, z1)) DEL(z0, cons(z1, z2)) -> c17(IF2(eq(z0, z1), z0, z1, z2), EQ(z0, z1)) K tuples:none Defined Rule Symbols: del_2, min_2, if1_4, le_2, if2_4, eq_2 Defined Pair Symbols: LE_2, EQ_2, IF1_4, IF2_4, MINSORT_1, MIN_2, DEL_2 Compound Symbols: c2_1, c6_1, c7_1, c8_1, c10_1, c12_1, c13_3, c15_2, c17_2 ---------------------------------------- (59) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1)) We considered the (Usable) Rules:none And the Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) IF1(true, z0, z1, z2) -> c7(MIN(z0, z2)) IF1(false, z0, z1, z2) -> c8(MIN(z1, z2)) IF2(false, z0, z1, z2) -> c10(DEL(z0, z2)) MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1)) MINSORT(cons(z0, z1)) -> c13(MINSORT(del(min(z0, z1), cons(z0, z1))), DEL(min(z0, z1), cons(z0, z1)), MIN(z0, z1)) MIN(z0, cons(z1, z2)) -> c15(IF1(le(z0, z1), z0, z1, z2), LE(z0, z1)) DEL(z0, cons(z1, z2)) -> c17(IF2(eq(z0, z1), z0, z1, z2), EQ(z0, z1)) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = [1] POL(DEL(x_1, x_2)) = 0 POL(EQ(x_1, x_2)) = 0 POL(IF1(x_1, x_2, x_3, x_4)) = 0 POL(IF2(x_1, x_2, x_3, x_4)) = 0 POL(LE(x_1, x_2)) = 0 POL(MIN(x_1, x_2)) = 0 POL(MINSORT(x_1)) = [1] POL(c10(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c13(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c15(x_1, x_2)) = x_1 + x_2 POL(c17(x_1, x_2)) = x_1 + x_2 POL(c2(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(cons(x_1, x_2)) = [1] + x_1 + x_2 POL(del(x_1, x_2)) = [1] + x_1 + x_2 POL(eq(x_1, x_2)) = [1] POL(false) = [1] POL(if1(x_1, x_2, x_3, x_4)) = [1] + x_1 + x_2 + x_3 + x_4 POL(if2(x_1, x_2, x_3, x_4)) = [1] + x_1 + x_2 + x_3 + x_4 POL(le(x_1, x_2)) = [1] POL(min(x_1, x_2)) = [1] + x_1 + x_2 POL(nil) = [1] POL(s(x_1)) = [1] + x_1 POL(true) = [1] ---------------------------------------- (60) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(z0, z2) if1(false, z0, z1, z2) -> min(z1, z2) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) 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) Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) IF1(true, z0, z1, z2) -> c7(MIN(z0, z2)) IF1(false, z0, z1, z2) -> c8(MIN(z1, z2)) IF2(false, z0, z1, z2) -> c10(DEL(z0, z2)) MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1)) MINSORT(cons(z0, z1)) -> c13(MINSORT(del(min(z0, z1), cons(z0, z1))), DEL(min(z0, z1), cons(z0, z1)), MIN(z0, z1)) MIN(z0, cons(z1, z2)) -> c15(IF1(le(z0, z1), z0, z1, z2), LE(z0, z1)) DEL(z0, cons(z1, z2)) -> c17(IF2(eq(z0, z1), z0, z1, z2), EQ(z0, z1)) S tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) IF1(true, z0, z1, z2) -> c7(MIN(z0, z2)) IF1(false, z0, z1, z2) -> c8(MIN(z1, z2)) IF2(false, z0, z1, z2) -> c10(DEL(z0, z2)) MINSORT(cons(z0, z1)) -> c13(MINSORT(del(min(z0, z1), cons(z0, z1))), DEL(min(z0, z1), cons(z0, z1)), MIN(z0, z1)) MIN(z0, cons(z1, z2)) -> c15(IF1(le(z0, z1), z0, z1, z2), LE(z0, z1)) DEL(z0, cons(z1, z2)) -> c17(IF2(eq(z0, z1), z0, z1, z2), EQ(z0, z1)) K tuples: MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1)) Defined Rule Symbols: del_2, min_2, if1_4, le_2, if2_4, eq_2 Defined Pair Symbols: LE_2, EQ_2, IF1_4, IF2_4, MINSORT_1, MIN_2, DEL_2 Compound Symbols: c2_1, c6_1, c7_1, c8_1, c10_1, c12_1, c13_3, c15_2, c17_2 ---------------------------------------- (61) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MINSORT(cons(z0, z1)) -> c13(MINSORT(del(min(z0, z1), cons(z0, z1))), DEL(min(z0, z1), cons(z0, z1)), MIN(z0, z1)) by MINSORT(cons(z1, z2)) -> c13(MINSORT(if2(eq(min(z1, z2), z1), min(z1, z2), z1, z2)), DEL(min(z1, z2), cons(z1, z2)), MIN(z1, z2)) MINSORT(cons(z0, nil)) -> c13(MINSORT(del(z0, cons(z0, nil))), DEL(min(z0, nil), cons(z0, nil)), MIN(z0, nil)) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(del(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) ---------------------------------------- (62) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(z0, z2) if1(false, z0, z1, z2) -> min(z1, z2) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) 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) Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) IF1(true, z0, z1, z2) -> c7(MIN(z0, z2)) IF1(false, z0, z1, z2) -> c8(MIN(z1, z2)) IF2(false, z0, z1, z2) -> c10(DEL(z0, z2)) MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1)) MIN(z0, cons(z1, z2)) -> c15(IF1(le(z0, z1), z0, z1, z2), LE(z0, z1)) DEL(z0, cons(z1, z2)) -> c17(IF2(eq(z0, z1), z0, z1, z2), EQ(z0, z1)) MINSORT(cons(z1, z2)) -> c13(MINSORT(if2(eq(min(z1, z2), z1), min(z1, z2), z1, z2)), DEL(min(z1, z2), cons(z1, z2)), MIN(z1, z2)) MINSORT(cons(z0, nil)) -> c13(MINSORT(del(z0, cons(z0, nil))), DEL(min(z0, nil), cons(z0, nil)), MIN(z0, nil)) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(del(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) S tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) IF1(true, z0, z1, z2) -> c7(MIN(z0, z2)) IF1(false, z0, z1, z2) -> c8(MIN(z1, z2)) IF2(false, z0, z1, z2) -> c10(DEL(z0, z2)) MIN(z0, cons(z1, z2)) -> c15(IF1(le(z0, z1), z0, z1, z2), LE(z0, z1)) DEL(z0, cons(z1, z2)) -> c17(IF2(eq(z0, z1), z0, z1, z2), EQ(z0, z1)) MINSORT(cons(z1, z2)) -> c13(MINSORT(if2(eq(min(z1, z2), z1), min(z1, z2), z1, z2)), DEL(min(z1, z2), cons(z1, z2)), MIN(z1, z2)) MINSORT(cons(z0, nil)) -> c13(MINSORT(del(z0, cons(z0, nil))), DEL(min(z0, nil), cons(z0, nil)), MIN(z0, nil)) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(del(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) K tuples: MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1)) Defined Rule Symbols: del_2, min_2, if1_4, le_2, if2_4, eq_2 Defined Pair Symbols: LE_2, EQ_2, IF1_4, IF2_4, MINSORT_1, MIN_2, DEL_2 Compound Symbols: c2_1, c6_1, c7_1, c8_1, c10_1, c12_1, c15_2, c17_2, c13_3 ---------------------------------------- (63) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (64) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(z0, z2) if1(false, z0, z1, z2) -> min(z1, z2) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) 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) Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) IF1(true, z0, z1, z2) -> c7(MIN(z0, z2)) IF1(false, z0, z1, z2) -> c8(MIN(z1, z2)) IF2(false, z0, z1, z2) -> c10(DEL(z0, z2)) MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1)) MIN(z0, cons(z1, z2)) -> c15(IF1(le(z0, z1), z0, z1, z2), LE(z0, z1)) DEL(z0, cons(z1, z2)) -> c17(IF2(eq(z0, z1), z0, z1, z2), EQ(z0, z1)) MINSORT(cons(z1, z2)) -> c13(MINSORT(if2(eq(min(z1, z2), z1), min(z1, z2), z1, z2)), DEL(min(z1, z2), cons(z1, z2)), MIN(z1, z2)) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(del(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(del(z0, cons(z0, nil))), DEL(min(z0, nil), cons(z0, nil))) S tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) IF1(true, z0, z1, z2) -> c7(MIN(z0, z2)) IF1(false, z0, z1, z2) -> c8(MIN(z1, z2)) IF2(false, z0, z1, z2) -> c10(DEL(z0, z2)) MIN(z0, cons(z1, z2)) -> c15(IF1(le(z0, z1), z0, z1, z2), LE(z0, z1)) DEL(z0, cons(z1, z2)) -> c17(IF2(eq(z0, z1), z0, z1, z2), EQ(z0, z1)) MINSORT(cons(z1, z2)) -> c13(MINSORT(if2(eq(min(z1, z2), z1), min(z1, z2), z1, z2)), DEL(min(z1, z2), cons(z1, z2)), MIN(z1, z2)) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(del(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(del(z0, cons(z0, nil))), DEL(min(z0, nil), cons(z0, nil))) K tuples: MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1)) Defined Rule Symbols: del_2, min_2, if1_4, le_2, if2_4, eq_2 Defined Pair Symbols: LE_2, EQ_2, IF1_4, IF2_4, MINSORT_1, MIN_2, DEL_2 Compound Symbols: c2_1, c6_1, c7_1, c8_1, c10_1, c12_1, c15_2, c17_2, c13_3, c13_2 ---------------------------------------- (65) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MIN(z0, cons(z1, z2)) -> c15(IF1(le(z0, z1), z0, z1, z2), LE(z0, z1)) by MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2), LE(0, z0)) MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2), LE(s(z0), 0)) MIN(s(z0), cons(s(z1), x2)) -> c15(IF1(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1))) ---------------------------------------- (66) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(z0, z2) if1(false, z0, z1, z2) -> min(z1, z2) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) 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) Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) IF1(true, z0, z1, z2) -> c7(MIN(z0, z2)) IF1(false, z0, z1, z2) -> c8(MIN(z1, z2)) IF2(false, z0, z1, z2) -> c10(DEL(z0, z2)) MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1)) DEL(z0, cons(z1, z2)) -> c17(IF2(eq(z0, z1), z0, z1, z2), EQ(z0, z1)) MINSORT(cons(z1, z2)) -> c13(MINSORT(if2(eq(min(z1, z2), z1), min(z1, z2), z1, z2)), DEL(min(z1, z2), cons(z1, z2)), MIN(z1, z2)) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(del(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(del(z0, cons(z0, nil))), DEL(min(z0, nil), cons(z0, nil))) MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2), LE(0, z0)) MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2), LE(s(z0), 0)) MIN(s(z0), cons(s(z1), x2)) -> c15(IF1(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1))) S tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) IF1(true, z0, z1, z2) -> c7(MIN(z0, z2)) IF1(false, z0, z1, z2) -> c8(MIN(z1, z2)) IF2(false, z0, z1, z2) -> c10(DEL(z0, z2)) DEL(z0, cons(z1, z2)) -> c17(IF2(eq(z0, z1), z0, z1, z2), EQ(z0, z1)) MINSORT(cons(z1, z2)) -> c13(MINSORT(if2(eq(min(z1, z2), z1), min(z1, z2), z1, z2)), DEL(min(z1, z2), cons(z1, z2)), MIN(z1, z2)) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(del(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(del(z0, cons(z0, nil))), DEL(min(z0, nil), cons(z0, nil))) MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2), LE(0, z0)) MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2), LE(s(z0), 0)) MIN(s(z0), cons(s(z1), x2)) -> c15(IF1(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1))) K tuples: MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1)) Defined Rule Symbols: del_2, min_2, if1_4, le_2, if2_4, eq_2 Defined Pair Symbols: LE_2, EQ_2, IF1_4, IF2_4, MINSORT_1, DEL_2, MIN_2 Compound Symbols: c2_1, c6_1, c7_1, c8_1, c10_1, c12_1, c17_2, c13_3, c13_2, c15_2 ---------------------------------------- (67) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (68) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(z0, z2) if1(false, z0, z1, z2) -> min(z1, z2) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) 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) Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) IF1(true, z0, z1, z2) -> c7(MIN(z0, z2)) IF1(false, z0, z1, z2) -> c8(MIN(z1, z2)) IF2(false, z0, z1, z2) -> c10(DEL(z0, z2)) MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1)) DEL(z0, cons(z1, z2)) -> c17(IF2(eq(z0, z1), z0, z1, z2), EQ(z0, z1)) MINSORT(cons(z1, z2)) -> c13(MINSORT(if2(eq(min(z1, z2), z1), min(z1, z2), z1, z2)), DEL(min(z1, z2), cons(z1, z2)), MIN(z1, z2)) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(del(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(del(z0, cons(z0, nil))), DEL(min(z0, nil), cons(z0, nil))) MIN(s(z0), cons(s(z1), x2)) -> c15(IF1(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1))) MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2)) MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2)) S tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) IF1(true, z0, z1, z2) -> c7(MIN(z0, z2)) IF1(false, z0, z1, z2) -> c8(MIN(z1, z2)) IF2(false, z0, z1, z2) -> c10(DEL(z0, z2)) DEL(z0, cons(z1, z2)) -> c17(IF2(eq(z0, z1), z0, z1, z2), EQ(z0, z1)) MINSORT(cons(z1, z2)) -> c13(MINSORT(if2(eq(min(z1, z2), z1), min(z1, z2), z1, z2)), DEL(min(z1, z2), cons(z1, z2)), MIN(z1, z2)) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(del(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(del(z0, cons(z0, nil))), DEL(min(z0, nil), cons(z0, nil))) MIN(s(z0), cons(s(z1), x2)) -> c15(IF1(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1))) MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2)) MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2)) K tuples: MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1)) Defined Rule Symbols: del_2, min_2, if1_4, le_2, if2_4, eq_2 Defined Pair Symbols: LE_2, EQ_2, IF1_4, IF2_4, MINSORT_1, DEL_2, MIN_2 Compound Symbols: c2_1, c6_1, c7_1, c8_1, c10_1, c12_1, c17_2, c13_3, c13_2, c15_2, c15_1 ---------------------------------------- (69) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace DEL(z0, cons(z1, z2)) -> c17(IF2(eq(z0, z1), z0, z1, z2), EQ(z0, z1)) by DEL(0, cons(0, x2)) -> c17(IF2(true, 0, 0, x2), EQ(0, 0)) DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2), EQ(0, s(z0))) DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2), EQ(s(z0), 0)) DEL(s(z0), cons(s(z1), x2)) -> c17(IF2(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1))) ---------------------------------------- (70) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(z0, z2) if1(false, z0, z1, z2) -> min(z1, z2) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) 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) Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) IF1(true, z0, z1, z2) -> c7(MIN(z0, z2)) IF1(false, z0, z1, z2) -> c8(MIN(z1, z2)) IF2(false, z0, z1, z2) -> c10(DEL(z0, z2)) MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1)) MINSORT(cons(z1, z2)) -> c13(MINSORT(if2(eq(min(z1, z2), z1), min(z1, z2), z1, z2)), DEL(min(z1, z2), cons(z1, z2)), MIN(z1, z2)) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(del(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(del(z0, cons(z0, nil))), DEL(min(z0, nil), cons(z0, nil))) MIN(s(z0), cons(s(z1), x2)) -> c15(IF1(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1))) MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2)) MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2)) DEL(0, cons(0, x2)) -> c17(IF2(true, 0, 0, x2), EQ(0, 0)) DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2), EQ(0, s(z0))) DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2), EQ(s(z0), 0)) DEL(s(z0), cons(s(z1), x2)) -> c17(IF2(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1))) S tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) IF1(true, z0, z1, z2) -> c7(MIN(z0, z2)) IF1(false, z0, z1, z2) -> c8(MIN(z1, z2)) IF2(false, z0, z1, z2) -> c10(DEL(z0, z2)) MINSORT(cons(z1, z2)) -> c13(MINSORT(if2(eq(min(z1, z2), z1), min(z1, z2), z1, z2)), DEL(min(z1, z2), cons(z1, z2)), MIN(z1, z2)) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(del(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(del(z0, cons(z0, nil))), DEL(min(z0, nil), cons(z0, nil))) MIN(s(z0), cons(s(z1), x2)) -> c15(IF1(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1))) MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2)) MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2)) DEL(0, cons(0, x2)) -> c17(IF2(true, 0, 0, x2), EQ(0, 0)) DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2), EQ(0, s(z0))) DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2), EQ(s(z0), 0)) DEL(s(z0), cons(s(z1), x2)) -> c17(IF2(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1))) K tuples: MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1)) Defined Rule Symbols: del_2, min_2, if1_4, le_2, if2_4, eq_2 Defined Pair Symbols: LE_2, EQ_2, IF1_4, IF2_4, MINSORT_1, MIN_2, DEL_2 Compound Symbols: c2_1, c6_1, c7_1, c8_1, c10_1, c12_1, c13_3, c13_2, c15_2, c15_1, c17_2 ---------------------------------------- (71) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: DEL(0, cons(0, x2)) -> c17(IF2(true, 0, 0, x2), EQ(0, 0)) ---------------------------------------- (72) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(z0, z2) if1(false, z0, z1, z2) -> min(z1, z2) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) 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) Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) IF1(true, z0, z1, z2) -> c7(MIN(z0, z2)) IF1(false, z0, z1, z2) -> c8(MIN(z1, z2)) IF2(false, z0, z1, z2) -> c10(DEL(z0, z2)) MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1)) MINSORT(cons(z1, z2)) -> c13(MINSORT(if2(eq(min(z1, z2), z1), min(z1, z2), z1, z2)), DEL(min(z1, z2), cons(z1, z2)), MIN(z1, z2)) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(del(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(del(z0, cons(z0, nil))), DEL(min(z0, nil), cons(z0, nil))) MIN(s(z0), cons(s(z1), x2)) -> c15(IF1(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1))) MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2)) MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2)) DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2), EQ(0, s(z0))) DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2), EQ(s(z0), 0)) DEL(s(z0), cons(s(z1), x2)) -> c17(IF2(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1))) S tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) IF1(true, z0, z1, z2) -> c7(MIN(z0, z2)) IF1(false, z0, z1, z2) -> c8(MIN(z1, z2)) IF2(false, z0, z1, z2) -> c10(DEL(z0, z2)) MINSORT(cons(z1, z2)) -> c13(MINSORT(if2(eq(min(z1, z2), z1), min(z1, z2), z1, z2)), DEL(min(z1, z2), cons(z1, z2)), MIN(z1, z2)) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(del(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(del(z0, cons(z0, nil))), DEL(min(z0, nil), cons(z0, nil))) MIN(s(z0), cons(s(z1), x2)) -> c15(IF1(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1))) MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2)) MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2)) DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2), EQ(0, s(z0))) DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2), EQ(s(z0), 0)) DEL(s(z0), cons(s(z1), x2)) -> c17(IF2(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1))) K tuples: MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1)) Defined Rule Symbols: del_2, min_2, if1_4, le_2, if2_4, eq_2 Defined Pair Symbols: LE_2, EQ_2, IF1_4, IF2_4, MINSORT_1, MIN_2, DEL_2 Compound Symbols: c2_1, c6_1, c7_1, c8_1, c10_1, c12_1, c13_3, c13_2, c15_2, c15_1, c17_2 ---------------------------------------- (73) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (74) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(z0, z2) if1(false, z0, z1, z2) -> min(z1, z2) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) 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) Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) IF1(true, z0, z1, z2) -> c7(MIN(z0, z2)) IF1(false, z0, z1, z2) -> c8(MIN(z1, z2)) IF2(false, z0, z1, z2) -> c10(DEL(z0, z2)) MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1)) MINSORT(cons(z1, z2)) -> c13(MINSORT(if2(eq(min(z1, z2), z1), min(z1, z2), z1, z2)), DEL(min(z1, z2), cons(z1, z2)), MIN(z1, z2)) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(del(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(del(z0, cons(z0, nil))), DEL(min(z0, nil), cons(z0, nil))) MIN(s(z0), cons(s(z1), x2)) -> c15(IF1(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1))) MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2)) MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2)) DEL(s(z0), cons(s(z1), x2)) -> c17(IF2(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1))) DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2)) S tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) IF1(true, z0, z1, z2) -> c7(MIN(z0, z2)) IF1(false, z0, z1, z2) -> c8(MIN(z1, z2)) IF2(false, z0, z1, z2) -> c10(DEL(z0, z2)) MINSORT(cons(z1, z2)) -> c13(MINSORT(if2(eq(min(z1, z2), z1), min(z1, z2), z1, z2)), DEL(min(z1, z2), cons(z1, z2)), MIN(z1, z2)) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(del(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(del(z0, cons(z0, nil))), DEL(min(z0, nil), cons(z0, nil))) MIN(s(z0), cons(s(z1), x2)) -> c15(IF1(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1))) MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2)) MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2)) DEL(s(z0), cons(s(z1), x2)) -> c17(IF2(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1))) DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2)) K tuples: MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1)) Defined Rule Symbols: del_2, min_2, if1_4, le_2, if2_4, eq_2 Defined Pair Symbols: LE_2, EQ_2, IF1_4, IF2_4, MINSORT_1, MIN_2, DEL_2 Compound Symbols: c2_1, c6_1, c7_1, c8_1, c10_1, c12_1, c13_3, c13_2, c15_2, c15_1, c17_2, c17_1 ---------------------------------------- (75) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MINSORT(cons(z1, z2)) -> c13(MINSORT(if2(eq(min(z1, z2), z1), min(z1, z2), z1, z2)), DEL(min(z1, z2), cons(z1, z2)), MIN(z1, z2)) by MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil)), MIN(z0, nil)) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(min(z0, nil), cons(z0, nil)), MIN(z0, nil)) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) ---------------------------------------- (76) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(z0, z2) if1(false, z0, z1, z2) -> min(z1, z2) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) 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) Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) IF1(true, z0, z1, z2) -> c7(MIN(z0, z2)) IF1(false, z0, z1, z2) -> c8(MIN(z1, z2)) IF2(false, z0, z1, z2) -> c10(DEL(z0, z2)) MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1)) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(del(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(del(z0, cons(z0, nil))), DEL(min(z0, nil), cons(z0, nil))) MIN(s(z0), cons(s(z1), x2)) -> c15(IF1(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1))) MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2)) MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2)) DEL(s(z0), cons(s(z1), x2)) -> c17(IF2(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1))) DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2)) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil)), MIN(z0, nil)) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(min(z0, nil), cons(z0, nil)), MIN(z0, nil)) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) S tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) IF1(true, z0, z1, z2) -> c7(MIN(z0, z2)) IF1(false, z0, z1, z2) -> c8(MIN(z1, z2)) IF2(false, z0, z1, z2) -> c10(DEL(z0, z2)) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(del(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(del(z0, cons(z0, nil))), DEL(min(z0, nil), cons(z0, nil))) MIN(s(z0), cons(s(z1), x2)) -> c15(IF1(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1))) MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2)) MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2)) DEL(s(z0), cons(s(z1), x2)) -> c17(IF2(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1))) DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2)) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil)), MIN(z0, nil)) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(min(z0, nil), cons(z0, nil)), MIN(z0, nil)) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) K tuples: MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1)) Defined Rule Symbols: del_2, min_2, if1_4, le_2, if2_4, eq_2 Defined Pair Symbols: LE_2, EQ_2, IF1_4, IF2_4, MINSORT_1, MIN_2, DEL_2 Compound Symbols: c2_1, c6_1, c7_1, c8_1, c10_1, c12_1, c13_3, c13_2, c15_2, c15_1, c17_2, c17_1 ---------------------------------------- (77) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (78) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(z0, z2) if1(false, z0, z1, z2) -> min(z1, z2) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) 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) Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) IF1(true, z0, z1, z2) -> c7(MIN(z0, z2)) IF1(false, z0, z1, z2) -> c8(MIN(z1, z2)) IF2(false, z0, z1, z2) -> c10(DEL(z0, z2)) MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1)) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(del(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(del(z0, cons(z0, nil))), DEL(min(z0, nil), cons(z0, nil))) MIN(s(z0), cons(s(z1), x2)) -> c15(IF1(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1))) MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2)) MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2)) DEL(s(z0), cons(s(z1), x2)) -> c17(IF2(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1))) DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2)) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(min(z0, nil), cons(z0, nil))) S tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) IF1(true, z0, z1, z2) -> c7(MIN(z0, z2)) IF1(false, z0, z1, z2) -> c8(MIN(z1, z2)) IF2(false, z0, z1, z2) -> c10(DEL(z0, z2)) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(del(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(del(z0, cons(z0, nil))), DEL(min(z0, nil), cons(z0, nil))) MIN(s(z0), cons(s(z1), x2)) -> c15(IF1(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1))) MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2)) MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2)) DEL(s(z0), cons(s(z1), x2)) -> c17(IF2(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1))) DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2)) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(min(z0, nil), cons(z0, nil))) K tuples: MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1)) Defined Rule Symbols: del_2, min_2, if1_4, le_2, if2_4, eq_2 Defined Pair Symbols: LE_2, EQ_2, IF1_4, IF2_4, MINSORT_1, MIN_2, DEL_2 Compound Symbols: c2_1, c6_1, c7_1, c8_1, c10_1, c12_1, c13_3, c13_2, c15_2, c15_1, c17_2, c17_1 ---------------------------------------- (79) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(del(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) by MINSORT(cons(z1, cons(x1, x2))) -> c13(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2))) MINSORT(cons(0, cons(z0, x2))) -> c13(MINSORT(del(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2))) MINSORT(cons(s(z0), cons(0, x2))) -> c13(MINSORT(del(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2))) MINSORT(cons(x0, cons(x1, x2))) -> c13(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) ---------------------------------------- (80) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(z0, z2) if1(false, z0, z1, z2) -> min(z1, z2) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) 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) Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) IF1(true, z0, z1, z2) -> c7(MIN(z0, z2)) IF1(false, z0, z1, z2) -> c8(MIN(z1, z2)) IF2(false, z0, z1, z2) -> c10(DEL(z0, z2)) MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1)) MINSORT(cons(z0, nil)) -> c13(MINSORT(del(z0, cons(z0, nil))), DEL(min(z0, nil), cons(z0, nil))) MIN(s(z0), cons(s(z1), x2)) -> c15(IF1(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1))) MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2)) MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2)) DEL(s(z0), cons(s(z1), x2)) -> c17(IF2(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1))) DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2)) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z1, cons(x1, x2))) -> c13(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2))) MINSORT(cons(0, cons(z0, x2))) -> c13(MINSORT(del(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2))) MINSORT(cons(s(z0), cons(0, x2))) -> c13(MINSORT(del(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2))) MINSORT(cons(x0, cons(x1, x2))) -> c13(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) S tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) IF1(true, z0, z1, z2) -> c7(MIN(z0, z2)) IF1(false, z0, z1, z2) -> c8(MIN(z1, z2)) IF2(false, z0, z1, z2) -> c10(DEL(z0, z2)) MINSORT(cons(z0, nil)) -> c13(MINSORT(del(z0, cons(z0, nil))), DEL(min(z0, nil), cons(z0, nil))) MIN(s(z0), cons(s(z1), x2)) -> c15(IF1(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1))) MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2)) MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2)) DEL(s(z0), cons(s(z1), x2)) -> c17(IF2(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1))) DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2)) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z1, cons(x1, x2))) -> c13(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2))) MINSORT(cons(0, cons(z0, x2))) -> c13(MINSORT(del(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2))) MINSORT(cons(s(z0), cons(0, x2))) -> c13(MINSORT(del(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2))) MINSORT(cons(x0, cons(x1, x2))) -> c13(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) K tuples: MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1)) Defined Rule Symbols: del_2, min_2, if1_4, le_2, if2_4, eq_2 Defined Pair Symbols: LE_2, EQ_2, IF1_4, IF2_4, MINSORT_1, MIN_2, DEL_2 Compound Symbols: c2_1, c6_1, c7_1, c8_1, c10_1, c12_1, c13_2, c15_2, c15_1, c17_2, c17_1, c13_3, c13_1 ---------------------------------------- (81) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. MINSORT(cons(x0, cons(x1, x2))) -> c13(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) We considered the (Usable) Rules:none And the Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) IF1(true, z0, z1, z2) -> c7(MIN(z0, z2)) IF1(false, z0, z1, z2) -> c8(MIN(z1, z2)) IF2(false, z0, z1, z2) -> c10(DEL(z0, z2)) MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1)) MINSORT(cons(z0, nil)) -> c13(MINSORT(del(z0, cons(z0, nil))), DEL(min(z0, nil), cons(z0, nil))) MIN(s(z0), cons(s(z1), x2)) -> c15(IF1(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1))) MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2)) MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2)) DEL(s(z0), cons(s(z1), x2)) -> c17(IF2(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1))) DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2)) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z1, cons(x1, x2))) -> c13(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2))) MINSORT(cons(0, cons(z0, x2))) -> c13(MINSORT(del(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2))) MINSORT(cons(s(z0), cons(0, x2))) -> c13(MINSORT(del(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2))) MINSORT(cons(x0, cons(x1, x2))) -> c13(DEL(min(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(IF1(x_1, x_2, x_3, x_4)) = 0 POL(IF2(x_1, x_2, x_3, x_4)) = x_3 POL(LE(x_1, x_2)) = 0 POL(MIN(x_1, x_2)) = 0 POL(MINSORT(x_1)) = [1] POL(c10(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c13(x_1)) = x_1 POL(c13(x_1, x_2)) = x_1 + x_2 POL(c13(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c15(x_1)) = x_1 POL(c15(x_1, x_2)) = x_1 + x_2 POL(c17(x_1)) = x_1 POL(c17(x_1, x_2)) = x_1 + x_2 POL(c2(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(cons(x_1, x_2)) = 0 POL(del(x_1, x_2)) = [1] + x_2 POL(eq(x_1, x_2)) = 0 POL(false) = 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(le(x_1, x_2)) = 0 POL(min(x_1, x_2)) = 0 POL(nil) = 0 POL(s(x_1)) = 0 POL(true) = 0 ---------------------------------------- (82) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(z0, z2) if1(false, z0, z1, z2) -> min(z1, z2) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) 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) Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) IF1(true, z0, z1, z2) -> c7(MIN(z0, z2)) IF1(false, z0, z1, z2) -> c8(MIN(z1, z2)) IF2(false, z0, z1, z2) -> c10(DEL(z0, z2)) MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1)) MINSORT(cons(z0, nil)) -> c13(MINSORT(del(z0, cons(z0, nil))), DEL(min(z0, nil), cons(z0, nil))) MIN(s(z0), cons(s(z1), x2)) -> c15(IF1(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1))) MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2)) MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2)) DEL(s(z0), cons(s(z1), x2)) -> c17(IF2(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1))) DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2)) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z1, cons(x1, x2))) -> c13(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2))) MINSORT(cons(0, cons(z0, x2))) -> c13(MINSORT(del(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2))) MINSORT(cons(s(z0), cons(0, x2))) -> c13(MINSORT(del(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2))) MINSORT(cons(x0, cons(x1, x2))) -> c13(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) S tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) IF1(true, z0, z1, z2) -> c7(MIN(z0, z2)) IF1(false, z0, z1, z2) -> c8(MIN(z1, z2)) IF2(false, z0, z1, z2) -> c10(DEL(z0, z2)) MINSORT(cons(z0, nil)) -> c13(MINSORT(del(z0, cons(z0, nil))), DEL(min(z0, nil), cons(z0, nil))) MIN(s(z0), cons(s(z1), x2)) -> c15(IF1(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1))) MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2)) MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2)) DEL(s(z0), cons(s(z1), x2)) -> c17(IF2(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1))) DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2)) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z1, cons(x1, x2))) -> c13(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2))) MINSORT(cons(0, cons(z0, x2))) -> c13(MINSORT(del(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2))) MINSORT(cons(s(z0), cons(0, x2))) -> c13(MINSORT(del(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2))) K tuples: MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1)) MINSORT(cons(x0, cons(x1, x2))) -> c13(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) Defined Rule Symbols: del_2, min_2, if1_4, le_2, if2_4, eq_2 Defined Pair Symbols: LE_2, EQ_2, IF1_4, IF2_4, MINSORT_1, MIN_2, DEL_2 Compound Symbols: c2_1, c6_1, c7_1, c8_1, c10_1, c12_1, c13_2, c15_2, c15_1, c17_2, c17_1, c13_3, c13_1 ---------------------------------------- (83) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MINSORT(cons(z0, nil)) -> c13(MINSORT(del(z0, cons(z0, nil))), DEL(min(z0, nil), cons(z0, nil))) by MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(x0, nil)) -> c13(DEL(min(x0, nil), cons(x0, nil))) ---------------------------------------- (84) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(z0, z2) if1(false, z0, z1, z2) -> min(z1, z2) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) 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) Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) IF1(true, z0, z1, z2) -> c7(MIN(z0, z2)) IF1(false, z0, z1, z2) -> c8(MIN(z1, z2)) IF2(false, z0, z1, z2) -> c10(DEL(z0, z2)) MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1)) MIN(s(z0), cons(s(z1), x2)) -> c15(IF1(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1))) MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2)) MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2)) DEL(s(z0), cons(s(z1), x2)) -> c17(IF2(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1))) DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2)) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z1, cons(x1, x2))) -> c13(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2))) MINSORT(cons(0, cons(z0, x2))) -> c13(MINSORT(del(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2))) MINSORT(cons(s(z0), cons(0, x2))) -> c13(MINSORT(del(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2))) MINSORT(cons(x0, cons(x1, x2))) -> c13(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(x0, nil)) -> c13(DEL(min(x0, nil), cons(x0, nil))) S tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) IF1(true, z0, z1, z2) -> c7(MIN(z0, z2)) IF1(false, z0, z1, z2) -> c8(MIN(z1, z2)) IF2(false, z0, z1, z2) -> c10(DEL(z0, z2)) MIN(s(z0), cons(s(z1), x2)) -> c15(IF1(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1))) MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2)) MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2)) DEL(s(z0), cons(s(z1), x2)) -> c17(IF2(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1))) DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2)) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z1, cons(x1, x2))) -> c13(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2))) MINSORT(cons(0, cons(z0, x2))) -> c13(MINSORT(del(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2))) MINSORT(cons(s(z0), cons(0, x2))) -> c13(MINSORT(del(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(x0, nil)) -> c13(DEL(min(x0, nil), cons(x0, nil))) K tuples: MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1)) MINSORT(cons(x0, cons(x1, x2))) -> c13(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) Defined Rule Symbols: del_2, min_2, if1_4, le_2, if2_4, eq_2 Defined Pair Symbols: LE_2, EQ_2, IF1_4, IF2_4, MINSORT_1, MIN_2, DEL_2 Compound Symbols: c2_1, c6_1, c7_1, c8_1, c10_1, c12_1, c15_2, c15_1, c17_2, c17_1, c13_3, c13_2, c13_1 ---------------------------------------- (85) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. MINSORT(cons(x0, nil)) -> c13(DEL(min(x0, nil), cons(x0, nil))) We considered the (Usable) Rules:none And the Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) IF1(true, z0, z1, z2) -> c7(MIN(z0, z2)) IF1(false, z0, z1, z2) -> c8(MIN(z1, z2)) IF2(false, z0, z1, z2) -> c10(DEL(z0, z2)) MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1)) MIN(s(z0), cons(s(z1), x2)) -> c15(IF1(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1))) MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2)) MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2)) DEL(s(z0), cons(s(z1), x2)) -> c17(IF2(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1))) DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2)) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z1, cons(x1, x2))) -> c13(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2))) MINSORT(cons(0, cons(z0, x2))) -> c13(MINSORT(del(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2))) MINSORT(cons(s(z0), cons(0, x2))) -> c13(MINSORT(del(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2))) MINSORT(cons(x0, cons(x1, x2))) -> c13(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(x0, nil)) -> c13(DEL(min(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(IF1(x_1, x_2, x_3, x_4)) = 0 POL(IF2(x_1, x_2, x_3, x_4)) = x_3 POL(LE(x_1, x_2)) = 0 POL(MIN(x_1, x_2)) = 0 POL(MINSORT(x_1)) = [1] POL(c10(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c13(x_1)) = x_1 POL(c13(x_1, x_2)) = x_1 + x_2 POL(c13(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c15(x_1)) = x_1 POL(c15(x_1, x_2)) = x_1 + x_2 POL(c17(x_1)) = x_1 POL(c17(x_1, x_2)) = x_1 + x_2 POL(c2(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(cons(x_1, x_2)) = 0 POL(del(x_1, x_2)) = x_2 POL(eq(x_1, x_2)) = 0 POL(false) = 0 POL(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(le(x_1, x_2)) = 0 POL(min(x_1, x_2)) = 0 POL(nil) = 0 POL(s(x_1)) = 0 POL(true) = 0 ---------------------------------------- (86) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(z0, z2) if1(false, z0, z1, z2) -> min(z1, z2) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) 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) Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) IF1(true, z0, z1, z2) -> c7(MIN(z0, z2)) IF1(false, z0, z1, z2) -> c8(MIN(z1, z2)) IF2(false, z0, z1, z2) -> c10(DEL(z0, z2)) MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1)) MIN(s(z0), cons(s(z1), x2)) -> c15(IF1(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1))) MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2)) MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2)) DEL(s(z0), cons(s(z1), x2)) -> c17(IF2(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1))) DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2)) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z1, cons(x1, x2))) -> c13(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2))) MINSORT(cons(0, cons(z0, x2))) -> c13(MINSORT(del(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2))) MINSORT(cons(s(z0), cons(0, x2))) -> c13(MINSORT(del(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2))) MINSORT(cons(x0, cons(x1, x2))) -> c13(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(x0, nil)) -> c13(DEL(min(x0, nil), cons(x0, nil))) S tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) IF1(true, z0, z1, z2) -> c7(MIN(z0, z2)) IF1(false, z0, z1, z2) -> c8(MIN(z1, z2)) IF2(false, z0, z1, z2) -> c10(DEL(z0, z2)) MIN(s(z0), cons(s(z1), x2)) -> c15(IF1(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1))) MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2)) MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2)) DEL(s(z0), cons(s(z1), x2)) -> c17(IF2(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1))) DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2)) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z1, cons(x1, x2))) -> c13(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2))) MINSORT(cons(0, cons(z0, x2))) -> c13(MINSORT(del(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2))) MINSORT(cons(s(z0), cons(0, x2))) -> c13(MINSORT(del(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil))) K tuples: MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1)) MINSORT(cons(x0, cons(x1, x2))) -> c13(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, nil)) -> c13(DEL(min(x0, nil), cons(x0, nil))) Defined Rule Symbols: del_2, min_2, if1_4, le_2, if2_4, eq_2 Defined Pair Symbols: LE_2, EQ_2, IF1_4, IF2_4, MINSORT_1, MIN_2, DEL_2 Compound Symbols: c2_1, c6_1, c7_1, c8_1, c10_1, c12_1, c15_2, c15_1, c17_2, c17_1, c13_3, c13_2, c13_1 ---------------------------------------- (87) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MIN(s(z0), cons(s(z1), x2)) -> c15(IF1(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1))) by MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2), LE(s(0), s(z0))) MIN(s(s(z0)), cons(s(0), x2)) -> c15(IF1(false, s(s(z0)), s(0), x2), LE(s(s(z0)), s(0))) MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1))) ---------------------------------------- (88) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(z0, z2) if1(false, z0, z1, z2) -> min(z1, z2) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) 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) Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) IF1(true, z0, z1, z2) -> c7(MIN(z0, z2)) IF1(false, z0, z1, z2) -> c8(MIN(z1, z2)) IF2(false, z0, z1, z2) -> c10(DEL(z0, z2)) MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1)) MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2)) MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2)) DEL(s(z0), cons(s(z1), x2)) -> c17(IF2(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1))) DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2)) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z1, cons(x1, x2))) -> c13(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2))) MINSORT(cons(0, cons(z0, x2))) -> c13(MINSORT(del(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2))) MINSORT(cons(s(z0), cons(0, x2))) -> c13(MINSORT(del(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2))) MINSORT(cons(x0, cons(x1, x2))) -> c13(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(x0, nil)) -> c13(DEL(min(x0, nil), cons(x0, nil))) MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2), LE(s(0), s(z0))) MIN(s(s(z0)), cons(s(0), x2)) -> c15(IF1(false, s(s(z0)), s(0), x2), LE(s(s(z0)), s(0))) MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1))) S tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) IF1(true, z0, z1, z2) -> c7(MIN(z0, z2)) IF1(false, z0, z1, z2) -> c8(MIN(z1, z2)) IF2(false, z0, z1, z2) -> c10(DEL(z0, z2)) MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2)) MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2)) DEL(s(z0), cons(s(z1), x2)) -> c17(IF2(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1))) DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2)) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z1, cons(x1, x2))) -> c13(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2))) MINSORT(cons(0, cons(z0, x2))) -> c13(MINSORT(del(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2))) MINSORT(cons(s(z0), cons(0, x2))) -> c13(MINSORT(del(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil))) MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2), LE(s(0), s(z0))) MIN(s(s(z0)), cons(s(0), x2)) -> c15(IF1(false, s(s(z0)), s(0), x2), LE(s(s(z0)), s(0))) MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1))) K tuples: MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1)) MINSORT(cons(x0, cons(x1, x2))) -> c13(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, nil)) -> c13(DEL(min(x0, nil), cons(x0, nil))) Defined Rule Symbols: del_2, min_2, if1_4, le_2, if2_4, eq_2 Defined Pair Symbols: LE_2, EQ_2, IF1_4, IF2_4, MINSORT_1, MIN_2, DEL_2 Compound Symbols: c2_1, c6_1, c7_1, c8_1, c10_1, c12_1, c15_1, c17_2, c17_1, c13_3, c13_2, c13_1, c15_2 ---------------------------------------- (89) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace DEL(s(z0), cons(s(z1), x2)) -> c17(IF2(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1))) by DEL(s(0), cons(s(0), x2)) -> c17(IF2(true, s(0), s(0), x2), EQ(s(0), s(0))) DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(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)) -> c17(EQ(s(x0), s(x1))) ---------------------------------------- (90) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(z0, z2) if1(false, z0, z1, z2) -> min(z1, z2) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) 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) Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) IF1(true, z0, z1, z2) -> c7(MIN(z0, z2)) IF1(false, z0, z1, z2) -> c8(MIN(z1, z2)) IF2(false, z0, z1, z2) -> c10(DEL(z0, z2)) MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1)) MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2)) MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2)) DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2)) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z1, cons(x1, x2))) -> c13(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2))) MINSORT(cons(0, cons(z0, x2))) -> c13(MINSORT(del(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2))) MINSORT(cons(s(z0), cons(0, x2))) -> c13(MINSORT(del(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2))) MINSORT(cons(x0, cons(x1, x2))) -> c13(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(x0, nil)) -> c13(DEL(min(x0, nil), cons(x0, nil))) MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2), LE(s(0), s(z0))) MIN(s(s(z0)), cons(s(0), x2)) -> c15(IF1(false, s(s(z0)), s(0), x2), LE(s(s(z0)), s(0))) MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c17(IF2(true, s(0), s(0), x2), EQ(s(0), s(0))) DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(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)) -> c17(EQ(s(x0), s(x1))) S tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) IF1(true, z0, z1, z2) -> c7(MIN(z0, z2)) IF1(false, z0, z1, z2) -> c8(MIN(z1, z2)) IF2(false, z0, z1, z2) -> c10(DEL(z0, z2)) MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2)) MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2)) DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2)) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z1, cons(x1, x2))) -> c13(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2))) MINSORT(cons(0, cons(z0, x2))) -> c13(MINSORT(del(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2))) MINSORT(cons(s(z0), cons(0, x2))) -> c13(MINSORT(del(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil))) MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2), LE(s(0), s(z0))) MIN(s(s(z0)), cons(s(0), x2)) -> c15(IF1(false, s(s(z0)), s(0), x2), LE(s(s(z0)), s(0))) MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c17(IF2(true, s(0), s(0), x2), EQ(s(0), s(0))) DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(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)) -> c17(EQ(s(x0), s(x1))) K tuples: MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1)) MINSORT(cons(x0, cons(x1, x2))) -> c13(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, nil)) -> c13(DEL(min(x0, nil), cons(x0, nil))) Defined Rule Symbols: del_2, min_2, if1_4, le_2, if2_4, eq_2 Defined Pair Symbols: LE_2, EQ_2, IF1_4, IF2_4, MINSORT_1, MIN_2, DEL_2 Compound Symbols: c2_1, c6_1, c7_1, c8_1, c10_1, c12_1, c15_1, c17_1, c13_3, c13_2, c13_1, c15_2, c17_2 ---------------------------------------- (91) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (92) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(z0, z2) if1(false, z0, z1, z2) -> min(z1, z2) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) 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) Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) IF1(true, z0, z1, z2) -> c7(MIN(z0, z2)) IF1(false, z0, z1, z2) -> c8(MIN(z1, z2)) IF2(false, z0, z1, z2) -> c10(DEL(z0, z2)) MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1)) MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2)) MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2)) DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2)) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z1, cons(x1, x2))) -> c13(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2))) MINSORT(cons(0, cons(z0, x2))) -> c13(MINSORT(del(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2))) MINSORT(cons(s(z0), cons(0, x2))) -> c13(MINSORT(del(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2))) MINSORT(cons(x0, cons(x1, x2))) -> c13(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(x0, nil)) -> c13(DEL(min(x0, nil), cons(x0, nil))) MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2), LE(s(0), s(z0))) MIN(s(s(z0)), cons(s(0), x2)) -> c15(IF1(false, s(s(z0)), s(0), x2), LE(s(s(z0)), s(0))) MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(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)) -> c17(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c17(EQ(s(0), s(0))) S tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) IF1(true, z0, z1, z2) -> c7(MIN(z0, z2)) IF1(false, z0, z1, z2) -> c8(MIN(z1, z2)) IF2(false, z0, z1, z2) -> c10(DEL(z0, z2)) MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2)) MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2)) DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2)) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z1, cons(x1, x2))) -> c13(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2))) MINSORT(cons(0, cons(z0, x2))) -> c13(MINSORT(del(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2))) MINSORT(cons(s(z0), cons(0, x2))) -> c13(MINSORT(del(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil))) MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2), LE(s(0), s(z0))) MIN(s(s(z0)), cons(s(0), x2)) -> c15(IF1(false, s(s(z0)), s(0), x2), LE(s(s(z0)), s(0))) MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(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)) -> c17(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c17(EQ(s(0), s(0))) K tuples: MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1)) MINSORT(cons(x0, cons(x1, x2))) -> c13(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, nil)) -> c13(DEL(min(x0, nil), cons(x0, nil))) Defined Rule Symbols: del_2, min_2, if1_4, le_2, if2_4, eq_2 Defined Pair Symbols: LE_2, EQ_2, IF1_4, IF2_4, MINSORT_1, MIN_2, DEL_2 Compound Symbols: c2_1, c6_1, c7_1, c8_1, c10_1, c12_1, c15_1, c17_1, c13_3, c13_2, c13_1, c15_2, c17_2 ---------------------------------------- (93) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IF1(true, z0, z1, z2) -> c7(MIN(z0, z2)) by IF1(true, 0, x0, x1) -> c7(MIN(0, x1)) IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2)) ---------------------------------------- (94) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(z0, z2) if1(false, z0, z1, z2) -> min(z1, z2) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) 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) Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) IF1(false, z0, z1, z2) -> c8(MIN(z1, z2)) IF2(false, z0, z1, z2) -> c10(DEL(z0, z2)) MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1)) MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2)) MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2)) DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2)) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z1, cons(x1, x2))) -> c13(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2))) MINSORT(cons(0, cons(z0, x2))) -> c13(MINSORT(del(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2))) MINSORT(cons(s(z0), cons(0, x2))) -> c13(MINSORT(del(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2))) MINSORT(cons(x0, cons(x1, x2))) -> c13(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(x0, nil)) -> c13(DEL(min(x0, nil), cons(x0, nil))) MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2), LE(s(0), s(z0))) MIN(s(s(z0)), cons(s(0), x2)) -> c15(IF1(false, s(s(z0)), s(0), x2), LE(s(s(z0)), s(0))) MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(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)) -> c17(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c17(EQ(s(0), s(0))) IF1(true, 0, x0, x1) -> c7(MIN(0, x1)) IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2)) S tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) IF1(false, z0, z1, z2) -> c8(MIN(z1, z2)) IF2(false, z0, z1, z2) -> c10(DEL(z0, z2)) MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2)) MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2)) DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2)) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z1, cons(x1, x2))) -> c13(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2))) MINSORT(cons(0, cons(z0, x2))) -> c13(MINSORT(del(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2))) MINSORT(cons(s(z0), cons(0, x2))) -> c13(MINSORT(del(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil))) MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2), LE(s(0), s(z0))) MIN(s(s(z0)), cons(s(0), x2)) -> c15(IF1(false, s(s(z0)), s(0), x2), LE(s(s(z0)), s(0))) MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(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)) -> c17(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c17(EQ(s(0), s(0))) IF1(true, 0, x0, x1) -> c7(MIN(0, x1)) IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2)) K tuples: MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1)) MINSORT(cons(x0, cons(x1, x2))) -> c13(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, nil)) -> c13(DEL(min(x0, nil), cons(x0, nil))) Defined Rule Symbols: del_2, min_2, if1_4, le_2, if2_4, eq_2 Defined Pair Symbols: LE_2, EQ_2, IF1_4, IF2_4, MINSORT_1, MIN_2, DEL_2 Compound Symbols: c2_1, c6_1, c8_1, c10_1, c12_1, c15_1, c17_1, c13_3, c13_2, c13_1, c15_2, c17_2, c7_1 ---------------------------------------- (95) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) by MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) ---------------------------------------- (96) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(z0, z2) if1(false, z0, z1, z2) -> min(z1, z2) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) 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) Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) IF1(false, z0, z1, z2) -> c8(MIN(z1, z2)) IF2(false, z0, z1, z2) -> c10(DEL(z0, z2)) MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1)) MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2)) MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2)) DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2)) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z1, cons(x1, x2))) -> c13(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2))) MINSORT(cons(0, cons(z0, x2))) -> c13(MINSORT(del(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2))) MINSORT(cons(s(z0), cons(0, x2))) -> c13(MINSORT(del(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2))) MINSORT(cons(x0, cons(x1, x2))) -> c13(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(x0, nil)) -> c13(DEL(min(x0, nil), cons(x0, nil))) MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2), LE(s(0), s(z0))) MIN(s(s(z0)), cons(s(0), x2)) -> c15(IF1(false, s(s(z0)), s(0), x2), LE(s(s(z0)), s(0))) MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(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)) -> c17(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c17(EQ(s(0), s(0))) IF1(true, 0, x0, x1) -> c7(MIN(0, x1)) IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2)) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) S tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) IF1(false, z0, z1, z2) -> c8(MIN(z1, z2)) IF2(false, z0, z1, z2) -> c10(DEL(z0, z2)) MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2)) MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2)) DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2)) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z1, cons(x1, x2))) -> c13(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2))) MINSORT(cons(0, cons(z0, x2))) -> c13(MINSORT(del(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2))) MINSORT(cons(s(z0), cons(0, x2))) -> c13(MINSORT(del(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil))) MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2), LE(s(0), s(z0))) MIN(s(s(z0)), cons(s(0), x2)) -> c15(IF1(false, s(s(z0)), s(0), x2), LE(s(s(z0)), s(0))) MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(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)) -> c17(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c17(EQ(s(0), s(0))) IF1(true, 0, x0, x1) -> c7(MIN(0, x1)) IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2)) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) K tuples: MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1)) MINSORT(cons(x0, cons(x1, x2))) -> c13(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, nil)) -> c13(DEL(min(x0, nil), cons(x0, nil))) Defined Rule Symbols: del_2, min_2, if1_4, le_2, if2_4, eq_2 Defined Pair Symbols: LE_2, EQ_2, IF1_4, IF2_4, MINSORT_1, MIN_2, DEL_2 Compound Symbols: c2_1, c6_1, c8_1, c10_1, c12_1, c15_1, c17_1, c13_3, c13_2, c13_1, c15_2, c17_2, c7_1 ---------------------------------------- (97) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) by MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) ---------------------------------------- (98) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(z0, z2) if1(false, z0, z1, z2) -> min(z1, z2) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) 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) Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) IF1(false, z0, z1, z2) -> c8(MIN(z1, z2)) IF2(false, z0, z1, z2) -> c10(DEL(z0, z2)) MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1)) MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2)) MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2)) DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2)) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z1, cons(x1, x2))) -> c13(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2))) MINSORT(cons(0, cons(z0, x2))) -> c13(MINSORT(del(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2))) MINSORT(cons(s(z0), cons(0, x2))) -> c13(MINSORT(del(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2))) MINSORT(cons(x0, cons(x1, x2))) -> c13(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(x0, nil)) -> c13(DEL(min(x0, nil), cons(x0, nil))) MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2), LE(s(0), s(z0))) MIN(s(s(z0)), cons(s(0), x2)) -> c15(IF1(false, s(s(z0)), s(0), x2), LE(s(s(z0)), s(0))) MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(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)) -> c17(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c17(EQ(s(0), s(0))) IF1(true, 0, x0, x1) -> c7(MIN(0, x1)) IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2)) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) S tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) IF1(false, z0, z1, z2) -> c8(MIN(z1, z2)) IF2(false, z0, z1, z2) -> c10(DEL(z0, z2)) MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2)) MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2)) DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2)) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z1, cons(x1, x2))) -> c13(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2))) MINSORT(cons(0, cons(z0, x2))) -> c13(MINSORT(del(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2))) MINSORT(cons(s(z0), cons(0, x2))) -> c13(MINSORT(del(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil))) MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2), LE(s(0), s(z0))) MIN(s(s(z0)), cons(s(0), x2)) -> c15(IF1(false, s(s(z0)), s(0), x2), LE(s(s(z0)), s(0))) MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(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)) -> c17(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c17(EQ(s(0), s(0))) IF1(true, 0, x0, x1) -> c7(MIN(0, x1)) IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2)) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) K tuples: MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1)) MINSORT(cons(x0, cons(x1, x2))) -> c13(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, nil)) -> c13(DEL(min(x0, nil), cons(x0, nil))) Defined Rule Symbols: del_2, min_2, if1_4, le_2, if2_4, eq_2 Defined Pair Symbols: LE_2, EQ_2, IF1_4, IF2_4, MINSORT_1, MIN_2, DEL_2 Compound Symbols: c2_1, c6_1, c8_1, c10_1, c12_1, c15_1, c17_1, c13_2, c13_3, c13_1, c15_2, c17_2, c7_1 ---------------------------------------- (99) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil))) by MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) ---------------------------------------- (100) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(z0, z2) if1(false, z0, z1, z2) -> min(z1, z2) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) 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) Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) IF1(false, z0, z1, z2) -> c8(MIN(z1, z2)) IF2(false, z0, z1, z2) -> c10(DEL(z0, z2)) MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1)) MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2)) MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2)) DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2)) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z1, cons(x1, x2))) -> c13(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2))) MINSORT(cons(0, cons(z0, x2))) -> c13(MINSORT(del(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2))) MINSORT(cons(s(z0), cons(0, x2))) -> c13(MINSORT(del(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2))) MINSORT(cons(x0, cons(x1, x2))) -> c13(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(x0, nil)) -> c13(DEL(min(x0, nil), cons(x0, nil))) MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2), LE(s(0), s(z0))) MIN(s(s(z0)), cons(s(0), x2)) -> c15(IF1(false, s(s(z0)), s(0), x2), LE(s(s(z0)), s(0))) MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(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)) -> c17(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c17(EQ(s(0), s(0))) IF1(true, 0, x0, x1) -> c7(MIN(0, x1)) IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2)) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) S tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) IF1(false, z0, z1, z2) -> c8(MIN(z1, z2)) IF2(false, z0, z1, z2) -> c10(DEL(z0, z2)) MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2)) MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2)) DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2)) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z1, cons(x1, x2))) -> c13(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2))) MINSORT(cons(0, cons(z0, x2))) -> c13(MINSORT(del(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2))) MINSORT(cons(s(z0), cons(0, x2))) -> c13(MINSORT(del(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil))) MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2), LE(s(0), s(z0))) MIN(s(s(z0)), cons(s(0), x2)) -> c15(IF1(false, s(s(z0)), s(0), x2), LE(s(s(z0)), s(0))) MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(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)) -> c17(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c17(EQ(s(0), s(0))) IF1(true, 0, x0, x1) -> c7(MIN(0, x1)) IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2)) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) K tuples: MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1)) MINSORT(cons(x0, cons(x1, x2))) -> c13(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, nil)) -> c13(DEL(min(x0, nil), cons(x0, nil))) Defined Rule Symbols: del_2, min_2, if1_4, le_2, if2_4, eq_2 Defined Pair Symbols: LE_2, EQ_2, IF1_4, IF2_4, MINSORT_1, MIN_2, DEL_2 Compound Symbols: c2_1, c6_1, c8_1, c10_1, c12_1, c15_1, c17_1, c13_2, c13_3, c13_1, c15_2, c17_2, c7_1 ---------------------------------------- (101) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IF1(false, z0, z1, z2) -> c8(MIN(z1, z2)) by IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1)) IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2)) ---------------------------------------- (102) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(z0, z2) if1(false, z0, z1, z2) -> min(z1, z2) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) 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) Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) IF2(false, z0, z1, z2) -> c10(DEL(z0, z2)) MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1)) MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2)) MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2)) DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2)) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z1, cons(x1, x2))) -> c13(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2))) MINSORT(cons(0, cons(z0, x2))) -> c13(MINSORT(del(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2))) MINSORT(cons(s(z0), cons(0, x2))) -> c13(MINSORT(del(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2))) MINSORT(cons(x0, cons(x1, x2))) -> c13(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(x0, nil)) -> c13(DEL(min(x0, nil), cons(x0, nil))) MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2), LE(s(0), s(z0))) MIN(s(s(z0)), cons(s(0), x2)) -> c15(IF1(false, s(s(z0)), s(0), x2), LE(s(s(z0)), s(0))) MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(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)) -> c17(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c17(EQ(s(0), s(0))) IF1(true, 0, x0, x1) -> c7(MIN(0, x1)) IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2)) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1)) IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2)) S tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) IF2(false, z0, z1, z2) -> c10(DEL(z0, z2)) MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2)) MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2)) DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2)) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z1, cons(x1, x2))) -> c13(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2))) MINSORT(cons(0, cons(z0, x2))) -> c13(MINSORT(del(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2))) MINSORT(cons(s(z0), cons(0, x2))) -> c13(MINSORT(del(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil))) MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2), LE(s(0), s(z0))) MIN(s(s(z0)), cons(s(0), x2)) -> c15(IF1(false, s(s(z0)), s(0), x2), LE(s(s(z0)), s(0))) MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(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)) -> c17(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c17(EQ(s(0), s(0))) IF1(true, 0, x0, x1) -> c7(MIN(0, x1)) IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2)) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1)) IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2)) K tuples: MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1)) MINSORT(cons(x0, cons(x1, x2))) -> c13(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, nil)) -> c13(DEL(min(x0, nil), cons(x0, nil))) Defined Rule Symbols: del_2, min_2, if1_4, le_2, if2_4, eq_2 Defined Pair Symbols: LE_2, EQ_2, IF2_4, MINSORT_1, MIN_2, DEL_2, IF1_4 Compound Symbols: c2_1, c6_1, c10_1, c12_1, c15_1, c17_1, c13_2, c13_3, c13_1, c15_2, c17_2, c7_1, c8_1 ---------------------------------------- (103) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (104) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(z0, z2) if1(false, z0, z1, z2) -> min(z1, z2) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) 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) Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) IF2(false, z0, z1, z2) -> c10(DEL(z0, z2)) MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1)) MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2)) MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2)) DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2)) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z1, cons(x1, x2))) -> c13(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2))) MINSORT(cons(0, cons(z0, x2))) -> c13(MINSORT(del(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2))) MINSORT(cons(s(z0), cons(0, x2))) -> c13(MINSORT(del(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2))) MINSORT(cons(x0, cons(x1, x2))) -> c13(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(x0, nil)) -> c13(DEL(min(x0, nil), cons(x0, nil))) MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2), LE(s(0), s(z0))) MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(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)) -> c17(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c17(EQ(s(0), s(0))) IF1(true, 0, x0, x1) -> c7(MIN(0, x1)) IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2)) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1)) IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2)) MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2)) MIN(s(s(z0)), cons(s(0), x2)) -> c(LE(s(s(z0)), s(0))) S tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) IF2(false, z0, z1, z2) -> c10(DEL(z0, z2)) MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2)) MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2)) DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2)) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z1, cons(x1, x2))) -> c13(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2))) MINSORT(cons(0, cons(z0, x2))) -> c13(MINSORT(del(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2))) MINSORT(cons(s(z0), cons(0, x2))) -> c13(MINSORT(del(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil))) MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2), LE(s(0), s(z0))) MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(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)) -> c17(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c17(EQ(s(0), s(0))) IF1(true, 0, x0, x1) -> c7(MIN(0, x1)) IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2)) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1)) IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2)) MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2)) MIN(s(s(z0)), cons(s(0), x2)) -> c(LE(s(s(z0)), s(0))) K tuples: MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1)) MINSORT(cons(x0, cons(x1, x2))) -> c13(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, nil)) -> c13(DEL(min(x0, nil), cons(x0, nil))) Defined Rule Symbols: del_2, min_2, if1_4, le_2, if2_4, eq_2 Defined Pair Symbols: LE_2, EQ_2, IF2_4, MINSORT_1, MIN_2, DEL_2, IF1_4 Compound Symbols: c2_1, c6_1, c10_1, c12_1, c15_1, c17_1, c13_2, c13_3, c13_1, c15_2, c17_2, c7_1, c8_1, c_1 ---------------------------------------- (105) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(min(z0, nil), cons(z0, nil))) by MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(z0, cons(z0, nil))) ---------------------------------------- (106) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(z0, z2) if1(false, z0, z1, z2) -> min(z1, z2) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) 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) Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) IF2(false, z0, z1, z2) -> c10(DEL(z0, z2)) MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1)) MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2)) MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2)) DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2)) MINSORT(cons(z1, cons(x1, x2))) -> c13(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2))) MINSORT(cons(0, cons(z0, x2))) -> c13(MINSORT(del(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2))) MINSORT(cons(s(z0), cons(0, x2))) -> c13(MINSORT(del(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2))) MINSORT(cons(x0, cons(x1, x2))) -> c13(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(x0, nil)) -> c13(DEL(min(x0, nil), cons(x0, nil))) MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2), LE(s(0), s(z0))) MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(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)) -> c17(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c17(EQ(s(0), s(0))) IF1(true, 0, x0, x1) -> c7(MIN(0, x1)) IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2)) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1)) IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2)) MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2)) MIN(s(s(z0)), cons(s(0), x2)) -> c(LE(s(s(z0)), s(0))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(z0, cons(z0, nil))) S tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) IF2(false, z0, z1, z2) -> c10(DEL(z0, z2)) MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2)) MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2)) DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2)) MINSORT(cons(z1, cons(x1, x2))) -> c13(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2))) MINSORT(cons(0, cons(z0, x2))) -> c13(MINSORT(del(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2))) MINSORT(cons(s(z0), cons(0, x2))) -> c13(MINSORT(del(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil))) MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2), LE(s(0), s(z0))) MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(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)) -> c17(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c17(EQ(s(0), s(0))) IF1(true, 0, x0, x1) -> c7(MIN(0, x1)) IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2)) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1)) IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2)) MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2)) MIN(s(s(z0)), cons(s(0), x2)) -> c(LE(s(s(z0)), s(0))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(z0, cons(z0, nil))) K tuples: MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1)) MINSORT(cons(x0, cons(x1, x2))) -> c13(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, nil)) -> c13(DEL(min(x0, nil), cons(x0, nil))) Defined Rule Symbols: del_2, min_2, if1_4, le_2, if2_4, eq_2 Defined Pair Symbols: LE_2, EQ_2, IF2_4, MINSORT_1, MIN_2, DEL_2, IF1_4 Compound Symbols: c2_1, c6_1, c10_1, c12_1, c15_1, c17_1, c13_3, c13_1, c13_2, c15_2, c17_2, c7_1, c8_1, c_1 ---------------------------------------- (107) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace MINSORT(cons(z1, cons(x1, x2))) -> c13(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2))) by MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) ---------------------------------------- (108) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(z0, z2) if1(false, z0, z1, z2) -> min(z1, z2) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) 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) Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) IF2(false, z0, z1, z2) -> c10(DEL(z0, z2)) MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1)) MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2)) MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2)) DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2)) MINSORT(cons(0, cons(z0, x2))) -> c13(MINSORT(del(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2))) MINSORT(cons(s(z0), cons(0, x2))) -> c13(MINSORT(del(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2))) MINSORT(cons(x0, cons(x1, x2))) -> c13(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(x0, nil)) -> c13(DEL(min(x0, nil), cons(x0, nil))) MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2), LE(s(0), s(z0))) MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(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)) -> c17(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c17(EQ(s(0), s(0))) IF1(true, 0, x0, x1) -> c7(MIN(0, x1)) IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2)) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1)) IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2)) MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2)) MIN(s(s(z0)), cons(s(0), x2)) -> c(LE(s(s(z0)), s(0))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) S tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) IF2(false, z0, z1, z2) -> c10(DEL(z0, z2)) MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2)) MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2)) DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2)) MINSORT(cons(0, cons(z0, x2))) -> c13(MINSORT(del(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2))) MINSORT(cons(s(z0), cons(0, x2))) -> c13(MINSORT(del(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil))) MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2), LE(s(0), s(z0))) MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(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)) -> c17(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c17(EQ(s(0), s(0))) IF1(true, 0, x0, x1) -> c7(MIN(0, x1)) IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2)) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1)) IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2)) MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2)) MIN(s(s(z0)), cons(s(0), x2)) -> c(LE(s(s(z0)), s(0))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) K tuples: MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1)) MINSORT(cons(x0, cons(x1, x2))) -> c13(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, nil)) -> c13(DEL(min(x0, nil), cons(x0, nil))) Defined Rule Symbols: del_2, min_2, if1_4, le_2, if2_4, eq_2 Defined Pair Symbols: LE_2, EQ_2, IF2_4, MINSORT_1, MIN_2, DEL_2, IF1_4 Compound Symbols: c2_1, c6_1, c10_1, c12_1, c15_1, c17_1, c13_3, c13_1, c13_2, c15_2, c17_2, c7_1, c8_1, c_1 ---------------------------------------- (109) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace MINSORT(cons(0, cons(z0, x2))) -> c13(MINSORT(del(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2))) by MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) ---------------------------------------- (110) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(z0, z2) if1(false, z0, z1, z2) -> min(z1, z2) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) 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) Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) IF2(false, z0, z1, z2) -> c10(DEL(z0, z2)) MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1)) MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2)) MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2)) DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2)) MINSORT(cons(s(z0), cons(0, x2))) -> c13(MINSORT(del(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2))) MINSORT(cons(x0, cons(x1, x2))) -> c13(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(x0, nil)) -> c13(DEL(min(x0, nil), cons(x0, nil))) MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2), LE(s(0), s(z0))) MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(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)) -> c17(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c17(EQ(s(0), s(0))) IF1(true, 0, x0, x1) -> c7(MIN(0, x1)) IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2)) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1)) IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2)) MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2)) MIN(s(s(z0)), cons(s(0), x2)) -> c(LE(s(s(z0)), s(0))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) S tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) IF2(false, z0, z1, z2) -> c10(DEL(z0, z2)) MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2)) MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2)) DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2)) MINSORT(cons(s(z0), cons(0, x2))) -> c13(MINSORT(del(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil))) MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2), LE(s(0), s(z0))) MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(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)) -> c17(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c17(EQ(s(0), s(0))) IF1(true, 0, x0, x1) -> c7(MIN(0, x1)) IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2)) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1)) IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2)) MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2)) MIN(s(s(z0)), cons(s(0), x2)) -> c(LE(s(s(z0)), s(0))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) K tuples: MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1)) MINSORT(cons(x0, cons(x1, x2))) -> c13(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, nil)) -> c13(DEL(min(x0, nil), cons(x0, nil))) Defined Rule Symbols: del_2, min_2, if1_4, le_2, if2_4, eq_2 Defined Pair Symbols: LE_2, EQ_2, IF2_4, MINSORT_1, MIN_2, DEL_2, IF1_4 Compound Symbols: c2_1, c6_1, c10_1, c12_1, c15_1, c17_1, c13_3, c13_1, c13_2, c15_2, c17_2, c7_1, c8_1, c_1 ---------------------------------------- (111) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace MINSORT(cons(s(z0), cons(0, x2))) -> c13(MINSORT(del(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2))) by MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) ---------------------------------------- (112) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(z0, z2) if1(false, z0, z1, z2) -> min(z1, z2) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) 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) Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) IF2(false, z0, z1, z2) -> c10(DEL(z0, z2)) MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1)) MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2)) MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2)) DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2)) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2))) MINSORT(cons(x0, cons(x1, x2))) -> c13(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(x0, nil)) -> c13(DEL(min(x0, nil), cons(x0, nil))) MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2), LE(s(0), s(z0))) MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(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)) -> c17(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c17(EQ(s(0), s(0))) IF1(true, 0, x0, x1) -> c7(MIN(0, x1)) IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2)) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1)) IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2)) MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2)) MIN(s(s(z0)), cons(s(0), x2)) -> c(LE(s(s(z0)), s(0))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) S tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) IF2(false, z0, z1, z2) -> c10(DEL(z0, z2)) MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2)) MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2)) DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2)) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil))) MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2), LE(s(0), s(z0))) MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(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)) -> c17(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c17(EQ(s(0), s(0))) IF1(true, 0, x0, x1) -> c7(MIN(0, x1)) IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2)) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1)) IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2)) MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2)) MIN(s(s(z0)), cons(s(0), x2)) -> c(LE(s(s(z0)), s(0))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) K tuples: MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1)) MINSORT(cons(x0, cons(x1, x2))) -> c13(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, nil)) -> c13(DEL(min(x0, nil), cons(x0, nil))) Defined Rule Symbols: del_2, min_2, if1_4, le_2, if2_4, eq_2 Defined Pair Symbols: LE_2, EQ_2, IF2_4, MINSORT_1, MIN_2, DEL_2, IF1_4 Compound Symbols: c2_1, c6_1, c10_1, c12_1, c15_1, c17_1, c13_3, c13_1, c13_2, c15_2, c17_2, c7_1, c8_1, c_1 ---------------------------------------- (113) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace MINSORT(cons(s(z0), cons(s(z1), x2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2))) by MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) ---------------------------------------- (114) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(z0, z2) if1(false, z0, z1, z2) -> min(z1, z2) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) 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) Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) IF2(false, z0, z1, z2) -> c10(DEL(z0, z2)) MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1)) MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2)) MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2)) DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2)) MINSORT(cons(x0, cons(x1, x2))) -> c13(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(x0, nil)) -> c13(DEL(min(x0, nil), cons(x0, nil))) MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2), LE(s(0), s(z0))) MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(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)) -> c17(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c17(EQ(s(0), s(0))) IF1(true, 0, x0, x1) -> c7(MIN(0, x1)) IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2)) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1)) IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2)) MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2)) MIN(s(s(z0)), cons(s(0), x2)) -> c(LE(s(s(z0)), s(0))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) S tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) IF2(false, z0, z1, z2) -> c10(DEL(z0, z2)) MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2)) MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2)) DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2)) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil))) MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2), LE(s(0), s(z0))) MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(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)) -> c17(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c17(EQ(s(0), s(0))) IF1(true, 0, x0, x1) -> c7(MIN(0, x1)) IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2)) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1)) IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2)) MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2)) MIN(s(s(z0)), cons(s(0), x2)) -> c(LE(s(s(z0)), s(0))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) K tuples: MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1)) MINSORT(cons(x0, cons(x1, x2))) -> c13(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, nil)) -> c13(DEL(min(x0, nil), cons(x0, nil))) Defined Rule Symbols: del_2, min_2, if1_4, le_2, if2_4, eq_2 Defined Pair Symbols: LE_2, EQ_2, IF2_4, MINSORT_1, MIN_2, DEL_2, IF1_4 Compound Symbols: c2_1, c6_1, c10_1, c12_1, c15_1, c17_1, c13_1, c13_2, c15_2, c17_2, c7_1, c13_3, c8_1, c_1 ---------------------------------------- (115) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace MINSORT(cons(x0, cons(x1, x2))) -> c13(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) by MINSORT(cons(z0, cons(z1, z2))) -> c13(DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))) ---------------------------------------- (116) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(z0, z2) if1(false, z0, z1, z2) -> min(z1, z2) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) 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) Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) IF2(false, z0, z1, z2) -> c10(DEL(z0, z2)) MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1)) MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2)) MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2)) DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2)) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(x0, nil)) -> c13(DEL(min(x0, nil), cons(x0, nil))) MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2), LE(s(0), s(z0))) MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(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)) -> c17(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c17(EQ(s(0), s(0))) IF1(true, 0, x0, x1) -> c7(MIN(0, x1)) IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2)) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1)) IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2)) MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2)) MIN(s(s(z0)), cons(s(0), x2)) -> c(LE(s(s(z0)), s(0))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) MINSORT(cons(z0, cons(z1, z2))) -> c13(DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))) S tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) IF2(false, z0, z1, z2) -> c10(DEL(z0, z2)) MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2)) MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2)) DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2)) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil))) MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2), LE(s(0), s(z0))) MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(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)) -> c17(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c17(EQ(s(0), s(0))) IF1(true, 0, x0, x1) -> c7(MIN(0, x1)) IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2)) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1)) IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2)) MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2)) MIN(s(s(z0)), cons(s(0), x2)) -> c(LE(s(s(z0)), s(0))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) K tuples: MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1)) MINSORT(cons(x0, cons(x1, x2))) -> c13(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, nil)) -> c13(DEL(min(x0, nil), cons(x0, nil))) Defined Rule Symbols: del_2, min_2, if1_4, le_2, if2_4, eq_2 Defined Pair Symbols: LE_2, EQ_2, IF2_4, MINSORT_1, MIN_2, DEL_2, IF1_4 Compound Symbols: c2_1, c6_1, c10_1, c12_1, c15_1, c17_1, c13_2, c13_1, c15_2, c17_2, c7_1, c13_3, c8_1, c_1 ---------------------------------------- (117) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(min(z0, nil), cons(z0, nil))) by MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) ---------------------------------------- (118) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(z0, z2) if1(false, z0, z1, z2) -> min(z1, z2) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) 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) Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) IF2(false, z0, z1, z2) -> c10(DEL(z0, z2)) MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1)) MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2)) MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2)) DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2)) MINSORT(cons(x0, nil)) -> c13(DEL(min(x0, nil), cons(x0, nil))) MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2), LE(s(0), s(z0))) MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(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)) -> c17(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c17(EQ(s(0), s(0))) IF1(true, 0, x0, x1) -> c7(MIN(0, x1)) IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2)) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1)) IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2)) MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2)) MIN(s(s(z0)), cons(s(0), x2)) -> c(LE(s(s(z0)), s(0))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) MINSORT(cons(z0, cons(z1, z2))) -> c13(DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) S tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) IF2(false, z0, z1, z2) -> c10(DEL(z0, z2)) MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2)) MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2)) DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2)) MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2), LE(s(0), s(z0))) MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(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)) -> c17(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c17(EQ(s(0), s(0))) IF1(true, 0, x0, x1) -> c7(MIN(0, x1)) IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2)) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1)) IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2)) MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2)) MIN(s(s(z0)), cons(s(0), x2)) -> c(LE(s(s(z0)), s(0))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) K tuples: MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1)) MINSORT(cons(x0, cons(x1, x2))) -> c13(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, nil)) -> c13(DEL(min(x0, nil), cons(x0, nil))) Defined Rule Symbols: del_2, min_2, if1_4, le_2, if2_4, eq_2 Defined Pair Symbols: LE_2, EQ_2, IF2_4, MINSORT_1, MIN_2, DEL_2, IF1_4 Compound Symbols: c2_1, c6_1, c10_1, c12_1, c15_1, c17_1, c13_1, c15_2, c17_2, c7_1, c13_3, c13_2, c8_1, c_1 ---------------------------------------- (119) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace MINSORT(cons(x0, nil)) -> c13(DEL(min(x0, nil), cons(x0, nil))) by MINSORT(cons(z0, nil)) -> c13(DEL(z0, cons(z0, nil))) ---------------------------------------- (120) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(z0, z2) if1(false, z0, z1, z2) -> min(z1, z2) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) 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) Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) IF2(false, z0, z1, z2) -> c10(DEL(z0, z2)) MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1)) MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2)) MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2)) DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2)) MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2), LE(s(0), s(z0))) MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(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)) -> c17(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c17(EQ(s(0), s(0))) IF1(true, 0, x0, x1) -> c7(MIN(0, x1)) IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2)) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1)) IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2)) MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2)) MIN(s(s(z0)), cons(s(0), x2)) -> c(LE(s(s(z0)), s(0))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) MINSORT(cons(z0, cons(z1, z2))) -> c13(DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c13(DEL(z0, cons(z0, nil))) S tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) IF2(false, z0, z1, z2) -> c10(DEL(z0, z2)) MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2)) MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2)) DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2)) MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2), LE(s(0), s(z0))) MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(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)) -> c17(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c17(EQ(s(0), s(0))) IF1(true, 0, x0, x1) -> c7(MIN(0, x1)) IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2)) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1)) IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2)) MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2)) MIN(s(s(z0)), cons(s(0), x2)) -> c(LE(s(s(z0)), s(0))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) K tuples: MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1)) MINSORT(cons(x0, cons(x1, x2))) -> c13(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, nil)) -> c13(DEL(min(x0, nil), cons(x0, nil))) Defined Rule Symbols: del_2, min_2, if1_4, le_2, if2_4, eq_2 Defined Pair Symbols: LE_2, EQ_2, IF2_4, MINSORT_1, MIN_2, DEL_2, IF1_4 Compound Symbols: c2_1, c6_1, c10_1, c12_1, c15_1, c17_1, c15_2, c17_2, c7_1, c13_3, c13_2, c8_1, c_1, c13_1 ---------------------------------------- (121) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IF2(false, z0, z1, z2) -> c10(DEL(z0, z2)) by IF2(false, 0, s(x0), x1) -> c10(DEL(0, x1)) IF2(false, s(x0), 0, x1) -> c10(DEL(s(x0), x1)) IF2(false, s(0), s(s(x0)), x1) -> c10(DEL(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c10(DEL(s(s(x0)), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(DEL(s(s(x0)), x2)) ---------------------------------------- (122) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(z0, z2) if1(false, z0, z1, z2) -> min(z1, z2) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) 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) Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1)) MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2)) MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2)) DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2)) MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2), LE(s(0), s(z0))) MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(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)) -> c17(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c17(EQ(s(0), s(0))) IF1(true, 0, x0, x1) -> c7(MIN(0, x1)) IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2)) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1)) IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2)) MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2)) MIN(s(s(z0)), cons(s(0), x2)) -> c(LE(s(s(z0)), s(0))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) MINSORT(cons(z0, cons(z1, z2))) -> c13(DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c13(DEL(z0, cons(z0, nil))) IF2(false, 0, s(x0), x1) -> c10(DEL(0, x1)) IF2(false, s(x0), 0, x1) -> c10(DEL(s(x0), x1)) IF2(false, s(0), s(s(x0)), x1) -> c10(DEL(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c10(DEL(s(s(x0)), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(DEL(s(s(x0)), x2)) S tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2)) MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2)) DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2)) MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2), LE(s(0), s(z0))) MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(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)) -> c17(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c17(EQ(s(0), s(0))) IF1(true, 0, x0, x1) -> c7(MIN(0, x1)) IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2)) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1)) IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2)) MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2)) MIN(s(s(z0)), cons(s(0), x2)) -> c(LE(s(s(z0)), s(0))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF2(false, 0, s(x0), x1) -> c10(DEL(0, x1)) IF2(false, s(x0), 0, x1) -> c10(DEL(s(x0), x1)) IF2(false, s(0), s(s(x0)), x1) -> c10(DEL(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c10(DEL(s(s(x0)), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(DEL(s(s(x0)), x2)) K tuples: MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1)) MINSORT(cons(x0, cons(x1, x2))) -> c13(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, nil)) -> c13(DEL(min(x0, nil), cons(x0, nil))) Defined Rule Symbols: del_2, min_2, if1_4, le_2, if2_4, eq_2 Defined Pair Symbols: LE_2, EQ_2, MINSORT_1, MIN_2, DEL_2, IF1_4, IF2_4 Compound Symbols: c2_1, c6_1, c12_1, c15_1, c17_1, c15_2, c17_2, c7_1, c13_3, c13_2, c8_1, c_1, c13_1, c10_1 ---------------------------------------- (123) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace LE(s(z0), s(z1)) -> c2(LE(z0, z1)) by LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) ---------------------------------------- (124) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(z0, z2) if1(false, z0, z1, z2) -> min(z1, z2) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) 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) Tuples: EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1)) MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2)) MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2)) DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2)) MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2), LE(s(0), s(z0))) MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(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)) -> c17(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c17(EQ(s(0), s(0))) IF1(true, 0, x0, x1) -> c7(MIN(0, x1)) IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2)) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1)) IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2)) MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2)) MIN(s(s(z0)), cons(s(0), x2)) -> c(LE(s(s(z0)), s(0))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) MINSORT(cons(z0, cons(z1, z2))) -> c13(DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c13(DEL(z0, cons(z0, nil))) IF2(false, 0, s(x0), x1) -> c10(DEL(0, x1)) IF2(false, s(x0), 0, x1) -> c10(DEL(s(x0), x1)) IF2(false, s(0), s(s(x0)), x1) -> c10(DEL(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c10(DEL(s(s(x0)), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(DEL(s(s(x0)), x2)) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) S tuples: EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2)) MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2)) DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2)) MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2), LE(s(0), s(z0))) MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(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)) -> c17(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c17(EQ(s(0), s(0))) IF1(true, 0, x0, x1) -> c7(MIN(0, x1)) IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2)) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1)) IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2)) MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2)) MIN(s(s(z0)), cons(s(0), x2)) -> c(LE(s(s(z0)), s(0))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF2(false, 0, s(x0), x1) -> c10(DEL(0, x1)) IF2(false, s(x0), 0, x1) -> c10(DEL(s(x0), x1)) IF2(false, s(0), s(s(x0)), x1) -> c10(DEL(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c10(DEL(s(s(x0)), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(DEL(s(s(x0)), x2)) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) K tuples: MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1)) MINSORT(cons(x0, cons(x1, x2))) -> c13(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, nil)) -> c13(DEL(min(x0, nil), cons(x0, nil))) Defined Rule Symbols: del_2, min_2, if1_4, le_2, if2_4, eq_2 Defined Pair Symbols: EQ_2, MINSORT_1, MIN_2, DEL_2, IF1_4, IF2_4, LE_2 Compound Symbols: c6_1, c12_1, c15_1, c17_1, c15_2, c17_2, c7_1, c13_3, c13_2, c8_1, c_1, c13_1, c10_1, c2_1 ---------------------------------------- (125) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MIN(s(s(z0)), cons(s(0), x2)) -> c(LE(s(s(z0)), s(0))) ---------------------------------------- (126) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(z0, z2) if1(false, z0, z1, z2) -> min(z1, z2) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) 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) Tuples: EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1)) MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2)) MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2)) DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2)) MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2), LE(s(0), s(z0))) MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(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)) -> c17(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c17(EQ(s(0), s(0))) IF1(true, 0, x0, x1) -> c7(MIN(0, x1)) IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2)) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1)) IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2)) MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2)) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) MINSORT(cons(z0, cons(z1, z2))) -> c13(DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c13(DEL(z0, cons(z0, nil))) IF2(false, 0, s(x0), x1) -> c10(DEL(0, x1)) IF2(false, s(x0), 0, x1) -> c10(DEL(s(x0), x1)) IF2(false, s(0), s(s(x0)), x1) -> c10(DEL(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c10(DEL(s(s(x0)), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(DEL(s(s(x0)), x2)) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) S tuples: EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2)) MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2)) DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2)) MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2), LE(s(0), s(z0))) MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(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)) -> c17(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c17(EQ(s(0), s(0))) IF1(true, 0, x0, x1) -> c7(MIN(0, x1)) IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2)) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1)) IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2)) MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2)) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF2(false, 0, s(x0), x1) -> c10(DEL(0, x1)) IF2(false, s(x0), 0, x1) -> c10(DEL(s(x0), x1)) IF2(false, s(0), s(s(x0)), x1) -> c10(DEL(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c10(DEL(s(s(x0)), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(DEL(s(s(x0)), x2)) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) K tuples: MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1)) Defined Rule Symbols: del_2, min_2, if1_4, le_2, if2_4, eq_2 Defined Pair Symbols: EQ_2, MINSORT_1, MIN_2, DEL_2, IF1_4, IF2_4, LE_2 Compound Symbols: c6_1, c12_1, c15_1, c17_1, c15_2, c17_2, c7_1, c13_3, c13_2, c8_1, c_1, c13_1, c10_1, c2_1 ---------------------------------------- (127) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (128) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(z0, z2) if1(false, z0, z1, z2) -> min(z1, z2) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) 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) Tuples: EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1)) MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2)) MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2)) DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2)) MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(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)) -> c17(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c17(EQ(s(0), s(0))) IF1(true, 0, x0, x1) -> c7(MIN(0, x1)) IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2)) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1)) IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2)) MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2)) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) MINSORT(cons(z0, cons(z1, z2))) -> c13(DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c13(DEL(z0, cons(z0, nil))) IF2(false, 0, s(x0), x1) -> c10(DEL(0, x1)) IF2(false, s(x0), 0, x1) -> c10(DEL(s(x0), x1)) IF2(false, s(0), s(s(x0)), x1) -> c10(DEL(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c10(DEL(s(s(x0)), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(DEL(s(s(x0)), x2)) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2)) S tuples: EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2)) MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2)) DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2)) MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(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)) -> c17(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c17(EQ(s(0), s(0))) IF1(true, 0, x0, x1) -> c7(MIN(0, x1)) IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2)) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1)) IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2)) MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2)) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF2(false, 0, s(x0), x1) -> c10(DEL(0, x1)) IF2(false, s(x0), 0, x1) -> c10(DEL(s(x0), x1)) IF2(false, s(0), s(s(x0)), x1) -> c10(DEL(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c10(DEL(s(s(x0)), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(DEL(s(s(x0)), x2)) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2)) K tuples: MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1)) Defined Rule Symbols: del_2, min_2, if1_4, le_2, if2_4, eq_2 Defined Pair Symbols: EQ_2, MINSORT_1, MIN_2, DEL_2, IF1_4, IF2_4, LE_2 Compound Symbols: c6_1, c12_1, c15_1, c17_1, c15_2, c17_2, c7_1, c13_3, c13_2, c8_1, c_1, c13_1, c10_1, c2_1 ---------------------------------------- (129) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(min(z0, cons(z1, z2)), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) by MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) ---------------------------------------- (130) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(z0, z2) if1(false, z0, z1, z2) -> min(z1, z2) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) 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) Tuples: EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1)) MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2)) MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2)) DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2)) MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(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)) -> c17(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c17(EQ(s(0), s(0))) IF1(true, 0, x0, x1) -> c7(MIN(0, x1)) IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2)) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1)) IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2)) MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2)) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) MINSORT(cons(z0, cons(z1, z2))) -> c13(DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c13(DEL(z0, cons(z0, nil))) IF2(false, 0, s(x0), x1) -> c10(DEL(0, x1)) IF2(false, s(x0), 0, x1) -> c10(DEL(s(x0), x1)) IF2(false, s(0), s(s(x0)), x1) -> c10(DEL(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c10(DEL(s(s(x0)), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(DEL(s(s(x0)), x2)) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2)) S tuples: EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2)) MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2)) DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2)) MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(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)) -> c17(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c17(EQ(s(0), s(0))) IF1(true, 0, x0, x1) -> c7(MIN(0, x1)) IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2)) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1)) IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2)) MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2)) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF2(false, 0, s(x0), x1) -> c10(DEL(0, x1)) IF2(false, s(x0), 0, x1) -> c10(DEL(s(x0), x1)) IF2(false, s(0), s(s(x0)), x1) -> c10(DEL(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c10(DEL(s(s(x0)), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(DEL(s(s(x0)), x2)) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2)) K tuples: MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1)) Defined Rule Symbols: del_2, min_2, if1_4, le_2, if2_4, eq_2 Defined Pair Symbols: EQ_2, MINSORT_1, MIN_2, DEL_2, IF1_4, IF2_4, LE_2 Compound Symbols: c6_1, c12_1, c15_1, c17_1, c15_2, c17_2, c7_1, c13_3, c13_2, c8_1, c_1, c13_1, c10_1, c2_1 ---------------------------------------- (131) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(z0, cons(z1, z2)), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) by MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) ---------------------------------------- (132) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(z0, z2) if1(false, z0, z1, z2) -> min(z1, z2) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) 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) Tuples: EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1)) MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2)) MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2)) DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2)) MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(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)) -> c17(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c17(EQ(s(0), s(0))) IF1(true, 0, x0, x1) -> c7(MIN(0, x1)) IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2)) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1)) IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2)) MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2)) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) MINSORT(cons(z0, cons(z1, z2))) -> c13(DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c13(DEL(z0, cons(z0, nil))) IF2(false, 0, s(x0), x1) -> c10(DEL(0, x1)) IF2(false, s(x0), 0, x1) -> c10(DEL(s(x0), x1)) IF2(false, s(0), s(s(x0)), x1) -> c10(DEL(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c10(DEL(s(s(x0)), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(DEL(s(s(x0)), x2)) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2)) S tuples: EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2)) MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2)) DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2)) MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(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)) -> c17(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c17(EQ(s(0), s(0))) IF1(true, 0, x0, x1) -> c7(MIN(0, x1)) IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2)) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1)) IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2)) MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2)) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF2(false, 0, s(x0), x1) -> c10(DEL(0, x1)) IF2(false, s(x0), 0, x1) -> c10(DEL(s(x0), x1)) IF2(false, s(0), s(s(x0)), x1) -> c10(DEL(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c10(DEL(s(s(x0)), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(DEL(s(s(x0)), x2)) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2)) K tuples: MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1)) Defined Rule Symbols: del_2, min_2, if1_4, le_2, if2_4, eq_2 Defined Pair Symbols: EQ_2, MINSORT_1, MIN_2, DEL_2, IF1_4, IF2_4, LE_2 Compound Symbols: c6_1, c12_1, c15_1, c17_1, c15_2, c17_2, c7_1, c13_2, c8_1, c_1, c13_3, c13_1, c10_1, c2_1 ---------------------------------------- (133) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(min(z0, nil), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) by MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) ---------------------------------------- (134) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(z0, z2) if1(false, z0, z1, z2) -> min(z1, z2) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) 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) Tuples: EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1)) MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2)) MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2)) DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2)) MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(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)) -> c17(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c17(EQ(s(0), s(0))) IF1(true, 0, x0, x1) -> c7(MIN(0, x1)) IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2)) IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1)) IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2)) MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2)) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) MINSORT(cons(z0, cons(z1, z2))) -> c13(DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c13(DEL(z0, cons(z0, nil))) IF2(false, 0, s(x0), x1) -> c10(DEL(0, x1)) IF2(false, s(x0), 0, x1) -> c10(DEL(s(x0), x1)) IF2(false, s(0), s(s(x0)), x1) -> c10(DEL(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c10(DEL(s(s(x0)), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(DEL(s(s(x0)), x2)) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2)) S tuples: EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2)) MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2)) DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2)) MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(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)) -> c17(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c17(EQ(s(0), s(0))) IF1(true, 0, x0, x1) -> c7(MIN(0, x1)) IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2)) IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1)) IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2)) MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2)) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF2(false, 0, s(x0), x1) -> c10(DEL(0, x1)) IF2(false, s(x0), 0, x1) -> c10(DEL(s(x0), x1)) IF2(false, s(0), s(s(x0)), x1) -> c10(DEL(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c10(DEL(s(s(x0)), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(DEL(s(s(x0)), x2)) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2)) K tuples: MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1)) Defined Rule Symbols: del_2, min_2, if1_4, le_2, if2_4, eq_2 Defined Pair Symbols: EQ_2, MINSORT_1, MIN_2, DEL_2, IF1_4, IF2_4, LE_2 Compound Symbols: c6_1, c12_1, c15_1, c17_1, c15_2, c17_2, c7_1, c8_1, c_1, c13_2, c13_3, c13_1, c10_1, c2_1 ---------------------------------------- (135) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), min(z0, nil), z0, nil)), DEL(z0, cons(z0, nil))) by MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) ---------------------------------------- (136) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(z0, z2) if1(false, z0, z1, z2) -> min(z1, z2) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) 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) Tuples: EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1)) MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2)) MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2)) DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2)) MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(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)) -> c17(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c17(EQ(s(0), s(0))) IF1(true, 0, x0, x1) -> c7(MIN(0, x1)) IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2)) IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1)) IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2)) MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2)) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) MINSORT(cons(z0, cons(z1, z2))) -> c13(DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c13(DEL(z0, cons(z0, nil))) IF2(false, 0, s(x0), x1) -> c10(DEL(0, x1)) IF2(false, s(x0), 0, x1) -> c10(DEL(s(x0), x1)) IF2(false, s(0), s(s(x0)), x1) -> c10(DEL(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c10(DEL(s(s(x0)), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(DEL(s(s(x0)), x2)) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2)) S tuples: EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2)) MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2)) DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2)) MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(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)) -> c17(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c17(EQ(s(0), s(0))) IF1(true, 0, x0, x1) -> c7(MIN(0, x1)) IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2)) IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1)) IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2)) MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2)) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF2(false, 0, s(x0), x1) -> c10(DEL(0, x1)) IF2(false, s(x0), 0, x1) -> c10(DEL(s(x0), x1)) IF2(false, s(0), s(s(x0)), x1) -> c10(DEL(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c10(DEL(s(s(x0)), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(DEL(s(s(x0)), x2)) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2)) K tuples: MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1)) Defined Rule Symbols: del_2, min_2, if1_4, le_2, if2_4, eq_2 Defined Pair Symbols: EQ_2, MINSORT_1, MIN_2, DEL_2, IF1_4, IF2_4, LE_2 Compound Symbols: c6_1, c12_1, c15_1, c17_1, c15_2, c17_2, c7_1, c8_1, c_1, c13_3, c13_1, c13_2, c10_1, c2_1 ---------------------------------------- (137) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) by EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) ---------------------------------------- (138) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(z0, z2) if1(false, z0, z1, z2) -> min(z1, z2) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) 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) Tuples: MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1)) MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2)) MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2)) DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2)) MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(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)) -> c17(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c17(EQ(s(0), s(0))) IF1(true, 0, x0, x1) -> c7(MIN(0, x1)) IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2)) IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1)) IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2)) MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2)) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) MINSORT(cons(z0, cons(z1, z2))) -> c13(DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c13(DEL(z0, cons(z0, nil))) IF2(false, 0, s(x0), x1) -> c10(DEL(0, x1)) IF2(false, s(x0), 0, x1) -> c10(DEL(s(x0), x1)) IF2(false, s(0), s(s(x0)), x1) -> c10(DEL(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c10(DEL(s(s(x0)), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(DEL(s(s(x0)), x2)) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2)) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) S tuples: MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2)) MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2)) DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2)) MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(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)) -> c17(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c17(EQ(s(0), s(0))) IF1(true, 0, x0, x1) -> c7(MIN(0, x1)) IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2)) IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1)) IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2)) MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2)) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF2(false, 0, s(x0), x1) -> c10(DEL(0, x1)) IF2(false, s(x0), 0, x1) -> c10(DEL(s(x0), x1)) IF2(false, s(0), s(s(x0)), x1) -> c10(DEL(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c10(DEL(s(s(x0)), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(DEL(s(s(x0)), x2)) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2)) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) K tuples: MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1)) Defined Rule Symbols: del_2, min_2, if1_4, le_2, if2_4, eq_2 Defined Pair Symbols: MINSORT_1, MIN_2, DEL_2, IF1_4, IF2_4, LE_2, EQ_2 Compound Symbols: c12_1, c15_1, c17_1, c15_2, c17_2, c7_1, c8_1, c_1, c13_3, c13_1, c13_2, c10_1, c2_1, c6_1 ---------------------------------------- (139) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (140) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(z0, z2) if1(false, z0, z1, z2) -> min(z1, z2) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) 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) Tuples: MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1)) MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2)) MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2)) DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2)) MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(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)) -> c17(EQ(s(x0), s(x1))) IF1(true, 0, x0, x1) -> c7(MIN(0, x1)) IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2)) IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1)) IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2)) MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2)) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) MINSORT(cons(z0, cons(z1, z2))) -> c13(DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c13(DEL(z0, cons(z0, nil))) IF2(false, 0, s(x0), x1) -> c10(DEL(0, x1)) IF2(false, s(x0), 0, x1) -> c10(DEL(s(x0), x1)) IF2(false, s(0), s(s(x0)), x1) -> c10(DEL(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c10(DEL(s(s(x0)), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(DEL(s(s(x0)), x2)) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2)) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c17 S tuples: MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2)) MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2)) DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2)) MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(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)) -> c17(EQ(s(x0), s(x1))) IF1(true, 0, x0, x1) -> c7(MIN(0, x1)) IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2)) IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1)) IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2)) MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2)) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF2(false, 0, s(x0), x1) -> c10(DEL(0, x1)) IF2(false, s(x0), 0, x1) -> c10(DEL(s(x0), x1)) IF2(false, s(0), s(s(x0)), x1) -> c10(DEL(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c10(DEL(s(s(x0)), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(DEL(s(s(x0)), x2)) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2)) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c17 K tuples: MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1)) Defined Rule Symbols: del_2, min_2, if1_4, le_2, if2_4, eq_2 Defined Pair Symbols: MINSORT_1, MIN_2, DEL_2, IF1_4, IF2_4, LE_2, EQ_2 Compound Symbols: c12_1, c15_1, c17_1, c15_2, c17_2, c7_1, c8_1, c_1, c13_3, c13_1, c13_2, c10_1, c2_1, c6_1, c17 ---------------------------------------- (141) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace MINSORT(cons(z0, z1)) -> c12(MIN(z0, z1)) by MINSORT(cons(0, cons(y0, y1))) -> c12(MIN(0, cons(y0, y1))) MINSORT(cons(s(y0), cons(0, y1))) -> c12(MIN(s(y0), cons(0, y1))) MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c12(MIN(s(s(y0)), cons(s(s(y1)), y2))) MINSORT(cons(s(y0), cons(s(y1), y2))) -> c12(MIN(s(y0), cons(s(y1), y2))) MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c12(MIN(s(s(y0)), cons(s(0), y1))) MINSORT(cons(s(0), cons(s(y0), y1))) -> c12(MIN(s(0), cons(s(y0), y1))) ---------------------------------------- (142) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(z0, z2) if1(false, z0, z1, z2) -> min(z1, z2) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) 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) Tuples: MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2)) MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2)) DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2)) MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(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)) -> c17(EQ(s(x0), s(x1))) IF1(true, 0, x0, x1) -> c7(MIN(0, x1)) IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2)) IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1)) IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2)) MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2)) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) MINSORT(cons(z0, cons(z1, z2))) -> c13(DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c13(DEL(z0, cons(z0, nil))) IF2(false, 0, s(x0), x1) -> c10(DEL(0, x1)) IF2(false, s(x0), 0, x1) -> c10(DEL(s(x0), x1)) IF2(false, s(0), s(s(x0)), x1) -> c10(DEL(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c10(DEL(s(s(x0)), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(DEL(s(s(x0)), x2)) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2)) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c17 MINSORT(cons(0, cons(y0, y1))) -> c12(MIN(0, cons(y0, y1))) MINSORT(cons(s(y0), cons(0, y1))) -> c12(MIN(s(y0), cons(0, y1))) MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c12(MIN(s(s(y0)), cons(s(s(y1)), y2))) MINSORT(cons(s(y0), cons(s(y1), y2))) -> c12(MIN(s(y0), cons(s(y1), y2))) MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c12(MIN(s(s(y0)), cons(s(0), y1))) MINSORT(cons(s(0), cons(s(y0), y1))) -> c12(MIN(s(0), cons(s(y0), y1))) S tuples: MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2)) MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2)) DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2)) MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(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)) -> c17(EQ(s(x0), s(x1))) IF1(true, 0, x0, x1) -> c7(MIN(0, x1)) IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2)) IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1)) IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2)) MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2)) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF2(false, 0, s(x0), x1) -> c10(DEL(0, x1)) IF2(false, s(x0), 0, x1) -> c10(DEL(s(x0), x1)) IF2(false, s(0), s(s(x0)), x1) -> c10(DEL(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c10(DEL(s(s(x0)), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(DEL(s(s(x0)), x2)) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2)) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c17 K tuples: MINSORT(cons(0, cons(y0, y1))) -> c12(MIN(0, cons(y0, y1))) MINSORT(cons(s(y0), cons(0, y1))) -> c12(MIN(s(y0), cons(0, y1))) MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c12(MIN(s(s(y0)), cons(s(s(y1)), y2))) MINSORT(cons(s(y0), cons(s(y1), y2))) -> c12(MIN(s(y0), cons(s(y1), y2))) MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c12(MIN(s(s(y0)), cons(s(0), y1))) MINSORT(cons(s(0), cons(s(y0), y1))) -> c12(MIN(s(0), cons(s(y0), y1))) Defined Rule Symbols: del_2, min_2, if1_4, le_2, if2_4, eq_2 Defined Pair Symbols: MIN_2, DEL_2, IF1_4, MINSORT_1, IF2_4, LE_2, EQ_2 Compound Symbols: c15_1, c17_1, c15_2, c17_2, c7_1, c8_1, c_1, c13_3, c13_1, c13_2, c10_1, c2_1, c6_1, c17, c12_1 ---------------------------------------- (143) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) by MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(true, 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) ---------------------------------------- (144) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(z0, z2) if1(false, z0, z1, z2) -> min(z1, z2) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) 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) Tuples: MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2)) MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2)) DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2)) MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(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)) -> c17(EQ(s(x0), s(x1))) IF1(true, 0, x0, x1) -> c7(MIN(0, x1)) IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2)) IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1)) IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2)) MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2)) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) MINSORT(cons(z0, cons(z1, z2))) -> c13(DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c13(DEL(z0, cons(z0, nil))) IF2(false, 0, s(x0), x1) -> c10(DEL(0, x1)) IF2(false, s(x0), 0, x1) -> c10(DEL(s(x0), x1)) IF2(false, s(0), s(s(x0)), x1) -> c10(DEL(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c10(DEL(s(s(x0)), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(DEL(s(s(x0)), x2)) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2)) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c17 MINSORT(cons(0, cons(y0, y1))) -> c12(MIN(0, cons(y0, y1))) MINSORT(cons(s(y0), cons(0, y1))) -> c12(MIN(s(y0), cons(0, y1))) MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c12(MIN(s(s(y0)), cons(s(s(y1)), y2))) MINSORT(cons(s(y0), cons(s(y1), y2))) -> c12(MIN(s(y0), cons(s(y1), y2))) MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c12(MIN(s(s(y0)), cons(s(0), y1))) MINSORT(cons(s(0), cons(s(y0), y1))) -> c12(MIN(s(0), cons(s(y0), y1))) MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(true, 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) S tuples: MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2)) MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2)) DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2)) MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(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)) -> c17(EQ(s(x0), s(x1))) IF1(true, 0, x0, x1) -> c7(MIN(0, x1)) IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2)) IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1)) IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2)) MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2)) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF2(false, 0, s(x0), x1) -> c10(DEL(0, x1)) IF2(false, s(x0), 0, x1) -> c10(DEL(s(x0), x1)) IF2(false, s(0), s(s(x0)), x1) -> c10(DEL(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c10(DEL(s(s(x0)), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(DEL(s(s(x0)), x2)) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2)) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c17 MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(true, 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) K tuples: MINSORT(cons(0, cons(y0, y1))) -> c12(MIN(0, cons(y0, y1))) MINSORT(cons(s(y0), cons(0, y1))) -> c12(MIN(s(y0), cons(0, y1))) MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c12(MIN(s(s(y0)), cons(s(s(y1)), y2))) MINSORT(cons(s(y0), cons(s(y1), y2))) -> c12(MIN(s(y0), cons(s(y1), y2))) MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c12(MIN(s(s(y0)), cons(s(0), y1))) MINSORT(cons(s(0), cons(s(y0), y1))) -> c12(MIN(s(0), cons(s(y0), y1))) Defined Rule Symbols: del_2, min_2, if1_4, le_2, if2_4, eq_2 Defined Pair Symbols: MIN_2, DEL_2, IF1_4, MINSORT_1, IF2_4, LE_2, EQ_2 Compound Symbols: c15_1, c17_1, c15_2, c17_2, c7_1, c8_1, c_1, c13_3, c13_1, c13_2, c10_1, c2_1, c6_1, c17, c12_1 ---------------------------------------- (145) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) by MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) ---------------------------------------- (146) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(z0, z2) if1(false, z0, z1, z2) -> min(z1, z2) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) 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) Tuples: MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2)) MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2)) DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2)) MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(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)) -> c17(EQ(s(x0), s(x1))) IF1(true, 0, x0, x1) -> c7(MIN(0, x1)) IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2)) IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1)) IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2)) MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2)) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) MINSORT(cons(z0, cons(z1, z2))) -> c13(DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c13(DEL(z0, cons(z0, nil))) IF2(false, 0, s(x0), x1) -> c10(DEL(0, x1)) IF2(false, s(x0), 0, x1) -> c10(DEL(s(x0), x1)) IF2(false, s(0), s(s(x0)), x1) -> c10(DEL(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c10(DEL(s(s(x0)), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(DEL(s(s(x0)), x2)) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2)) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c17 MINSORT(cons(0, cons(y0, y1))) -> c12(MIN(0, cons(y0, y1))) MINSORT(cons(s(y0), cons(0, y1))) -> c12(MIN(s(y0), cons(0, y1))) MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c12(MIN(s(s(y0)), cons(s(s(y1)), y2))) MINSORT(cons(s(y0), cons(s(y1), y2))) -> c12(MIN(s(y0), cons(s(y1), y2))) MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c12(MIN(s(s(y0)), cons(s(0), y1))) MINSORT(cons(s(0), cons(s(y0), y1))) -> c12(MIN(s(0), cons(s(y0), y1))) MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(true, 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) S tuples: MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2)) MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2)) DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2)) MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(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)) -> c17(EQ(s(x0), s(x1))) IF1(true, 0, x0, x1) -> c7(MIN(0, x1)) IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2)) IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1)) IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2)) MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2)) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF2(false, 0, s(x0), x1) -> c10(DEL(0, x1)) IF2(false, s(x0), 0, x1) -> c10(DEL(s(x0), x1)) IF2(false, s(0), s(s(x0)), x1) -> c10(DEL(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c10(DEL(s(s(x0)), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(DEL(s(s(x0)), x2)) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2)) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c17 MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(true, 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) K tuples: MINSORT(cons(0, cons(y0, y1))) -> c12(MIN(0, cons(y0, y1))) MINSORT(cons(s(y0), cons(0, y1))) -> c12(MIN(s(y0), cons(0, y1))) MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c12(MIN(s(s(y0)), cons(s(s(y1)), y2))) MINSORT(cons(s(y0), cons(s(y1), y2))) -> c12(MIN(s(y0), cons(s(y1), y2))) MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c12(MIN(s(s(y0)), cons(s(0), y1))) MINSORT(cons(s(0), cons(s(y0), y1))) -> c12(MIN(s(0), cons(s(y0), y1))) Defined Rule Symbols: del_2, min_2, if1_4, le_2, if2_4, eq_2 Defined Pair Symbols: MIN_2, DEL_2, IF1_4, MINSORT_1, IF2_4, LE_2, EQ_2 Compound Symbols: c15_1, c17_1, c15_2, c17_2, c7_1, c8_1, c_1, c13_3, c13_1, c13_2, c10_1, c2_1, c6_1, c17, c12_1 ---------------------------------------- (147) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace MIN(s(x0), cons(s(x1), x2)) -> c15(LE(s(x0), s(x1))) by MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c15(LE(s(s(y0)), s(s(y1)))) ---------------------------------------- (148) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(z0, z2) if1(false, z0, z1, z2) -> min(z1, z2) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) 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) Tuples: MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2)) MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2)) DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2)) MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(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)) -> c17(EQ(s(x0), s(x1))) IF1(true, 0, x0, x1) -> c7(MIN(0, x1)) IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2)) IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1)) IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2)) MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2)) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) MINSORT(cons(z0, cons(z1, z2))) -> c13(DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c13(DEL(z0, cons(z0, nil))) IF2(false, 0, s(x0), x1) -> c10(DEL(0, x1)) IF2(false, s(x0), 0, x1) -> c10(DEL(s(x0), x1)) IF2(false, s(0), s(s(x0)), x1) -> c10(DEL(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c10(DEL(s(s(x0)), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(DEL(s(s(x0)), x2)) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2)) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c17 MINSORT(cons(0, cons(y0, y1))) -> c12(MIN(0, cons(y0, y1))) MINSORT(cons(s(y0), cons(0, y1))) -> c12(MIN(s(y0), cons(0, y1))) MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c12(MIN(s(s(y0)), cons(s(s(y1)), y2))) MINSORT(cons(s(y0), cons(s(y1), y2))) -> c12(MIN(s(y0), cons(s(y1), y2))) MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c12(MIN(s(s(y0)), cons(s(0), y1))) MINSORT(cons(s(0), cons(s(y0), y1))) -> c12(MIN(s(0), cons(s(y0), y1))) MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(true, 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c15(LE(s(s(y0)), s(s(y1)))) S tuples: MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2)) MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2)) DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2)) MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(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)) -> c17(EQ(s(x0), s(x1))) IF1(true, 0, x0, x1) -> c7(MIN(0, x1)) IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2)) IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1)) IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2)) MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2)) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF2(false, 0, s(x0), x1) -> c10(DEL(0, x1)) IF2(false, s(x0), 0, x1) -> c10(DEL(s(x0), x1)) IF2(false, s(0), s(s(x0)), x1) -> c10(DEL(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c10(DEL(s(s(x0)), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(DEL(s(s(x0)), x2)) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2)) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c17 MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(true, 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c15(LE(s(s(y0)), s(s(y1)))) K tuples: MINSORT(cons(0, cons(y0, y1))) -> c12(MIN(0, cons(y0, y1))) MINSORT(cons(s(y0), cons(0, y1))) -> c12(MIN(s(y0), cons(0, y1))) MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c12(MIN(s(s(y0)), cons(s(s(y1)), y2))) MINSORT(cons(s(y0), cons(s(y1), y2))) -> c12(MIN(s(y0), cons(s(y1), y2))) MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c12(MIN(s(s(y0)), cons(s(0), y1))) MINSORT(cons(s(0), cons(s(y0), y1))) -> c12(MIN(s(0), cons(s(y0), y1))) Defined Rule Symbols: del_2, min_2, if1_4, le_2, if2_4, eq_2 Defined Pair Symbols: MIN_2, DEL_2, IF1_4, MINSORT_1, IF2_4, LE_2, EQ_2 Compound Symbols: c15_1, c17_1, c15_2, c17_2, c7_1, c8_1, c_1, c13_3, c13_1, c13_2, c10_1, c2_1, c6_1, c17, c12_1 ---------------------------------------- (149) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace DEL(s(x0), cons(s(x1), x2)) -> c17(EQ(s(x0), s(x1))) by DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c17(EQ(s(s(y0)), s(s(y1)))) ---------------------------------------- (150) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(z0, z2) if1(false, z0, z1, z2) -> min(z1, z2) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) 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) Tuples: MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2)) MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2)) DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2)) MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) IF1(true, 0, x0, x1) -> c7(MIN(0, x1)) IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2)) IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1)) IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2)) MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2)) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) MINSORT(cons(z0, cons(z1, z2))) -> c13(DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c13(DEL(z0, cons(z0, nil))) IF2(false, 0, s(x0), x1) -> c10(DEL(0, x1)) IF2(false, s(x0), 0, x1) -> c10(DEL(s(x0), x1)) IF2(false, s(0), s(s(x0)), x1) -> c10(DEL(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c10(DEL(s(s(x0)), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(DEL(s(s(x0)), x2)) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2)) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c17 MINSORT(cons(0, cons(y0, y1))) -> c12(MIN(0, cons(y0, y1))) MINSORT(cons(s(y0), cons(0, y1))) -> c12(MIN(s(y0), cons(0, y1))) MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c12(MIN(s(s(y0)), cons(s(s(y1)), y2))) MINSORT(cons(s(y0), cons(s(y1), y2))) -> c12(MIN(s(y0), cons(s(y1), y2))) MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c12(MIN(s(s(y0)), cons(s(0), y1))) MINSORT(cons(s(0), cons(s(y0), y1))) -> c12(MIN(s(0), cons(s(y0), y1))) MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(true, 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c15(LE(s(s(y0)), s(s(y1)))) DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c17(EQ(s(s(y0)), s(s(y1)))) S tuples: MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2)) MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2)) DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2)) MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) IF1(true, 0, x0, x1) -> c7(MIN(0, x1)) IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2)) IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1)) IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2)) MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2)) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF2(false, 0, s(x0), x1) -> c10(DEL(0, x1)) IF2(false, s(x0), 0, x1) -> c10(DEL(s(x0), x1)) IF2(false, s(0), s(s(x0)), x1) -> c10(DEL(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c10(DEL(s(s(x0)), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(DEL(s(s(x0)), x2)) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2)) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c17 MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(true, 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c15(LE(s(s(y0)), s(s(y1)))) DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c17(EQ(s(s(y0)), s(s(y1)))) K tuples: MINSORT(cons(0, cons(y0, y1))) -> c12(MIN(0, cons(y0, y1))) MINSORT(cons(s(y0), cons(0, y1))) -> c12(MIN(s(y0), cons(0, y1))) MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c12(MIN(s(s(y0)), cons(s(s(y1)), y2))) MINSORT(cons(s(y0), cons(s(y1), y2))) -> c12(MIN(s(y0), cons(s(y1), y2))) MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c12(MIN(s(s(y0)), cons(s(0), y1))) MINSORT(cons(s(0), cons(s(y0), y1))) -> c12(MIN(s(0), cons(s(y0), y1))) Defined Rule Symbols: del_2, min_2, if1_4, le_2, if2_4, eq_2 Defined Pair Symbols: MIN_2, DEL_2, IF1_4, MINSORT_1, IF2_4, LE_2, EQ_2 Compound Symbols: c15_1, c17_1, c15_2, c17_2, c7_1, c8_1, c_1, c13_3, c13_1, c13_2, c10_1, c2_1, c6_1, c17, c12_1 ---------------------------------------- (151) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace IF1(true, 0, x0, x1) -> c7(MIN(0, x1)) by IF1(true, 0, z0, cons(y0, y1)) -> c7(MIN(0, cons(y0, y1))) ---------------------------------------- (152) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(z0, z2) if1(false, z0, z1, z2) -> min(z1, z2) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) 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) Tuples: MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2)) MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2)) DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2)) MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2)) IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1)) IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2)) MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2)) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) MINSORT(cons(z0, cons(z1, z2))) -> c13(DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c13(DEL(z0, cons(z0, nil))) IF2(false, 0, s(x0), x1) -> c10(DEL(0, x1)) IF2(false, s(x0), 0, x1) -> c10(DEL(s(x0), x1)) IF2(false, s(0), s(s(x0)), x1) -> c10(DEL(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c10(DEL(s(s(x0)), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(DEL(s(s(x0)), x2)) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2)) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c17 MINSORT(cons(0, cons(y0, y1))) -> c12(MIN(0, cons(y0, y1))) MINSORT(cons(s(y0), cons(0, y1))) -> c12(MIN(s(y0), cons(0, y1))) MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c12(MIN(s(s(y0)), cons(s(s(y1)), y2))) MINSORT(cons(s(y0), cons(s(y1), y2))) -> c12(MIN(s(y0), cons(s(y1), y2))) MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c12(MIN(s(s(y0)), cons(s(0), y1))) MINSORT(cons(s(0), cons(s(y0), y1))) -> c12(MIN(s(0), cons(s(y0), y1))) MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(true, 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c15(LE(s(s(y0)), s(s(y1)))) DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c17(EQ(s(s(y0)), s(s(y1)))) IF1(true, 0, z0, cons(y0, y1)) -> c7(MIN(0, cons(y0, y1))) S tuples: MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2)) MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2)) DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2)) MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2)) IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1)) IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2)) MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2)) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF2(false, 0, s(x0), x1) -> c10(DEL(0, x1)) IF2(false, s(x0), 0, x1) -> c10(DEL(s(x0), x1)) IF2(false, s(0), s(s(x0)), x1) -> c10(DEL(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c10(DEL(s(s(x0)), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(DEL(s(s(x0)), x2)) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2)) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c17 MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(true, 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c15(LE(s(s(y0)), s(s(y1)))) DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c17(EQ(s(s(y0)), s(s(y1)))) IF1(true, 0, z0, cons(y0, y1)) -> c7(MIN(0, cons(y0, y1))) K tuples: MINSORT(cons(0, cons(y0, y1))) -> c12(MIN(0, cons(y0, y1))) MINSORT(cons(s(y0), cons(0, y1))) -> c12(MIN(s(y0), cons(0, y1))) MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c12(MIN(s(s(y0)), cons(s(s(y1)), y2))) MINSORT(cons(s(y0), cons(s(y1), y2))) -> c12(MIN(s(y0), cons(s(y1), y2))) MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c12(MIN(s(s(y0)), cons(s(0), y1))) MINSORT(cons(s(0), cons(s(y0), y1))) -> c12(MIN(s(0), cons(s(y0), y1))) Defined Rule Symbols: del_2, min_2, if1_4, le_2, if2_4, eq_2 Defined Pair Symbols: MIN_2, DEL_2, IF1_4, MINSORT_1, IF2_4, LE_2, EQ_2 Compound Symbols: c15_1, c17_1, c15_2, c17_2, c7_1, c8_1, c_1, c13_3, c13_1, c13_2, c10_1, c2_1, c6_1, c17, c12_1 ---------------------------------------- (153) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace MIN(0, cons(z0, x2)) -> c15(IF1(true, 0, z0, x2)) by MIN(0, cons(z0, cons(y1, y2))) -> c15(IF1(true, 0, z0, cons(y1, y2))) ---------------------------------------- (154) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(z0, z2) if1(false, z0, z1, z2) -> min(z1, z2) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) 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) Tuples: MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2)) DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2)) MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2)) IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1)) IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2)) MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2)) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) MINSORT(cons(z0, cons(z1, z2))) -> c13(DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c13(DEL(z0, cons(z0, nil))) IF2(false, 0, s(x0), x1) -> c10(DEL(0, x1)) IF2(false, s(x0), 0, x1) -> c10(DEL(s(x0), x1)) IF2(false, s(0), s(s(x0)), x1) -> c10(DEL(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c10(DEL(s(s(x0)), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(DEL(s(s(x0)), x2)) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2)) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c17 MINSORT(cons(0, cons(y0, y1))) -> c12(MIN(0, cons(y0, y1))) MINSORT(cons(s(y0), cons(0, y1))) -> c12(MIN(s(y0), cons(0, y1))) MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c12(MIN(s(s(y0)), cons(s(s(y1)), y2))) MINSORT(cons(s(y0), cons(s(y1), y2))) -> c12(MIN(s(y0), cons(s(y1), y2))) MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c12(MIN(s(s(y0)), cons(s(0), y1))) MINSORT(cons(s(0), cons(s(y0), y1))) -> c12(MIN(s(0), cons(s(y0), y1))) MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(true, 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c15(LE(s(s(y0)), s(s(y1)))) DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c17(EQ(s(s(y0)), s(s(y1)))) IF1(true, 0, z0, cons(y0, y1)) -> c7(MIN(0, cons(y0, y1))) MIN(0, cons(z0, cons(y1, y2))) -> c15(IF1(true, 0, z0, cons(y1, y2))) S tuples: MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2)) DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2)) MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1)) IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2)) IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1)) IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2)) MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2)) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF2(false, 0, s(x0), x1) -> c10(DEL(0, x1)) IF2(false, s(x0), 0, x1) -> c10(DEL(s(x0), x1)) IF2(false, s(0), s(s(x0)), x1) -> c10(DEL(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c10(DEL(s(s(x0)), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(DEL(s(s(x0)), x2)) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2)) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c17 MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(true, 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c15(LE(s(s(y0)), s(s(y1)))) DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c17(EQ(s(s(y0)), s(s(y1)))) IF1(true, 0, z0, cons(y0, y1)) -> c7(MIN(0, cons(y0, y1))) MIN(0, cons(z0, cons(y1, y2))) -> c15(IF1(true, 0, z0, cons(y1, y2))) K tuples: MINSORT(cons(0, cons(y0, y1))) -> c12(MIN(0, cons(y0, y1))) MINSORT(cons(s(y0), cons(0, y1))) -> c12(MIN(s(y0), cons(0, y1))) MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c12(MIN(s(s(y0)), cons(s(s(y1)), y2))) MINSORT(cons(s(y0), cons(s(y1), y2))) -> c12(MIN(s(y0), cons(s(y1), y2))) MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c12(MIN(s(s(y0)), cons(s(0), y1))) MINSORT(cons(s(0), cons(s(y0), y1))) -> c12(MIN(s(0), cons(s(y0), y1))) Defined Rule Symbols: del_2, min_2, if1_4, le_2, if2_4, eq_2 Defined Pair Symbols: MIN_2, DEL_2, IF1_4, MINSORT_1, IF2_4, LE_2, EQ_2 Compound Symbols: c15_1, c17_1, c15_2, c17_2, c7_1, c8_1, c_1, c13_3, c13_1, c13_2, c10_1, c2_1, c6_1, c17, c12_1 ---------------------------------------- (155) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace IF1(true, s(0), s(x0), x1) -> c7(MIN(s(0), x1)) by IF1(true, s(0), s(z0), cons(0, y1)) -> c7(MIN(s(0), cons(0, y1))) IF1(true, s(0), s(z0), cons(s(y0), y1)) -> c7(MIN(s(0), cons(s(y0), y1))) ---------------------------------------- (156) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(z0, z2) if1(false, z0, z1, z2) -> min(z1, z2) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) 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) Tuples: MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2)) DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2)) MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(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) -> c7(MIN(s(s(x0)), x2)) IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1)) IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2)) MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2)) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) MINSORT(cons(z0, cons(z1, z2))) -> c13(DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c13(DEL(z0, cons(z0, nil))) IF2(false, 0, s(x0), x1) -> c10(DEL(0, x1)) IF2(false, s(x0), 0, x1) -> c10(DEL(s(x0), x1)) IF2(false, s(0), s(s(x0)), x1) -> c10(DEL(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c10(DEL(s(s(x0)), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(DEL(s(s(x0)), x2)) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2)) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c17 MINSORT(cons(0, cons(y0, y1))) -> c12(MIN(0, cons(y0, y1))) MINSORT(cons(s(y0), cons(0, y1))) -> c12(MIN(s(y0), cons(0, y1))) MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c12(MIN(s(s(y0)), cons(s(s(y1)), y2))) MINSORT(cons(s(y0), cons(s(y1), y2))) -> c12(MIN(s(y0), cons(s(y1), y2))) MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c12(MIN(s(s(y0)), cons(s(0), y1))) MINSORT(cons(s(0), cons(s(y0), y1))) -> c12(MIN(s(0), cons(s(y0), y1))) MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(true, 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c15(LE(s(s(y0)), s(s(y1)))) DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c17(EQ(s(s(y0)), s(s(y1)))) IF1(true, 0, z0, cons(y0, y1)) -> c7(MIN(0, cons(y0, y1))) MIN(0, cons(z0, cons(y1, y2))) -> c15(IF1(true, 0, z0, cons(y1, y2))) IF1(true, s(0), s(z0), cons(0, y1)) -> c7(MIN(s(0), cons(0, y1))) IF1(true, s(0), s(z0), cons(s(y0), y1)) -> c7(MIN(s(0), cons(s(y0), y1))) S tuples: MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2)) DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2)) MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(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) -> c7(MIN(s(s(x0)), x2)) IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1)) IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2)) MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2)) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF2(false, 0, s(x0), x1) -> c10(DEL(0, x1)) IF2(false, s(x0), 0, x1) -> c10(DEL(s(x0), x1)) IF2(false, s(0), s(s(x0)), x1) -> c10(DEL(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c10(DEL(s(s(x0)), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(DEL(s(s(x0)), x2)) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2)) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c17 MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(true, 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c15(LE(s(s(y0)), s(s(y1)))) DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c17(EQ(s(s(y0)), s(s(y1)))) IF1(true, 0, z0, cons(y0, y1)) -> c7(MIN(0, cons(y0, y1))) MIN(0, cons(z0, cons(y1, y2))) -> c15(IF1(true, 0, z0, cons(y1, y2))) IF1(true, s(0), s(z0), cons(0, y1)) -> c7(MIN(s(0), cons(0, y1))) IF1(true, s(0), s(z0), cons(s(y0), y1)) -> c7(MIN(s(0), cons(s(y0), y1))) K tuples: MINSORT(cons(0, cons(y0, y1))) -> c12(MIN(0, cons(y0, y1))) MINSORT(cons(s(y0), cons(0, y1))) -> c12(MIN(s(y0), cons(0, y1))) MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c12(MIN(s(s(y0)), cons(s(s(y1)), y2))) MINSORT(cons(s(y0), cons(s(y1), y2))) -> c12(MIN(s(y0), cons(s(y1), y2))) MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c12(MIN(s(s(y0)), cons(s(0), y1))) MINSORT(cons(s(0), cons(s(y0), y1))) -> c12(MIN(s(0), cons(s(y0), y1))) Defined Rule Symbols: del_2, min_2, if1_4, le_2, if2_4, eq_2 Defined Pair Symbols: MIN_2, DEL_2, IF1_4, MINSORT_1, IF2_4, LE_2, EQ_2 Compound Symbols: c15_1, c17_1, c15_2, c17_2, c7_1, c8_1, c_1, c13_3, c13_1, c13_2, c10_1, c2_1, c6_1, c17, c12_1 ---------------------------------------- (157) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) by MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) ---------------------------------------- (158) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(z0, z2) if1(false, z0, z1, z2) -> min(z1, z2) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) 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) Tuples: MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2)) DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2)) MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(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) -> c7(MIN(s(s(x0)), x2)) IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1)) IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2)) MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2)) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c13(DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c13(DEL(z0, cons(z0, nil))) IF2(false, 0, s(x0), x1) -> c10(DEL(0, x1)) IF2(false, s(x0), 0, x1) -> c10(DEL(s(x0), x1)) IF2(false, s(0), s(s(x0)), x1) -> c10(DEL(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c10(DEL(s(s(x0)), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(DEL(s(s(x0)), x2)) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2)) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c17 MINSORT(cons(0, cons(y0, y1))) -> c12(MIN(0, cons(y0, y1))) MINSORT(cons(s(y0), cons(0, y1))) -> c12(MIN(s(y0), cons(0, y1))) MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c12(MIN(s(s(y0)), cons(s(s(y1)), y2))) MINSORT(cons(s(y0), cons(s(y1), y2))) -> c12(MIN(s(y0), cons(s(y1), y2))) MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c12(MIN(s(s(y0)), cons(s(0), y1))) MINSORT(cons(s(0), cons(s(y0), y1))) -> c12(MIN(s(0), cons(s(y0), y1))) MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(true, 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c15(LE(s(s(y0)), s(s(y1)))) DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c17(EQ(s(s(y0)), s(s(y1)))) IF1(true, 0, z0, cons(y0, y1)) -> c7(MIN(0, cons(y0, y1))) MIN(0, cons(z0, cons(y1, y2))) -> c15(IF1(true, 0, z0, cons(y1, y2))) IF1(true, s(0), s(z0), cons(0, y1)) -> c7(MIN(s(0), cons(0, y1))) IF1(true, s(0), s(z0), cons(s(y0), y1)) -> c7(MIN(s(0), cons(s(y0), y1))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) S tuples: MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2)) DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2)) MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(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) -> c7(MIN(s(s(x0)), x2)) IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1)) IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2)) MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2)) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF2(false, 0, s(x0), x1) -> c10(DEL(0, x1)) IF2(false, s(x0), 0, x1) -> c10(DEL(s(x0), x1)) IF2(false, s(0), s(s(x0)), x1) -> c10(DEL(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c10(DEL(s(s(x0)), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(DEL(s(s(x0)), x2)) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2)) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c17 MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(true, 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c15(LE(s(s(y0)), s(s(y1)))) DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c17(EQ(s(s(y0)), s(s(y1)))) IF1(true, 0, z0, cons(y0, y1)) -> c7(MIN(0, cons(y0, y1))) MIN(0, cons(z0, cons(y1, y2))) -> c15(IF1(true, 0, z0, cons(y1, y2))) IF1(true, s(0), s(z0), cons(0, y1)) -> c7(MIN(s(0), cons(0, y1))) IF1(true, s(0), s(z0), cons(s(y0), y1)) -> c7(MIN(s(0), cons(s(y0), y1))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) K tuples: MINSORT(cons(0, cons(y0, y1))) -> c12(MIN(0, cons(y0, y1))) MINSORT(cons(s(y0), cons(0, y1))) -> c12(MIN(s(y0), cons(0, y1))) MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c12(MIN(s(s(y0)), cons(s(s(y1)), y2))) MINSORT(cons(s(y0), cons(s(y1), y2))) -> c12(MIN(s(y0), cons(s(y1), y2))) MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c12(MIN(s(s(y0)), cons(s(0), y1))) MINSORT(cons(s(0), cons(s(y0), y1))) -> c12(MIN(s(0), cons(s(y0), y1))) Defined Rule Symbols: del_2, min_2, if1_4, le_2, if2_4, eq_2 Defined Pair Symbols: MIN_2, DEL_2, IF1_4, MINSORT_1, IF2_4, LE_2, EQ_2 Compound Symbols: c15_1, c17_1, c15_2, c17_2, c7_1, c8_1, c_1, c13_3, c13_1, c13_2, c10_1, c2_1, c6_1, c17, c12_1 ---------------------------------------- (159) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace IF1(true, s(s(x0)), s(s(x1)), x2) -> c7(MIN(s(s(x0)), x2)) by IF1(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c7(MIN(s(s(z0)), cons(0, y1))) IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c7(MIN(s(s(z0)), cons(s(s(y1)), y2))) IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c7(MIN(s(s(z0)), cons(s(0), y1))) ---------------------------------------- (160) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(z0, z2) if1(false, z0, z1, z2) -> min(z1, z2) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) 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) Tuples: MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2)) DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2)) MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1)) IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2)) MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2)) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c13(DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c13(DEL(z0, cons(z0, nil))) IF2(false, 0, s(x0), x1) -> c10(DEL(0, x1)) IF2(false, s(x0), 0, x1) -> c10(DEL(s(x0), x1)) IF2(false, s(0), s(s(x0)), x1) -> c10(DEL(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c10(DEL(s(s(x0)), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(DEL(s(s(x0)), x2)) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2)) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c17 MINSORT(cons(0, cons(y0, y1))) -> c12(MIN(0, cons(y0, y1))) MINSORT(cons(s(y0), cons(0, y1))) -> c12(MIN(s(y0), cons(0, y1))) MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c12(MIN(s(s(y0)), cons(s(s(y1)), y2))) MINSORT(cons(s(y0), cons(s(y1), y2))) -> c12(MIN(s(y0), cons(s(y1), y2))) MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c12(MIN(s(s(y0)), cons(s(0), y1))) MINSORT(cons(s(0), cons(s(y0), y1))) -> c12(MIN(s(0), cons(s(y0), y1))) MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(true, 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c15(LE(s(s(y0)), s(s(y1)))) DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c17(EQ(s(s(y0)), s(s(y1)))) IF1(true, 0, z0, cons(y0, y1)) -> c7(MIN(0, cons(y0, y1))) MIN(0, cons(z0, cons(y1, y2))) -> c15(IF1(true, 0, z0, cons(y1, y2))) IF1(true, s(0), s(z0), cons(0, y1)) -> c7(MIN(s(0), cons(0, y1))) IF1(true, s(0), s(z0), cons(s(y0), y1)) -> c7(MIN(s(0), cons(s(y0), y1))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) IF1(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c7(MIN(s(s(z0)), cons(0, y1))) IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c7(MIN(s(s(z0)), cons(s(s(y1)), y2))) IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c7(MIN(s(s(z0)), cons(s(0), y1))) S tuples: MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2)) DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2)) MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1)) IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2)) MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2)) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF2(false, 0, s(x0), x1) -> c10(DEL(0, x1)) IF2(false, s(x0), 0, x1) -> c10(DEL(s(x0), x1)) IF2(false, s(0), s(s(x0)), x1) -> c10(DEL(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c10(DEL(s(s(x0)), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(DEL(s(s(x0)), x2)) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2)) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c17 MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(true, 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c15(LE(s(s(y0)), s(s(y1)))) DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c17(EQ(s(s(y0)), s(s(y1)))) IF1(true, 0, z0, cons(y0, y1)) -> c7(MIN(0, cons(y0, y1))) MIN(0, cons(z0, cons(y1, y2))) -> c15(IF1(true, 0, z0, cons(y1, y2))) IF1(true, s(0), s(z0), cons(0, y1)) -> c7(MIN(s(0), cons(0, y1))) IF1(true, s(0), s(z0), cons(s(y0), y1)) -> c7(MIN(s(0), cons(s(y0), y1))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) IF1(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c7(MIN(s(s(z0)), cons(0, y1))) IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c7(MIN(s(s(z0)), cons(s(s(y1)), y2))) IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c7(MIN(s(s(z0)), cons(s(0), y1))) K tuples: MINSORT(cons(0, cons(y0, y1))) -> c12(MIN(0, cons(y0, y1))) MINSORT(cons(s(y0), cons(0, y1))) -> c12(MIN(s(y0), cons(0, y1))) MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c12(MIN(s(s(y0)), cons(s(s(y1)), y2))) MINSORT(cons(s(y0), cons(s(y1), y2))) -> c12(MIN(s(y0), cons(s(y1), y2))) MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c12(MIN(s(s(y0)), cons(s(0), y1))) MINSORT(cons(s(0), cons(s(y0), y1))) -> c12(MIN(s(0), cons(s(y0), y1))) Defined Rule Symbols: del_2, min_2, if1_4, le_2, if2_4, eq_2 Defined Pair Symbols: MIN_2, DEL_2, IF1_4, MINSORT_1, IF2_4, LE_2, EQ_2 Compound Symbols: c15_1, c17_1, c15_2, c17_2, c8_1, c_1, c13_3, c13_1, c13_2, c10_1, c2_1, c6_1, c17, c12_1, c7_1 ---------------------------------------- (161) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace IF1(false, s(x0), 0, x1) -> c8(MIN(0, x1)) by IF1(false, s(z0), 0, cons(y0, cons(y1, y2))) -> c8(MIN(0, cons(y0, cons(y1, y2)))) ---------------------------------------- (162) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(z0, z2) if1(false, z0, z1, z2) -> min(z1, z2) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) 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) Tuples: MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2)) DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2)) MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2)) MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2)) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c13(DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c13(DEL(z0, cons(z0, nil))) IF2(false, 0, s(x0), x1) -> c10(DEL(0, x1)) IF2(false, s(x0), 0, x1) -> c10(DEL(s(x0), x1)) IF2(false, s(0), s(s(x0)), x1) -> c10(DEL(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c10(DEL(s(s(x0)), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(DEL(s(s(x0)), x2)) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2)) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c17 MINSORT(cons(0, cons(y0, y1))) -> c12(MIN(0, cons(y0, y1))) MINSORT(cons(s(y0), cons(0, y1))) -> c12(MIN(s(y0), cons(0, y1))) MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c12(MIN(s(s(y0)), cons(s(s(y1)), y2))) MINSORT(cons(s(y0), cons(s(y1), y2))) -> c12(MIN(s(y0), cons(s(y1), y2))) MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c12(MIN(s(s(y0)), cons(s(0), y1))) MINSORT(cons(s(0), cons(s(y0), y1))) -> c12(MIN(s(0), cons(s(y0), y1))) MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(true, 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c15(LE(s(s(y0)), s(s(y1)))) DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c17(EQ(s(s(y0)), s(s(y1)))) IF1(true, 0, z0, cons(y0, y1)) -> c7(MIN(0, cons(y0, y1))) MIN(0, cons(z0, cons(y1, y2))) -> c15(IF1(true, 0, z0, cons(y1, y2))) IF1(true, s(0), s(z0), cons(0, y1)) -> c7(MIN(s(0), cons(0, y1))) IF1(true, s(0), s(z0), cons(s(y0), y1)) -> c7(MIN(s(0), cons(s(y0), y1))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) IF1(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c7(MIN(s(s(z0)), cons(0, y1))) IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c7(MIN(s(s(z0)), cons(s(s(y1)), y2))) IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c7(MIN(s(s(z0)), cons(s(0), y1))) IF1(false, s(z0), 0, cons(y0, cons(y1, y2))) -> c8(MIN(0, cons(y0, cons(y1, y2)))) S tuples: MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2)) DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2)) MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2)) MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2)) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF2(false, 0, s(x0), x1) -> c10(DEL(0, x1)) IF2(false, s(x0), 0, x1) -> c10(DEL(s(x0), x1)) IF2(false, s(0), s(s(x0)), x1) -> c10(DEL(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c10(DEL(s(s(x0)), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(DEL(s(s(x0)), x2)) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2)) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c17 MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(true, 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c15(LE(s(s(y0)), s(s(y1)))) DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c17(EQ(s(s(y0)), s(s(y1)))) IF1(true, 0, z0, cons(y0, y1)) -> c7(MIN(0, cons(y0, y1))) MIN(0, cons(z0, cons(y1, y2))) -> c15(IF1(true, 0, z0, cons(y1, y2))) IF1(true, s(0), s(z0), cons(0, y1)) -> c7(MIN(s(0), cons(0, y1))) IF1(true, s(0), s(z0), cons(s(y0), y1)) -> c7(MIN(s(0), cons(s(y0), y1))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) IF1(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c7(MIN(s(s(z0)), cons(0, y1))) IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c7(MIN(s(s(z0)), cons(s(s(y1)), y2))) IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c7(MIN(s(s(z0)), cons(s(0), y1))) IF1(false, s(z0), 0, cons(y0, cons(y1, y2))) -> c8(MIN(0, cons(y0, cons(y1, y2)))) K tuples: MINSORT(cons(0, cons(y0, y1))) -> c12(MIN(0, cons(y0, y1))) MINSORT(cons(s(y0), cons(0, y1))) -> c12(MIN(s(y0), cons(0, y1))) MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c12(MIN(s(s(y0)), cons(s(s(y1)), y2))) MINSORT(cons(s(y0), cons(s(y1), y2))) -> c12(MIN(s(y0), cons(s(y1), y2))) MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c12(MIN(s(s(y0)), cons(s(0), y1))) MINSORT(cons(s(0), cons(s(y0), y1))) -> c12(MIN(s(0), cons(s(y0), y1))) Defined Rule Symbols: del_2, min_2, if1_4, le_2, if2_4, eq_2 Defined Pair Symbols: MIN_2, DEL_2, IF1_4, MINSORT_1, IF2_4, LE_2, EQ_2 Compound Symbols: c15_1, c17_1, c15_2, c17_2, c8_1, c_1, c13_3, c13_1, c13_2, c10_1, c2_1, c6_1, c17, c12_1, c7_1 ---------------------------------------- (163) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace MIN(s(z0), cons(0, x2)) -> c15(IF1(false, s(z0), 0, x2)) by MIN(s(z0), cons(0, cons(y1, cons(y2, y3)))) -> c15(IF1(false, s(z0), 0, cons(y1, cons(y2, y3)))) ---------------------------------------- (164) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(z0, z2) if1(false, z0, z1, z2) -> min(z1, z2) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) 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) Tuples: DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2)) MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2)) MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2)) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c13(DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c13(DEL(z0, cons(z0, nil))) IF2(false, 0, s(x0), x1) -> c10(DEL(0, x1)) IF2(false, s(x0), 0, x1) -> c10(DEL(s(x0), x1)) IF2(false, s(0), s(s(x0)), x1) -> c10(DEL(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c10(DEL(s(s(x0)), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(DEL(s(s(x0)), x2)) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2)) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c17 MINSORT(cons(0, cons(y0, y1))) -> c12(MIN(0, cons(y0, y1))) MINSORT(cons(s(y0), cons(0, y1))) -> c12(MIN(s(y0), cons(0, y1))) MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c12(MIN(s(s(y0)), cons(s(s(y1)), y2))) MINSORT(cons(s(y0), cons(s(y1), y2))) -> c12(MIN(s(y0), cons(s(y1), y2))) MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c12(MIN(s(s(y0)), cons(s(0), y1))) MINSORT(cons(s(0), cons(s(y0), y1))) -> c12(MIN(s(0), cons(s(y0), y1))) MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(true, 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c15(LE(s(s(y0)), s(s(y1)))) DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c17(EQ(s(s(y0)), s(s(y1)))) IF1(true, 0, z0, cons(y0, y1)) -> c7(MIN(0, cons(y0, y1))) MIN(0, cons(z0, cons(y1, y2))) -> c15(IF1(true, 0, z0, cons(y1, y2))) IF1(true, s(0), s(z0), cons(0, y1)) -> c7(MIN(s(0), cons(0, y1))) IF1(true, s(0), s(z0), cons(s(y0), y1)) -> c7(MIN(s(0), cons(s(y0), y1))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) IF1(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c7(MIN(s(s(z0)), cons(0, y1))) IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c7(MIN(s(s(z0)), cons(s(s(y1)), y2))) IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c7(MIN(s(s(z0)), cons(s(0), y1))) IF1(false, s(z0), 0, cons(y0, cons(y1, y2))) -> c8(MIN(0, cons(y0, cons(y1, y2)))) MIN(s(z0), cons(0, cons(y1, cons(y2, y3)))) -> c15(IF1(false, s(z0), 0, cons(y1, cons(y2, y3)))) S tuples: DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2)) MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2)) MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2)) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF2(false, 0, s(x0), x1) -> c10(DEL(0, x1)) IF2(false, s(x0), 0, x1) -> c10(DEL(s(x0), x1)) IF2(false, s(0), s(s(x0)), x1) -> c10(DEL(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c10(DEL(s(s(x0)), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(DEL(s(s(x0)), x2)) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2)) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c17 MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(true, 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c15(LE(s(s(y0)), s(s(y1)))) DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c17(EQ(s(s(y0)), s(s(y1)))) IF1(true, 0, z0, cons(y0, y1)) -> c7(MIN(0, cons(y0, y1))) MIN(0, cons(z0, cons(y1, y2))) -> c15(IF1(true, 0, z0, cons(y1, y2))) IF1(true, s(0), s(z0), cons(0, y1)) -> c7(MIN(s(0), cons(0, y1))) IF1(true, s(0), s(z0), cons(s(y0), y1)) -> c7(MIN(s(0), cons(s(y0), y1))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) IF1(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c7(MIN(s(s(z0)), cons(0, y1))) IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c7(MIN(s(s(z0)), cons(s(s(y1)), y2))) IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c7(MIN(s(s(z0)), cons(s(0), y1))) IF1(false, s(z0), 0, cons(y0, cons(y1, y2))) -> c8(MIN(0, cons(y0, cons(y1, y2)))) MIN(s(z0), cons(0, cons(y1, cons(y2, y3)))) -> c15(IF1(false, s(z0), 0, cons(y1, cons(y2, y3)))) K tuples: MINSORT(cons(0, cons(y0, y1))) -> c12(MIN(0, cons(y0, y1))) MINSORT(cons(s(y0), cons(0, y1))) -> c12(MIN(s(y0), cons(0, y1))) MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c12(MIN(s(s(y0)), cons(s(s(y1)), y2))) MINSORT(cons(s(y0), cons(s(y1), y2))) -> c12(MIN(s(y0), cons(s(y1), y2))) MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c12(MIN(s(s(y0)), cons(s(0), y1))) MINSORT(cons(s(0), cons(s(y0), y1))) -> c12(MIN(s(0), cons(s(y0), y1))) Defined Rule Symbols: del_2, min_2, if1_4, le_2, if2_4, eq_2 Defined Pair Symbols: DEL_2, MIN_2, IF1_4, MINSORT_1, IF2_4, LE_2, EQ_2 Compound Symbols: c17_1, c15_2, c17_2, c8_1, c_1, c13_3, c13_1, c13_2, c10_1, c2_1, c15_1, c6_1, c17, c12_1, c7_1 ---------------------------------------- (165) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace IF1(false, s(s(x0)), s(0), x1) -> c8(MIN(s(0), x1)) by IF1(false, s(s(z0)), s(0), cons(s(y0), y1)) -> c8(MIN(s(0), cons(s(y0), y1))) IF1(false, s(s(z0)), s(0), cons(0, cons(y1, cons(y2, y3)))) -> c8(MIN(s(0), cons(0, cons(y1, cons(y2, y3))))) ---------------------------------------- (166) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(z0, z2) if1(false, z0, z1, z2) -> min(z1, z2) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) 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) Tuples: DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2)) MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(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) -> c8(MIN(s(s(x1)), x2)) MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2)) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c13(DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c13(DEL(z0, cons(z0, nil))) IF2(false, 0, s(x0), x1) -> c10(DEL(0, x1)) IF2(false, s(x0), 0, x1) -> c10(DEL(s(x0), x1)) IF2(false, s(0), s(s(x0)), x1) -> c10(DEL(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c10(DEL(s(s(x0)), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(DEL(s(s(x0)), x2)) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2)) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c17 MINSORT(cons(0, cons(y0, y1))) -> c12(MIN(0, cons(y0, y1))) MINSORT(cons(s(y0), cons(0, y1))) -> c12(MIN(s(y0), cons(0, y1))) MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c12(MIN(s(s(y0)), cons(s(s(y1)), y2))) MINSORT(cons(s(y0), cons(s(y1), y2))) -> c12(MIN(s(y0), cons(s(y1), y2))) MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c12(MIN(s(s(y0)), cons(s(0), y1))) MINSORT(cons(s(0), cons(s(y0), y1))) -> c12(MIN(s(0), cons(s(y0), y1))) MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(true, 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c15(LE(s(s(y0)), s(s(y1)))) DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c17(EQ(s(s(y0)), s(s(y1)))) IF1(true, 0, z0, cons(y0, y1)) -> c7(MIN(0, cons(y0, y1))) MIN(0, cons(z0, cons(y1, y2))) -> c15(IF1(true, 0, z0, cons(y1, y2))) IF1(true, s(0), s(z0), cons(0, y1)) -> c7(MIN(s(0), cons(0, y1))) IF1(true, s(0), s(z0), cons(s(y0), y1)) -> c7(MIN(s(0), cons(s(y0), y1))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) IF1(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c7(MIN(s(s(z0)), cons(0, y1))) IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c7(MIN(s(s(z0)), cons(s(s(y1)), y2))) IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c7(MIN(s(s(z0)), cons(s(0), y1))) IF1(false, s(z0), 0, cons(y0, cons(y1, y2))) -> c8(MIN(0, cons(y0, cons(y1, y2)))) MIN(s(z0), cons(0, cons(y1, cons(y2, y3)))) -> c15(IF1(false, s(z0), 0, cons(y1, cons(y2, y3)))) IF1(false, s(s(z0)), s(0), cons(s(y0), y1)) -> c8(MIN(s(0), cons(s(y0), y1))) IF1(false, s(s(z0)), s(0), cons(0, cons(y1, cons(y2, y3)))) -> c8(MIN(s(0), cons(0, cons(y1, cons(y2, y3))))) S tuples: DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2)) MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(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) -> c8(MIN(s(s(x1)), x2)) MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2)) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF2(false, 0, s(x0), x1) -> c10(DEL(0, x1)) IF2(false, s(x0), 0, x1) -> c10(DEL(s(x0), x1)) IF2(false, s(0), s(s(x0)), x1) -> c10(DEL(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c10(DEL(s(s(x0)), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(DEL(s(s(x0)), x2)) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2)) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c17 MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(true, 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c15(LE(s(s(y0)), s(s(y1)))) DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c17(EQ(s(s(y0)), s(s(y1)))) IF1(true, 0, z0, cons(y0, y1)) -> c7(MIN(0, cons(y0, y1))) MIN(0, cons(z0, cons(y1, y2))) -> c15(IF1(true, 0, z0, cons(y1, y2))) IF1(true, s(0), s(z0), cons(0, y1)) -> c7(MIN(s(0), cons(0, y1))) IF1(true, s(0), s(z0), cons(s(y0), y1)) -> c7(MIN(s(0), cons(s(y0), y1))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) IF1(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c7(MIN(s(s(z0)), cons(0, y1))) IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c7(MIN(s(s(z0)), cons(s(s(y1)), y2))) IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c7(MIN(s(s(z0)), cons(s(0), y1))) IF1(false, s(z0), 0, cons(y0, cons(y1, y2))) -> c8(MIN(0, cons(y0, cons(y1, y2)))) MIN(s(z0), cons(0, cons(y1, cons(y2, y3)))) -> c15(IF1(false, s(z0), 0, cons(y1, cons(y2, y3)))) IF1(false, s(s(z0)), s(0), cons(s(y0), y1)) -> c8(MIN(s(0), cons(s(y0), y1))) IF1(false, s(s(z0)), s(0), cons(0, cons(y1, cons(y2, y3)))) -> c8(MIN(s(0), cons(0, cons(y1, cons(y2, y3))))) K tuples: MINSORT(cons(0, cons(y0, y1))) -> c12(MIN(0, cons(y0, y1))) MINSORT(cons(s(y0), cons(0, y1))) -> c12(MIN(s(y0), cons(0, y1))) MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c12(MIN(s(s(y0)), cons(s(s(y1)), y2))) MINSORT(cons(s(y0), cons(s(y1), y2))) -> c12(MIN(s(y0), cons(s(y1), y2))) MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c12(MIN(s(s(y0)), cons(s(0), y1))) MINSORT(cons(s(0), cons(s(y0), y1))) -> c12(MIN(s(0), cons(s(y0), y1))) Defined Rule Symbols: del_2, min_2, if1_4, le_2, if2_4, eq_2 Defined Pair Symbols: DEL_2, MIN_2, IF1_4, MINSORT_1, IF2_4, LE_2, EQ_2 Compound Symbols: c17_1, c15_2, c17_2, c8_1, c_1, c13_3, c13_1, c13_2, c10_1, c2_1, c15_1, c6_1, c17, c12_1, c7_1 ---------------------------------------- (167) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace IF1(false, s(s(x0)), s(s(x1)), x2) -> c8(MIN(s(s(x1)), x2)) by IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c8(MIN(s(s(z1)), cons(s(s(y1)), y2))) IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c8(MIN(s(s(z1)), cons(s(0), y1))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(y1, cons(y2, y3)))) -> c8(MIN(s(s(z1)), cons(0, cons(y1, cons(y2, y3))))) ---------------------------------------- (168) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(z0, z2) if1(false, z0, z1, z2) -> min(z1, z2) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) 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) Tuples: DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2)) MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2)) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c13(DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c13(DEL(z0, cons(z0, nil))) IF2(false, 0, s(x0), x1) -> c10(DEL(0, x1)) IF2(false, s(x0), 0, x1) -> c10(DEL(s(x0), x1)) IF2(false, s(0), s(s(x0)), x1) -> c10(DEL(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c10(DEL(s(s(x0)), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(DEL(s(s(x0)), x2)) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2)) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c17 MINSORT(cons(0, cons(y0, y1))) -> c12(MIN(0, cons(y0, y1))) MINSORT(cons(s(y0), cons(0, y1))) -> c12(MIN(s(y0), cons(0, y1))) MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c12(MIN(s(s(y0)), cons(s(s(y1)), y2))) MINSORT(cons(s(y0), cons(s(y1), y2))) -> c12(MIN(s(y0), cons(s(y1), y2))) MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c12(MIN(s(s(y0)), cons(s(0), y1))) MINSORT(cons(s(0), cons(s(y0), y1))) -> c12(MIN(s(0), cons(s(y0), y1))) MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(true, 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c15(LE(s(s(y0)), s(s(y1)))) DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c17(EQ(s(s(y0)), s(s(y1)))) IF1(true, 0, z0, cons(y0, y1)) -> c7(MIN(0, cons(y0, y1))) MIN(0, cons(z0, cons(y1, y2))) -> c15(IF1(true, 0, z0, cons(y1, y2))) IF1(true, s(0), s(z0), cons(0, y1)) -> c7(MIN(s(0), cons(0, y1))) IF1(true, s(0), s(z0), cons(s(y0), y1)) -> c7(MIN(s(0), cons(s(y0), y1))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) IF1(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c7(MIN(s(s(z0)), cons(0, y1))) IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c7(MIN(s(s(z0)), cons(s(s(y1)), y2))) IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c7(MIN(s(s(z0)), cons(s(0), y1))) IF1(false, s(z0), 0, cons(y0, cons(y1, y2))) -> c8(MIN(0, cons(y0, cons(y1, y2)))) MIN(s(z0), cons(0, cons(y1, cons(y2, y3)))) -> c15(IF1(false, s(z0), 0, cons(y1, cons(y2, y3)))) IF1(false, s(s(z0)), s(0), cons(s(y0), y1)) -> c8(MIN(s(0), cons(s(y0), y1))) IF1(false, s(s(z0)), s(0), cons(0, cons(y1, cons(y2, y3)))) -> c8(MIN(s(0), cons(0, cons(y1, cons(y2, y3))))) IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c8(MIN(s(s(z1)), cons(s(s(y1)), y2))) IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c8(MIN(s(s(z1)), cons(s(0), y1))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(y1, cons(y2, y3)))) -> c8(MIN(s(s(z1)), cons(0, cons(y1, cons(y2, y3))))) S tuples: DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2)) MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2)) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF2(false, 0, s(x0), x1) -> c10(DEL(0, x1)) IF2(false, s(x0), 0, x1) -> c10(DEL(s(x0), x1)) IF2(false, s(0), s(s(x0)), x1) -> c10(DEL(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c10(DEL(s(s(x0)), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(DEL(s(s(x0)), x2)) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2)) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c17 MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(true, 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c15(LE(s(s(y0)), s(s(y1)))) DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c17(EQ(s(s(y0)), s(s(y1)))) IF1(true, 0, z0, cons(y0, y1)) -> c7(MIN(0, cons(y0, y1))) MIN(0, cons(z0, cons(y1, y2))) -> c15(IF1(true, 0, z0, cons(y1, y2))) IF1(true, s(0), s(z0), cons(0, y1)) -> c7(MIN(s(0), cons(0, y1))) IF1(true, s(0), s(z0), cons(s(y0), y1)) -> c7(MIN(s(0), cons(s(y0), y1))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) IF1(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c7(MIN(s(s(z0)), cons(0, y1))) IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c7(MIN(s(s(z0)), cons(s(s(y1)), y2))) IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c7(MIN(s(s(z0)), cons(s(0), y1))) IF1(false, s(z0), 0, cons(y0, cons(y1, y2))) -> c8(MIN(0, cons(y0, cons(y1, y2)))) MIN(s(z0), cons(0, cons(y1, cons(y2, y3)))) -> c15(IF1(false, s(z0), 0, cons(y1, cons(y2, y3)))) IF1(false, s(s(z0)), s(0), cons(s(y0), y1)) -> c8(MIN(s(0), cons(s(y0), y1))) IF1(false, s(s(z0)), s(0), cons(0, cons(y1, cons(y2, y3)))) -> c8(MIN(s(0), cons(0, cons(y1, cons(y2, y3))))) IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c8(MIN(s(s(z1)), cons(s(s(y1)), y2))) IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c8(MIN(s(s(z1)), cons(s(0), y1))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(y1, cons(y2, y3)))) -> c8(MIN(s(s(z1)), cons(0, cons(y1, cons(y2, y3))))) K tuples: MINSORT(cons(0, cons(y0, y1))) -> c12(MIN(0, cons(y0, y1))) MINSORT(cons(s(y0), cons(0, y1))) -> c12(MIN(s(y0), cons(0, y1))) MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c12(MIN(s(s(y0)), cons(s(s(y1)), y2))) MINSORT(cons(s(y0), cons(s(y1), y2))) -> c12(MIN(s(y0), cons(s(y1), y2))) MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c12(MIN(s(s(y0)), cons(s(0), y1))) MINSORT(cons(s(0), cons(s(y0), y1))) -> c12(MIN(s(0), cons(s(y0), y1))) Defined Rule Symbols: del_2, min_2, if1_4, le_2, if2_4, eq_2 Defined Pair Symbols: DEL_2, MIN_2, MINSORT_1, IF2_4, LE_2, EQ_2, IF1_4 Compound Symbols: c17_1, c15_2, c17_2, c_1, c13_3, c13_1, c13_2, c10_1, c2_1, c15_1, c6_1, c17, c12_1, c7_1, c8_1 ---------------------------------------- (169) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace MIN(s(s(z0)), cons(s(0), x2)) -> c(IF1(false, s(s(z0)), s(0), x2)) by MIN(s(s(z0)), cons(s(0), cons(s(y1), y2))) -> c(IF1(false, s(s(z0)), s(0), cons(s(y1), y2))) MIN(s(s(z0)), cons(s(0), cons(0, cons(y1, cons(y2, y3))))) -> c(IF1(false, s(s(z0)), s(0), cons(0, cons(y1, cons(y2, y3))))) ---------------------------------------- (170) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(z0, z2) if1(false, z0, z1, z2) -> min(z1, z2) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) 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) Tuples: DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2)) MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c13(DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c13(DEL(z0, cons(z0, nil))) IF2(false, 0, s(x0), x1) -> c10(DEL(0, x1)) IF2(false, s(x0), 0, x1) -> c10(DEL(s(x0), x1)) IF2(false, s(0), s(s(x0)), x1) -> c10(DEL(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c10(DEL(s(s(x0)), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(DEL(s(s(x0)), x2)) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2)) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c17 MINSORT(cons(0, cons(y0, y1))) -> c12(MIN(0, cons(y0, y1))) MINSORT(cons(s(y0), cons(0, y1))) -> c12(MIN(s(y0), cons(0, y1))) MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c12(MIN(s(s(y0)), cons(s(s(y1)), y2))) MINSORT(cons(s(y0), cons(s(y1), y2))) -> c12(MIN(s(y0), cons(s(y1), y2))) MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c12(MIN(s(s(y0)), cons(s(0), y1))) MINSORT(cons(s(0), cons(s(y0), y1))) -> c12(MIN(s(0), cons(s(y0), y1))) MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(true, 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c15(LE(s(s(y0)), s(s(y1)))) DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c17(EQ(s(s(y0)), s(s(y1)))) IF1(true, 0, z0, cons(y0, y1)) -> c7(MIN(0, cons(y0, y1))) MIN(0, cons(z0, cons(y1, y2))) -> c15(IF1(true, 0, z0, cons(y1, y2))) IF1(true, s(0), s(z0), cons(0, y1)) -> c7(MIN(s(0), cons(0, y1))) IF1(true, s(0), s(z0), cons(s(y0), y1)) -> c7(MIN(s(0), cons(s(y0), y1))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) IF1(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c7(MIN(s(s(z0)), cons(0, y1))) IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c7(MIN(s(s(z0)), cons(s(s(y1)), y2))) IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c7(MIN(s(s(z0)), cons(s(0), y1))) IF1(false, s(z0), 0, cons(y0, cons(y1, y2))) -> c8(MIN(0, cons(y0, cons(y1, y2)))) MIN(s(z0), cons(0, cons(y1, cons(y2, y3)))) -> c15(IF1(false, s(z0), 0, cons(y1, cons(y2, y3)))) IF1(false, s(s(z0)), s(0), cons(s(y0), y1)) -> c8(MIN(s(0), cons(s(y0), y1))) IF1(false, s(s(z0)), s(0), cons(0, cons(y1, cons(y2, y3)))) -> c8(MIN(s(0), cons(0, cons(y1, cons(y2, y3))))) IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c8(MIN(s(s(z1)), cons(s(s(y1)), y2))) IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c8(MIN(s(s(z1)), cons(s(0), y1))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(y1, cons(y2, y3)))) -> c8(MIN(s(s(z1)), cons(0, cons(y1, cons(y2, y3))))) MIN(s(s(z0)), cons(s(0), cons(s(y1), y2))) -> c(IF1(false, s(s(z0)), s(0), cons(s(y1), y2))) MIN(s(s(z0)), cons(s(0), cons(0, cons(y1, cons(y2, y3))))) -> c(IF1(false, s(s(z0)), s(0), cons(0, cons(y1, cons(y2, y3))))) S tuples: DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2)) MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF2(false, 0, s(x0), x1) -> c10(DEL(0, x1)) IF2(false, s(x0), 0, x1) -> c10(DEL(s(x0), x1)) IF2(false, s(0), s(s(x0)), x1) -> c10(DEL(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c10(DEL(s(s(x0)), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(DEL(s(s(x0)), x2)) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2)) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c17 MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(true, 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c15(LE(s(s(y0)), s(s(y1)))) DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c17(EQ(s(s(y0)), s(s(y1)))) IF1(true, 0, z0, cons(y0, y1)) -> c7(MIN(0, cons(y0, y1))) MIN(0, cons(z0, cons(y1, y2))) -> c15(IF1(true, 0, z0, cons(y1, y2))) IF1(true, s(0), s(z0), cons(0, y1)) -> c7(MIN(s(0), cons(0, y1))) IF1(true, s(0), s(z0), cons(s(y0), y1)) -> c7(MIN(s(0), cons(s(y0), y1))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) IF1(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c7(MIN(s(s(z0)), cons(0, y1))) IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c7(MIN(s(s(z0)), cons(s(s(y1)), y2))) IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c7(MIN(s(s(z0)), cons(s(0), y1))) IF1(false, s(z0), 0, cons(y0, cons(y1, y2))) -> c8(MIN(0, cons(y0, cons(y1, y2)))) MIN(s(z0), cons(0, cons(y1, cons(y2, y3)))) -> c15(IF1(false, s(z0), 0, cons(y1, cons(y2, y3)))) IF1(false, s(s(z0)), s(0), cons(s(y0), y1)) -> c8(MIN(s(0), cons(s(y0), y1))) IF1(false, s(s(z0)), s(0), cons(0, cons(y1, cons(y2, y3)))) -> c8(MIN(s(0), cons(0, cons(y1, cons(y2, y3))))) IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c8(MIN(s(s(z1)), cons(s(s(y1)), y2))) IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c8(MIN(s(s(z1)), cons(s(0), y1))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(y1, cons(y2, y3)))) -> c8(MIN(s(s(z1)), cons(0, cons(y1, cons(y2, y3))))) MIN(s(s(z0)), cons(s(0), cons(s(y1), y2))) -> c(IF1(false, s(s(z0)), s(0), cons(s(y1), y2))) MIN(s(s(z0)), cons(s(0), cons(0, cons(y1, cons(y2, y3))))) -> c(IF1(false, s(s(z0)), s(0), cons(0, cons(y1, cons(y2, y3))))) K tuples: MINSORT(cons(0, cons(y0, y1))) -> c12(MIN(0, cons(y0, y1))) MINSORT(cons(s(y0), cons(0, y1))) -> c12(MIN(s(y0), cons(0, y1))) MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c12(MIN(s(s(y0)), cons(s(s(y1)), y2))) MINSORT(cons(s(y0), cons(s(y1), y2))) -> c12(MIN(s(y0), cons(s(y1), y2))) MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c12(MIN(s(s(y0)), cons(s(0), y1))) MINSORT(cons(s(0), cons(s(y0), y1))) -> c12(MIN(s(0), cons(s(y0), y1))) Defined Rule Symbols: del_2, min_2, if1_4, le_2, if2_4, eq_2 Defined Pair Symbols: DEL_2, MIN_2, MINSORT_1, IF2_4, LE_2, EQ_2, IF1_4 Compound Symbols: c17_1, c15_2, c17_2, c13_3, c13_1, c13_2, c10_1, c2_1, c15_1, c6_1, c17, c12_1, c7_1, c8_1, c_1 ---------------------------------------- (171) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace MINSORT(cons(z0, nil)) -> c13(DEL(z0, cons(z0, nil))) by MINSORT(cons(s(s(y0)), nil)) -> c13(DEL(s(s(y0)), cons(s(s(y0)), nil))) MINSORT(cons(s(0), nil)) -> c13(DEL(s(0), cons(s(0), nil))) ---------------------------------------- (172) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(z0, z2) if1(false, z0, z1, z2) -> min(z1, z2) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) 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) Tuples: DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2)) MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c13(DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF2(false, 0, s(x0), x1) -> c10(DEL(0, x1)) IF2(false, s(x0), 0, x1) -> c10(DEL(s(x0), x1)) IF2(false, s(0), s(s(x0)), x1) -> c10(DEL(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c10(DEL(s(s(x0)), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(DEL(s(s(x0)), x2)) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2)) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c17 MINSORT(cons(0, cons(y0, y1))) -> c12(MIN(0, cons(y0, y1))) MINSORT(cons(s(y0), cons(0, y1))) -> c12(MIN(s(y0), cons(0, y1))) MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c12(MIN(s(s(y0)), cons(s(s(y1)), y2))) MINSORT(cons(s(y0), cons(s(y1), y2))) -> c12(MIN(s(y0), cons(s(y1), y2))) MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c12(MIN(s(s(y0)), cons(s(0), y1))) MINSORT(cons(s(0), cons(s(y0), y1))) -> c12(MIN(s(0), cons(s(y0), y1))) MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(true, 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c15(LE(s(s(y0)), s(s(y1)))) DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c17(EQ(s(s(y0)), s(s(y1)))) IF1(true, 0, z0, cons(y0, y1)) -> c7(MIN(0, cons(y0, y1))) MIN(0, cons(z0, cons(y1, y2))) -> c15(IF1(true, 0, z0, cons(y1, y2))) IF1(true, s(0), s(z0), cons(0, y1)) -> c7(MIN(s(0), cons(0, y1))) IF1(true, s(0), s(z0), cons(s(y0), y1)) -> c7(MIN(s(0), cons(s(y0), y1))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) IF1(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c7(MIN(s(s(z0)), cons(0, y1))) IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c7(MIN(s(s(z0)), cons(s(s(y1)), y2))) IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c7(MIN(s(s(z0)), cons(s(0), y1))) IF1(false, s(z0), 0, cons(y0, cons(y1, y2))) -> c8(MIN(0, cons(y0, cons(y1, y2)))) MIN(s(z0), cons(0, cons(y1, cons(y2, y3)))) -> c15(IF1(false, s(z0), 0, cons(y1, cons(y2, y3)))) IF1(false, s(s(z0)), s(0), cons(s(y0), y1)) -> c8(MIN(s(0), cons(s(y0), y1))) IF1(false, s(s(z0)), s(0), cons(0, cons(y1, cons(y2, y3)))) -> c8(MIN(s(0), cons(0, cons(y1, cons(y2, y3))))) IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c8(MIN(s(s(z1)), cons(s(s(y1)), y2))) IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c8(MIN(s(s(z1)), cons(s(0), y1))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(y1, cons(y2, y3)))) -> c8(MIN(s(s(z1)), cons(0, cons(y1, cons(y2, y3))))) MIN(s(s(z0)), cons(s(0), cons(s(y1), y2))) -> c(IF1(false, s(s(z0)), s(0), cons(s(y1), y2))) MIN(s(s(z0)), cons(s(0), cons(0, cons(y1, cons(y2, y3))))) -> c(IF1(false, s(s(z0)), s(0), cons(0, cons(y1, cons(y2, y3))))) MINSORT(cons(s(s(y0)), nil)) -> c13(DEL(s(s(y0)), cons(s(s(y0)), nil))) MINSORT(cons(s(0), nil)) -> c13(DEL(s(0), cons(s(0), nil))) S tuples: DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2)) MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF2(false, 0, s(x0), x1) -> c10(DEL(0, x1)) IF2(false, s(x0), 0, x1) -> c10(DEL(s(x0), x1)) IF2(false, s(0), s(s(x0)), x1) -> c10(DEL(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c10(DEL(s(s(x0)), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(DEL(s(s(x0)), x2)) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2)) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c17 MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(true, 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c15(LE(s(s(y0)), s(s(y1)))) DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c17(EQ(s(s(y0)), s(s(y1)))) IF1(true, 0, z0, cons(y0, y1)) -> c7(MIN(0, cons(y0, y1))) MIN(0, cons(z0, cons(y1, y2))) -> c15(IF1(true, 0, z0, cons(y1, y2))) IF1(true, s(0), s(z0), cons(0, y1)) -> c7(MIN(s(0), cons(0, y1))) IF1(true, s(0), s(z0), cons(s(y0), y1)) -> c7(MIN(s(0), cons(s(y0), y1))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) IF1(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c7(MIN(s(s(z0)), cons(0, y1))) IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c7(MIN(s(s(z0)), cons(s(s(y1)), y2))) IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c7(MIN(s(s(z0)), cons(s(0), y1))) IF1(false, s(z0), 0, cons(y0, cons(y1, y2))) -> c8(MIN(0, cons(y0, cons(y1, y2)))) MIN(s(z0), cons(0, cons(y1, cons(y2, y3)))) -> c15(IF1(false, s(z0), 0, cons(y1, cons(y2, y3)))) IF1(false, s(s(z0)), s(0), cons(s(y0), y1)) -> c8(MIN(s(0), cons(s(y0), y1))) IF1(false, s(s(z0)), s(0), cons(0, cons(y1, cons(y2, y3)))) -> c8(MIN(s(0), cons(0, cons(y1, cons(y2, y3))))) IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c8(MIN(s(s(z1)), cons(s(s(y1)), y2))) IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c8(MIN(s(s(z1)), cons(s(0), y1))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(y1, cons(y2, y3)))) -> c8(MIN(s(s(z1)), cons(0, cons(y1, cons(y2, y3))))) MIN(s(s(z0)), cons(s(0), cons(s(y1), y2))) -> c(IF1(false, s(s(z0)), s(0), cons(s(y1), y2))) MIN(s(s(z0)), cons(s(0), cons(0, cons(y1, cons(y2, y3))))) -> c(IF1(false, s(s(z0)), s(0), cons(0, cons(y1, cons(y2, y3))))) K tuples: MINSORT(cons(0, cons(y0, y1))) -> c12(MIN(0, cons(y0, y1))) MINSORT(cons(s(y0), cons(0, y1))) -> c12(MIN(s(y0), cons(0, y1))) MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c12(MIN(s(s(y0)), cons(s(s(y1)), y2))) MINSORT(cons(s(y0), cons(s(y1), y2))) -> c12(MIN(s(y0), cons(s(y1), y2))) MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c12(MIN(s(s(y0)), cons(s(0), y1))) MINSORT(cons(s(0), cons(s(y0), y1))) -> c12(MIN(s(0), cons(s(y0), y1))) Defined Rule Symbols: del_2, min_2, if1_4, le_2, if2_4, eq_2 Defined Pair Symbols: DEL_2, MIN_2, MINSORT_1, IF2_4, LE_2, EQ_2, IF1_4 Compound Symbols: c17_1, c15_2, c17_2, c13_3, c13_1, c13_2, c10_1, c2_1, c15_1, c6_1, c17, c12_1, c7_1, c8_1, c_1 ---------------------------------------- (173) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace IF2(false, 0, s(x0), x1) -> c10(DEL(0, x1)) by IF2(false, 0, s(z0), cons(s(y0), y1)) -> c10(DEL(0, cons(s(y0), y1))) ---------------------------------------- (174) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(z0, z2) if1(false, z0, z1, z2) -> min(z1, z2) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) 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) Tuples: DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2)) MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c13(DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF2(false, s(x0), 0, x1) -> c10(DEL(s(x0), x1)) IF2(false, s(0), s(s(x0)), x1) -> c10(DEL(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c10(DEL(s(s(x0)), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(DEL(s(s(x0)), x2)) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2)) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c17 MINSORT(cons(0, cons(y0, y1))) -> c12(MIN(0, cons(y0, y1))) MINSORT(cons(s(y0), cons(0, y1))) -> c12(MIN(s(y0), cons(0, y1))) MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c12(MIN(s(s(y0)), cons(s(s(y1)), y2))) MINSORT(cons(s(y0), cons(s(y1), y2))) -> c12(MIN(s(y0), cons(s(y1), y2))) MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c12(MIN(s(s(y0)), cons(s(0), y1))) MINSORT(cons(s(0), cons(s(y0), y1))) -> c12(MIN(s(0), cons(s(y0), y1))) MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(true, 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c15(LE(s(s(y0)), s(s(y1)))) DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c17(EQ(s(s(y0)), s(s(y1)))) IF1(true, 0, z0, cons(y0, y1)) -> c7(MIN(0, cons(y0, y1))) MIN(0, cons(z0, cons(y1, y2))) -> c15(IF1(true, 0, z0, cons(y1, y2))) IF1(true, s(0), s(z0), cons(0, y1)) -> c7(MIN(s(0), cons(0, y1))) IF1(true, s(0), s(z0), cons(s(y0), y1)) -> c7(MIN(s(0), cons(s(y0), y1))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) IF1(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c7(MIN(s(s(z0)), cons(0, y1))) IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c7(MIN(s(s(z0)), cons(s(s(y1)), y2))) IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c7(MIN(s(s(z0)), cons(s(0), y1))) IF1(false, s(z0), 0, cons(y0, cons(y1, y2))) -> c8(MIN(0, cons(y0, cons(y1, y2)))) MIN(s(z0), cons(0, cons(y1, cons(y2, y3)))) -> c15(IF1(false, s(z0), 0, cons(y1, cons(y2, y3)))) IF1(false, s(s(z0)), s(0), cons(s(y0), y1)) -> c8(MIN(s(0), cons(s(y0), y1))) IF1(false, s(s(z0)), s(0), cons(0, cons(y1, cons(y2, y3)))) -> c8(MIN(s(0), cons(0, cons(y1, cons(y2, y3))))) IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c8(MIN(s(s(z1)), cons(s(s(y1)), y2))) IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c8(MIN(s(s(z1)), cons(s(0), y1))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(y1, cons(y2, y3)))) -> c8(MIN(s(s(z1)), cons(0, cons(y1, cons(y2, y3))))) MIN(s(s(z0)), cons(s(0), cons(s(y1), y2))) -> c(IF1(false, s(s(z0)), s(0), cons(s(y1), y2))) MIN(s(s(z0)), cons(s(0), cons(0, cons(y1, cons(y2, y3))))) -> c(IF1(false, s(s(z0)), s(0), cons(0, cons(y1, cons(y2, y3))))) MINSORT(cons(s(s(y0)), nil)) -> c13(DEL(s(s(y0)), cons(s(s(y0)), nil))) MINSORT(cons(s(0), nil)) -> c13(DEL(s(0), cons(s(0), nil))) IF2(false, 0, s(z0), cons(s(y0), y1)) -> c10(DEL(0, cons(s(y0), y1))) S tuples: DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2)) MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF2(false, s(x0), 0, x1) -> c10(DEL(s(x0), x1)) IF2(false, s(0), s(s(x0)), x1) -> c10(DEL(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c10(DEL(s(s(x0)), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(DEL(s(s(x0)), x2)) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2)) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c17 MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(true, 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c15(LE(s(s(y0)), s(s(y1)))) DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c17(EQ(s(s(y0)), s(s(y1)))) IF1(true, 0, z0, cons(y0, y1)) -> c7(MIN(0, cons(y0, y1))) MIN(0, cons(z0, cons(y1, y2))) -> c15(IF1(true, 0, z0, cons(y1, y2))) IF1(true, s(0), s(z0), cons(0, y1)) -> c7(MIN(s(0), cons(0, y1))) IF1(true, s(0), s(z0), cons(s(y0), y1)) -> c7(MIN(s(0), cons(s(y0), y1))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) IF1(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c7(MIN(s(s(z0)), cons(0, y1))) IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c7(MIN(s(s(z0)), cons(s(s(y1)), y2))) IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c7(MIN(s(s(z0)), cons(s(0), y1))) IF1(false, s(z0), 0, cons(y0, cons(y1, y2))) -> c8(MIN(0, cons(y0, cons(y1, y2)))) MIN(s(z0), cons(0, cons(y1, cons(y2, y3)))) -> c15(IF1(false, s(z0), 0, cons(y1, cons(y2, y3)))) IF1(false, s(s(z0)), s(0), cons(s(y0), y1)) -> c8(MIN(s(0), cons(s(y0), y1))) IF1(false, s(s(z0)), s(0), cons(0, cons(y1, cons(y2, y3)))) -> c8(MIN(s(0), cons(0, cons(y1, cons(y2, y3))))) IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c8(MIN(s(s(z1)), cons(s(s(y1)), y2))) IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c8(MIN(s(s(z1)), cons(s(0), y1))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(y1, cons(y2, y3)))) -> c8(MIN(s(s(z1)), cons(0, cons(y1, cons(y2, y3))))) MIN(s(s(z0)), cons(s(0), cons(s(y1), y2))) -> c(IF1(false, s(s(z0)), s(0), cons(s(y1), y2))) MIN(s(s(z0)), cons(s(0), cons(0, cons(y1, cons(y2, y3))))) -> c(IF1(false, s(s(z0)), s(0), cons(0, cons(y1, cons(y2, y3))))) IF2(false, 0, s(z0), cons(s(y0), y1)) -> c10(DEL(0, cons(s(y0), y1))) K tuples: MINSORT(cons(0, cons(y0, y1))) -> c12(MIN(0, cons(y0, y1))) MINSORT(cons(s(y0), cons(0, y1))) -> c12(MIN(s(y0), cons(0, y1))) MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c12(MIN(s(s(y0)), cons(s(s(y1)), y2))) MINSORT(cons(s(y0), cons(s(y1), y2))) -> c12(MIN(s(y0), cons(s(y1), y2))) MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c12(MIN(s(s(y0)), cons(s(0), y1))) MINSORT(cons(s(0), cons(s(y0), y1))) -> c12(MIN(s(0), cons(s(y0), y1))) Defined Rule Symbols: del_2, min_2, if1_4, le_2, if2_4, eq_2 Defined Pair Symbols: DEL_2, MIN_2, MINSORT_1, IF2_4, LE_2, EQ_2, IF1_4 Compound Symbols: c17_1, c15_2, c17_2, c13_3, c13_1, c13_2, c10_1, c2_1, c15_1, c6_1, c17, c12_1, c7_1, c8_1, c_1 ---------------------------------------- (175) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace DEL(0, cons(s(z0), x2)) -> c17(IF2(false, 0, s(z0), x2)) by DEL(0, cons(s(z0), cons(s(y1), y2))) -> c17(IF2(false, 0, s(z0), cons(s(y1), y2))) ---------------------------------------- (176) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(z0, z2) if1(false, z0, z1, z2) -> min(z1, z2) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) 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) Tuples: DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2)) MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c13(DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF2(false, s(x0), 0, x1) -> c10(DEL(s(x0), x1)) IF2(false, s(0), s(s(x0)), x1) -> c10(DEL(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c10(DEL(s(s(x0)), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(DEL(s(s(x0)), x2)) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2)) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c17 MINSORT(cons(0, cons(y0, y1))) -> c12(MIN(0, cons(y0, y1))) MINSORT(cons(s(y0), cons(0, y1))) -> c12(MIN(s(y0), cons(0, y1))) MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c12(MIN(s(s(y0)), cons(s(s(y1)), y2))) MINSORT(cons(s(y0), cons(s(y1), y2))) -> c12(MIN(s(y0), cons(s(y1), y2))) MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c12(MIN(s(s(y0)), cons(s(0), y1))) MINSORT(cons(s(0), cons(s(y0), y1))) -> c12(MIN(s(0), cons(s(y0), y1))) MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(true, 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c15(LE(s(s(y0)), s(s(y1)))) DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c17(EQ(s(s(y0)), s(s(y1)))) IF1(true, 0, z0, cons(y0, y1)) -> c7(MIN(0, cons(y0, y1))) MIN(0, cons(z0, cons(y1, y2))) -> c15(IF1(true, 0, z0, cons(y1, y2))) IF1(true, s(0), s(z0), cons(0, y1)) -> c7(MIN(s(0), cons(0, y1))) IF1(true, s(0), s(z0), cons(s(y0), y1)) -> c7(MIN(s(0), cons(s(y0), y1))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) IF1(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c7(MIN(s(s(z0)), cons(0, y1))) IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c7(MIN(s(s(z0)), cons(s(s(y1)), y2))) IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c7(MIN(s(s(z0)), cons(s(0), y1))) IF1(false, s(z0), 0, cons(y0, cons(y1, y2))) -> c8(MIN(0, cons(y0, cons(y1, y2)))) MIN(s(z0), cons(0, cons(y1, cons(y2, y3)))) -> c15(IF1(false, s(z0), 0, cons(y1, cons(y2, y3)))) IF1(false, s(s(z0)), s(0), cons(s(y0), y1)) -> c8(MIN(s(0), cons(s(y0), y1))) IF1(false, s(s(z0)), s(0), cons(0, cons(y1, cons(y2, y3)))) -> c8(MIN(s(0), cons(0, cons(y1, cons(y2, y3))))) IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c8(MIN(s(s(z1)), cons(s(s(y1)), y2))) IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c8(MIN(s(s(z1)), cons(s(0), y1))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(y1, cons(y2, y3)))) -> c8(MIN(s(s(z1)), cons(0, cons(y1, cons(y2, y3))))) MIN(s(s(z0)), cons(s(0), cons(s(y1), y2))) -> c(IF1(false, s(s(z0)), s(0), cons(s(y1), y2))) MIN(s(s(z0)), cons(s(0), cons(0, cons(y1, cons(y2, y3))))) -> c(IF1(false, s(s(z0)), s(0), cons(0, cons(y1, cons(y2, y3))))) MINSORT(cons(s(s(y0)), nil)) -> c13(DEL(s(s(y0)), cons(s(s(y0)), nil))) MINSORT(cons(s(0), nil)) -> c13(DEL(s(0), cons(s(0), nil))) IF2(false, 0, s(z0), cons(s(y0), y1)) -> c10(DEL(0, cons(s(y0), y1))) DEL(0, cons(s(z0), cons(s(y1), y2))) -> c17(IF2(false, 0, s(z0), cons(s(y1), y2))) S tuples: DEL(s(z0), cons(0, x2)) -> c17(IF2(false, s(z0), 0, x2)) MIN(s(s(z0)), cons(s(s(z1)), x2)) -> c15(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c17(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) MINSORT(cons(z0, cons(z1, z2))) -> c13(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), DEL(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c13(MINSORT(if2(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) IF2(false, s(x0), 0, x1) -> c10(DEL(s(x0), x1)) IF2(false, s(0), s(s(x0)), x1) -> c10(DEL(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c10(DEL(s(s(x0)), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c10(DEL(s(s(x0)), x2)) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MIN(s(0), cons(s(z0), x2)) -> c15(IF1(true, s(0), s(z0), x2)) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) DEL(s(0), cons(s(s(z0)), x2)) -> c17(IF2(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c17(IF2(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c17 MINSORT(cons(0, cons(z0, z1))) -> c13(MINSORT(del(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), DEL(if1(true, 0, z0, z1), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) MINSORT(cons(s(z0), cons(0, z1))) -> c13(MINSORT(del(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), DEL(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c15(LE(s(s(y0)), s(s(y1)))) DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c17(EQ(s(s(y0)), s(s(y1)))) IF1(true, 0, z0, cons(y0, y1)) -> c7(MIN(0, cons(y0, y1))) MIN(0, cons(z0, cons(y1, y2))) -> c15(IF1(true, 0, z0, cons(y1, y2))) IF1(true, s(0), s(z0), cons(0, y1)) -> c7(MIN(s(0), cons(0, y1))) IF1(true, s(0), s(z0), cons(s(y0), y1)) -> c7(MIN(s(0), cons(s(y0), y1))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c13(MINSORT(del(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), DEL(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) IF1(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c7(MIN(s(s(z0)), cons(0, y1))) IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c7(MIN(s(s(z0)), cons(s(s(y1)), y2))) IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c7(MIN(s(s(z0)), cons(s(0), y1))) IF1(false, s(z0), 0, cons(y0, cons(y1, y2))) -> c8(MIN(0, cons(y0, cons(y1, y2)))) MIN(s(z0), cons(0, cons(y1, cons(y2, y3)))) -> c15(IF1(false, s(z0), 0, cons(y1, cons(y2, y3)))) IF1(false, s(s(z0)), s(0), cons(s(y0), y1)) -> c8(MIN(s(0), cons(s(y0), y1))) IF1(false, s(s(z0)), s(0), cons(0, cons(y1, cons(y2, y3)))) -> c8(MIN(s(0), cons(0, cons(y1, cons(y2, y3))))) IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c8(MIN(s(s(z1)), cons(s(s(y1)), y2))) IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c8(MIN(s(s(z1)), cons(s(0), y1))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(y1, cons(y2, y3)))) -> c8(MIN(s(s(z1)), cons(0, cons(y1, cons(y2, y3))))) MIN(s(s(z0)), cons(s(0), cons(s(y1), y2))) -> c(IF1(false, s(s(z0)), s(0), cons(s(y1), y2))) MIN(s(s(z0)), cons(s(0), cons(0, cons(y1, cons(y2, y3))))) -> c(IF1(false, s(s(z0)), s(0), cons(0, cons(y1, cons(y2, y3))))) IF2(false, 0, s(z0), cons(s(y0), y1)) -> c10(DEL(0, cons(s(y0), y1))) DEL(0, cons(s(z0), cons(s(y1), y2))) -> c17(IF2(false, 0, s(z0), cons(s(y1), y2))) K tuples: MINSORT(cons(0, cons(y0, y1))) -> c12(MIN(0, cons(y0, y1))) MINSORT(cons(s(y0), cons(0, y1))) -> c12(MIN(s(y0), cons(0, y1))) MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c12(MIN(s(s(y0)), cons(s(s(y1)), y2))) MINSORT(cons(s(y0), cons(s(y1), y2))) -> c12(MIN(s(y0), cons(s(y1), y2))) MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c12(MIN(s(s(y0)), cons(s(0), y1))) MINSORT(cons(s(0), cons(s(y0), y1))) -> c12(MIN(s(0), cons(s(y0), y1))) Defined Rule Symbols: del_2, min_2, if1_4, le_2, if2_4, eq_2 Defined Pair Symbols: DEL_2, MIN_2, MINSORT_1, IF2_4, LE_2, EQ_2, IF1_4 Compound Symbols: c17_1, c15_2, c17_2, c13_3, c13_1, c13_2, c10_1, c2_1, c15_1, c6_1, c17, c12_1, c7_1, c8_1, c_1