KILLED proof of input_DTXjXGLI28.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).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), 6 ms] (20) CpxRNTS (21) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms] (22) CpxRNTS (23) ResultPropagationProof [UPPER BOUND(ID), 2 ms] (24) CpxRNTS (25) IntTrsBoundProof [UPPER BOUND(ID), 343 ms] (26) CpxRNTS (27) IntTrsBoundProof [UPPER BOUND(ID), 161 ms] (28) CpxRNTS (29) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (30) CpxRNTS (31) IntTrsBoundProof [UPPER BOUND(ID), 144 ms] (32) CpxRNTS (33) IntTrsBoundProof [UPPER BOUND(ID), 168 ms] (34) CpxRNTS (35) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (36) CpxRNTS (37) IntTrsBoundProof [UPPER BOUND(ID), 718 ms] (38) CpxRNTS (39) IntTrsBoundProof [UPPER BOUND(ID), 415 ms] (40) CpxRNTS (41) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (42) CpxRNTS (43) IntTrsBoundProof [UPPER BOUND(ID), 2176 ms] (44) CpxRNTS (45) IntTrsBoundProof [UPPER BOUND(ID), 1054 ms] (46) CpxRNTS (47) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (48) CpxRNTS (49) IntTrsBoundProof [UPPER BOUND(ID), 1365 ms] (50) CpxRNTS (51) IntTrsBoundProof [UPPER BOUND(ID), 165 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)), 181 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) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (72) CdtProblem (73) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (74) CdtProblem (75) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (76) CdtProblem (77) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (78) CdtProblem (79) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 78 ms] (80) CdtProblem (81) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (82) CdtProblem (83) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 100 ms] (84) CdtProblem (85) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (86) CdtProblem (87) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (88) CdtProblem (89) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (90) CdtProblem (91) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (92) CdtProblem (93) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (94) CdtProblem (95) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (96) CdtProblem (97) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (98) CdtProblem (99) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (100) CdtProblem (101) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (102) CdtProblem (103) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (104) CdtProblem (105) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (106) CdtProblem (107) CdtInstantiationProof [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) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (118) CdtProblem (119) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (120) CdtProblem (121) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (122) CdtProblem (123) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (124) CdtProblem (125) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (126) CdtProblem (127) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (128) CdtProblem (129) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (130) CdtProblem (131) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (132) CdtProblem (133) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (134) CdtProblem (135) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (136) CdtProblem (137) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (138) CdtProblem (139) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (140) CdtProblem (141) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (142) CdtProblem (143) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (144) CdtProblem (145) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (146) CdtProblem (147) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (148) CdtProblem (149) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (150) CdtProblem (151) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (152) CdtProblem (153) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (154) CdtProblem (155) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (156) CdtProblem (157) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (158) CdtProblem (159) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (160) CdtProblem (161) CdtRewritingProof [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), 4 ms] (168) CdtProblem (169) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (170) 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) minsort(nil) -> nil minsort(cons(x, xs)) -> cons(min(cons(x, xs)), minsort(rm(min(cons(x, xs)), cons(x, xs)))) min(nil) -> 0 min(cons(x, nil)) -> x min(cons(x, cons(y, xs))) -> if1(le(x, y), x, y, xs) if1(true, x, y, xs) -> min(cons(x, xs)) if1(false, x, y, xs) -> min(cons(y, xs)) rm(x, nil) -> nil rm(x, cons(y, xs)) -> if2(eq(x, y), x, y, xs) if2(true, x, y, xs) -> rm(x, xs) if2(false, x, y, xs) -> cons(y, rm(x, xs)) 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) minsort(nil) -> nil minsort(cons(x, xs)) -> cons(min(cons(x, xs)), minsort(rm(min(cons(x, xs)), cons(x, xs)))) min(nil) -> 0' min(cons(x, nil)) -> x min(cons(x, cons(y, xs))) -> if1(le(x, y), x, y, xs) if1(true, x, y, xs) -> min(cons(x, xs)) if1(false, x, y, xs) -> min(cons(y, xs)) rm(x, nil) -> nil rm(x, cons(y, xs)) -> if2(eq(x, y), x, y, xs) if2(true, x, y, xs) -> rm(x, xs) if2(false, x, y, xs) -> cons(y, rm(x, xs)) 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) minsort(nil) -> nil minsort(cons(x, xs)) -> cons(min(cons(x, xs)), minsort(rm(min(cons(x, xs)), cons(x, xs)))) min(nil) -> 0 min(cons(x, nil)) -> x min(cons(x, cons(y, xs))) -> if1(le(x, y), x, y, xs) if1(true, x, y, xs) -> min(cons(x, xs)) if1(false, x, y, xs) -> min(cons(y, xs)) rm(x, nil) -> nil rm(x, cons(y, xs)) -> if2(eq(x, y), x, y, xs) if2(true, x, y, xs) -> rm(x, xs) if2(false, x, y, xs) -> cons(y, rm(x, xs)) 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] minsort(nil) -> nil [1] minsort(cons(x, xs)) -> cons(min(cons(x, xs)), minsort(rm(min(cons(x, xs)), cons(x, xs)))) [1] min(nil) -> 0 [1] min(cons(x, nil)) -> x [1] min(cons(x, cons(y, xs))) -> if1(le(x, y), x, y, xs) [1] if1(true, x, y, xs) -> min(cons(x, xs)) [1] if1(false, x, y, xs) -> min(cons(y, xs)) [1] rm(x, nil) -> nil [1] rm(x, cons(y, xs)) -> if2(eq(x, y), x, y, xs) [1] if2(true, x, y, xs) -> rm(x, xs) [1] if2(false, x, y, xs) -> cons(y, rm(x, xs)) [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] minsort(nil) -> nil [1] minsort(cons(x, xs)) -> cons(min(cons(x, xs)), minsort(rm(min(cons(x, xs)), cons(x, xs)))) [1] min(nil) -> 0 [1] min(cons(x, nil)) -> x [1] min(cons(x, cons(y, xs))) -> if1(le(x, y), x, y, xs) [1] if1(true, x, y, xs) -> min(cons(x, xs)) [1] if1(false, x, y, xs) -> min(cons(y, xs)) [1] rm(x, nil) -> nil [1] rm(x, cons(y, xs)) -> if2(eq(x, y), x, y, xs) [1] if2(true, x, y, xs) -> rm(x, xs) [1] if2(false, x, y, xs) -> cons(y, rm(x, xs)) [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 minsort :: nil:cons -> nil:cons nil :: nil:cons cons :: 0:s -> nil:cons -> nil:cons min :: nil:cons -> 0:s rm :: 0:s -> nil:cons -> nil:cons if1 :: true:false -> 0:s -> 0:s -> nil:cons -> 0:s if2 :: true:false -> 0:s -> 0:s -> nil:cons -> nil:cons Rewrite Strategy: INNERMOST ---------------------------------------- (9) CompletionProof (UPPER BOUND(ID)) The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added: none And the following fresh constants: none ---------------------------------------- (10) Obligation: Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: 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] minsort(nil) -> nil [1] minsort(cons(x, xs)) -> cons(min(cons(x, xs)), minsort(rm(min(cons(x, xs)), cons(x, xs)))) [1] min(nil) -> 0 [1] min(cons(x, nil)) -> x [1] min(cons(x, cons(y, xs))) -> if1(le(x, y), x, y, xs) [1] if1(true, x, y, xs) -> min(cons(x, xs)) [1] if1(false, x, y, xs) -> min(cons(y, xs)) [1] rm(x, nil) -> nil [1] rm(x, cons(y, xs)) -> if2(eq(x, y), x, y, xs) [1] if2(true, x, y, xs) -> rm(x, xs) [1] if2(false, x, y, xs) -> cons(y, rm(x, xs)) [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 minsort :: nil:cons -> nil:cons nil :: nil:cons cons :: 0:s -> nil:cons -> nil:cons min :: nil:cons -> 0:s rm :: 0:s -> nil:cons -> nil:cons if1 :: true:false -> 0:s -> 0:s -> nil:cons -> 0:s if2 :: true:false -> 0:s -> 0:s -> nil:cons -> nil:cons Rewrite Strategy: INNERMOST ---------------------------------------- (11) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: 0 => 0 true => 1 false => 0 nil => 0 ---------------------------------------- (12) Obligation: Complexity RNTS consisting of the following rules: eq(z, z') -{ 1 }-> eq(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 eq(z, z') -{ 1 }-> 0 :|: z' = 1 + y, y >= 0, z = 0 eq(z, z') -{ 1 }-> 0 :|: x >= 0, z = 1 + x, z' = 0 if1(z, z', z'', z1) -{ 1 }-> min(1 + x + xs) :|: xs >= 0, z' = x, z'' = y, z = 1, x >= 0, y >= 0, z1 = xs if1(z, z', z'', z1) -{ 1 }-> min(1 + y + xs) :|: xs >= 0, z' = x, z'' = y, x >= 0, y >= 0, z = 0, z1 = xs if2(z, z', z'', z1) -{ 1 }-> rm(x, xs) :|: xs >= 0, z' = x, z'' = y, z = 1, x >= 0, y >= 0, z1 = xs if2(z, z', z'', z1) -{ 1 }-> 1 + y + rm(x, xs) :|: xs >= 0, z' = x, z'' = y, x >= 0, y >= 0, z = 0, z1 = xs le(z, z') -{ 1 }-> le(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x le(z, z') -{ 1 }-> 1 :|: y >= 0, z = 0, z' = y le(z, z') -{ 1 }-> 0 :|: x >= 0, z = 1 + x, z' = 0 min(z) -{ 1 }-> x :|: x >= 0, z = 1 + x + 0 min(z) -{ 1 }-> if1(le(x, y), x, y, xs) :|: xs >= 0, x >= 0, y >= 0, z = 1 + x + (1 + y + xs) min(z) -{ 1 }-> 0 :|: z = 0 minsort(z) -{ 1 }-> 0 :|: z = 0 minsort(z) -{ 1 }-> 1 + min(1 + x + xs) + minsort(rm(min(1 + x + xs), 1 + x + xs)) :|: z = 1 + x + xs, xs >= 0, x >= 0 rm(z, z') -{ 1 }-> if2(eq(x, y), x, y, xs) :|: xs >= 0, z' = 1 + y + xs, x >= 0, y >= 0, z = x rm(z, z') -{ 1 }-> 0 :|: x >= 0, z = x, z' = 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: rm_2 min_1 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] minsort(nil) -> nil [1] minsort(cons(x, xs)) -> cons(min(cons(x, xs)), minsort(rm(min(cons(x, xs)), cons(x, xs)))) [1] min(nil) -> 0 [1] min(cons(x, nil)) -> x [1] min(cons(x, cons(y, xs))) -> if1(le(x, y), x, y, xs) [1] if1(true, x, y, xs) -> min(cons(x, xs)) [1] if1(false, x, y, xs) -> min(cons(y, xs)) [1] rm(x, nil) -> nil [1] rm(x, cons(y, xs)) -> if2(eq(x, y), x, y, xs) [1] if2(true, x, y, xs) -> rm(x, xs) [1] if2(false, x, y, xs) -> cons(y, rm(x, xs)) [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 minsort :: nil:cons -> nil:cons nil :: nil:cons cons :: 0:s -> nil:cons -> nil:cons min :: nil:cons -> 0:s rm :: 0:s -> nil:cons -> nil:cons if1 :: true:false -> 0:s -> 0:s -> nil:cons -> 0:s if2 :: true:false -> 0:s -> 0:s -> nil:cons -> nil:cons Rewrite Strategy: INNERMOST ---------------------------------------- (15) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (16) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: 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] minsort(nil) -> nil [1] minsort(cons(x, nil)) -> cons(min(cons(x, nil)), minsort(rm(x, cons(x, nil)))) [2] minsort(cons(x, cons(y', xs'))) -> cons(min(cons(x, cons(y', xs'))), minsort(rm(if1(le(x, y'), x, y', xs'), cons(x, cons(y', xs'))))) [2] min(nil) -> 0 [1] min(cons(x, nil)) -> x [1] min(cons(0, cons(y, xs))) -> if1(true, 0, y, xs) [2] min(cons(s(x'), cons(0, xs))) -> if1(false, s(x'), 0, xs) [2] min(cons(s(x''), cons(s(y''), xs))) -> if1(le(x'', y''), s(x''), s(y''), xs) [2] if1(true, x, y, xs) -> min(cons(x, xs)) [1] if1(false, x, y, xs) -> min(cons(y, xs)) [1] rm(x, nil) -> nil [1] rm(0, cons(0, xs)) -> if2(true, 0, 0, xs) [2] rm(0, cons(s(y1), xs)) -> if2(false, 0, s(y1), xs) [2] rm(s(x1), cons(0, xs)) -> if2(false, s(x1), 0, xs) [2] rm(s(x2), cons(s(y2), xs)) -> if2(eq(x2, y2), s(x2), s(y2), xs) [2] if2(true, x, y, xs) -> rm(x, xs) [1] if2(false, x, y, xs) -> cons(y, rm(x, xs)) [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 minsort :: nil:cons -> nil:cons nil :: nil:cons cons :: 0:s -> nil:cons -> nil:cons min :: nil:cons -> 0:s rm :: 0:s -> nil:cons -> nil:cons if1 :: true:false -> 0:s -> 0:s -> nil:cons -> 0:s if2 :: true:false -> 0:s -> 0:s -> nil:cons -> nil:cons Rewrite Strategy: INNERMOST ---------------------------------------- (17) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: 0 => 0 true => 1 false => 0 nil => 0 ---------------------------------------- (18) Obligation: Complexity RNTS consisting of the following rules: eq(z, z') -{ 1 }-> eq(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 eq(z, z') -{ 1 }-> 0 :|: z' = 1 + y, y >= 0, z = 0 eq(z, z') -{ 1 }-> 0 :|: x >= 0, z = 1 + x, z' = 0 if1(z, z', z'', z1) -{ 1 }-> min(1 + x + xs) :|: xs >= 0, z' = x, z'' = y, z = 1, x >= 0, y >= 0, z1 = xs if1(z, z', z'', z1) -{ 1 }-> min(1 + y + xs) :|: xs >= 0, z' = x, z'' = y, x >= 0, y >= 0, z = 0, z1 = xs if2(z, z', z'', z1) -{ 1 }-> rm(x, xs) :|: xs >= 0, z' = x, z'' = y, z = 1, x >= 0, y >= 0, z1 = xs if2(z, z', z'', z1) -{ 1 }-> 1 + y + rm(x, xs) :|: xs >= 0, z' = x, z'' = y, x >= 0, y >= 0, z = 0, z1 = xs le(z, z') -{ 1 }-> le(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x le(z, z') -{ 1 }-> 1 :|: y >= 0, z = 0, z' = y le(z, z') -{ 1 }-> 0 :|: x >= 0, z = 1 + x, z' = 0 min(z) -{ 1 }-> x :|: x >= 0, z = 1 + x + 0 min(z) -{ 2 }-> if1(le(x'', y''), 1 + x'', 1 + y'', xs) :|: xs >= 0, y'' >= 0, x'' >= 0, z = 1 + (1 + x'') + (1 + (1 + y'') + xs) min(z) -{ 2 }-> if1(1, 0, y, xs) :|: xs >= 0, z = 1 + 0 + (1 + y + xs), y >= 0 min(z) -{ 2 }-> if1(0, 1 + x', 0, xs) :|: xs >= 0, x' >= 0, z = 1 + (1 + x') + (1 + 0 + xs) min(z) -{ 1 }-> 0 :|: z = 0 minsort(z) -{ 1 }-> 0 :|: z = 0 minsort(z) -{ 2 }-> 1 + min(1 + x + 0) + minsort(rm(x, 1 + x + 0)) :|: x >= 0, z = 1 + x + 0 minsort(z) -{ 2 }-> 1 + min(1 + x + (1 + y' + xs')) + minsort(rm(if1(le(x, y'), x, y', xs'), 1 + x + (1 + y' + xs'))) :|: x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs') rm(z, z') -{ 2 }-> if2(eq(x2, y2), 1 + x2, 1 + y2, xs) :|: xs >= 0, z = 1 + x2, z' = 1 + (1 + y2) + xs, y2 >= 0, x2 >= 0 rm(z, z') -{ 2 }-> if2(1, 0, 0, xs) :|: xs >= 0, z' = 1 + 0 + xs, z = 0 rm(z, z') -{ 2 }-> if2(0, 0, 1 + y1, xs) :|: y1 >= 0, xs >= 0, z' = 1 + (1 + y1) + xs, z = 0 rm(z, z') -{ 2 }-> if2(0, 1 + x1, 0, xs) :|: xs >= 0, x1 >= 0, z' = 1 + 0 + xs, z = 1 + x1 rm(z, z') -{ 1 }-> 0 :|: x >= 0, z = x, z' = 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: eq(z, z') -{ 1 }-> eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 eq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 eq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 if1(z, z', z'', z1) -{ 1 }-> min(1 + z' + z1) :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0 if1(z, z', z'', z1) -{ 1 }-> min(1 + z'' + z1) :|: z1 >= 0, z' >= 0, z'' >= 0, z = 0 if2(z, z', z'', z1) -{ 1 }-> rm(z', z1) :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0 if2(z, z', z'', z1) -{ 1 }-> 1 + z'' + rm(z', z1) :|: z1 >= 0, z' >= 0, z'' >= 0, z = 0 le(z, z') -{ 1 }-> le(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 min(z) -{ 2 }-> if1(le(x'', y''), 1 + x'', 1 + y'', xs) :|: xs >= 0, y'' >= 0, x'' >= 0, z = 1 + (1 + x'') + (1 + (1 + y'') + xs) min(z) -{ 2 }-> if1(1, 0, y, xs) :|: xs >= 0, z = 1 + 0 + (1 + y + xs), y >= 0 min(z) -{ 2 }-> if1(0, 1 + x', 0, xs) :|: xs >= 0, x' >= 0, z = 1 + (1 + x') + (1 + 0 + xs) min(z) -{ 1 }-> 0 :|: z = 0 min(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 minsort(z) -{ 1 }-> 0 :|: z = 0 minsort(z) -{ 2 }-> 1 + min(1 + x + (1 + y' + xs')) + minsort(rm(if1(le(x, y'), x, y', xs'), 1 + x + (1 + y' + xs'))) :|: x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs') minsort(z) -{ 2 }-> 1 + min(1 + (z - 1) + 0) + minsort(rm(z - 1, 1 + (z - 1) + 0)) :|: z - 1 >= 0 rm(z, z') -{ 2 }-> if2(eq(z - 1, y2), 1 + (z - 1), 1 + y2, xs) :|: xs >= 0, z' = 1 + (1 + y2) + xs, y2 >= 0, z - 1 >= 0 rm(z, z') -{ 2 }-> if2(1, 0, 0, z' - 1) :|: z' - 1 >= 0, z = 0 rm(z, z') -{ 2 }-> if2(0, 0, 1 + y1, xs) :|: y1 >= 0, xs >= 0, z' = 1 + (1 + y1) + xs, z = 0 rm(z, z') -{ 2 }-> if2(0, 1 + (z - 1), 0, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 rm(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 ---------------------------------------- (21) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { le } { eq } { min, if1 } { if2, rm } { minsort } ---------------------------------------- (22) Obligation: Complexity RNTS consisting of the following rules: eq(z, z') -{ 1 }-> eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 eq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 eq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 if1(z, z', z'', z1) -{ 1 }-> min(1 + z' + z1) :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0 if1(z, z', z'', z1) -{ 1 }-> min(1 + z'' + z1) :|: z1 >= 0, z' >= 0, z'' >= 0, z = 0 if2(z, z', z'', z1) -{ 1 }-> rm(z', z1) :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0 if2(z, z', z'', z1) -{ 1 }-> 1 + z'' + rm(z', z1) :|: z1 >= 0, z' >= 0, z'' >= 0, z = 0 le(z, z') -{ 1 }-> le(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 min(z) -{ 2 }-> if1(le(x'', y''), 1 + x'', 1 + y'', xs) :|: xs >= 0, y'' >= 0, x'' >= 0, z = 1 + (1 + x'') + (1 + (1 + y'') + xs) min(z) -{ 2 }-> if1(1, 0, y, xs) :|: xs >= 0, z = 1 + 0 + (1 + y + xs), y >= 0 min(z) -{ 2 }-> if1(0, 1 + x', 0, xs) :|: xs >= 0, x' >= 0, z = 1 + (1 + x') + (1 + 0 + xs) min(z) -{ 1 }-> 0 :|: z = 0 min(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 minsort(z) -{ 1 }-> 0 :|: z = 0 minsort(z) -{ 2 }-> 1 + min(1 + x + (1 + y' + xs')) + minsort(rm(if1(le(x, y'), x, y', xs'), 1 + x + (1 + y' + xs'))) :|: x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs') minsort(z) -{ 2 }-> 1 + min(1 + (z - 1) + 0) + minsort(rm(z - 1, 1 + (z - 1) + 0)) :|: z - 1 >= 0 rm(z, z') -{ 2 }-> if2(eq(z - 1, y2), 1 + (z - 1), 1 + y2, xs) :|: xs >= 0, z' = 1 + (1 + y2) + xs, y2 >= 0, z - 1 >= 0 rm(z, z') -{ 2 }-> if2(1, 0, 0, z' - 1) :|: z' - 1 >= 0, z = 0 rm(z, z') -{ 2 }-> if2(0, 0, 1 + y1, xs) :|: y1 >= 0, xs >= 0, z' = 1 + (1 + y1) + xs, z = 0 rm(z, z') -{ 2 }-> if2(0, 1 + (z - 1), 0, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 rm(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {le}, {eq}, {min,if1}, {if2,rm}, {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: eq(z, z') -{ 1 }-> eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 eq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 eq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 if1(z, z', z'', z1) -{ 1 }-> min(1 + z' + z1) :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0 if1(z, z', z'', z1) -{ 1 }-> min(1 + z'' + z1) :|: z1 >= 0, z' >= 0, z'' >= 0, z = 0 if2(z, z', z'', z1) -{ 1 }-> rm(z', z1) :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0 if2(z, z', z'', z1) -{ 1 }-> 1 + z'' + rm(z', z1) :|: z1 >= 0, z' >= 0, z'' >= 0, z = 0 le(z, z') -{ 1 }-> le(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 min(z) -{ 2 }-> if1(le(x'', y''), 1 + x'', 1 + y'', xs) :|: xs >= 0, y'' >= 0, x'' >= 0, z = 1 + (1 + x'') + (1 + (1 + y'') + xs) min(z) -{ 2 }-> if1(1, 0, y, xs) :|: xs >= 0, z = 1 + 0 + (1 + y + xs), y >= 0 min(z) -{ 2 }-> if1(0, 1 + x', 0, xs) :|: xs >= 0, x' >= 0, z = 1 + (1 + x') + (1 + 0 + xs) min(z) -{ 1 }-> 0 :|: z = 0 min(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 minsort(z) -{ 1 }-> 0 :|: z = 0 minsort(z) -{ 2 }-> 1 + min(1 + x + (1 + y' + xs')) + minsort(rm(if1(le(x, y'), x, y', xs'), 1 + x + (1 + y' + xs'))) :|: x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs') minsort(z) -{ 2 }-> 1 + min(1 + (z - 1) + 0) + minsort(rm(z - 1, 1 + (z - 1) + 0)) :|: z - 1 >= 0 rm(z, z') -{ 2 }-> if2(eq(z - 1, y2), 1 + (z - 1), 1 + y2, xs) :|: xs >= 0, z' = 1 + (1 + y2) + xs, y2 >= 0, z - 1 >= 0 rm(z, z') -{ 2 }-> if2(1, 0, 0, z' - 1) :|: z' - 1 >= 0, z = 0 rm(z, z') -{ 2 }-> if2(0, 0, 1 + y1, xs) :|: y1 >= 0, xs >= 0, z' = 1 + (1 + y1) + xs, z = 0 rm(z, z') -{ 2 }-> if2(0, 1 + (z - 1), 0, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 rm(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {le}, {eq}, {min,if1}, {if2,rm}, {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: eq(z, z') -{ 1 }-> eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 eq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 eq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 if1(z, z', z'', z1) -{ 1 }-> min(1 + z' + z1) :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0 if1(z, z', z'', z1) -{ 1 }-> min(1 + z'' + z1) :|: z1 >= 0, z' >= 0, z'' >= 0, z = 0 if2(z, z', z'', z1) -{ 1 }-> rm(z', z1) :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0 if2(z, z', z'', z1) -{ 1 }-> 1 + z'' + rm(z', z1) :|: z1 >= 0, z' >= 0, z'' >= 0, z = 0 le(z, z') -{ 1 }-> le(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 min(z) -{ 2 }-> if1(le(x'', y''), 1 + x'', 1 + y'', xs) :|: xs >= 0, y'' >= 0, x'' >= 0, z = 1 + (1 + x'') + (1 + (1 + y'') + xs) min(z) -{ 2 }-> if1(1, 0, y, xs) :|: xs >= 0, z = 1 + 0 + (1 + y + xs), y >= 0 min(z) -{ 2 }-> if1(0, 1 + x', 0, xs) :|: xs >= 0, x' >= 0, z = 1 + (1 + x') + (1 + 0 + xs) min(z) -{ 1 }-> 0 :|: z = 0 min(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 minsort(z) -{ 1 }-> 0 :|: z = 0 minsort(z) -{ 2 }-> 1 + min(1 + x + (1 + y' + xs')) + minsort(rm(if1(le(x, y'), x, y', xs'), 1 + x + (1 + y' + xs'))) :|: x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs') minsort(z) -{ 2 }-> 1 + min(1 + (z - 1) + 0) + minsort(rm(z - 1, 1 + (z - 1) + 0)) :|: z - 1 >= 0 rm(z, z') -{ 2 }-> if2(eq(z - 1, y2), 1 + (z - 1), 1 + y2, xs) :|: xs >= 0, z' = 1 + (1 + y2) + xs, y2 >= 0, z - 1 >= 0 rm(z, z') -{ 2 }-> if2(1, 0, 0, z' - 1) :|: z' - 1 >= 0, z = 0 rm(z, z') -{ 2 }-> if2(0, 0, 1 + y1, xs) :|: y1 >= 0, xs >= 0, z' = 1 + (1 + y1) + xs, z = 0 rm(z, z') -{ 2 }-> if2(0, 1 + (z - 1), 0, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 rm(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {le}, {eq}, {min,if1}, {if2,rm}, {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 + z' ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: eq(z, z') -{ 1 }-> eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 eq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 eq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 if1(z, z', z'', z1) -{ 1 }-> min(1 + z' + z1) :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0 if1(z, z', z'', z1) -{ 1 }-> min(1 + z'' + z1) :|: z1 >= 0, z' >= 0, z'' >= 0, z = 0 if2(z, z', z'', z1) -{ 1 }-> rm(z', z1) :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0 if2(z, z', z'', z1) -{ 1 }-> 1 + z'' + rm(z', z1) :|: z1 >= 0, z' >= 0, z'' >= 0, z = 0 le(z, z') -{ 1 }-> le(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 min(z) -{ 2 }-> if1(le(x'', y''), 1 + x'', 1 + y'', xs) :|: xs >= 0, y'' >= 0, x'' >= 0, z = 1 + (1 + x'') + (1 + (1 + y'') + xs) min(z) -{ 2 }-> if1(1, 0, y, xs) :|: xs >= 0, z = 1 + 0 + (1 + y + xs), y >= 0 min(z) -{ 2 }-> if1(0, 1 + x', 0, xs) :|: xs >= 0, x' >= 0, z = 1 + (1 + x') + (1 + 0 + xs) min(z) -{ 1 }-> 0 :|: z = 0 min(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 minsort(z) -{ 1 }-> 0 :|: z = 0 minsort(z) -{ 2 }-> 1 + min(1 + x + (1 + y' + xs')) + minsort(rm(if1(le(x, y'), x, y', xs'), 1 + x + (1 + y' + xs'))) :|: x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs') minsort(z) -{ 2 }-> 1 + min(1 + (z - 1) + 0) + minsort(rm(z - 1, 1 + (z - 1) + 0)) :|: z - 1 >= 0 rm(z, z') -{ 2 }-> if2(eq(z - 1, y2), 1 + (z - 1), 1 + y2, xs) :|: xs >= 0, z' = 1 + (1 + y2) + xs, y2 >= 0, z - 1 >= 0 rm(z, z') -{ 2 }-> if2(1, 0, 0, z' - 1) :|: z' - 1 >= 0, z = 0 rm(z, z') -{ 2 }-> if2(0, 0, 1 + y1, xs) :|: y1 >= 0, xs >= 0, z' = 1 + (1 + y1) + xs, z = 0 rm(z, z') -{ 2 }-> if2(0, 1 + (z - 1), 0, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 rm(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {eq}, {min,if1}, {if2,rm}, {minsort} Previous analysis results are: le: runtime: O(n^1) [2 + z'], size: O(1) [1] ---------------------------------------- (29) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (30) Obligation: Complexity RNTS consisting of the following rules: eq(z, z') -{ 1 }-> eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 eq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 eq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 if1(z, z', z'', z1) -{ 1 }-> min(1 + z' + z1) :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0 if1(z, z', z'', z1) -{ 1 }-> min(1 + z'' + z1) :|: z1 >= 0, z' >= 0, z'' >= 0, z = 0 if2(z, z', z'', z1) -{ 1 }-> rm(z', z1) :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0 if2(z, z', z'', z1) -{ 1 }-> 1 + z'' + rm(z', z1) :|: z1 >= 0, z' >= 0, z'' >= 0, z = 0 le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0 le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 min(z) -{ 4 + y'' }-> if1(s'', 1 + x'', 1 + y'', xs) :|: s'' >= 0, s'' <= 1, xs >= 0, y'' >= 0, x'' >= 0, z = 1 + (1 + x'') + (1 + (1 + y'') + xs) min(z) -{ 2 }-> if1(1, 0, y, xs) :|: xs >= 0, z = 1 + 0 + (1 + y + xs), y >= 0 min(z) -{ 2 }-> if1(0, 1 + x', 0, xs) :|: xs >= 0, x' >= 0, z = 1 + (1 + x') + (1 + 0 + xs) min(z) -{ 1 }-> 0 :|: z = 0 min(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 minsort(z) -{ 1 }-> 0 :|: z = 0 minsort(z) -{ 4 + y' }-> 1 + min(1 + x + (1 + y' + xs')) + minsort(rm(if1(s', x, y', xs'), 1 + x + (1 + y' + xs'))) :|: s' >= 0, s' <= 1, x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs') minsort(z) -{ 2 }-> 1 + min(1 + (z - 1) + 0) + minsort(rm(z - 1, 1 + (z - 1) + 0)) :|: z - 1 >= 0 rm(z, z') -{ 2 }-> if2(eq(z - 1, y2), 1 + (z - 1), 1 + y2, xs) :|: xs >= 0, z' = 1 + (1 + y2) + xs, y2 >= 0, z - 1 >= 0 rm(z, z') -{ 2 }-> if2(1, 0, 0, z' - 1) :|: z' - 1 >= 0, z = 0 rm(z, z') -{ 2 }-> if2(0, 0, 1 + y1, xs) :|: y1 >= 0, xs >= 0, z' = 1 + (1 + y1) + xs, z = 0 rm(z, z') -{ 2 }-> if2(0, 1 + (z - 1), 0, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 rm(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {eq}, {min,if1}, {if2,rm}, {minsort} Previous analysis results are: le: runtime: O(n^1) [2 + z'], size: O(1) [1] ---------------------------------------- (31) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: eq after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (32) Obligation: Complexity RNTS consisting of the following rules: eq(z, z') -{ 1 }-> eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 eq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 eq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 if1(z, z', z'', z1) -{ 1 }-> min(1 + z' + z1) :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0 if1(z, z', z'', z1) -{ 1 }-> min(1 + z'' + z1) :|: z1 >= 0, z' >= 0, z'' >= 0, z = 0 if2(z, z', z'', z1) -{ 1 }-> rm(z', z1) :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0 if2(z, z', z'', z1) -{ 1 }-> 1 + z'' + rm(z', z1) :|: z1 >= 0, z' >= 0, z'' >= 0, z = 0 le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0 le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 min(z) -{ 4 + y'' }-> if1(s'', 1 + x'', 1 + y'', xs) :|: s'' >= 0, s'' <= 1, xs >= 0, y'' >= 0, x'' >= 0, z = 1 + (1 + x'') + (1 + (1 + y'') + xs) min(z) -{ 2 }-> if1(1, 0, y, xs) :|: xs >= 0, z = 1 + 0 + (1 + y + xs), y >= 0 min(z) -{ 2 }-> if1(0, 1 + x', 0, xs) :|: xs >= 0, x' >= 0, z = 1 + (1 + x') + (1 + 0 + xs) min(z) -{ 1 }-> 0 :|: z = 0 min(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 minsort(z) -{ 1 }-> 0 :|: z = 0 minsort(z) -{ 4 + y' }-> 1 + min(1 + x + (1 + y' + xs')) + minsort(rm(if1(s', x, y', xs'), 1 + x + (1 + y' + xs'))) :|: s' >= 0, s' <= 1, x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs') minsort(z) -{ 2 }-> 1 + min(1 + (z - 1) + 0) + minsort(rm(z - 1, 1 + (z - 1) + 0)) :|: z - 1 >= 0 rm(z, z') -{ 2 }-> if2(eq(z - 1, y2), 1 + (z - 1), 1 + y2, xs) :|: xs >= 0, z' = 1 + (1 + y2) + xs, y2 >= 0, z - 1 >= 0 rm(z, z') -{ 2 }-> if2(1, 0, 0, z' - 1) :|: z' - 1 >= 0, z = 0 rm(z, z') -{ 2 }-> if2(0, 0, 1 + y1, xs) :|: y1 >= 0, xs >= 0, z' = 1 + (1 + y1) + xs, z = 0 rm(z, z') -{ 2 }-> if2(0, 1 + (z - 1), 0, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 rm(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {eq}, {min,if1}, {if2,rm}, {minsort} Previous analysis results are: le: runtime: O(n^1) [2 + z'], size: O(1) [1] eq: runtime: ?, size: O(1) [1] ---------------------------------------- (33) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: eq after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 3 + z' ---------------------------------------- (34) Obligation: Complexity RNTS consisting of the following rules: eq(z, z') -{ 1 }-> eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 eq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 eq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 if1(z, z', z'', z1) -{ 1 }-> min(1 + z' + z1) :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0 if1(z, z', z'', z1) -{ 1 }-> min(1 + z'' + z1) :|: z1 >= 0, z' >= 0, z'' >= 0, z = 0 if2(z, z', z'', z1) -{ 1 }-> rm(z', z1) :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0 if2(z, z', z'', z1) -{ 1 }-> 1 + z'' + rm(z', z1) :|: z1 >= 0, z' >= 0, z'' >= 0, z = 0 le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0 le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 min(z) -{ 4 + y'' }-> if1(s'', 1 + x'', 1 + y'', xs) :|: s'' >= 0, s'' <= 1, xs >= 0, y'' >= 0, x'' >= 0, z = 1 + (1 + x'') + (1 + (1 + y'') + xs) min(z) -{ 2 }-> if1(1, 0, y, xs) :|: xs >= 0, z = 1 + 0 + (1 + y + xs), y >= 0 min(z) -{ 2 }-> if1(0, 1 + x', 0, xs) :|: xs >= 0, x' >= 0, z = 1 + (1 + x') + (1 + 0 + xs) min(z) -{ 1 }-> 0 :|: z = 0 min(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 minsort(z) -{ 1 }-> 0 :|: z = 0 minsort(z) -{ 4 + y' }-> 1 + min(1 + x + (1 + y' + xs')) + minsort(rm(if1(s', x, y', xs'), 1 + x + (1 + y' + xs'))) :|: s' >= 0, s' <= 1, x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs') minsort(z) -{ 2 }-> 1 + min(1 + (z - 1) + 0) + minsort(rm(z - 1, 1 + (z - 1) + 0)) :|: z - 1 >= 0 rm(z, z') -{ 2 }-> if2(eq(z - 1, y2), 1 + (z - 1), 1 + y2, xs) :|: xs >= 0, z' = 1 + (1 + y2) + xs, y2 >= 0, z - 1 >= 0 rm(z, z') -{ 2 }-> if2(1, 0, 0, z' - 1) :|: z' - 1 >= 0, z = 0 rm(z, z') -{ 2 }-> if2(0, 0, 1 + y1, xs) :|: y1 >= 0, xs >= 0, z' = 1 + (1 + y1) + xs, z = 0 rm(z, z') -{ 2 }-> if2(0, 1 + (z - 1), 0, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 rm(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {min,if1}, {if2,rm}, {minsort} Previous analysis results are: le: runtime: O(n^1) [2 + z'], size: O(1) [1] eq: runtime: O(n^1) [3 + z'], size: O(1) [1] ---------------------------------------- (35) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (36) Obligation: Complexity RNTS consisting of the following rules: eq(z, z') -{ 3 + z' }-> s1 :|: s1 >= 0, s1 <= 1, z - 1 >= 0, z' - 1 >= 0 eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 eq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 eq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 if1(z, z', z'', z1) -{ 1 }-> min(1 + z' + z1) :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0 if1(z, z', z'', z1) -{ 1 }-> min(1 + z'' + z1) :|: z1 >= 0, z' >= 0, z'' >= 0, z = 0 if2(z, z', z'', z1) -{ 1 }-> rm(z', z1) :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0 if2(z, z', z'', z1) -{ 1 }-> 1 + z'' + rm(z', z1) :|: z1 >= 0, z' >= 0, z'' >= 0, z = 0 le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0 le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 min(z) -{ 4 + y'' }-> if1(s'', 1 + x'', 1 + y'', xs) :|: s'' >= 0, s'' <= 1, xs >= 0, y'' >= 0, x'' >= 0, z = 1 + (1 + x'') + (1 + (1 + y'') + xs) min(z) -{ 2 }-> if1(1, 0, y, xs) :|: xs >= 0, z = 1 + 0 + (1 + y + xs), y >= 0 min(z) -{ 2 }-> if1(0, 1 + x', 0, xs) :|: xs >= 0, x' >= 0, z = 1 + (1 + x') + (1 + 0 + xs) min(z) -{ 1 }-> 0 :|: z = 0 min(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 minsort(z) -{ 1 }-> 0 :|: z = 0 minsort(z) -{ 4 + y' }-> 1 + min(1 + x + (1 + y' + xs')) + minsort(rm(if1(s', x, y', xs'), 1 + x + (1 + y' + xs'))) :|: s' >= 0, s' <= 1, x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs') minsort(z) -{ 2 }-> 1 + min(1 + (z - 1) + 0) + minsort(rm(z - 1, 1 + (z - 1) + 0)) :|: z - 1 >= 0 rm(z, z') -{ 5 + y2 }-> if2(s2, 1 + (z - 1), 1 + y2, xs) :|: s2 >= 0, s2 <= 1, xs >= 0, z' = 1 + (1 + y2) + xs, y2 >= 0, z - 1 >= 0 rm(z, z') -{ 2 }-> if2(1, 0, 0, z' - 1) :|: z' - 1 >= 0, z = 0 rm(z, z') -{ 2 }-> if2(0, 0, 1 + y1, xs) :|: y1 >= 0, xs >= 0, z' = 1 + (1 + y1) + xs, z = 0 rm(z, z') -{ 2 }-> if2(0, 1 + (z - 1), 0, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 rm(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {min,if1}, {if2,rm}, {minsort} Previous analysis results are: le: runtime: O(n^1) [2 + z'], size: O(1) [1] eq: runtime: O(n^1) [3 + z'], size: O(1) [1] ---------------------------------------- (37) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: min after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z Computed SIZE bound using CoFloCo for: if1 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z' + z'' + z1 ---------------------------------------- (38) Obligation: Complexity RNTS consisting of the following rules: eq(z, z') -{ 3 + z' }-> s1 :|: s1 >= 0, s1 <= 1, z - 1 >= 0, z' - 1 >= 0 eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 eq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 eq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 if1(z, z', z'', z1) -{ 1 }-> min(1 + z' + z1) :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0 if1(z, z', z'', z1) -{ 1 }-> min(1 + z'' + z1) :|: z1 >= 0, z' >= 0, z'' >= 0, z = 0 if2(z, z', z'', z1) -{ 1 }-> rm(z', z1) :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0 if2(z, z', z'', z1) -{ 1 }-> 1 + z'' + rm(z', z1) :|: z1 >= 0, z' >= 0, z'' >= 0, z = 0 le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0 le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 min(z) -{ 4 + y'' }-> if1(s'', 1 + x'', 1 + y'', xs) :|: s'' >= 0, s'' <= 1, xs >= 0, y'' >= 0, x'' >= 0, z = 1 + (1 + x'') + (1 + (1 + y'') + xs) min(z) -{ 2 }-> if1(1, 0, y, xs) :|: xs >= 0, z = 1 + 0 + (1 + y + xs), y >= 0 min(z) -{ 2 }-> if1(0, 1 + x', 0, xs) :|: xs >= 0, x' >= 0, z = 1 + (1 + x') + (1 + 0 + xs) min(z) -{ 1 }-> 0 :|: z = 0 min(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 minsort(z) -{ 1 }-> 0 :|: z = 0 minsort(z) -{ 4 + y' }-> 1 + min(1 + x + (1 + y' + xs')) + minsort(rm(if1(s', x, y', xs'), 1 + x + (1 + y' + xs'))) :|: s' >= 0, s' <= 1, x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs') minsort(z) -{ 2 }-> 1 + min(1 + (z - 1) + 0) + minsort(rm(z - 1, 1 + (z - 1) + 0)) :|: z - 1 >= 0 rm(z, z') -{ 5 + y2 }-> if2(s2, 1 + (z - 1), 1 + y2, xs) :|: s2 >= 0, s2 <= 1, xs >= 0, z' = 1 + (1 + y2) + xs, y2 >= 0, z - 1 >= 0 rm(z, z') -{ 2 }-> if2(1, 0, 0, z' - 1) :|: z' - 1 >= 0, z = 0 rm(z, z') -{ 2 }-> if2(0, 0, 1 + y1, xs) :|: y1 >= 0, xs >= 0, z' = 1 + (1 + y1) + xs, z = 0 rm(z, z') -{ 2 }-> if2(0, 1 + (z - 1), 0, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 rm(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {min,if1}, {if2,rm}, {minsort} Previous analysis results are: le: runtime: O(n^1) [2 + z'], size: O(1) [1] eq: runtime: O(n^1) [3 + z'], size: O(1) [1] min: runtime: ?, size: O(n^1) [z] if1: runtime: ?, size: O(n^1) [1 + z' + z'' + z1] ---------------------------------------- (39) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: min after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 12 + 5*z + z^2 Computed RUNTIME bound using KoAT for: if1 after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 38 + 7*z' + 2*z'*z1 + z'^2 + 7*z'' + 2*z''*z1 + z''^2 + 14*z1 + 2*z1^2 ---------------------------------------- (40) Obligation: Complexity RNTS consisting of the following rules: eq(z, z') -{ 3 + z' }-> s1 :|: s1 >= 0, s1 <= 1, z - 1 >= 0, z' - 1 >= 0 eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 eq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 eq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 if1(z, z', z'', z1) -{ 1 }-> min(1 + z' + z1) :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0 if1(z, z', z'', z1) -{ 1 }-> min(1 + z'' + z1) :|: z1 >= 0, z' >= 0, z'' >= 0, z = 0 if2(z, z', z'', z1) -{ 1 }-> rm(z', z1) :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0 if2(z, z', z'', z1) -{ 1 }-> 1 + z'' + rm(z', z1) :|: z1 >= 0, z' >= 0, z'' >= 0, z = 0 le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0 le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 min(z) -{ 4 + y'' }-> if1(s'', 1 + x'', 1 + y'', xs) :|: s'' >= 0, s'' <= 1, xs >= 0, y'' >= 0, x'' >= 0, z = 1 + (1 + x'') + (1 + (1 + y'') + xs) min(z) -{ 2 }-> if1(1, 0, y, xs) :|: xs >= 0, z = 1 + 0 + (1 + y + xs), y >= 0 min(z) -{ 2 }-> if1(0, 1 + x', 0, xs) :|: xs >= 0, x' >= 0, z = 1 + (1 + x') + (1 + 0 + xs) min(z) -{ 1 }-> 0 :|: z = 0 min(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 minsort(z) -{ 1 }-> 0 :|: z = 0 minsort(z) -{ 4 + y' }-> 1 + min(1 + x + (1 + y' + xs')) + minsort(rm(if1(s', x, y', xs'), 1 + x + (1 + y' + xs'))) :|: s' >= 0, s' <= 1, x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs') minsort(z) -{ 2 }-> 1 + min(1 + (z - 1) + 0) + minsort(rm(z - 1, 1 + (z - 1) + 0)) :|: z - 1 >= 0 rm(z, z') -{ 5 + y2 }-> if2(s2, 1 + (z - 1), 1 + y2, xs) :|: s2 >= 0, s2 <= 1, xs >= 0, z' = 1 + (1 + y2) + xs, y2 >= 0, z - 1 >= 0 rm(z, z') -{ 2 }-> if2(1, 0, 0, z' - 1) :|: z' - 1 >= 0, z = 0 rm(z, z') -{ 2 }-> if2(0, 0, 1 + y1, xs) :|: y1 >= 0, xs >= 0, z' = 1 + (1 + y1) + xs, z = 0 rm(z, z') -{ 2 }-> if2(0, 1 + (z - 1), 0, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 rm(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {if2,rm}, {minsort} Previous analysis results are: le: runtime: O(n^1) [2 + z'], size: O(1) [1] eq: runtime: O(n^1) [3 + z'], size: O(1) [1] min: runtime: O(n^2) [12 + 5*z + z^2], size: O(n^1) [z] if1: runtime: O(n^2) [38 + 7*z' + 2*z'*z1 + z'^2 + 7*z'' + 2*z''*z1 + z''^2 + 14*z1 + 2*z1^2], size: O(n^1) [1 + z' + z'' + z1] ---------------------------------------- (41) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (42) Obligation: Complexity RNTS consisting of the following rules: eq(z, z') -{ 3 + z' }-> s1 :|: s1 >= 0, s1 <= 1, z - 1 >= 0, z' - 1 >= 0 eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 eq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 eq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 if1(z, z', z'', z1) -{ 19 + 7*z'' + 2*z''*z1 + z''^2 + 7*z1 + z1^2 }-> s10 :|: s10 >= 0, s10 <= 1 + z'' + z1, z1 >= 0, z' >= 0, z'' >= 0, z = 0 if1(z, z', z'', z1) -{ 19 + 7*z' + 2*z'*z1 + z'^2 + 7*z1 + z1^2 }-> s9 :|: s9 >= 0, s9 <= 1 + z' + z1, z1 >= 0, z = 1, z' >= 0, z'' >= 0 if2(z, z', z'', z1) -{ 1 }-> rm(z', z1) :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0 if2(z, z', z'', z1) -{ 1 }-> 1 + z'' + rm(z', z1) :|: z1 >= 0, z' >= 0, z'' >= 0, z = 0 le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0 le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 min(z) -{ 40 + 14*xs + 2*xs*y + 2*xs^2 + 7*y + y^2 }-> s6 :|: s6 >= 0, s6 <= y + xs + 1 + 0, xs >= 0, z = 1 + 0 + (1 + y + xs), y >= 0 min(z) -{ 48 + 9*x' + 2*x'*xs + x'^2 + 16*xs + 2*xs^2 }-> s7 :|: s7 >= 0, s7 <= 0 + xs + 1 + (1 + x'), xs >= 0, x' >= 0, z = 1 + (1 + x') + (1 + 0 + xs) min(z) -{ 58 + 9*x'' + 2*x''*xs + x''^2 + 18*xs + 2*xs*y'' + 2*xs^2 + 10*y'' + y''^2 }-> s8 :|: s8 >= 0, s8 <= 1 + y'' + xs + 1 + (1 + x''), s'' >= 0, s'' <= 1, xs >= 0, y'' >= 0, x'' >= 0, z = 1 + (1 + x'') + (1 + (1 + y'') + xs) min(z) -{ 1 }-> 0 :|: z = 0 min(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 minsort(z) -{ 1 }-> 0 :|: z = 0 minsort(z) -{ 14 + 5*z + z^2 }-> 1 + s3 + minsort(rm(z - 1, 1 + (z - 1) + 0)) :|: s3 >= 0, s3 <= 1 + (z - 1) + 0, z - 1 >= 0 minsort(z) -{ 68 + 16*x + 4*x*xs' + 2*x*y' + 2*x^2 + 23*xs' + 4*xs'*y' + 3*xs'^2 + 17*y' + 2*y'^2 }-> 1 + s4 + minsort(rm(s5, 1 + x + (1 + y' + xs'))) :|: s4 >= 0, s4 <= 1 + x + (1 + y' + xs'), s5 >= 0, s5 <= y' + xs' + 1 + x, s' >= 0, s' <= 1, x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs') rm(z, z') -{ 5 + y2 }-> if2(s2, 1 + (z - 1), 1 + y2, xs) :|: s2 >= 0, s2 <= 1, xs >= 0, z' = 1 + (1 + y2) + xs, y2 >= 0, z - 1 >= 0 rm(z, z') -{ 2 }-> if2(1, 0, 0, z' - 1) :|: z' - 1 >= 0, z = 0 rm(z, z') -{ 2 }-> if2(0, 0, 1 + y1, xs) :|: y1 >= 0, xs >= 0, z' = 1 + (1 + y1) + xs, z = 0 rm(z, z') -{ 2 }-> if2(0, 1 + (z - 1), 0, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 rm(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {if2,rm}, {minsort} Previous analysis results are: le: runtime: O(n^1) [2 + z'], size: O(1) [1] eq: runtime: O(n^1) [3 + z'], size: O(1) [1] min: runtime: O(n^2) [12 + 5*z + z^2], size: O(n^1) [z] if1: runtime: O(n^2) [38 + 7*z' + 2*z'*z1 + z'^2 + 7*z'' + 2*z''*z1 + z''^2 + 14*z1 + 2*z1^2], size: O(n^1) [1 + z' + z'' + z1] ---------------------------------------- (43) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: if2 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z'' + z1 Computed SIZE bound using CoFloCo for: rm after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z' ---------------------------------------- (44) Obligation: Complexity RNTS consisting of the following rules: eq(z, z') -{ 3 + z' }-> s1 :|: s1 >= 0, s1 <= 1, z - 1 >= 0, z' - 1 >= 0 eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 eq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 eq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 if1(z, z', z'', z1) -{ 19 + 7*z'' + 2*z''*z1 + z''^2 + 7*z1 + z1^2 }-> s10 :|: s10 >= 0, s10 <= 1 + z'' + z1, z1 >= 0, z' >= 0, z'' >= 0, z = 0 if1(z, z', z'', z1) -{ 19 + 7*z' + 2*z'*z1 + z'^2 + 7*z1 + z1^2 }-> s9 :|: s9 >= 0, s9 <= 1 + z' + z1, z1 >= 0, z = 1, z' >= 0, z'' >= 0 if2(z, z', z'', z1) -{ 1 }-> rm(z', z1) :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0 if2(z, z', z'', z1) -{ 1 }-> 1 + z'' + rm(z', z1) :|: z1 >= 0, z' >= 0, z'' >= 0, z = 0 le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0 le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 min(z) -{ 40 + 14*xs + 2*xs*y + 2*xs^2 + 7*y + y^2 }-> s6 :|: s6 >= 0, s6 <= y + xs + 1 + 0, xs >= 0, z = 1 + 0 + (1 + y + xs), y >= 0 min(z) -{ 48 + 9*x' + 2*x'*xs + x'^2 + 16*xs + 2*xs^2 }-> s7 :|: s7 >= 0, s7 <= 0 + xs + 1 + (1 + x'), xs >= 0, x' >= 0, z = 1 + (1 + x') + (1 + 0 + xs) min(z) -{ 58 + 9*x'' + 2*x''*xs + x''^2 + 18*xs + 2*xs*y'' + 2*xs^2 + 10*y'' + y''^2 }-> s8 :|: s8 >= 0, s8 <= 1 + y'' + xs + 1 + (1 + x''), s'' >= 0, s'' <= 1, xs >= 0, y'' >= 0, x'' >= 0, z = 1 + (1 + x'') + (1 + (1 + y'') + xs) min(z) -{ 1 }-> 0 :|: z = 0 min(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 minsort(z) -{ 1 }-> 0 :|: z = 0 minsort(z) -{ 14 + 5*z + z^2 }-> 1 + s3 + minsort(rm(z - 1, 1 + (z - 1) + 0)) :|: s3 >= 0, s3 <= 1 + (z - 1) + 0, z - 1 >= 0 minsort(z) -{ 68 + 16*x + 4*x*xs' + 2*x*y' + 2*x^2 + 23*xs' + 4*xs'*y' + 3*xs'^2 + 17*y' + 2*y'^2 }-> 1 + s4 + minsort(rm(s5, 1 + x + (1 + y' + xs'))) :|: s4 >= 0, s4 <= 1 + x + (1 + y' + xs'), s5 >= 0, s5 <= y' + xs' + 1 + x, s' >= 0, s' <= 1, x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs') rm(z, z') -{ 5 + y2 }-> if2(s2, 1 + (z - 1), 1 + y2, xs) :|: s2 >= 0, s2 <= 1, xs >= 0, z' = 1 + (1 + y2) + xs, y2 >= 0, z - 1 >= 0 rm(z, z') -{ 2 }-> if2(1, 0, 0, z' - 1) :|: z' - 1 >= 0, z = 0 rm(z, z') -{ 2 }-> if2(0, 0, 1 + y1, xs) :|: y1 >= 0, xs >= 0, z' = 1 + (1 + y1) + xs, z = 0 rm(z, z') -{ 2 }-> if2(0, 1 + (z - 1), 0, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 rm(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {if2,rm}, {minsort} Previous analysis results are: le: runtime: O(n^1) [2 + z'], size: O(1) [1] eq: runtime: O(n^1) [3 + z'], size: O(1) [1] min: runtime: O(n^2) [12 + 5*z + z^2], size: O(n^1) [z] if1: runtime: O(n^2) [38 + 7*z' + 2*z'*z1 + z'^2 + 7*z'' + 2*z''*z1 + z''^2 + 14*z1 + 2*z1^2], size: O(n^1) [1 + z' + z'' + z1] if2: runtime: ?, size: O(n^1) [1 + z'' + z1] rm: runtime: ?, size: O(n^1) [z'] ---------------------------------------- (45) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: if2 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 5 + 3*z + 8*z1 Computed RUNTIME bound using CoFloCo for: rm after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 9 + 8*z' ---------------------------------------- (46) Obligation: Complexity RNTS consisting of the following rules: eq(z, z') -{ 3 + z' }-> s1 :|: s1 >= 0, s1 <= 1, z - 1 >= 0, z' - 1 >= 0 eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 eq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 eq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 if1(z, z', z'', z1) -{ 19 + 7*z'' + 2*z''*z1 + z''^2 + 7*z1 + z1^2 }-> s10 :|: s10 >= 0, s10 <= 1 + z'' + z1, z1 >= 0, z' >= 0, z'' >= 0, z = 0 if1(z, z', z'', z1) -{ 19 + 7*z' + 2*z'*z1 + z'^2 + 7*z1 + z1^2 }-> s9 :|: s9 >= 0, s9 <= 1 + z' + z1, z1 >= 0, z = 1, z' >= 0, z'' >= 0 if2(z, z', z'', z1) -{ 1 }-> rm(z', z1) :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0 if2(z, z', z'', z1) -{ 1 }-> 1 + z'' + rm(z', z1) :|: z1 >= 0, z' >= 0, z'' >= 0, z = 0 le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0 le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 min(z) -{ 40 + 14*xs + 2*xs*y + 2*xs^2 + 7*y + y^2 }-> s6 :|: s6 >= 0, s6 <= y + xs + 1 + 0, xs >= 0, z = 1 + 0 + (1 + y + xs), y >= 0 min(z) -{ 48 + 9*x' + 2*x'*xs + x'^2 + 16*xs + 2*xs^2 }-> s7 :|: s7 >= 0, s7 <= 0 + xs + 1 + (1 + x'), xs >= 0, x' >= 0, z = 1 + (1 + x') + (1 + 0 + xs) min(z) -{ 58 + 9*x'' + 2*x''*xs + x''^2 + 18*xs + 2*xs*y'' + 2*xs^2 + 10*y'' + y''^2 }-> s8 :|: s8 >= 0, s8 <= 1 + y'' + xs + 1 + (1 + x''), s'' >= 0, s'' <= 1, xs >= 0, y'' >= 0, x'' >= 0, z = 1 + (1 + x'') + (1 + (1 + y'') + xs) min(z) -{ 1 }-> 0 :|: z = 0 min(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 minsort(z) -{ 1 }-> 0 :|: z = 0 minsort(z) -{ 14 + 5*z + z^2 }-> 1 + s3 + minsort(rm(z - 1, 1 + (z - 1) + 0)) :|: s3 >= 0, s3 <= 1 + (z - 1) + 0, z - 1 >= 0 minsort(z) -{ 68 + 16*x + 4*x*xs' + 2*x*y' + 2*x^2 + 23*xs' + 4*xs'*y' + 3*xs'^2 + 17*y' + 2*y'^2 }-> 1 + s4 + minsort(rm(s5, 1 + x + (1 + y' + xs'))) :|: s4 >= 0, s4 <= 1 + x + (1 + y' + xs'), s5 >= 0, s5 <= y' + xs' + 1 + x, s' >= 0, s' <= 1, x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs') rm(z, z') -{ 5 + y2 }-> if2(s2, 1 + (z - 1), 1 + y2, xs) :|: s2 >= 0, s2 <= 1, xs >= 0, z' = 1 + (1 + y2) + xs, y2 >= 0, z - 1 >= 0 rm(z, z') -{ 2 }-> if2(1, 0, 0, z' - 1) :|: z' - 1 >= 0, z = 0 rm(z, z') -{ 2 }-> if2(0, 0, 1 + y1, xs) :|: y1 >= 0, xs >= 0, z' = 1 + (1 + y1) + xs, z = 0 rm(z, z') -{ 2 }-> if2(0, 1 + (z - 1), 0, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 rm(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {minsort} Previous analysis results are: le: runtime: O(n^1) [2 + z'], size: O(1) [1] eq: runtime: O(n^1) [3 + z'], size: O(1) [1] min: runtime: O(n^2) [12 + 5*z + z^2], size: O(n^1) [z] if1: runtime: O(n^2) [38 + 7*z' + 2*z'*z1 + z'^2 + 7*z'' + 2*z''*z1 + z''^2 + 14*z1 + 2*z1^2], size: O(n^1) [1 + z' + z'' + z1] if2: runtime: O(n^1) [5 + 3*z + 8*z1], size: O(n^1) [1 + z'' + z1] rm: runtime: O(n^1) [9 + 8*z'], size: O(n^1) [z'] ---------------------------------------- (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: eq(z, z') -{ 3 + z' }-> s1 :|: s1 >= 0, s1 <= 1, z - 1 >= 0, z' - 1 >= 0 eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 eq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 eq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 if1(z, z', z'', z1) -{ 19 + 7*z'' + 2*z''*z1 + z''^2 + 7*z1 + z1^2 }-> s10 :|: s10 >= 0, s10 <= 1 + z'' + z1, z1 >= 0, z' >= 0, z'' >= 0, z = 0 if1(z, z', z'', z1) -{ 19 + 7*z' + 2*z'*z1 + z'^2 + 7*z1 + z1^2 }-> s9 :|: s9 >= 0, s9 <= 1 + z' + z1, z1 >= 0, z = 1, z' >= 0, z'' >= 0 if2(z, z', z'', z1) -{ 10 + 8*z1 }-> s17 :|: s17 >= 0, s17 <= z1, z1 >= 0, z = 1, z' >= 0, z'' >= 0 if2(z, z', z'', z1) -{ 10 + 8*z1 }-> 1 + z'' + s18 :|: s18 >= 0, s18 <= z1, z1 >= 0, z' >= 0, z'' >= 0, z = 0 le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0 le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 min(z) -{ 40 + 14*xs + 2*xs*y + 2*xs^2 + 7*y + y^2 }-> s6 :|: s6 >= 0, s6 <= y + xs + 1 + 0, xs >= 0, z = 1 + 0 + (1 + y + xs), y >= 0 min(z) -{ 48 + 9*x' + 2*x'*xs + x'^2 + 16*xs + 2*xs^2 }-> s7 :|: s7 >= 0, s7 <= 0 + xs + 1 + (1 + x'), xs >= 0, x' >= 0, z = 1 + (1 + x') + (1 + 0 + xs) min(z) -{ 58 + 9*x'' + 2*x''*xs + x''^2 + 18*xs + 2*xs*y'' + 2*xs^2 + 10*y'' + y''^2 }-> s8 :|: s8 >= 0, s8 <= 1 + y'' + xs + 1 + (1 + x''), s'' >= 0, s'' <= 1, xs >= 0, y'' >= 0, x'' >= 0, z = 1 + (1 + x'') + (1 + (1 + y'') + xs) min(z) -{ 1 }-> 0 :|: z = 0 min(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 minsort(z) -{ 1 }-> 0 :|: z = 0 minsort(z) -{ 23 + 13*z + z^2 }-> 1 + s3 + minsort(s11) :|: s11 >= 0, s11 <= 1 + (z - 1) + 0, s3 >= 0, s3 <= 1 + (z - 1) + 0, z - 1 >= 0 minsort(z) -{ 93 + 24*x + 4*x*xs' + 2*x*y' + 2*x^2 + 31*xs' + 4*xs'*y' + 3*xs'^2 + 25*y' + 2*y'^2 }-> 1 + s4 + minsort(s12) :|: s12 >= 0, s12 <= 1 + x + (1 + y' + xs'), s4 >= 0, s4 <= 1 + x + (1 + y' + xs'), s5 >= 0, s5 <= y' + xs' + 1 + x, s' >= 0, s' <= 1, x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs') rm(z, z') -{ 2 + 8*z' }-> s13 :|: s13 >= 0, s13 <= 0 + (z' - 1) + 1, z' - 1 >= 0, z = 0 rm(z, z') -{ 7 + 8*xs }-> s14 :|: s14 >= 0, s14 <= 1 + y1 + xs + 1, y1 >= 0, xs >= 0, z' = 1 + (1 + y1) + xs, z = 0 rm(z, z') -{ -1 + 8*z' }-> s15 :|: s15 >= 0, s15 <= 0 + (z' - 1) + 1, z' - 1 >= 0, z - 1 >= 0 rm(z, z') -{ 10 + 3*s2 + 8*xs + y2 }-> s16 :|: s16 >= 0, s16 <= 1 + y2 + xs + 1, s2 >= 0, s2 <= 1, xs >= 0, z' = 1 + (1 + y2) + xs, y2 >= 0, z - 1 >= 0 rm(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {minsort} Previous analysis results are: le: runtime: O(n^1) [2 + z'], size: O(1) [1] eq: runtime: O(n^1) [3 + z'], size: O(1) [1] min: runtime: O(n^2) [12 + 5*z + z^2], size: O(n^1) [z] if1: runtime: O(n^2) [38 + 7*z' + 2*z'*z1 + z'^2 + 7*z'' + 2*z''*z1 + z''^2 + 14*z1 + 2*z1^2], size: O(n^1) [1 + z' + z'' + z1] if2: runtime: O(n^1) [5 + 3*z + 8*z1], size: O(n^1) [1 + z'' + z1] rm: runtime: O(n^1) [9 + 8*z'], size: O(n^1) [z'] ---------------------------------------- (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: eq(z, z') -{ 3 + z' }-> s1 :|: s1 >= 0, s1 <= 1, z - 1 >= 0, z' - 1 >= 0 eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 eq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 eq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 if1(z, z', z'', z1) -{ 19 + 7*z'' + 2*z''*z1 + z''^2 + 7*z1 + z1^2 }-> s10 :|: s10 >= 0, s10 <= 1 + z'' + z1, z1 >= 0, z' >= 0, z'' >= 0, z = 0 if1(z, z', z'', z1) -{ 19 + 7*z' + 2*z'*z1 + z'^2 + 7*z1 + z1^2 }-> s9 :|: s9 >= 0, s9 <= 1 + z' + z1, z1 >= 0, z = 1, z' >= 0, z'' >= 0 if2(z, z', z'', z1) -{ 10 + 8*z1 }-> s17 :|: s17 >= 0, s17 <= z1, z1 >= 0, z = 1, z' >= 0, z'' >= 0 if2(z, z', z'', z1) -{ 10 + 8*z1 }-> 1 + z'' + s18 :|: s18 >= 0, s18 <= z1, z1 >= 0, z' >= 0, z'' >= 0, z = 0 le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0 le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 min(z) -{ 40 + 14*xs + 2*xs*y + 2*xs^2 + 7*y + y^2 }-> s6 :|: s6 >= 0, s6 <= y + xs + 1 + 0, xs >= 0, z = 1 + 0 + (1 + y + xs), y >= 0 min(z) -{ 48 + 9*x' + 2*x'*xs + x'^2 + 16*xs + 2*xs^2 }-> s7 :|: s7 >= 0, s7 <= 0 + xs + 1 + (1 + x'), xs >= 0, x' >= 0, z = 1 + (1 + x') + (1 + 0 + xs) min(z) -{ 58 + 9*x'' + 2*x''*xs + x''^2 + 18*xs + 2*xs*y'' + 2*xs^2 + 10*y'' + y''^2 }-> s8 :|: s8 >= 0, s8 <= 1 + y'' + xs + 1 + (1 + x''), s'' >= 0, s'' <= 1, xs >= 0, y'' >= 0, x'' >= 0, z = 1 + (1 + x'') + (1 + (1 + y'') + xs) min(z) -{ 1 }-> 0 :|: z = 0 min(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 minsort(z) -{ 1 }-> 0 :|: z = 0 minsort(z) -{ 23 + 13*z + z^2 }-> 1 + s3 + minsort(s11) :|: s11 >= 0, s11 <= 1 + (z - 1) + 0, s3 >= 0, s3 <= 1 + (z - 1) + 0, z - 1 >= 0 minsort(z) -{ 93 + 24*x + 4*x*xs' + 2*x*y' + 2*x^2 + 31*xs' + 4*xs'*y' + 3*xs'^2 + 25*y' + 2*y'^2 }-> 1 + s4 + minsort(s12) :|: s12 >= 0, s12 <= 1 + x + (1 + y' + xs'), s4 >= 0, s4 <= 1 + x + (1 + y' + xs'), s5 >= 0, s5 <= y' + xs' + 1 + x, s' >= 0, s' <= 1, x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs') rm(z, z') -{ 2 + 8*z' }-> s13 :|: s13 >= 0, s13 <= 0 + (z' - 1) + 1, z' - 1 >= 0, z = 0 rm(z, z') -{ 7 + 8*xs }-> s14 :|: s14 >= 0, s14 <= 1 + y1 + xs + 1, y1 >= 0, xs >= 0, z' = 1 + (1 + y1) + xs, z = 0 rm(z, z') -{ -1 + 8*z' }-> s15 :|: s15 >= 0, s15 <= 0 + (z' - 1) + 1, z' - 1 >= 0, z - 1 >= 0 rm(z, z') -{ 10 + 3*s2 + 8*xs + y2 }-> s16 :|: s16 >= 0, s16 <= 1 + y2 + xs + 1, s2 >= 0, s2 <= 1, xs >= 0, z' = 1 + (1 + y2) + xs, y2 >= 0, z - 1 >= 0 rm(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {minsort} Previous analysis results are: le: runtime: O(n^1) [2 + z'], size: O(1) [1] eq: runtime: O(n^1) [3 + z'], size: O(1) [1] min: runtime: O(n^2) [12 + 5*z + z^2], size: O(n^1) [z] if1: runtime: O(n^2) [38 + 7*z' + 2*z'*z1 + z'^2 + 7*z'' + 2*z''*z1 + z''^2 + 14*z1 + 2*z1^2], size: O(n^1) [1 + z' + z'' + z1] if2: runtime: O(n^1) [5 + 3*z + 8*z1], size: O(n^1) [1 + z'' + z1] rm: runtime: O(n^1) [9 + 8*z'], size: O(n^1) [z'] 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: eq(z, z') -{ 3 + z' }-> s1 :|: s1 >= 0, s1 <= 1, z - 1 >= 0, z' - 1 >= 0 eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 eq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 eq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 if1(z, z', z'', z1) -{ 19 + 7*z'' + 2*z''*z1 + z''^2 + 7*z1 + z1^2 }-> s10 :|: s10 >= 0, s10 <= 1 + z'' + z1, z1 >= 0, z' >= 0, z'' >= 0, z = 0 if1(z, z', z'', z1) -{ 19 + 7*z' + 2*z'*z1 + z'^2 + 7*z1 + z1^2 }-> s9 :|: s9 >= 0, s9 <= 1 + z' + z1, z1 >= 0, z = 1, z' >= 0, z'' >= 0 if2(z, z', z'', z1) -{ 10 + 8*z1 }-> s17 :|: s17 >= 0, s17 <= z1, z1 >= 0, z = 1, z' >= 0, z'' >= 0 if2(z, z', z'', z1) -{ 10 + 8*z1 }-> 1 + z'' + s18 :|: s18 >= 0, s18 <= z1, z1 >= 0, z' >= 0, z'' >= 0, z = 0 le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0 le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 min(z) -{ 40 + 14*xs + 2*xs*y + 2*xs^2 + 7*y + y^2 }-> s6 :|: s6 >= 0, s6 <= y + xs + 1 + 0, xs >= 0, z = 1 + 0 + (1 + y + xs), y >= 0 min(z) -{ 48 + 9*x' + 2*x'*xs + x'^2 + 16*xs + 2*xs^2 }-> s7 :|: s7 >= 0, s7 <= 0 + xs + 1 + (1 + x'), xs >= 0, x' >= 0, z = 1 + (1 + x') + (1 + 0 + xs) min(z) -{ 58 + 9*x'' + 2*x''*xs + x''^2 + 18*xs + 2*xs*y'' + 2*xs^2 + 10*y'' + y''^2 }-> s8 :|: s8 >= 0, s8 <= 1 + y'' + xs + 1 + (1 + x''), s'' >= 0, s'' <= 1, xs >= 0, y'' >= 0, x'' >= 0, z = 1 + (1 + x'') + (1 + (1 + y'') + xs) min(z) -{ 1 }-> 0 :|: z = 0 min(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 minsort(z) -{ 1 }-> 0 :|: z = 0 minsort(z) -{ 23 + 13*z + z^2 }-> 1 + s3 + minsort(s11) :|: s11 >= 0, s11 <= 1 + (z - 1) + 0, s3 >= 0, s3 <= 1 + (z - 1) + 0, z - 1 >= 0 minsort(z) -{ 93 + 24*x + 4*x*xs' + 2*x*y' + 2*x^2 + 31*xs' + 4*xs'*y' + 3*xs'^2 + 25*y' + 2*y'^2 }-> 1 + s4 + minsort(s12) :|: s12 >= 0, s12 <= 1 + x + (1 + y' + xs'), s4 >= 0, s4 <= 1 + x + (1 + y' + xs'), s5 >= 0, s5 <= y' + xs' + 1 + x, s' >= 0, s' <= 1, x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs') rm(z, z') -{ 2 + 8*z' }-> s13 :|: s13 >= 0, s13 <= 0 + (z' - 1) + 1, z' - 1 >= 0, z = 0 rm(z, z') -{ 7 + 8*xs }-> s14 :|: s14 >= 0, s14 <= 1 + y1 + xs + 1, y1 >= 0, xs >= 0, z' = 1 + (1 + y1) + xs, z = 0 rm(z, z') -{ -1 + 8*z' }-> s15 :|: s15 >= 0, s15 <= 0 + (z' - 1) + 1, z' - 1 >= 0, z - 1 >= 0 rm(z, z') -{ 10 + 3*s2 + 8*xs + y2 }-> s16 :|: s16 >= 0, s16 <= 1 + y2 + xs + 1, s2 >= 0, s2 <= 1, xs >= 0, z' = 1 + (1 + y2) + xs, y2 >= 0, z - 1 >= 0 rm(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {minsort} Previous analysis results are: le: runtime: O(n^1) [2 + z'], size: O(1) [1] eq: runtime: O(n^1) [3 + z'], size: O(1) [1] min: runtime: O(n^2) [12 + 5*z + z^2], size: O(n^1) [z] if1: runtime: O(n^2) [38 + 7*z' + 2*z'*z1 + z'^2 + 7*z'' + 2*z''*z1 + z''^2 + 14*z1 + 2*z1^2], size: O(n^1) [1 + z' + z'' + z1] if2: runtime: O(n^1) [5 + 3*z + 8*z1], size: O(n^1) [1 + z'' + z1] rm: runtime: O(n^1) [9 + 8*z'], size: O(n^1) [z'] 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) minsort(nil) -> nil minsort(cons(z0, z1)) -> cons(min(cons(z0, z1)), minsort(rm(min(cons(z0, z1)), cons(z0, z1)))) min(nil) -> 0 min(cons(z0, nil)) -> z0 min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(cons(z0, z2)) if1(false, z0, z1, z2) -> min(cons(z1, z2)) rm(z0, nil) -> nil rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) if2(true, z0, z1, z2) -> rm(z0, z2) if2(false, z0, z1, z2) -> cons(z1, rm(z0, 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)) MINSORT(nil) -> c7 MINSORT(cons(z0, z1)) -> c8(MIN(cons(z0, z1))) MINSORT(cons(z0, z1)) -> c9(MINSORT(rm(min(cons(z0, z1)), cons(z0, z1))), RM(min(cons(z0, z1)), cons(z0, z1)), MIN(cons(z0, z1))) MIN(nil) -> c10 MIN(cons(z0, nil)) -> c11 MIN(cons(z0, cons(z1, z2))) -> c12(IF1(le(z0, z1), z0, z1, z2), LE(z0, z1)) IF1(true, z0, z1, z2) -> c13(MIN(cons(z0, z2))) IF1(false, z0, z1, z2) -> c14(MIN(cons(z1, z2))) RM(z0, nil) -> c15 RM(z0, cons(z1, z2)) -> c16(IF2(eq(z0, z1), z0, z1, z2), EQ(z0, z1)) IF2(true, z0, z1, z2) -> c17(RM(z0, z2)) IF2(false, z0, z1, z2) -> c18(RM(z0, z2)) 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)) MINSORT(nil) -> c7 MINSORT(cons(z0, z1)) -> c8(MIN(cons(z0, z1))) MINSORT(cons(z0, z1)) -> c9(MINSORT(rm(min(cons(z0, z1)), cons(z0, z1))), RM(min(cons(z0, z1)), cons(z0, z1)), MIN(cons(z0, z1))) MIN(nil) -> c10 MIN(cons(z0, nil)) -> c11 MIN(cons(z0, cons(z1, z2))) -> c12(IF1(le(z0, z1), z0, z1, z2), LE(z0, z1)) IF1(true, z0, z1, z2) -> c13(MIN(cons(z0, z2))) IF1(false, z0, z1, z2) -> c14(MIN(cons(z1, z2))) RM(z0, nil) -> c15 RM(z0, cons(z1, z2)) -> c16(IF2(eq(z0, z1), z0, z1, z2), EQ(z0, z1)) IF2(true, z0, z1, z2) -> c17(RM(z0, z2)) IF2(false, z0, z1, z2) -> c18(RM(z0, z2)) K tuples:none Defined Rule Symbols: le_2, eq_2, minsort_1, min_1, if1_4, rm_2, if2_4 Defined Pair Symbols: LE_2, EQ_2, MINSORT_1, MIN_1, IF1_4, RM_2, IF2_4 Compound Symbols: c, c1, c2_1, c3, c4, c5, c6_1, c7, c8_1, c9_3, c10, c11, c12_2, c13_1, c14_1, c15, c16_2, c17_1, c18_1 ---------------------------------------- (55) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 9 trailing nodes: RM(z0, nil) -> c15 MINSORT(nil) -> c7 LE(s(z0), 0) -> c1 EQ(0, 0) -> c3 MIN(cons(z0, nil)) -> c11 LE(0, z0) -> c EQ(0, s(z0)) -> c4 MIN(nil) -> c10 EQ(s(z0), 0) -> c5 ---------------------------------------- (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) minsort(nil) -> nil minsort(cons(z0, z1)) -> cons(min(cons(z0, z1)), minsort(rm(min(cons(z0, z1)), cons(z0, z1)))) min(nil) -> 0 min(cons(z0, nil)) -> z0 min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(cons(z0, z2)) if1(false, z0, z1, z2) -> min(cons(z1, z2)) rm(z0, nil) -> nil rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) if2(true, z0, z1, z2) -> rm(z0, z2) if2(false, z0, z1, z2) -> cons(z1, rm(z0, z2)) Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) MINSORT(cons(z0, z1)) -> c8(MIN(cons(z0, z1))) MINSORT(cons(z0, z1)) -> c9(MINSORT(rm(min(cons(z0, z1)), cons(z0, z1))), RM(min(cons(z0, z1)), cons(z0, z1)), MIN(cons(z0, z1))) MIN(cons(z0, cons(z1, z2))) -> c12(IF1(le(z0, z1), z0, z1, z2), LE(z0, z1)) IF1(true, z0, z1, z2) -> c13(MIN(cons(z0, z2))) IF1(false, z0, z1, z2) -> c14(MIN(cons(z1, z2))) RM(z0, cons(z1, z2)) -> c16(IF2(eq(z0, z1), z0, z1, z2), EQ(z0, z1)) IF2(true, z0, z1, z2) -> c17(RM(z0, z2)) IF2(false, z0, z1, z2) -> c18(RM(z0, z2)) S tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) MINSORT(cons(z0, z1)) -> c8(MIN(cons(z0, z1))) MINSORT(cons(z0, z1)) -> c9(MINSORT(rm(min(cons(z0, z1)), cons(z0, z1))), RM(min(cons(z0, z1)), cons(z0, z1)), MIN(cons(z0, z1))) MIN(cons(z0, cons(z1, z2))) -> c12(IF1(le(z0, z1), z0, z1, z2), LE(z0, z1)) IF1(true, z0, z1, z2) -> c13(MIN(cons(z0, z2))) IF1(false, z0, z1, z2) -> c14(MIN(cons(z1, z2))) RM(z0, cons(z1, z2)) -> c16(IF2(eq(z0, z1), z0, z1, z2), EQ(z0, z1)) IF2(true, z0, z1, z2) -> c17(RM(z0, z2)) IF2(false, z0, z1, z2) -> c18(RM(z0, z2)) K tuples:none Defined Rule Symbols: le_2, eq_2, minsort_1, min_1, if1_4, rm_2, if2_4 Defined Pair Symbols: LE_2, EQ_2, MINSORT_1, MIN_1, IF1_4, RM_2, IF2_4 Compound Symbols: c2_1, c6_1, c8_1, c9_3, c12_2, c13_1, c14_1, c16_2, c17_1, c18_1 ---------------------------------------- (57) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: minsort(nil) -> nil minsort(cons(z0, z1)) -> cons(min(cons(z0, z1)), minsort(rm(min(cons(z0, z1)), cons(z0, z1)))) min(nil) -> 0 ---------------------------------------- (58) Obligation: Complexity Dependency Tuples Problem Rules: rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) rm(z0, nil) -> nil min(cons(z0, nil)) -> z0 min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(cons(z0, z2)) if1(false, z0, z1, z2) -> min(cons(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) if2(true, z0, z1, z2) -> rm(z0, z2) if2(false, z0, z1, z2) -> cons(z1, rm(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)) -> c8(MIN(cons(z0, z1))) MINSORT(cons(z0, z1)) -> c9(MINSORT(rm(min(cons(z0, z1)), cons(z0, z1))), RM(min(cons(z0, z1)), cons(z0, z1)), MIN(cons(z0, z1))) MIN(cons(z0, cons(z1, z2))) -> c12(IF1(le(z0, z1), z0, z1, z2), LE(z0, z1)) IF1(true, z0, z1, z2) -> c13(MIN(cons(z0, z2))) IF1(false, z0, z1, z2) -> c14(MIN(cons(z1, z2))) RM(z0, cons(z1, z2)) -> c16(IF2(eq(z0, z1), z0, z1, z2), EQ(z0, z1)) IF2(true, z0, z1, z2) -> c17(RM(z0, z2)) IF2(false, z0, z1, z2) -> c18(RM(z0, z2)) S tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) MINSORT(cons(z0, z1)) -> c8(MIN(cons(z0, z1))) MINSORT(cons(z0, z1)) -> c9(MINSORT(rm(min(cons(z0, z1)), cons(z0, z1))), RM(min(cons(z0, z1)), cons(z0, z1)), MIN(cons(z0, z1))) MIN(cons(z0, cons(z1, z2))) -> c12(IF1(le(z0, z1), z0, z1, z2), LE(z0, z1)) IF1(true, z0, z1, z2) -> c13(MIN(cons(z0, z2))) IF1(false, z0, z1, z2) -> c14(MIN(cons(z1, z2))) RM(z0, cons(z1, z2)) -> c16(IF2(eq(z0, z1), z0, z1, z2), EQ(z0, z1)) IF2(true, z0, z1, z2) -> c17(RM(z0, z2)) IF2(false, z0, z1, z2) -> c18(RM(z0, z2)) K tuples:none Defined Rule Symbols: rm_2, min_1, if1_4, le_2, if2_4, eq_2 Defined Pair Symbols: LE_2, EQ_2, MINSORT_1, MIN_1, IF1_4, RM_2, IF2_4 Compound Symbols: c2_1, c6_1, c8_1, c9_3, c12_2, c13_1, c14_1, c16_2, c17_1, c18_1 ---------------------------------------- (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)) -> c8(MIN(cons(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)) MINSORT(cons(z0, z1)) -> c8(MIN(cons(z0, z1))) MINSORT(cons(z0, z1)) -> c9(MINSORT(rm(min(cons(z0, z1)), cons(z0, z1))), RM(min(cons(z0, z1)), cons(z0, z1)), MIN(cons(z0, z1))) MIN(cons(z0, cons(z1, z2))) -> c12(IF1(le(z0, z1), z0, z1, z2), LE(z0, z1)) IF1(true, z0, z1, z2) -> c13(MIN(cons(z0, z2))) IF1(false, z0, z1, z2) -> c14(MIN(cons(z1, z2))) RM(z0, cons(z1, z2)) -> c16(IF2(eq(z0, z1), z0, z1, z2), EQ(z0, z1)) IF2(true, z0, z1, z2) -> c17(RM(z0, z2)) IF2(false, z0, z1, z2) -> c18(RM(z0, z2)) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = [1] 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_1 POL(MINSORT(x_1)) = [1] POL(RM(x_1, x_2)) = 0 POL(c12(x_1, x_2)) = x_1 + x_2 POL(c13(x_1)) = x_1 POL(c14(x_1)) = x_1 POL(c16(x_1, x_2)) = x_1 + x_2 POL(c17(x_1)) = x_1 POL(c18(x_1)) = x_1 POL(c2(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c9(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(cons(x_1, x_2)) = 0 POL(eq(x_1, x_2)) = [1] + x_1 + x_2 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_2 POL(le(x_1, x_2)) = [1] + x_1 + x_2 POL(min(x_1)) = [1] + x_1 POL(nil) = [1] POL(rm(x_1, x_2)) = [1] + x_1 POL(s(x_1)) = [1] + x_1 POL(true) = [1] ---------------------------------------- (60) Obligation: Complexity Dependency Tuples Problem Rules: rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) rm(z0, nil) -> nil min(cons(z0, nil)) -> z0 min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(cons(z0, z2)) if1(false, z0, z1, z2) -> min(cons(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) if2(true, z0, z1, z2) -> rm(z0, z2) if2(false, z0, z1, z2) -> cons(z1, rm(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)) -> c8(MIN(cons(z0, z1))) MINSORT(cons(z0, z1)) -> c9(MINSORT(rm(min(cons(z0, z1)), cons(z0, z1))), RM(min(cons(z0, z1)), cons(z0, z1)), MIN(cons(z0, z1))) MIN(cons(z0, cons(z1, z2))) -> c12(IF1(le(z0, z1), z0, z1, z2), LE(z0, z1)) IF1(true, z0, z1, z2) -> c13(MIN(cons(z0, z2))) IF1(false, z0, z1, z2) -> c14(MIN(cons(z1, z2))) RM(z0, cons(z1, z2)) -> c16(IF2(eq(z0, z1), z0, z1, z2), EQ(z0, z1)) IF2(true, z0, z1, z2) -> c17(RM(z0, z2)) IF2(false, z0, z1, z2) -> c18(RM(z0, z2)) S tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) MINSORT(cons(z0, z1)) -> c9(MINSORT(rm(min(cons(z0, z1)), cons(z0, z1))), RM(min(cons(z0, z1)), cons(z0, z1)), MIN(cons(z0, z1))) MIN(cons(z0, cons(z1, z2))) -> c12(IF1(le(z0, z1), z0, z1, z2), LE(z0, z1)) IF1(true, z0, z1, z2) -> c13(MIN(cons(z0, z2))) IF1(false, z0, z1, z2) -> c14(MIN(cons(z1, z2))) RM(z0, cons(z1, z2)) -> c16(IF2(eq(z0, z1), z0, z1, z2), EQ(z0, z1)) IF2(true, z0, z1, z2) -> c17(RM(z0, z2)) IF2(false, z0, z1, z2) -> c18(RM(z0, z2)) K tuples: MINSORT(cons(z0, z1)) -> c8(MIN(cons(z0, z1))) Defined Rule Symbols: rm_2, min_1, if1_4, le_2, if2_4, eq_2 Defined Pair Symbols: LE_2, EQ_2, MINSORT_1, MIN_1, IF1_4, RM_2, IF2_4 Compound Symbols: c2_1, c6_1, c8_1, c9_3, c12_2, c13_1, c14_1, c16_2, c17_1, c18_1 ---------------------------------------- (61) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MINSORT(cons(z0, z1)) -> c9(MINSORT(rm(min(cons(z0, z1)), cons(z0, z1))), RM(min(cons(z0, z1)), cons(z0, z1)), MIN(cons(z0, z1))) by MINSORT(cons(z1, z2)) -> c9(MINSORT(if2(eq(min(cons(z1, z2)), z1), min(cons(z1, z2)), z1, z2)), RM(min(cons(z1, z2)), cons(z1, z2)), MIN(cons(z1, z2))) MINSORT(cons(z0, nil)) -> c9(MINSORT(rm(z0, cons(z0, nil))), RM(min(cons(z0, nil)), cons(z0, nil)), MIN(cons(z0, nil))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(rm(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) ---------------------------------------- (62) Obligation: Complexity Dependency Tuples Problem Rules: rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) rm(z0, nil) -> nil min(cons(z0, nil)) -> z0 min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(cons(z0, z2)) if1(false, z0, z1, z2) -> min(cons(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) if2(true, z0, z1, z2) -> rm(z0, z2) if2(false, z0, z1, z2) -> cons(z1, rm(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)) -> c8(MIN(cons(z0, z1))) MIN(cons(z0, cons(z1, z2))) -> c12(IF1(le(z0, z1), z0, z1, z2), LE(z0, z1)) IF1(true, z0, z1, z2) -> c13(MIN(cons(z0, z2))) IF1(false, z0, z1, z2) -> c14(MIN(cons(z1, z2))) RM(z0, cons(z1, z2)) -> c16(IF2(eq(z0, z1), z0, z1, z2), EQ(z0, z1)) IF2(true, z0, z1, z2) -> c17(RM(z0, z2)) IF2(false, z0, z1, z2) -> c18(RM(z0, z2)) MINSORT(cons(z1, z2)) -> c9(MINSORT(if2(eq(min(cons(z1, z2)), z1), min(cons(z1, z2)), z1, z2)), RM(min(cons(z1, z2)), cons(z1, z2)), MIN(cons(z1, z2))) MINSORT(cons(z0, nil)) -> c9(MINSORT(rm(z0, cons(z0, nil))), RM(min(cons(z0, nil)), cons(z0, nil)), MIN(cons(z0, nil))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(rm(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(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)) MIN(cons(z0, cons(z1, z2))) -> c12(IF1(le(z0, z1), z0, z1, z2), LE(z0, z1)) IF1(true, z0, z1, z2) -> c13(MIN(cons(z0, z2))) IF1(false, z0, z1, z2) -> c14(MIN(cons(z1, z2))) RM(z0, cons(z1, z2)) -> c16(IF2(eq(z0, z1), z0, z1, z2), EQ(z0, z1)) IF2(true, z0, z1, z2) -> c17(RM(z0, z2)) IF2(false, z0, z1, z2) -> c18(RM(z0, z2)) MINSORT(cons(z1, z2)) -> c9(MINSORT(if2(eq(min(cons(z1, z2)), z1), min(cons(z1, z2)), z1, z2)), RM(min(cons(z1, z2)), cons(z1, z2)), MIN(cons(z1, z2))) MINSORT(cons(z0, nil)) -> c9(MINSORT(rm(z0, cons(z0, nil))), RM(min(cons(z0, nil)), cons(z0, nil)), MIN(cons(z0, nil))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(rm(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) K tuples: MINSORT(cons(z0, z1)) -> c8(MIN(cons(z0, z1))) Defined Rule Symbols: rm_2, min_1, if1_4, le_2, if2_4, eq_2 Defined Pair Symbols: LE_2, EQ_2, MINSORT_1, MIN_1, IF1_4, RM_2, IF2_4 Compound Symbols: c2_1, c6_1, c8_1, c12_2, c13_1, c14_1, c16_2, c17_1, c18_1, c9_3 ---------------------------------------- (63) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (64) Obligation: Complexity Dependency Tuples Problem Rules: rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) rm(z0, nil) -> nil min(cons(z0, nil)) -> z0 min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(cons(z0, z2)) if1(false, z0, z1, z2) -> min(cons(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) if2(true, z0, z1, z2) -> rm(z0, z2) if2(false, z0, z1, z2) -> cons(z1, rm(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)) -> c8(MIN(cons(z0, z1))) MIN(cons(z0, cons(z1, z2))) -> c12(IF1(le(z0, z1), z0, z1, z2), LE(z0, z1)) IF1(true, z0, z1, z2) -> c13(MIN(cons(z0, z2))) IF1(false, z0, z1, z2) -> c14(MIN(cons(z1, z2))) RM(z0, cons(z1, z2)) -> c16(IF2(eq(z0, z1), z0, z1, z2), EQ(z0, z1)) IF2(true, z0, z1, z2) -> c17(RM(z0, z2)) IF2(false, z0, z1, z2) -> c18(RM(z0, z2)) MINSORT(cons(z1, z2)) -> c9(MINSORT(if2(eq(min(cons(z1, z2)), z1), min(cons(z1, z2)), z1, z2)), RM(min(cons(z1, z2)), cons(z1, z2)), MIN(cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(rm(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(rm(z0, cons(z0, nil))), RM(min(cons(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)) MIN(cons(z0, cons(z1, z2))) -> c12(IF1(le(z0, z1), z0, z1, z2), LE(z0, z1)) IF1(true, z0, z1, z2) -> c13(MIN(cons(z0, z2))) IF1(false, z0, z1, z2) -> c14(MIN(cons(z1, z2))) RM(z0, cons(z1, z2)) -> c16(IF2(eq(z0, z1), z0, z1, z2), EQ(z0, z1)) IF2(true, z0, z1, z2) -> c17(RM(z0, z2)) IF2(false, z0, z1, z2) -> c18(RM(z0, z2)) MINSORT(cons(z1, z2)) -> c9(MINSORT(if2(eq(min(cons(z1, z2)), z1), min(cons(z1, z2)), z1, z2)), RM(min(cons(z1, z2)), cons(z1, z2)), MIN(cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(rm(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(rm(z0, cons(z0, nil))), RM(min(cons(z0, nil)), cons(z0, nil))) K tuples: MINSORT(cons(z0, z1)) -> c8(MIN(cons(z0, z1))) Defined Rule Symbols: rm_2, min_1, if1_4, le_2, if2_4, eq_2 Defined Pair Symbols: LE_2, EQ_2, MINSORT_1, MIN_1, IF1_4, RM_2, IF2_4 Compound Symbols: c2_1, c6_1, c8_1, c12_2, c13_1, c14_1, c16_2, c17_1, c18_1, c9_3, c9_2 ---------------------------------------- (65) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MIN(cons(z0, cons(z1, z2))) -> c12(IF1(le(z0, z1), z0, z1, z2), LE(z0, z1)) by MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2), LE(0, z0)) MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2), LE(s(z0), 0)) MIN(cons(s(z0), cons(s(z1), x2))) -> c12(IF1(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1))) ---------------------------------------- (66) Obligation: Complexity Dependency Tuples Problem Rules: rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) rm(z0, nil) -> nil min(cons(z0, nil)) -> z0 min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(cons(z0, z2)) if1(false, z0, z1, z2) -> min(cons(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) if2(true, z0, z1, z2) -> rm(z0, z2) if2(false, z0, z1, z2) -> cons(z1, rm(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)) -> c8(MIN(cons(z0, z1))) IF1(true, z0, z1, z2) -> c13(MIN(cons(z0, z2))) IF1(false, z0, z1, z2) -> c14(MIN(cons(z1, z2))) RM(z0, cons(z1, z2)) -> c16(IF2(eq(z0, z1), z0, z1, z2), EQ(z0, z1)) IF2(true, z0, z1, z2) -> c17(RM(z0, z2)) IF2(false, z0, z1, z2) -> c18(RM(z0, z2)) MINSORT(cons(z1, z2)) -> c9(MINSORT(if2(eq(min(cons(z1, z2)), z1), min(cons(z1, z2)), z1, z2)), RM(min(cons(z1, z2)), cons(z1, z2)), MIN(cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(rm(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(rm(z0, cons(z0, nil))), RM(min(cons(z0, nil)), cons(z0, nil))) MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2), LE(0, z0)) MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2), LE(s(z0), 0)) MIN(cons(s(z0), cons(s(z1), x2))) -> c12(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) -> c13(MIN(cons(z0, z2))) IF1(false, z0, z1, z2) -> c14(MIN(cons(z1, z2))) RM(z0, cons(z1, z2)) -> c16(IF2(eq(z0, z1), z0, z1, z2), EQ(z0, z1)) IF2(true, z0, z1, z2) -> c17(RM(z0, z2)) IF2(false, z0, z1, z2) -> c18(RM(z0, z2)) MINSORT(cons(z1, z2)) -> c9(MINSORT(if2(eq(min(cons(z1, z2)), z1), min(cons(z1, z2)), z1, z2)), RM(min(cons(z1, z2)), cons(z1, z2)), MIN(cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(rm(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(rm(z0, cons(z0, nil))), RM(min(cons(z0, nil)), cons(z0, nil))) MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2), LE(0, z0)) MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2), LE(s(z0), 0)) MIN(cons(s(z0), cons(s(z1), x2))) -> c12(IF1(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1))) K tuples: MINSORT(cons(z0, z1)) -> c8(MIN(cons(z0, z1))) Defined Rule Symbols: rm_2, min_1, if1_4, le_2, if2_4, eq_2 Defined Pair Symbols: LE_2, EQ_2, MINSORT_1, IF1_4, RM_2, IF2_4, MIN_1 Compound Symbols: c2_1, c6_1, c8_1, c13_1, c14_1, c16_2, c17_1, c18_1, c9_3, c9_2, c12_2 ---------------------------------------- (67) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (68) Obligation: Complexity Dependency Tuples Problem Rules: rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) rm(z0, nil) -> nil min(cons(z0, nil)) -> z0 min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(cons(z0, z2)) if1(false, z0, z1, z2) -> min(cons(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) if2(true, z0, z1, z2) -> rm(z0, z2) if2(false, z0, z1, z2) -> cons(z1, rm(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)) -> c8(MIN(cons(z0, z1))) IF1(true, z0, z1, z2) -> c13(MIN(cons(z0, z2))) IF1(false, z0, z1, z2) -> c14(MIN(cons(z1, z2))) RM(z0, cons(z1, z2)) -> c16(IF2(eq(z0, z1), z0, z1, z2), EQ(z0, z1)) IF2(true, z0, z1, z2) -> c17(RM(z0, z2)) IF2(false, z0, z1, z2) -> c18(RM(z0, z2)) MINSORT(cons(z1, z2)) -> c9(MINSORT(if2(eq(min(cons(z1, z2)), z1), min(cons(z1, z2)), z1, z2)), RM(min(cons(z1, z2)), cons(z1, z2)), MIN(cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(rm(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(rm(z0, cons(z0, nil))), RM(min(cons(z0, nil)), cons(z0, nil))) MIN(cons(s(z0), cons(s(z1), x2))) -> c12(IF1(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1))) MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2)) MIN(cons(s(z0), cons(0, x2))) -> c12(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) -> c13(MIN(cons(z0, z2))) IF1(false, z0, z1, z2) -> c14(MIN(cons(z1, z2))) RM(z0, cons(z1, z2)) -> c16(IF2(eq(z0, z1), z0, z1, z2), EQ(z0, z1)) IF2(true, z0, z1, z2) -> c17(RM(z0, z2)) IF2(false, z0, z1, z2) -> c18(RM(z0, z2)) MINSORT(cons(z1, z2)) -> c9(MINSORT(if2(eq(min(cons(z1, z2)), z1), min(cons(z1, z2)), z1, z2)), RM(min(cons(z1, z2)), cons(z1, z2)), MIN(cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(rm(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(rm(z0, cons(z0, nil))), RM(min(cons(z0, nil)), cons(z0, nil))) MIN(cons(s(z0), cons(s(z1), x2))) -> c12(IF1(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1))) MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2)) MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2)) K tuples: MINSORT(cons(z0, z1)) -> c8(MIN(cons(z0, z1))) Defined Rule Symbols: rm_2, min_1, if1_4, le_2, if2_4, eq_2 Defined Pair Symbols: LE_2, EQ_2, MINSORT_1, IF1_4, RM_2, IF2_4, MIN_1 Compound Symbols: c2_1, c6_1, c8_1, c13_1, c14_1, c16_2, c17_1, c18_1, c9_3, c9_2, c12_2, c12_1 ---------------------------------------- (69) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace RM(z0, cons(z1, z2)) -> c16(IF2(eq(z0, z1), z0, z1, z2), EQ(z0, z1)) by RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2), EQ(0, 0)) RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2), EQ(0, s(z0))) RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2), EQ(s(z0), 0)) RM(s(z0), cons(s(z1), x2)) -> c16(IF2(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1))) ---------------------------------------- (70) Obligation: Complexity Dependency Tuples Problem Rules: rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) rm(z0, nil) -> nil min(cons(z0, nil)) -> z0 min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(cons(z0, z2)) if1(false, z0, z1, z2) -> min(cons(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) if2(true, z0, z1, z2) -> rm(z0, z2) if2(false, z0, z1, z2) -> cons(z1, rm(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)) -> c8(MIN(cons(z0, z1))) IF1(true, z0, z1, z2) -> c13(MIN(cons(z0, z2))) IF1(false, z0, z1, z2) -> c14(MIN(cons(z1, z2))) IF2(true, z0, z1, z2) -> c17(RM(z0, z2)) IF2(false, z0, z1, z2) -> c18(RM(z0, z2)) MINSORT(cons(z1, z2)) -> c9(MINSORT(if2(eq(min(cons(z1, z2)), z1), min(cons(z1, z2)), z1, z2)), RM(min(cons(z1, z2)), cons(z1, z2)), MIN(cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(rm(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(rm(z0, cons(z0, nil))), RM(min(cons(z0, nil)), cons(z0, nil))) MIN(cons(s(z0), cons(s(z1), x2))) -> c12(IF1(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1))) MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2)) MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2)) RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2), EQ(0, 0)) RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2), EQ(0, s(z0))) RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2), EQ(s(z0), 0)) RM(s(z0), cons(s(z1), x2)) -> c16(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) -> c13(MIN(cons(z0, z2))) IF1(false, z0, z1, z2) -> c14(MIN(cons(z1, z2))) IF2(true, z0, z1, z2) -> c17(RM(z0, z2)) IF2(false, z0, z1, z2) -> c18(RM(z0, z2)) MINSORT(cons(z1, z2)) -> c9(MINSORT(if2(eq(min(cons(z1, z2)), z1), min(cons(z1, z2)), z1, z2)), RM(min(cons(z1, z2)), cons(z1, z2)), MIN(cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(rm(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(rm(z0, cons(z0, nil))), RM(min(cons(z0, nil)), cons(z0, nil))) MIN(cons(s(z0), cons(s(z1), x2))) -> c12(IF1(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1))) MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2)) MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2)) RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2), EQ(0, 0)) RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2), EQ(0, s(z0))) RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2), EQ(s(z0), 0)) RM(s(z0), cons(s(z1), x2)) -> c16(IF2(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1))) K tuples: MINSORT(cons(z0, z1)) -> c8(MIN(cons(z0, z1))) Defined Rule Symbols: rm_2, min_1, if1_4, le_2, if2_4, eq_2 Defined Pair Symbols: LE_2, EQ_2, MINSORT_1, IF1_4, IF2_4, MIN_1, RM_2 Compound Symbols: c2_1, c6_1, c8_1, c13_1, c14_1, c17_1, c18_1, c9_3, c9_2, c12_2, c12_1, c16_2 ---------------------------------------- (71) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (72) Obligation: Complexity Dependency Tuples Problem Rules: rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) rm(z0, nil) -> nil min(cons(z0, nil)) -> z0 min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(cons(z0, z2)) if1(false, z0, z1, z2) -> min(cons(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) if2(true, z0, z1, z2) -> rm(z0, z2) if2(false, z0, z1, z2) -> cons(z1, rm(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)) -> c8(MIN(cons(z0, z1))) IF1(true, z0, z1, z2) -> c13(MIN(cons(z0, z2))) IF1(false, z0, z1, z2) -> c14(MIN(cons(z1, z2))) IF2(true, z0, z1, z2) -> c17(RM(z0, z2)) IF2(false, z0, z1, z2) -> c18(RM(z0, z2)) MINSORT(cons(z1, z2)) -> c9(MINSORT(if2(eq(min(cons(z1, z2)), z1), min(cons(z1, z2)), z1, z2)), RM(min(cons(z1, z2)), cons(z1, z2)), MIN(cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(rm(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(rm(z0, cons(z0, nil))), RM(min(cons(z0, nil)), cons(z0, nil))) MIN(cons(s(z0), cons(s(z1), x2))) -> c12(IF1(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1))) MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2)) MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2)) RM(s(z0), cons(s(z1), x2)) -> c16(IF2(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1))) RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2)) RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2)) RM(s(z0), cons(0, x2)) -> c16(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) -> c13(MIN(cons(z0, z2))) IF1(false, z0, z1, z2) -> c14(MIN(cons(z1, z2))) IF2(true, z0, z1, z2) -> c17(RM(z0, z2)) IF2(false, z0, z1, z2) -> c18(RM(z0, z2)) MINSORT(cons(z1, z2)) -> c9(MINSORT(if2(eq(min(cons(z1, z2)), z1), min(cons(z1, z2)), z1, z2)), RM(min(cons(z1, z2)), cons(z1, z2)), MIN(cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(rm(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(rm(z0, cons(z0, nil))), RM(min(cons(z0, nil)), cons(z0, nil))) MIN(cons(s(z0), cons(s(z1), x2))) -> c12(IF1(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1))) MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2)) MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2)) RM(s(z0), cons(s(z1), x2)) -> c16(IF2(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1))) RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2)) RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2)) RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2)) K tuples: MINSORT(cons(z0, z1)) -> c8(MIN(cons(z0, z1))) Defined Rule Symbols: rm_2, min_1, if1_4, le_2, if2_4, eq_2 Defined Pair Symbols: LE_2, EQ_2, MINSORT_1, IF1_4, IF2_4, MIN_1, RM_2 Compound Symbols: c2_1, c6_1, c8_1, c13_1, c14_1, c17_1, c18_1, c9_3, c9_2, c12_2, c12_1, c16_2, c16_1 ---------------------------------------- (73) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MINSORT(cons(z1, z2)) -> c9(MINSORT(if2(eq(min(cons(z1, z2)), z1), min(cons(z1, z2)), z1, z2)), RM(min(cons(z1, z2)), cons(z1, z2)), MIN(cons(z1, z2))) by MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)), MIN(cons(z0, nil))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)), MIN(cons(z0, nil))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) ---------------------------------------- (74) Obligation: Complexity Dependency Tuples Problem Rules: rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) rm(z0, nil) -> nil min(cons(z0, nil)) -> z0 min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(cons(z0, z2)) if1(false, z0, z1, z2) -> min(cons(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) if2(true, z0, z1, z2) -> rm(z0, z2) if2(false, z0, z1, z2) -> cons(z1, rm(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)) -> c8(MIN(cons(z0, z1))) IF1(true, z0, z1, z2) -> c13(MIN(cons(z0, z2))) IF1(false, z0, z1, z2) -> c14(MIN(cons(z1, z2))) IF2(true, z0, z1, z2) -> c17(RM(z0, z2)) IF2(false, z0, z1, z2) -> c18(RM(z0, z2)) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(rm(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(rm(z0, cons(z0, nil))), RM(min(cons(z0, nil)), cons(z0, nil))) MIN(cons(s(z0), cons(s(z1), x2))) -> c12(IF1(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1))) MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2)) MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2)) RM(s(z0), cons(s(z1), x2)) -> c16(IF2(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1))) RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2)) RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2)) RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2)) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)), MIN(cons(z0, nil))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)), MIN(cons(z0, nil))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(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)) IF1(true, z0, z1, z2) -> c13(MIN(cons(z0, z2))) IF1(false, z0, z1, z2) -> c14(MIN(cons(z1, z2))) IF2(true, z0, z1, z2) -> c17(RM(z0, z2)) IF2(false, z0, z1, z2) -> c18(RM(z0, z2)) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(rm(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(rm(z0, cons(z0, nil))), RM(min(cons(z0, nil)), cons(z0, nil))) MIN(cons(s(z0), cons(s(z1), x2))) -> c12(IF1(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1))) MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2)) MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2)) RM(s(z0), cons(s(z1), x2)) -> c16(IF2(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1))) RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2)) RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2)) RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2)) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)), MIN(cons(z0, nil))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil)), MIN(cons(z0, nil))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) K tuples: MINSORT(cons(z0, z1)) -> c8(MIN(cons(z0, z1))) Defined Rule Symbols: rm_2, min_1, if1_4, le_2, if2_4, eq_2 Defined Pair Symbols: LE_2, EQ_2, MINSORT_1, IF1_4, IF2_4, MIN_1, RM_2 Compound Symbols: c2_1, c6_1, c8_1, c13_1, c14_1, c17_1, c18_1, c9_3, c9_2, c12_2, c12_1, c16_2, c16_1 ---------------------------------------- (75) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (76) Obligation: Complexity Dependency Tuples Problem Rules: rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) rm(z0, nil) -> nil min(cons(z0, nil)) -> z0 min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(cons(z0, z2)) if1(false, z0, z1, z2) -> min(cons(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) if2(true, z0, z1, z2) -> rm(z0, z2) if2(false, z0, z1, z2) -> cons(z1, rm(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)) -> c8(MIN(cons(z0, z1))) IF1(true, z0, z1, z2) -> c13(MIN(cons(z0, z2))) IF1(false, z0, z1, z2) -> c14(MIN(cons(z1, z2))) IF2(true, z0, z1, z2) -> c17(RM(z0, z2)) IF2(false, z0, z1, z2) -> c18(RM(z0, z2)) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(rm(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(rm(z0, cons(z0, nil))), RM(min(cons(z0, nil)), cons(z0, nil))) MIN(cons(s(z0), cons(s(z1), x2))) -> c12(IF1(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1))) MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2)) MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2)) RM(s(z0), cons(s(z1), x2)) -> c16(IF2(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1))) RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2)) RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2)) RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2)) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(min(cons(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) -> c13(MIN(cons(z0, z2))) IF1(false, z0, z1, z2) -> c14(MIN(cons(z1, z2))) IF2(true, z0, z1, z2) -> c17(RM(z0, z2)) IF2(false, z0, z1, z2) -> c18(RM(z0, z2)) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(rm(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(rm(z0, cons(z0, nil))), RM(min(cons(z0, nil)), cons(z0, nil))) MIN(cons(s(z0), cons(s(z1), x2))) -> c12(IF1(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1))) MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2)) MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2)) RM(s(z0), cons(s(z1), x2)) -> c16(IF2(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1))) RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2)) RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2)) RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2)) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) K tuples: MINSORT(cons(z0, z1)) -> c8(MIN(cons(z0, z1))) Defined Rule Symbols: rm_2, min_1, if1_4, le_2, if2_4, eq_2 Defined Pair Symbols: LE_2, EQ_2, MINSORT_1, IF1_4, IF2_4, MIN_1, RM_2 Compound Symbols: c2_1, c6_1, c8_1, c13_1, c14_1, c17_1, c18_1, c9_3, c9_2, c12_2, c12_1, c16_2, c16_1 ---------------------------------------- (77) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(rm(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) by MINSORT(cons(z1, cons(x1, x2))) -> c9(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), RM(min(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), MIN(cons(z1, cons(x1, x2)))) MINSORT(cons(0, cons(z0, x2))) -> c9(MINSORT(rm(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), RM(min(cons(0, cons(z0, x2))), cons(0, cons(z0, x2))), MIN(cons(0, cons(z0, x2)))) MINSORT(cons(s(z0), cons(0, x2))) -> c9(MINSORT(rm(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), RM(min(cons(s(z0), cons(0, x2))), cons(s(z0), cons(0, x2))), MIN(cons(s(z0), cons(0, x2)))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), RM(min(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MIN(cons(s(z0), cons(s(z1), x2)))) MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) ---------------------------------------- (78) Obligation: Complexity Dependency Tuples Problem Rules: rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) rm(z0, nil) -> nil min(cons(z0, nil)) -> z0 min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(cons(z0, z2)) if1(false, z0, z1, z2) -> min(cons(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) if2(true, z0, z1, z2) -> rm(z0, z2) if2(false, z0, z1, z2) -> cons(z1, rm(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)) -> c8(MIN(cons(z0, z1))) IF1(true, z0, z1, z2) -> c13(MIN(cons(z0, z2))) IF1(false, z0, z1, z2) -> c14(MIN(cons(z1, z2))) IF2(true, z0, z1, z2) -> c17(RM(z0, z2)) IF2(false, z0, z1, z2) -> c18(RM(z0, z2)) MINSORT(cons(z0, nil)) -> c9(MINSORT(rm(z0, cons(z0, nil))), RM(min(cons(z0, nil)), cons(z0, nil))) MIN(cons(s(z0), cons(s(z1), x2))) -> c12(IF1(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1))) MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2)) MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2)) RM(s(z0), cons(s(z1), x2)) -> c16(IF2(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1))) RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2)) RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2)) RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2)) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) MINSORT(cons(z1, cons(x1, x2))) -> c9(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), RM(min(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), MIN(cons(z1, cons(x1, x2)))) MINSORT(cons(0, cons(z0, x2))) -> c9(MINSORT(rm(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), RM(min(cons(0, cons(z0, x2))), cons(0, cons(z0, x2))), MIN(cons(0, cons(z0, x2)))) MINSORT(cons(s(z0), cons(0, x2))) -> c9(MINSORT(rm(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), RM(min(cons(s(z0), cons(0, x2))), cons(s(z0), cons(0, x2))), MIN(cons(s(z0), cons(0, x2)))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), RM(min(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MIN(cons(s(z0), cons(s(z1), x2)))) MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(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) -> c13(MIN(cons(z0, z2))) IF1(false, z0, z1, z2) -> c14(MIN(cons(z1, z2))) IF2(true, z0, z1, z2) -> c17(RM(z0, z2)) IF2(false, z0, z1, z2) -> c18(RM(z0, z2)) MINSORT(cons(z0, nil)) -> c9(MINSORT(rm(z0, cons(z0, nil))), RM(min(cons(z0, nil)), cons(z0, nil))) MIN(cons(s(z0), cons(s(z1), x2))) -> c12(IF1(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1))) MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2)) MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2)) RM(s(z0), cons(s(z1), x2)) -> c16(IF2(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1))) RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2)) RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2)) RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2)) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) MINSORT(cons(z1, cons(x1, x2))) -> c9(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), RM(min(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), MIN(cons(z1, cons(x1, x2)))) MINSORT(cons(0, cons(z0, x2))) -> c9(MINSORT(rm(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), RM(min(cons(0, cons(z0, x2))), cons(0, cons(z0, x2))), MIN(cons(0, cons(z0, x2)))) MINSORT(cons(s(z0), cons(0, x2))) -> c9(MINSORT(rm(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), RM(min(cons(s(z0), cons(0, x2))), cons(s(z0), cons(0, x2))), MIN(cons(s(z0), cons(0, x2)))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), RM(min(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MIN(cons(s(z0), cons(s(z1), x2)))) MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) K tuples: MINSORT(cons(z0, z1)) -> c8(MIN(cons(z0, z1))) Defined Rule Symbols: rm_2, min_1, if1_4, le_2, if2_4, eq_2 Defined Pair Symbols: LE_2, EQ_2, MINSORT_1, IF1_4, IF2_4, MIN_1, RM_2 Compound Symbols: c2_1, c6_1, c8_1, c13_1, c14_1, c17_1, c18_1, c9_2, c12_2, c12_1, c16_2, c16_1, c9_3, c9_1 ---------------------------------------- (79) 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))) -> c9(RM(min(cons(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)) MINSORT(cons(z0, z1)) -> c8(MIN(cons(z0, z1))) IF1(true, z0, z1, z2) -> c13(MIN(cons(z0, z2))) IF1(false, z0, z1, z2) -> c14(MIN(cons(z1, z2))) IF2(true, z0, z1, z2) -> c17(RM(z0, z2)) IF2(false, z0, z1, z2) -> c18(RM(z0, z2)) MINSORT(cons(z0, nil)) -> c9(MINSORT(rm(z0, cons(z0, nil))), RM(min(cons(z0, nil)), cons(z0, nil))) MIN(cons(s(z0), cons(s(z1), x2))) -> c12(IF1(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1))) MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2)) MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2)) RM(s(z0), cons(s(z1), x2)) -> c16(IF2(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1))) RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2)) RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2)) RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2)) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) MINSORT(cons(z1, cons(x1, x2))) -> c9(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), RM(min(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), MIN(cons(z1, cons(x1, x2)))) MINSORT(cons(0, cons(z0, x2))) -> c9(MINSORT(rm(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), RM(min(cons(0, cons(z0, x2))), cons(0, cons(z0, x2))), MIN(cons(0, cons(z0, x2)))) MINSORT(cons(s(z0), cons(0, x2))) -> c9(MINSORT(rm(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), RM(min(cons(s(z0), cons(0, x2))), cons(s(z0), cons(0, x2))), MIN(cons(s(z0), cons(0, x2)))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), RM(min(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MIN(cons(s(z0), cons(s(z1), x2)))) MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(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_1 POL(MINSORT(x_1)) = [1] POL(RM(x_1, x_2)) = 0 POL(c12(x_1)) = x_1 POL(c12(x_1, x_2)) = x_1 + x_2 POL(c13(x_1)) = x_1 POL(c14(x_1)) = x_1 POL(c16(x_1)) = x_1 POL(c16(x_1, x_2)) = x_1 + x_2 POL(c17(x_1)) = x_1 POL(c18(x_1)) = x_1 POL(c2(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c9(x_1)) = x_1 POL(c9(x_1, x_2)) = x_1 + x_2 POL(c9(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(cons(x_1, x_2)) = 0 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)) = 0 POL(nil) = 0 POL(rm(x_1, x_2)) = [1] + x_2 POL(s(x_1)) = 0 POL(true) = 0 ---------------------------------------- (80) Obligation: Complexity Dependency Tuples Problem Rules: rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) rm(z0, nil) -> nil min(cons(z0, nil)) -> z0 min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(cons(z0, z2)) if1(false, z0, z1, z2) -> min(cons(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) if2(true, z0, z1, z2) -> rm(z0, z2) if2(false, z0, z1, z2) -> cons(z1, rm(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)) -> c8(MIN(cons(z0, z1))) IF1(true, z0, z1, z2) -> c13(MIN(cons(z0, z2))) IF1(false, z0, z1, z2) -> c14(MIN(cons(z1, z2))) IF2(true, z0, z1, z2) -> c17(RM(z0, z2)) IF2(false, z0, z1, z2) -> c18(RM(z0, z2)) MINSORT(cons(z0, nil)) -> c9(MINSORT(rm(z0, cons(z0, nil))), RM(min(cons(z0, nil)), cons(z0, nil))) MIN(cons(s(z0), cons(s(z1), x2))) -> c12(IF1(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1))) MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2)) MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2)) RM(s(z0), cons(s(z1), x2)) -> c16(IF2(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1))) RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2)) RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2)) RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2)) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) MINSORT(cons(z1, cons(x1, x2))) -> c9(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), RM(min(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), MIN(cons(z1, cons(x1, x2)))) MINSORT(cons(0, cons(z0, x2))) -> c9(MINSORT(rm(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), RM(min(cons(0, cons(z0, x2))), cons(0, cons(z0, x2))), MIN(cons(0, cons(z0, x2)))) MINSORT(cons(s(z0), cons(0, x2))) -> c9(MINSORT(rm(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), RM(min(cons(s(z0), cons(0, x2))), cons(s(z0), cons(0, x2))), MIN(cons(s(z0), cons(0, x2)))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), RM(min(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MIN(cons(s(z0), cons(s(z1), x2)))) MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(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) -> c13(MIN(cons(z0, z2))) IF1(false, z0, z1, z2) -> c14(MIN(cons(z1, z2))) IF2(true, z0, z1, z2) -> c17(RM(z0, z2)) IF2(false, z0, z1, z2) -> c18(RM(z0, z2)) MINSORT(cons(z0, nil)) -> c9(MINSORT(rm(z0, cons(z0, nil))), RM(min(cons(z0, nil)), cons(z0, nil))) MIN(cons(s(z0), cons(s(z1), x2))) -> c12(IF1(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1))) MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2)) MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2)) RM(s(z0), cons(s(z1), x2)) -> c16(IF2(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1))) RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2)) RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2)) RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2)) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) MINSORT(cons(z1, cons(x1, x2))) -> c9(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), RM(min(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), MIN(cons(z1, cons(x1, x2)))) MINSORT(cons(0, cons(z0, x2))) -> c9(MINSORT(rm(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), RM(min(cons(0, cons(z0, x2))), cons(0, cons(z0, x2))), MIN(cons(0, cons(z0, x2)))) MINSORT(cons(s(z0), cons(0, x2))) -> c9(MINSORT(rm(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), RM(min(cons(s(z0), cons(0, x2))), cons(s(z0), cons(0, x2))), MIN(cons(s(z0), cons(0, x2)))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), RM(min(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MIN(cons(s(z0), cons(s(z1), x2)))) K tuples: MINSORT(cons(z0, z1)) -> c8(MIN(cons(z0, z1))) MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) Defined Rule Symbols: rm_2, min_1, if1_4, le_2, if2_4, eq_2 Defined Pair Symbols: LE_2, EQ_2, MINSORT_1, IF1_4, IF2_4, MIN_1, RM_2 Compound Symbols: c2_1, c6_1, c8_1, c13_1, c14_1, c17_1, c18_1, c9_2, c12_2, c12_1, c16_2, c16_1, c9_3, c9_1 ---------------------------------------- (81) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MINSORT(cons(z0, nil)) -> c9(MINSORT(rm(z0, cons(z0, nil))), RM(min(cons(z0, nil)), cons(z0, nil))) by MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil))) ---------------------------------------- (82) Obligation: Complexity Dependency Tuples Problem Rules: rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) rm(z0, nil) -> nil min(cons(z0, nil)) -> z0 min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(cons(z0, z2)) if1(false, z0, z1, z2) -> min(cons(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) if2(true, z0, z1, z2) -> rm(z0, z2) if2(false, z0, z1, z2) -> cons(z1, rm(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)) -> c8(MIN(cons(z0, z1))) IF1(true, z0, z1, z2) -> c13(MIN(cons(z0, z2))) IF1(false, z0, z1, z2) -> c14(MIN(cons(z1, z2))) IF2(true, z0, z1, z2) -> c17(RM(z0, z2)) IF2(false, z0, z1, z2) -> c18(RM(z0, z2)) MIN(cons(s(z0), cons(s(z1), x2))) -> c12(IF1(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1))) MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2)) MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2)) RM(s(z0), cons(s(z1), x2)) -> c16(IF2(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1))) RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2)) RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2)) RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2)) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) MINSORT(cons(z1, cons(x1, x2))) -> c9(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), RM(min(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), MIN(cons(z1, cons(x1, x2)))) MINSORT(cons(0, cons(z0, x2))) -> c9(MINSORT(rm(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), RM(min(cons(0, cons(z0, x2))), cons(0, cons(z0, x2))), MIN(cons(0, cons(z0, x2)))) MINSORT(cons(s(z0), cons(0, x2))) -> c9(MINSORT(rm(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), RM(min(cons(s(z0), cons(0, x2))), cons(s(z0), cons(0, x2))), MIN(cons(s(z0), cons(0, x2)))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), RM(min(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MIN(cons(s(z0), cons(s(z1), x2)))) MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) MINSORT(cons(x0, nil)) -> c9(RM(min(cons(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) -> c13(MIN(cons(z0, z2))) IF1(false, z0, z1, z2) -> c14(MIN(cons(z1, z2))) IF2(true, z0, z1, z2) -> c17(RM(z0, z2)) IF2(false, z0, z1, z2) -> c18(RM(z0, z2)) MIN(cons(s(z0), cons(s(z1), x2))) -> c12(IF1(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1))) MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2)) MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2)) RM(s(z0), cons(s(z1), x2)) -> c16(IF2(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1))) RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2)) RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2)) RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2)) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) MINSORT(cons(z1, cons(x1, x2))) -> c9(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), RM(min(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), MIN(cons(z1, cons(x1, x2)))) MINSORT(cons(0, cons(z0, x2))) -> c9(MINSORT(rm(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), RM(min(cons(0, cons(z0, x2))), cons(0, cons(z0, x2))), MIN(cons(0, cons(z0, x2)))) MINSORT(cons(s(z0), cons(0, x2))) -> c9(MINSORT(rm(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), RM(min(cons(s(z0), cons(0, x2))), cons(s(z0), cons(0, x2))), MIN(cons(s(z0), cons(0, x2)))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), RM(min(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MIN(cons(s(z0), cons(s(z1), x2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil))) K tuples: MINSORT(cons(z0, z1)) -> c8(MIN(cons(z0, z1))) MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) Defined Rule Symbols: rm_2, min_1, if1_4, le_2, if2_4, eq_2 Defined Pair Symbols: LE_2, EQ_2, MINSORT_1, IF1_4, IF2_4, MIN_1, RM_2 Compound Symbols: c2_1, c6_1, c8_1, c13_1, c14_1, c17_1, c18_1, c12_2, c12_1, c16_2, c16_1, c9_3, c9_2, c9_1 ---------------------------------------- (83) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. MINSORT(cons(x0, nil)) -> c9(RM(min(cons(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)) MINSORT(cons(z0, z1)) -> c8(MIN(cons(z0, z1))) IF1(true, z0, z1, z2) -> c13(MIN(cons(z0, z2))) IF1(false, z0, z1, z2) -> c14(MIN(cons(z1, z2))) IF2(true, z0, z1, z2) -> c17(RM(z0, z2)) IF2(false, z0, z1, z2) -> c18(RM(z0, z2)) MIN(cons(s(z0), cons(s(z1), x2))) -> c12(IF1(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1))) MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2)) MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2)) RM(s(z0), cons(s(z1), x2)) -> c16(IF2(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1))) RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2)) RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2)) RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2)) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) MINSORT(cons(z1, cons(x1, x2))) -> c9(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), RM(min(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), MIN(cons(z1, cons(x1, x2)))) MINSORT(cons(0, cons(z0, x2))) -> c9(MINSORT(rm(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), RM(min(cons(0, cons(z0, x2))), cons(0, cons(z0, x2))), MIN(cons(0, cons(z0, x2)))) MINSORT(cons(s(z0), cons(0, x2))) -> c9(MINSORT(rm(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), RM(min(cons(s(z0), cons(0, x2))), cons(s(z0), cons(0, x2))), MIN(cons(s(z0), cons(0, x2)))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), RM(min(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MIN(cons(s(z0), cons(s(z1), x2)))) MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 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_1 POL(MINSORT(x_1)) = [1] POL(RM(x_1, x_2)) = 0 POL(c12(x_1)) = x_1 POL(c12(x_1, x_2)) = x_1 + x_2 POL(c13(x_1)) = x_1 POL(c14(x_1)) = x_1 POL(c16(x_1)) = x_1 POL(c16(x_1, x_2)) = x_1 + x_2 POL(c17(x_1)) = x_1 POL(c18(x_1)) = x_1 POL(c2(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c9(x_1)) = x_1 POL(c9(x_1, x_2)) = x_1 + x_2 POL(c9(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(cons(x_1, x_2)) = 0 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)) = 0 POL(nil) = 0 POL(rm(x_1, x_2)) = [1] + x_2 POL(s(x_1)) = 0 POL(true) = 0 ---------------------------------------- (84) Obligation: Complexity Dependency Tuples Problem Rules: rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) rm(z0, nil) -> nil min(cons(z0, nil)) -> z0 min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(cons(z0, z2)) if1(false, z0, z1, z2) -> min(cons(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) if2(true, z0, z1, z2) -> rm(z0, z2) if2(false, z0, z1, z2) -> cons(z1, rm(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)) -> c8(MIN(cons(z0, z1))) IF1(true, z0, z1, z2) -> c13(MIN(cons(z0, z2))) IF1(false, z0, z1, z2) -> c14(MIN(cons(z1, z2))) IF2(true, z0, z1, z2) -> c17(RM(z0, z2)) IF2(false, z0, z1, z2) -> c18(RM(z0, z2)) MIN(cons(s(z0), cons(s(z1), x2))) -> c12(IF1(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1))) MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2)) MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2)) RM(s(z0), cons(s(z1), x2)) -> c16(IF2(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1))) RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2)) RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2)) RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2)) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) MINSORT(cons(z1, cons(x1, x2))) -> c9(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), RM(min(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), MIN(cons(z1, cons(x1, x2)))) MINSORT(cons(0, cons(z0, x2))) -> c9(MINSORT(rm(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), RM(min(cons(0, cons(z0, x2))), cons(0, cons(z0, x2))), MIN(cons(0, cons(z0, x2)))) MINSORT(cons(s(z0), cons(0, x2))) -> c9(MINSORT(rm(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), RM(min(cons(s(z0), cons(0, x2))), cons(s(z0), cons(0, x2))), MIN(cons(s(z0), cons(0, x2)))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), RM(min(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MIN(cons(s(z0), cons(s(z1), x2)))) MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) MINSORT(cons(x0, nil)) -> c9(RM(min(cons(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) -> c13(MIN(cons(z0, z2))) IF1(false, z0, z1, z2) -> c14(MIN(cons(z1, z2))) IF2(true, z0, z1, z2) -> c17(RM(z0, z2)) IF2(false, z0, z1, z2) -> c18(RM(z0, z2)) MIN(cons(s(z0), cons(s(z1), x2))) -> c12(IF1(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1))) MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2)) MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2)) RM(s(z0), cons(s(z1), x2)) -> c16(IF2(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1))) RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2)) RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2)) RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2)) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) MINSORT(cons(z1, cons(x1, x2))) -> c9(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), RM(min(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), MIN(cons(z1, cons(x1, x2)))) MINSORT(cons(0, cons(z0, x2))) -> c9(MINSORT(rm(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), RM(min(cons(0, cons(z0, x2))), cons(0, cons(z0, x2))), MIN(cons(0, cons(z0, x2)))) MINSORT(cons(s(z0), cons(0, x2))) -> c9(MINSORT(rm(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), RM(min(cons(s(z0), cons(0, x2))), cons(s(z0), cons(0, x2))), MIN(cons(s(z0), cons(0, x2)))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), RM(min(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MIN(cons(s(z0), cons(s(z1), x2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) K tuples: MINSORT(cons(z0, z1)) -> c8(MIN(cons(z0, z1))) MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil))) Defined Rule Symbols: rm_2, min_1, if1_4, le_2, if2_4, eq_2 Defined Pair Symbols: LE_2, EQ_2, MINSORT_1, IF1_4, IF2_4, MIN_1, RM_2 Compound Symbols: c2_1, c6_1, c8_1, c13_1, c14_1, c17_1, c18_1, c12_2, c12_1, c16_2, c16_1, c9_3, c9_2, c9_1 ---------------------------------------- (85) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MIN(cons(s(z0), cons(s(z1), x2))) -> c12(IF1(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1))) by MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2), LE(s(0), s(z0))) MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2), LE(s(s(z0)), s(0))) MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) MIN(cons(s(x0), cons(s(x1), x2))) -> c12(LE(s(x0), s(x1))) ---------------------------------------- (86) Obligation: Complexity Dependency Tuples Problem Rules: rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) rm(z0, nil) -> nil min(cons(z0, nil)) -> z0 min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(cons(z0, z2)) if1(false, z0, z1, z2) -> min(cons(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) if2(true, z0, z1, z2) -> rm(z0, z2) if2(false, z0, z1, z2) -> cons(z1, rm(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)) -> c8(MIN(cons(z0, z1))) IF1(true, z0, z1, z2) -> c13(MIN(cons(z0, z2))) IF1(false, z0, z1, z2) -> c14(MIN(cons(z1, z2))) IF2(true, z0, z1, z2) -> c17(RM(z0, z2)) IF2(false, z0, z1, z2) -> c18(RM(z0, z2)) MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2)) MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2)) RM(s(z0), cons(s(z1), x2)) -> c16(IF2(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1))) RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2)) RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2)) RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2)) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) MINSORT(cons(z1, cons(x1, x2))) -> c9(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), RM(min(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), MIN(cons(z1, cons(x1, x2)))) MINSORT(cons(0, cons(z0, x2))) -> c9(MINSORT(rm(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), RM(min(cons(0, cons(z0, x2))), cons(0, cons(z0, x2))), MIN(cons(0, cons(z0, x2)))) MINSORT(cons(s(z0), cons(0, x2))) -> c9(MINSORT(rm(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), RM(min(cons(s(z0), cons(0, x2))), cons(s(z0), cons(0, x2))), MIN(cons(s(z0), cons(0, x2)))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), RM(min(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MIN(cons(s(z0), cons(s(z1), x2)))) MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil))) MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2), LE(s(0), s(z0))) MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2), LE(s(s(z0)), s(0))) MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) MIN(cons(s(x0), cons(s(x1), x2))) -> c12(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) -> c13(MIN(cons(z0, z2))) IF1(false, z0, z1, z2) -> c14(MIN(cons(z1, z2))) IF2(true, z0, z1, z2) -> c17(RM(z0, z2)) IF2(false, z0, z1, z2) -> c18(RM(z0, z2)) MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2)) MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2)) RM(s(z0), cons(s(z1), x2)) -> c16(IF2(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1))) RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2)) RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2)) RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2)) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) MINSORT(cons(z1, cons(x1, x2))) -> c9(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), RM(min(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), MIN(cons(z1, cons(x1, x2)))) MINSORT(cons(0, cons(z0, x2))) -> c9(MINSORT(rm(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), RM(min(cons(0, cons(z0, x2))), cons(0, cons(z0, x2))), MIN(cons(0, cons(z0, x2)))) MINSORT(cons(s(z0), cons(0, x2))) -> c9(MINSORT(rm(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), RM(min(cons(s(z0), cons(0, x2))), cons(s(z0), cons(0, x2))), MIN(cons(s(z0), cons(0, x2)))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), RM(min(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MIN(cons(s(z0), cons(s(z1), x2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2), LE(s(0), s(z0))) MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2), LE(s(s(z0)), s(0))) MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) MIN(cons(s(x0), cons(s(x1), x2))) -> c12(LE(s(x0), s(x1))) K tuples: MINSORT(cons(z0, z1)) -> c8(MIN(cons(z0, z1))) MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil))) Defined Rule Symbols: rm_2, min_1, if1_4, le_2, if2_4, eq_2 Defined Pair Symbols: LE_2, EQ_2, MINSORT_1, IF1_4, IF2_4, MIN_1, RM_2 Compound Symbols: c2_1, c6_1, c8_1, c13_1, c14_1, c17_1, c18_1, c12_1, c16_2, c16_1, c9_3, c9_2, c9_1, c12_2 ---------------------------------------- (87) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace RM(s(z0), cons(s(z1), x2)) -> c16(IF2(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1))) by RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2), EQ(s(0), s(0))) RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), cons(s(x1), x2)) -> c16(EQ(s(x0), s(x1))) ---------------------------------------- (88) Obligation: Complexity Dependency Tuples Problem Rules: rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) rm(z0, nil) -> nil min(cons(z0, nil)) -> z0 min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(cons(z0, z2)) if1(false, z0, z1, z2) -> min(cons(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) if2(true, z0, z1, z2) -> rm(z0, z2) if2(false, z0, z1, z2) -> cons(z1, rm(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)) -> c8(MIN(cons(z0, z1))) IF1(true, z0, z1, z2) -> c13(MIN(cons(z0, z2))) IF1(false, z0, z1, z2) -> c14(MIN(cons(z1, z2))) IF2(true, z0, z1, z2) -> c17(RM(z0, z2)) IF2(false, z0, z1, z2) -> c18(RM(z0, z2)) MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2)) MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2)) RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2)) RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2)) RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2)) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) MINSORT(cons(z1, cons(x1, x2))) -> c9(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), RM(min(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), MIN(cons(z1, cons(x1, x2)))) MINSORT(cons(0, cons(z0, x2))) -> c9(MINSORT(rm(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), RM(min(cons(0, cons(z0, x2))), cons(0, cons(z0, x2))), MIN(cons(0, cons(z0, x2)))) MINSORT(cons(s(z0), cons(0, x2))) -> c9(MINSORT(rm(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), RM(min(cons(s(z0), cons(0, x2))), cons(s(z0), cons(0, x2))), MIN(cons(s(z0), cons(0, x2)))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), RM(min(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MIN(cons(s(z0), cons(s(z1), x2)))) MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil))) MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2), LE(s(0), s(z0))) MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2), LE(s(s(z0)), s(0))) MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) MIN(cons(s(x0), cons(s(x1), x2))) -> c12(LE(s(x0), s(x1))) RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2), EQ(s(0), s(0))) RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), cons(s(x1), x2)) -> c16(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) -> c13(MIN(cons(z0, z2))) IF1(false, z0, z1, z2) -> c14(MIN(cons(z1, z2))) IF2(true, z0, z1, z2) -> c17(RM(z0, z2)) IF2(false, z0, z1, z2) -> c18(RM(z0, z2)) MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2)) MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2)) RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2)) RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2)) RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2)) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) MINSORT(cons(z1, cons(x1, x2))) -> c9(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), RM(min(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), MIN(cons(z1, cons(x1, x2)))) MINSORT(cons(0, cons(z0, x2))) -> c9(MINSORT(rm(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), RM(min(cons(0, cons(z0, x2))), cons(0, cons(z0, x2))), MIN(cons(0, cons(z0, x2)))) MINSORT(cons(s(z0), cons(0, x2))) -> c9(MINSORT(rm(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), RM(min(cons(s(z0), cons(0, x2))), cons(s(z0), cons(0, x2))), MIN(cons(s(z0), cons(0, x2)))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), RM(min(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MIN(cons(s(z0), cons(s(z1), x2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2), LE(s(0), s(z0))) MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2), LE(s(s(z0)), s(0))) MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) MIN(cons(s(x0), cons(s(x1), x2))) -> c12(LE(s(x0), s(x1))) RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2), EQ(s(0), s(0))) RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), cons(s(x1), x2)) -> c16(EQ(s(x0), s(x1))) K tuples: MINSORT(cons(z0, z1)) -> c8(MIN(cons(z0, z1))) MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil))) Defined Rule Symbols: rm_2, min_1, if1_4, le_2, if2_4, eq_2 Defined Pair Symbols: LE_2, EQ_2, MINSORT_1, IF1_4, IF2_4, MIN_1, RM_2 Compound Symbols: c2_1, c6_1, c8_1, c13_1, c14_1, c17_1, c18_1, c12_1, c16_1, c9_3, c9_2, c9_1, c12_2, c16_2 ---------------------------------------- (89) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) by MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) ---------------------------------------- (90) Obligation: Complexity Dependency Tuples Problem Rules: rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) rm(z0, nil) -> nil min(cons(z0, nil)) -> z0 min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(cons(z0, z2)) if1(false, z0, z1, z2) -> min(cons(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) if2(true, z0, z1, z2) -> rm(z0, z2) if2(false, z0, z1, z2) -> cons(z1, rm(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)) -> c8(MIN(cons(z0, z1))) IF1(true, z0, z1, z2) -> c13(MIN(cons(z0, z2))) IF1(false, z0, z1, z2) -> c14(MIN(cons(z1, z2))) IF2(true, z0, z1, z2) -> c17(RM(z0, z2)) IF2(false, z0, z1, z2) -> c18(RM(z0, z2)) MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2)) MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2)) RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2)) RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2)) RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2)) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) MINSORT(cons(z1, cons(x1, x2))) -> c9(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), RM(min(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), MIN(cons(z1, cons(x1, x2)))) MINSORT(cons(0, cons(z0, x2))) -> c9(MINSORT(rm(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), RM(min(cons(0, cons(z0, x2))), cons(0, cons(z0, x2))), MIN(cons(0, cons(z0, x2)))) MINSORT(cons(s(z0), cons(0, x2))) -> c9(MINSORT(rm(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), RM(min(cons(s(z0), cons(0, x2))), cons(s(z0), cons(0, x2))), MIN(cons(s(z0), cons(0, x2)))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), RM(min(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MIN(cons(s(z0), cons(s(z1), x2)))) MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil))) MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2), LE(s(0), s(z0))) MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2), LE(s(s(z0)), s(0))) MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) MIN(cons(s(x0), cons(s(x1), x2))) -> c12(LE(s(x0), s(x1))) RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2), EQ(s(0), s(0))) RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), cons(s(x1), x2)) -> c16(EQ(s(x0), s(x1))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(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)) IF1(true, z0, z1, z2) -> c13(MIN(cons(z0, z2))) IF1(false, z0, z1, z2) -> c14(MIN(cons(z1, z2))) IF2(true, z0, z1, z2) -> c17(RM(z0, z2)) IF2(false, z0, z1, z2) -> c18(RM(z0, z2)) MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2)) MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2)) RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2)) RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2)) RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2)) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) MINSORT(cons(z1, cons(x1, x2))) -> c9(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), RM(min(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), MIN(cons(z1, cons(x1, x2)))) MINSORT(cons(0, cons(z0, x2))) -> c9(MINSORT(rm(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), RM(min(cons(0, cons(z0, x2))), cons(0, cons(z0, x2))), MIN(cons(0, cons(z0, x2)))) MINSORT(cons(s(z0), cons(0, x2))) -> c9(MINSORT(rm(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), RM(min(cons(s(z0), cons(0, x2))), cons(s(z0), cons(0, x2))), MIN(cons(s(z0), cons(0, x2)))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), RM(min(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MIN(cons(s(z0), cons(s(z1), x2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2), LE(s(0), s(z0))) MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2), LE(s(s(z0)), s(0))) MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) MIN(cons(s(x0), cons(s(x1), x2))) -> c12(LE(s(x0), s(x1))) RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2), EQ(s(0), s(0))) RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), cons(s(x1), x2)) -> c16(EQ(s(x0), s(x1))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) K tuples: MINSORT(cons(z0, z1)) -> c8(MIN(cons(z0, z1))) MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil))) Defined Rule Symbols: rm_2, min_1, if1_4, le_2, if2_4, eq_2 Defined Pair Symbols: LE_2, EQ_2, MINSORT_1, IF1_4, IF2_4, MIN_1, RM_2 Compound Symbols: c2_1, c6_1, c8_1, c13_1, c14_1, c17_1, c18_1, c12_1, c16_1, c9_3, c9_2, c9_1, c12_2, c16_2 ---------------------------------------- (91) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IF1(true, z0, z1, z2) -> c13(MIN(cons(z0, z2))) by IF1(true, 0, x0, x1) -> c13(MIN(cons(0, x1))) IF1(true, s(0), s(x0), x1) -> c13(MIN(cons(s(0), x1))) IF1(true, s(s(x0)), s(s(x1)), x2) -> c13(MIN(cons(s(s(x0)), x2))) ---------------------------------------- (92) Obligation: Complexity Dependency Tuples Problem Rules: rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) rm(z0, nil) -> nil min(cons(z0, nil)) -> z0 min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(cons(z0, z2)) if1(false, z0, z1, z2) -> min(cons(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) if2(true, z0, z1, z2) -> rm(z0, z2) if2(false, z0, z1, z2) -> cons(z1, rm(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)) -> c8(MIN(cons(z0, z1))) IF1(false, z0, z1, z2) -> c14(MIN(cons(z1, z2))) IF2(true, z0, z1, z2) -> c17(RM(z0, z2)) IF2(false, z0, z1, z2) -> c18(RM(z0, z2)) MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2)) MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2)) RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2)) RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2)) RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2)) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) MINSORT(cons(z1, cons(x1, x2))) -> c9(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), RM(min(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), MIN(cons(z1, cons(x1, x2)))) MINSORT(cons(0, cons(z0, x2))) -> c9(MINSORT(rm(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), RM(min(cons(0, cons(z0, x2))), cons(0, cons(z0, x2))), MIN(cons(0, cons(z0, x2)))) MINSORT(cons(s(z0), cons(0, x2))) -> c9(MINSORT(rm(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), RM(min(cons(s(z0), cons(0, x2))), cons(s(z0), cons(0, x2))), MIN(cons(s(z0), cons(0, x2)))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), RM(min(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MIN(cons(s(z0), cons(s(z1), x2)))) MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil))) MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2), LE(s(0), s(z0))) MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2), LE(s(s(z0)), s(0))) MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) MIN(cons(s(x0), cons(s(x1), x2))) -> c12(LE(s(x0), s(x1))) RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2), EQ(s(0), s(0))) RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), cons(s(x1), x2)) -> c16(EQ(s(x0), s(x1))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) IF1(true, 0, x0, x1) -> c13(MIN(cons(0, x1))) IF1(true, s(0), s(x0), x1) -> c13(MIN(cons(s(0), x1))) IF1(true, s(s(x0)), s(s(x1)), x2) -> c13(MIN(cons(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) -> c14(MIN(cons(z1, z2))) IF2(true, z0, z1, z2) -> c17(RM(z0, z2)) IF2(false, z0, z1, z2) -> c18(RM(z0, z2)) MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2)) MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2)) RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2)) RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2)) RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2)) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) MINSORT(cons(z1, cons(x1, x2))) -> c9(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), RM(min(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), MIN(cons(z1, cons(x1, x2)))) MINSORT(cons(0, cons(z0, x2))) -> c9(MINSORT(rm(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), RM(min(cons(0, cons(z0, x2))), cons(0, cons(z0, x2))), MIN(cons(0, cons(z0, x2)))) MINSORT(cons(s(z0), cons(0, x2))) -> c9(MINSORT(rm(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), RM(min(cons(s(z0), cons(0, x2))), cons(s(z0), cons(0, x2))), MIN(cons(s(z0), cons(0, x2)))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), RM(min(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MIN(cons(s(z0), cons(s(z1), x2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2), LE(s(0), s(z0))) MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2), LE(s(s(z0)), s(0))) MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) MIN(cons(s(x0), cons(s(x1), x2))) -> c12(LE(s(x0), s(x1))) RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2), EQ(s(0), s(0))) RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), cons(s(x1), x2)) -> c16(EQ(s(x0), s(x1))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) IF1(true, 0, x0, x1) -> c13(MIN(cons(0, x1))) IF1(true, s(0), s(x0), x1) -> c13(MIN(cons(s(0), x1))) IF1(true, s(s(x0)), s(s(x1)), x2) -> c13(MIN(cons(s(s(x0)), x2))) K tuples: MINSORT(cons(z0, z1)) -> c8(MIN(cons(z0, z1))) MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil))) Defined Rule Symbols: rm_2, min_1, if1_4, le_2, if2_4, eq_2 Defined Pair Symbols: LE_2, EQ_2, MINSORT_1, IF1_4, IF2_4, MIN_1, RM_2 Compound Symbols: c2_1, c6_1, c8_1, c14_1, c17_1, c18_1, c12_1, c16_1, c9_3, c9_2, c9_1, c12_2, c16_2, c13_1 ---------------------------------------- (93) 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))) ---------------------------------------- (94) Obligation: Complexity Dependency Tuples Problem Rules: rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) rm(z0, nil) -> nil min(cons(z0, nil)) -> z0 min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(cons(z0, z2)) if1(false, z0, z1, z2) -> min(cons(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) if2(true, z0, z1, z2) -> rm(z0, z2) if2(false, z0, z1, z2) -> cons(z1, rm(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)) -> c8(MIN(cons(z0, z1))) IF1(false, z0, z1, z2) -> c14(MIN(cons(z1, z2))) IF2(true, z0, z1, z2) -> c17(RM(z0, z2)) IF2(false, z0, z1, z2) -> c18(RM(z0, z2)) MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2)) MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2)) RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2)) RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2)) RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2)) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) MINSORT(cons(z1, cons(x1, x2))) -> c9(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), RM(min(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), MIN(cons(z1, cons(x1, x2)))) MINSORT(cons(0, cons(z0, x2))) -> c9(MINSORT(rm(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), RM(min(cons(0, cons(z0, x2))), cons(0, cons(z0, x2))), MIN(cons(0, cons(z0, x2)))) MINSORT(cons(s(z0), cons(0, x2))) -> c9(MINSORT(rm(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), RM(min(cons(s(z0), cons(0, x2))), cons(s(z0), cons(0, x2))), MIN(cons(s(z0), cons(0, x2)))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), RM(min(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MIN(cons(s(z0), cons(s(z1), x2)))) MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil))) MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2), LE(s(0), s(z0))) MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2), LE(s(s(z0)), s(0))) MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) MIN(cons(s(x0), cons(s(x1), x2))) -> c12(LE(s(x0), s(x1))) RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2), EQ(s(0), s(0))) RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), cons(s(x1), x2)) -> c16(EQ(s(x0), s(x1))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) IF1(true, 0, x0, x1) -> c13(MIN(cons(0, x1))) IF1(true, s(0), s(x0), x1) -> c13(MIN(cons(s(0), x1))) IF1(true, s(s(x0)), s(s(x1)), x2) -> c13(MIN(cons(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)) IF1(false, z0, z1, z2) -> c14(MIN(cons(z1, z2))) IF2(true, z0, z1, z2) -> c17(RM(z0, z2)) IF2(false, z0, z1, z2) -> c18(RM(z0, z2)) MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2)) MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2)) RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2)) RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2)) RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2)) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) MINSORT(cons(z1, cons(x1, x2))) -> c9(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), RM(min(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), MIN(cons(z1, cons(x1, x2)))) MINSORT(cons(0, cons(z0, x2))) -> c9(MINSORT(rm(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), RM(min(cons(0, cons(z0, x2))), cons(0, cons(z0, x2))), MIN(cons(0, cons(z0, x2)))) MINSORT(cons(s(z0), cons(0, x2))) -> c9(MINSORT(rm(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), RM(min(cons(s(z0), cons(0, x2))), cons(s(z0), cons(0, x2))), MIN(cons(s(z0), cons(0, x2)))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), RM(min(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MIN(cons(s(z0), cons(s(z1), x2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2), LE(s(0), s(z0))) MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2), LE(s(s(z0)), s(0))) MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) MIN(cons(s(x0), cons(s(x1), x2))) -> c12(LE(s(x0), s(x1))) RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2), EQ(s(0), s(0))) RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), cons(s(x1), x2)) -> c16(EQ(s(x0), s(x1))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) IF1(true, 0, x0, x1) -> c13(MIN(cons(0, x1))) IF1(true, s(0), s(x0), x1) -> c13(MIN(cons(s(0), x1))) IF1(true, s(s(x0)), s(s(x1)), x2) -> c13(MIN(cons(s(s(x0)), x2))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) K tuples: MINSORT(cons(z0, z1)) -> c8(MIN(cons(z0, z1))) MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil))) Defined Rule Symbols: rm_2, min_1, if1_4, le_2, if2_4, eq_2 Defined Pair Symbols: EQ_2, MINSORT_1, IF1_4, IF2_4, MIN_1, RM_2, LE_2 Compound Symbols: c6_1, c8_1, c14_1, c17_1, c18_1, c12_1, c16_1, c9_3, c9_2, c9_1, c12_2, c16_2, c13_1, c2_1 ---------------------------------------- (95) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (96) Obligation: Complexity Dependency Tuples Problem Rules: rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) rm(z0, nil) -> nil min(cons(z0, nil)) -> z0 min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(cons(z0, z2)) if1(false, z0, z1, z2) -> min(cons(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) if2(true, z0, z1, z2) -> rm(z0, z2) if2(false, z0, z1, z2) -> cons(z1, rm(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)) -> c8(MIN(cons(z0, z1))) IF1(false, z0, z1, z2) -> c14(MIN(cons(z1, z2))) IF2(true, z0, z1, z2) -> c17(RM(z0, z2)) IF2(false, z0, z1, z2) -> c18(RM(z0, z2)) MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2)) MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2)) RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2)) RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2)) RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2)) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) MINSORT(cons(z1, cons(x1, x2))) -> c9(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), RM(min(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), MIN(cons(z1, cons(x1, x2)))) MINSORT(cons(0, cons(z0, x2))) -> c9(MINSORT(rm(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), RM(min(cons(0, cons(z0, x2))), cons(0, cons(z0, x2))), MIN(cons(0, cons(z0, x2)))) MINSORT(cons(s(z0), cons(0, x2))) -> c9(MINSORT(rm(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), RM(min(cons(s(z0), cons(0, x2))), cons(s(z0), cons(0, x2))), MIN(cons(s(z0), cons(0, x2)))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), RM(min(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MIN(cons(s(z0), cons(s(z1), x2)))) MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil))) MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) MIN(cons(s(x0), cons(s(x1), x2))) -> c12(LE(s(x0), s(x1))) RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2), EQ(s(0), s(0))) RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), cons(s(x1), x2)) -> c16(EQ(s(x0), s(x1))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) IF1(true, 0, x0, x1) -> c13(MIN(cons(0, x1))) IF1(true, s(0), s(x0), x1) -> c13(MIN(cons(s(0), x1))) IF1(true, s(s(x0)), s(s(x1)), x2) -> c13(MIN(cons(s(s(x0)), x2))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2)) MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2)) S tuples: EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) IF1(false, z0, z1, z2) -> c14(MIN(cons(z1, z2))) IF2(true, z0, z1, z2) -> c17(RM(z0, z2)) IF2(false, z0, z1, z2) -> c18(RM(z0, z2)) MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2)) MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2)) RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2)) RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2)) RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2)) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) MINSORT(cons(z1, cons(x1, x2))) -> c9(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), RM(min(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), MIN(cons(z1, cons(x1, x2)))) MINSORT(cons(0, cons(z0, x2))) -> c9(MINSORT(rm(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), RM(min(cons(0, cons(z0, x2))), cons(0, cons(z0, x2))), MIN(cons(0, cons(z0, x2)))) MINSORT(cons(s(z0), cons(0, x2))) -> c9(MINSORT(rm(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), RM(min(cons(s(z0), cons(0, x2))), cons(s(z0), cons(0, x2))), MIN(cons(s(z0), cons(0, x2)))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), RM(min(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MIN(cons(s(z0), cons(s(z1), x2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) MIN(cons(s(x0), cons(s(x1), x2))) -> c12(LE(s(x0), s(x1))) RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2), EQ(s(0), s(0))) RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), cons(s(x1), x2)) -> c16(EQ(s(x0), s(x1))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) IF1(true, 0, x0, x1) -> c13(MIN(cons(0, x1))) IF1(true, s(0), s(x0), x1) -> c13(MIN(cons(s(0), x1))) IF1(true, s(s(x0)), s(s(x1)), x2) -> c13(MIN(cons(s(s(x0)), x2))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2)) MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2)) K tuples: MINSORT(cons(z0, z1)) -> c8(MIN(cons(z0, z1))) MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil))) Defined Rule Symbols: rm_2, min_1, if1_4, le_2, if2_4, eq_2 Defined Pair Symbols: EQ_2, MINSORT_1, IF1_4, IF2_4, MIN_1, RM_2, LE_2 Compound Symbols: c6_1, c8_1, c14_1, c17_1, c18_1, c12_1, c16_1, c9_3, c9_2, c9_1, c12_2, c16_2, c13_1, c2_1 ---------------------------------------- (97) 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))) ---------------------------------------- (98) Obligation: Complexity Dependency Tuples Problem Rules: rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) rm(z0, nil) -> nil min(cons(z0, nil)) -> z0 min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(cons(z0, z2)) if1(false, z0, z1, z2) -> min(cons(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) if2(true, z0, z1, z2) -> rm(z0, z2) if2(false, z0, z1, z2) -> cons(z1, rm(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)) -> c8(MIN(cons(z0, z1))) IF1(false, z0, z1, z2) -> c14(MIN(cons(z1, z2))) IF2(true, z0, z1, z2) -> c17(RM(z0, z2)) IF2(false, z0, z1, z2) -> c18(RM(z0, z2)) MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2)) MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2)) RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2)) RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2)) RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2)) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) MINSORT(cons(z1, cons(x1, x2))) -> c9(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), RM(min(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), MIN(cons(z1, cons(x1, x2)))) MINSORT(cons(0, cons(z0, x2))) -> c9(MINSORT(rm(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), RM(min(cons(0, cons(z0, x2))), cons(0, cons(z0, x2))), MIN(cons(0, cons(z0, x2)))) MINSORT(cons(s(z0), cons(0, x2))) -> c9(MINSORT(rm(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), RM(min(cons(s(z0), cons(0, x2))), cons(s(z0), cons(0, x2))), MIN(cons(s(z0), cons(0, x2)))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), RM(min(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MIN(cons(s(z0), cons(s(z1), x2)))) MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil))) MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) MIN(cons(s(x0), cons(s(x1), x2))) -> c12(LE(s(x0), s(x1))) RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2), EQ(s(0), s(0))) RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), cons(s(x1), x2)) -> c16(EQ(s(x0), s(x1))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) IF1(true, 0, x0, x1) -> c13(MIN(cons(0, x1))) IF1(true, s(0), s(x0), x1) -> c13(MIN(cons(s(0), x1))) IF1(true, s(s(x0)), s(s(x1)), x2) -> c13(MIN(cons(s(s(x0)), x2))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2)) MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2)) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) S tuples: IF1(false, z0, z1, z2) -> c14(MIN(cons(z1, z2))) IF2(true, z0, z1, z2) -> c17(RM(z0, z2)) IF2(false, z0, z1, z2) -> c18(RM(z0, z2)) MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2)) MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2)) RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2)) RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2)) RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2)) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) MINSORT(cons(z1, cons(x1, x2))) -> c9(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), RM(min(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), MIN(cons(z1, cons(x1, x2)))) MINSORT(cons(0, cons(z0, x2))) -> c9(MINSORT(rm(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), RM(min(cons(0, cons(z0, x2))), cons(0, cons(z0, x2))), MIN(cons(0, cons(z0, x2)))) MINSORT(cons(s(z0), cons(0, x2))) -> c9(MINSORT(rm(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), RM(min(cons(s(z0), cons(0, x2))), cons(s(z0), cons(0, x2))), MIN(cons(s(z0), cons(0, x2)))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), RM(min(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MIN(cons(s(z0), cons(s(z1), x2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) MIN(cons(s(x0), cons(s(x1), x2))) -> c12(LE(s(x0), s(x1))) RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2), EQ(s(0), s(0))) RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), cons(s(x1), x2)) -> c16(EQ(s(x0), s(x1))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) IF1(true, 0, x0, x1) -> c13(MIN(cons(0, x1))) IF1(true, s(0), s(x0), x1) -> c13(MIN(cons(s(0), x1))) IF1(true, s(s(x0)), s(s(x1)), x2) -> c13(MIN(cons(s(s(x0)), x2))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2)) MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2)) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) K tuples: MINSORT(cons(z0, z1)) -> c8(MIN(cons(z0, z1))) MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil))) Defined Rule Symbols: rm_2, min_1, if1_4, le_2, if2_4, eq_2 Defined Pair Symbols: MINSORT_1, IF1_4, IF2_4, MIN_1, RM_2, LE_2, EQ_2 Compound Symbols: c8_1, c14_1, c17_1, c18_1, c12_1, c16_1, c9_3, c9_2, c9_1, c12_2, c16_2, c13_1, c2_1, c6_1 ---------------------------------------- (99) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (100) Obligation: Complexity Dependency Tuples Problem Rules: rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) rm(z0, nil) -> nil min(cons(z0, nil)) -> z0 min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(cons(z0, z2)) if1(false, z0, z1, z2) -> min(cons(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) if2(true, z0, z1, z2) -> rm(z0, z2) if2(false, z0, z1, z2) -> cons(z1, rm(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)) -> c8(MIN(cons(z0, z1))) IF1(false, z0, z1, z2) -> c14(MIN(cons(z1, z2))) IF2(true, z0, z1, z2) -> c17(RM(z0, z2)) IF2(false, z0, z1, z2) -> c18(RM(z0, z2)) MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2)) MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2)) RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2)) RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2)) RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2)) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) MINSORT(cons(z1, cons(x1, x2))) -> c9(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), RM(min(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), MIN(cons(z1, cons(x1, x2)))) MINSORT(cons(0, cons(z0, x2))) -> c9(MINSORT(rm(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), RM(min(cons(0, cons(z0, x2))), cons(0, cons(z0, x2))), MIN(cons(0, cons(z0, x2)))) MINSORT(cons(s(z0), cons(0, x2))) -> c9(MINSORT(rm(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), RM(min(cons(s(z0), cons(0, x2))), cons(s(z0), cons(0, x2))), MIN(cons(s(z0), cons(0, x2)))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), RM(min(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MIN(cons(s(z0), cons(s(z1), x2)))) MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil))) MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) MIN(cons(s(x0), cons(s(x1), x2))) -> c12(LE(s(x0), s(x1))) RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), cons(s(x1), x2)) -> c16(EQ(s(x0), s(x1))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) IF1(true, 0, x0, x1) -> c13(MIN(cons(0, x1))) IF1(true, s(0), s(x0), x1) -> c13(MIN(cons(s(0), x1))) IF1(true, s(s(x0)), s(s(x1)), x2) -> c13(MIN(cons(s(s(x0)), x2))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2)) MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2)) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2)) RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2)) RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2)) S tuples: IF1(false, z0, z1, z2) -> c14(MIN(cons(z1, z2))) IF2(true, z0, z1, z2) -> c17(RM(z0, z2)) IF2(false, z0, z1, z2) -> c18(RM(z0, z2)) MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2)) MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2)) RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2)) RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2)) RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2)) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) MINSORT(cons(z1, cons(x1, x2))) -> c9(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), RM(min(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), MIN(cons(z1, cons(x1, x2)))) MINSORT(cons(0, cons(z0, x2))) -> c9(MINSORT(rm(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), RM(min(cons(0, cons(z0, x2))), cons(0, cons(z0, x2))), MIN(cons(0, cons(z0, x2)))) MINSORT(cons(s(z0), cons(0, x2))) -> c9(MINSORT(rm(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), RM(min(cons(s(z0), cons(0, x2))), cons(s(z0), cons(0, x2))), MIN(cons(s(z0), cons(0, x2)))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), RM(min(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MIN(cons(s(z0), cons(s(z1), x2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) MIN(cons(s(x0), cons(s(x1), x2))) -> c12(LE(s(x0), s(x1))) RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), cons(s(x1), x2)) -> c16(EQ(s(x0), s(x1))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) IF1(true, 0, x0, x1) -> c13(MIN(cons(0, x1))) IF1(true, s(0), s(x0), x1) -> c13(MIN(cons(s(0), x1))) IF1(true, s(s(x0)), s(s(x1)), x2) -> c13(MIN(cons(s(s(x0)), x2))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2)) MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2)) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2)) RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2)) RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2)) K tuples: MINSORT(cons(z0, z1)) -> c8(MIN(cons(z0, z1))) MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil))) Defined Rule Symbols: rm_2, min_1, if1_4, le_2, if2_4, eq_2 Defined Pair Symbols: MINSORT_1, IF1_4, IF2_4, MIN_1, RM_2, LE_2, EQ_2 Compound Symbols: c8_1, c14_1, c17_1, c18_1, c12_1, c16_1, c9_3, c9_2, c9_1, c12_2, c16_2, c13_1, c2_1, c6_1 ---------------------------------------- (101) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IF1(false, z0, z1, z2) -> c14(MIN(cons(z1, z2))) by IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1))) IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2))) IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1))) ---------------------------------------- (102) Obligation: Complexity Dependency Tuples Problem Rules: rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) rm(z0, nil) -> nil min(cons(z0, nil)) -> z0 min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(cons(z0, z2)) if1(false, z0, z1, z2) -> min(cons(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) if2(true, z0, z1, z2) -> rm(z0, z2) if2(false, z0, z1, z2) -> cons(z1, rm(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)) -> c8(MIN(cons(z0, z1))) IF2(true, z0, z1, z2) -> c17(RM(z0, z2)) IF2(false, z0, z1, z2) -> c18(RM(z0, z2)) MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2)) MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2)) RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2)) RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2)) RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2)) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) MINSORT(cons(z1, cons(x1, x2))) -> c9(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), RM(min(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), MIN(cons(z1, cons(x1, x2)))) MINSORT(cons(0, cons(z0, x2))) -> c9(MINSORT(rm(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), RM(min(cons(0, cons(z0, x2))), cons(0, cons(z0, x2))), MIN(cons(0, cons(z0, x2)))) MINSORT(cons(s(z0), cons(0, x2))) -> c9(MINSORT(rm(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), RM(min(cons(s(z0), cons(0, x2))), cons(s(z0), cons(0, x2))), MIN(cons(s(z0), cons(0, x2)))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), RM(min(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MIN(cons(s(z0), cons(s(z1), x2)))) MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil))) MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) MIN(cons(s(x0), cons(s(x1), x2))) -> c12(LE(s(x0), s(x1))) RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), cons(s(x1), x2)) -> c16(EQ(s(x0), s(x1))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) IF1(true, 0, x0, x1) -> c13(MIN(cons(0, x1))) IF1(true, s(0), s(x0), x1) -> c13(MIN(cons(s(0), x1))) IF1(true, s(s(x0)), s(s(x1)), x2) -> c13(MIN(cons(s(s(x0)), x2))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2)) MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2)) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2)) RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2)) RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2)) IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1))) IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2))) IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1))) S tuples: IF2(true, z0, z1, z2) -> c17(RM(z0, z2)) IF2(false, z0, z1, z2) -> c18(RM(z0, z2)) MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2)) MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2)) RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2)) RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2)) RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2)) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) MINSORT(cons(z1, cons(x1, x2))) -> c9(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), RM(min(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), MIN(cons(z1, cons(x1, x2)))) MINSORT(cons(0, cons(z0, x2))) -> c9(MINSORT(rm(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), RM(min(cons(0, cons(z0, x2))), cons(0, cons(z0, x2))), MIN(cons(0, cons(z0, x2)))) MINSORT(cons(s(z0), cons(0, x2))) -> c9(MINSORT(rm(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), RM(min(cons(s(z0), cons(0, x2))), cons(s(z0), cons(0, x2))), MIN(cons(s(z0), cons(0, x2)))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), RM(min(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MIN(cons(s(z0), cons(s(z1), x2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) MIN(cons(s(x0), cons(s(x1), x2))) -> c12(LE(s(x0), s(x1))) RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), cons(s(x1), x2)) -> c16(EQ(s(x0), s(x1))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) IF1(true, 0, x0, x1) -> c13(MIN(cons(0, x1))) IF1(true, s(0), s(x0), x1) -> c13(MIN(cons(s(0), x1))) IF1(true, s(s(x0)), s(s(x1)), x2) -> c13(MIN(cons(s(s(x0)), x2))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2)) MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2)) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2)) RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2)) RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2)) IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1))) IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2))) IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1))) K tuples: MINSORT(cons(z0, z1)) -> c8(MIN(cons(z0, z1))) MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil))) Defined Rule Symbols: rm_2, min_1, if1_4, le_2, if2_4, eq_2 Defined Pair Symbols: MINSORT_1, IF2_4, MIN_1, RM_2, IF1_4, LE_2, EQ_2 Compound Symbols: c8_1, c17_1, c18_1, c12_1, c16_1, c9_3, c9_2, c9_1, c12_2, c16_2, c13_1, c2_1, c6_1, c14_1 ---------------------------------------- (103) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(min(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) by MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) ---------------------------------------- (104) Obligation: Complexity Dependency Tuples Problem Rules: rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) rm(z0, nil) -> nil min(cons(z0, nil)) -> z0 min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(cons(z0, z2)) if1(false, z0, z1, z2) -> min(cons(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) if2(true, z0, z1, z2) -> rm(z0, z2) if2(false, z0, z1, z2) -> cons(z1, rm(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)) -> c8(MIN(cons(z0, z1))) IF2(true, z0, z1, z2) -> c17(RM(z0, z2)) IF2(false, z0, z1, z2) -> c18(RM(z0, z2)) MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2)) MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2)) RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2)) RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2)) RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2)) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) MINSORT(cons(z1, cons(x1, x2))) -> c9(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), RM(min(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), MIN(cons(z1, cons(x1, x2)))) MINSORT(cons(0, cons(z0, x2))) -> c9(MINSORT(rm(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), RM(min(cons(0, cons(z0, x2))), cons(0, cons(z0, x2))), MIN(cons(0, cons(z0, x2)))) MINSORT(cons(s(z0), cons(0, x2))) -> c9(MINSORT(rm(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), RM(min(cons(s(z0), cons(0, x2))), cons(s(z0), cons(0, x2))), MIN(cons(s(z0), cons(0, x2)))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), RM(min(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MIN(cons(s(z0), cons(s(z1), x2)))) MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil))) MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) MIN(cons(s(x0), cons(s(x1), x2))) -> c12(LE(s(x0), s(x1))) RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), cons(s(x1), x2)) -> c16(EQ(s(x0), s(x1))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) IF1(true, 0, x0, x1) -> c13(MIN(cons(0, x1))) IF1(true, s(0), s(x0), x1) -> c13(MIN(cons(s(0), x1))) IF1(true, s(s(x0)), s(s(x1)), x2) -> c13(MIN(cons(s(s(x0)), x2))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2)) MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2)) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2)) RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2)) RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2)) IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1))) IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2))) IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) S tuples: IF2(true, z0, z1, z2) -> c17(RM(z0, z2)) IF2(false, z0, z1, z2) -> c18(RM(z0, z2)) MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2)) MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2)) RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2)) RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2)) RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2)) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) MINSORT(cons(z1, cons(x1, x2))) -> c9(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), RM(min(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), MIN(cons(z1, cons(x1, x2)))) MINSORT(cons(0, cons(z0, x2))) -> c9(MINSORT(rm(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), RM(min(cons(0, cons(z0, x2))), cons(0, cons(z0, x2))), MIN(cons(0, cons(z0, x2)))) MINSORT(cons(s(z0), cons(0, x2))) -> c9(MINSORT(rm(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), RM(min(cons(s(z0), cons(0, x2))), cons(s(z0), cons(0, x2))), MIN(cons(s(z0), cons(0, x2)))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), RM(min(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MIN(cons(s(z0), cons(s(z1), x2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) MIN(cons(s(x0), cons(s(x1), x2))) -> c12(LE(s(x0), s(x1))) RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), cons(s(x1), x2)) -> c16(EQ(s(x0), s(x1))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) IF1(true, 0, x0, x1) -> c13(MIN(cons(0, x1))) IF1(true, s(0), s(x0), x1) -> c13(MIN(cons(s(0), x1))) IF1(true, s(s(x0)), s(s(x1)), x2) -> c13(MIN(cons(s(s(x0)), x2))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2)) MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2)) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2)) RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2)) RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2)) IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1))) IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2))) IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) K tuples: MINSORT(cons(z0, z1)) -> c8(MIN(cons(z0, z1))) MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil))) Defined Rule Symbols: rm_2, min_1, if1_4, le_2, if2_4, eq_2 Defined Pair Symbols: MINSORT_1, IF2_4, MIN_1, RM_2, IF1_4, LE_2, EQ_2 Compound Symbols: c8_1, c17_1, c18_1, c12_1, c16_1, c9_2, c9_3, c9_1, c12_2, c16_2, c13_1, c2_1, c6_1, c14_1 ---------------------------------------- (105) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IF2(true, z0, z1, z2) -> c17(RM(z0, z2)) by IF2(true, 0, 0, x0) -> c17(RM(0, x0)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2)) IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0)) ---------------------------------------- (106) Obligation: Complexity Dependency Tuples Problem Rules: rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) rm(z0, nil) -> nil min(cons(z0, nil)) -> z0 min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(cons(z0, z2)) if1(false, z0, z1, z2) -> min(cons(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) if2(true, z0, z1, z2) -> rm(z0, z2) if2(false, z0, z1, z2) -> cons(z1, rm(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)) -> c8(MIN(cons(z0, z1))) IF2(false, z0, z1, z2) -> c18(RM(z0, z2)) MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2)) MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2)) RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2)) RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2)) RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2)) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) MINSORT(cons(z1, cons(x1, x2))) -> c9(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), RM(min(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), MIN(cons(z1, cons(x1, x2)))) MINSORT(cons(0, cons(z0, x2))) -> c9(MINSORT(rm(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), RM(min(cons(0, cons(z0, x2))), cons(0, cons(z0, x2))), MIN(cons(0, cons(z0, x2)))) MINSORT(cons(s(z0), cons(0, x2))) -> c9(MINSORT(rm(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), RM(min(cons(s(z0), cons(0, x2))), cons(s(z0), cons(0, x2))), MIN(cons(s(z0), cons(0, x2)))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), RM(min(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MIN(cons(s(z0), cons(s(z1), x2)))) MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil))) MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) MIN(cons(s(x0), cons(s(x1), x2))) -> c12(LE(s(x0), s(x1))) RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), cons(s(x1), x2)) -> c16(EQ(s(x0), s(x1))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) IF1(true, 0, x0, x1) -> c13(MIN(cons(0, x1))) IF1(true, s(0), s(x0), x1) -> c13(MIN(cons(s(0), x1))) IF1(true, s(s(x0)), s(s(x1)), x2) -> c13(MIN(cons(s(s(x0)), x2))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2)) MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2)) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2)) RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2)) RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2)) IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1))) IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2))) IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) IF2(true, 0, 0, x0) -> c17(RM(0, x0)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2)) IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0)) S tuples: IF2(false, z0, z1, z2) -> c18(RM(z0, z2)) MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2)) MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2)) RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2)) RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2)) RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2)) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) MINSORT(cons(z1, cons(x1, x2))) -> c9(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), RM(min(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), MIN(cons(z1, cons(x1, x2)))) MINSORT(cons(0, cons(z0, x2))) -> c9(MINSORT(rm(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), RM(min(cons(0, cons(z0, x2))), cons(0, cons(z0, x2))), MIN(cons(0, cons(z0, x2)))) MINSORT(cons(s(z0), cons(0, x2))) -> c9(MINSORT(rm(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), RM(min(cons(s(z0), cons(0, x2))), cons(s(z0), cons(0, x2))), MIN(cons(s(z0), cons(0, x2)))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), RM(min(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MIN(cons(s(z0), cons(s(z1), x2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) MIN(cons(s(x0), cons(s(x1), x2))) -> c12(LE(s(x0), s(x1))) RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), cons(s(x1), x2)) -> c16(EQ(s(x0), s(x1))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) IF1(true, 0, x0, x1) -> c13(MIN(cons(0, x1))) IF1(true, s(0), s(x0), x1) -> c13(MIN(cons(s(0), x1))) IF1(true, s(s(x0)), s(s(x1)), x2) -> c13(MIN(cons(s(s(x0)), x2))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2)) MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2)) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2)) RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2)) RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2)) IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1))) IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2))) IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) IF2(true, 0, 0, x0) -> c17(RM(0, x0)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2)) IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0)) K tuples: MINSORT(cons(z0, z1)) -> c8(MIN(cons(z0, z1))) MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil))) Defined Rule Symbols: rm_2, min_1, if1_4, le_2, if2_4, eq_2 Defined Pair Symbols: MINSORT_1, IF2_4, MIN_1, RM_2, IF1_4, LE_2, EQ_2 Compound Symbols: c8_1, c18_1, c12_1, c16_1, c9_2, c9_3, c9_1, c12_2, c16_2, c13_1, c2_1, c6_1, c14_1, c17_1 ---------------------------------------- (107) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IF2(false, z0, z1, z2) -> c18(RM(z0, z2)) by IF2(false, 0, s(x0), x1) -> c18(RM(0, x1)) IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2)) IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1)) ---------------------------------------- (108) Obligation: Complexity Dependency Tuples Problem Rules: rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) rm(z0, nil) -> nil min(cons(z0, nil)) -> z0 min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(cons(z0, z2)) if1(false, z0, z1, z2) -> min(cons(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) if2(true, z0, z1, z2) -> rm(z0, z2) if2(false, z0, z1, z2) -> cons(z1, rm(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)) -> c8(MIN(cons(z0, z1))) MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2)) MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2)) RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2)) RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2)) RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2)) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) MINSORT(cons(z1, cons(x1, x2))) -> c9(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), RM(min(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), MIN(cons(z1, cons(x1, x2)))) MINSORT(cons(0, cons(z0, x2))) -> c9(MINSORT(rm(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), RM(min(cons(0, cons(z0, x2))), cons(0, cons(z0, x2))), MIN(cons(0, cons(z0, x2)))) MINSORT(cons(s(z0), cons(0, x2))) -> c9(MINSORT(rm(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), RM(min(cons(s(z0), cons(0, x2))), cons(s(z0), cons(0, x2))), MIN(cons(s(z0), cons(0, x2)))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), RM(min(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MIN(cons(s(z0), cons(s(z1), x2)))) MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil))) MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) MIN(cons(s(x0), cons(s(x1), x2))) -> c12(LE(s(x0), s(x1))) RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), cons(s(x1), x2)) -> c16(EQ(s(x0), s(x1))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) IF1(true, 0, x0, x1) -> c13(MIN(cons(0, x1))) IF1(true, s(0), s(x0), x1) -> c13(MIN(cons(s(0), x1))) IF1(true, s(s(x0)), s(s(x1)), x2) -> c13(MIN(cons(s(s(x0)), x2))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2)) MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2)) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2)) RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2)) RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2)) IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1))) IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2))) IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) IF2(true, 0, 0, x0) -> c17(RM(0, x0)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2)) IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0)) IF2(false, 0, s(x0), x1) -> c18(RM(0, x1)) IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2)) IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1)) S tuples: MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2)) MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2)) RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2)) RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2)) RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2)) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) MINSORT(cons(z1, cons(x1, x2))) -> c9(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), RM(min(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), MIN(cons(z1, cons(x1, x2)))) MINSORT(cons(0, cons(z0, x2))) -> c9(MINSORT(rm(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), RM(min(cons(0, cons(z0, x2))), cons(0, cons(z0, x2))), MIN(cons(0, cons(z0, x2)))) MINSORT(cons(s(z0), cons(0, x2))) -> c9(MINSORT(rm(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), RM(min(cons(s(z0), cons(0, x2))), cons(s(z0), cons(0, x2))), MIN(cons(s(z0), cons(0, x2)))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), RM(min(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MIN(cons(s(z0), cons(s(z1), x2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) MIN(cons(s(x0), cons(s(x1), x2))) -> c12(LE(s(x0), s(x1))) RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), cons(s(x1), x2)) -> c16(EQ(s(x0), s(x1))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) IF1(true, 0, x0, x1) -> c13(MIN(cons(0, x1))) IF1(true, s(0), s(x0), x1) -> c13(MIN(cons(s(0), x1))) IF1(true, s(s(x0)), s(s(x1)), x2) -> c13(MIN(cons(s(s(x0)), x2))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2)) MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2)) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2)) RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2)) RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2)) IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1))) IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2))) IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) IF2(true, 0, 0, x0) -> c17(RM(0, x0)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2)) IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0)) IF2(false, 0, s(x0), x1) -> c18(RM(0, x1)) IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2)) IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1)) K tuples: MINSORT(cons(z0, z1)) -> c8(MIN(cons(z0, z1))) MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil))) Defined Rule Symbols: rm_2, min_1, if1_4, le_2, if2_4, eq_2 Defined Pair Symbols: MINSORT_1, MIN_1, RM_2, IF1_4, LE_2, EQ_2, IF2_4 Compound Symbols: c8_1, c12_1, c16_1, c9_2, c9_3, c9_1, c12_2, c16_2, c13_1, c2_1, c6_1, c14_1, c17_1, c18_1 ---------------------------------------- (109) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) by MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(z0, cons(z0, nil))) ---------------------------------------- (110) Obligation: Complexity Dependency Tuples Problem Rules: rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) rm(z0, nil) -> nil min(cons(z0, nil)) -> z0 min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(cons(z0, z2)) if1(false, z0, z1, z2) -> min(cons(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) if2(true, z0, z1, z2) -> rm(z0, z2) if2(false, z0, z1, z2) -> cons(z1, rm(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)) -> c8(MIN(cons(z0, z1))) MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2)) MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2)) RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2)) RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2)) RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2)) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) MINSORT(cons(z1, cons(x1, x2))) -> c9(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), RM(min(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), MIN(cons(z1, cons(x1, x2)))) MINSORT(cons(0, cons(z0, x2))) -> c9(MINSORT(rm(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), RM(min(cons(0, cons(z0, x2))), cons(0, cons(z0, x2))), MIN(cons(0, cons(z0, x2)))) MINSORT(cons(s(z0), cons(0, x2))) -> c9(MINSORT(rm(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), RM(min(cons(s(z0), cons(0, x2))), cons(s(z0), cons(0, x2))), MIN(cons(s(z0), cons(0, x2)))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), RM(min(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MIN(cons(s(z0), cons(s(z1), x2)))) MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil))) MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) MIN(cons(s(x0), cons(s(x1), x2))) -> c12(LE(s(x0), s(x1))) RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), cons(s(x1), x2)) -> c16(EQ(s(x0), s(x1))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) IF1(true, 0, x0, x1) -> c13(MIN(cons(0, x1))) IF1(true, s(0), s(x0), x1) -> c13(MIN(cons(s(0), x1))) IF1(true, s(s(x0)), s(s(x1)), x2) -> c13(MIN(cons(s(s(x0)), x2))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2)) MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2)) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2)) RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2)) RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2)) IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1))) IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2))) IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) IF2(true, 0, 0, x0) -> c17(RM(0, x0)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2)) IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0)) IF2(false, 0, s(x0), x1) -> c18(RM(0, x1)) IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2)) IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1)) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(z0, cons(z0, nil))) S tuples: MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2)) MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2)) RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2)) RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2)) RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2)) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) MINSORT(cons(z1, cons(x1, x2))) -> c9(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), RM(min(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), MIN(cons(z1, cons(x1, x2)))) MINSORT(cons(0, cons(z0, x2))) -> c9(MINSORT(rm(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), RM(min(cons(0, cons(z0, x2))), cons(0, cons(z0, x2))), MIN(cons(0, cons(z0, x2)))) MINSORT(cons(s(z0), cons(0, x2))) -> c9(MINSORT(rm(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), RM(min(cons(s(z0), cons(0, x2))), cons(s(z0), cons(0, x2))), MIN(cons(s(z0), cons(0, x2)))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), RM(min(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MIN(cons(s(z0), cons(s(z1), x2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) MIN(cons(s(x0), cons(s(x1), x2))) -> c12(LE(s(x0), s(x1))) RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), cons(s(x1), x2)) -> c16(EQ(s(x0), s(x1))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) IF1(true, 0, x0, x1) -> c13(MIN(cons(0, x1))) IF1(true, s(0), s(x0), x1) -> c13(MIN(cons(s(0), x1))) IF1(true, s(s(x0)), s(s(x1)), x2) -> c13(MIN(cons(s(s(x0)), x2))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2)) MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2)) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2)) RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2)) RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2)) IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1))) IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2))) IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) IF2(true, 0, 0, x0) -> c17(RM(0, x0)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2)) IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0)) IF2(false, 0, s(x0), x1) -> c18(RM(0, x1)) IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2)) IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1)) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(z0, cons(z0, nil))) K tuples: MINSORT(cons(z0, z1)) -> c8(MIN(cons(z0, z1))) MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil))) Defined Rule Symbols: rm_2, min_1, if1_4, le_2, if2_4, eq_2 Defined Pair Symbols: MINSORT_1, MIN_1, RM_2, IF1_4, LE_2, EQ_2, IF2_4 Compound Symbols: c8_1, c12_1, c16_1, c9_2, c9_3, c9_1, c12_2, c16_2, c13_1, c2_1, c6_1, c14_1, c17_1, c18_1 ---------------------------------------- (111) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) by MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(z0, cons(z0, nil))) ---------------------------------------- (112) Obligation: Complexity Dependency Tuples Problem Rules: rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) rm(z0, nil) -> nil min(cons(z0, nil)) -> z0 min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(cons(z0, z2)) if1(false, z0, z1, z2) -> min(cons(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) if2(true, z0, z1, z2) -> rm(z0, z2) if2(false, z0, z1, z2) -> cons(z1, rm(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)) -> c8(MIN(cons(z0, z1))) MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2)) MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2)) RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2)) RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2)) RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2)) MINSORT(cons(z1, cons(x1, x2))) -> c9(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), RM(min(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), MIN(cons(z1, cons(x1, x2)))) MINSORT(cons(0, cons(z0, x2))) -> c9(MINSORT(rm(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), RM(min(cons(0, cons(z0, x2))), cons(0, cons(z0, x2))), MIN(cons(0, cons(z0, x2)))) MINSORT(cons(s(z0), cons(0, x2))) -> c9(MINSORT(rm(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), RM(min(cons(s(z0), cons(0, x2))), cons(s(z0), cons(0, x2))), MIN(cons(s(z0), cons(0, x2)))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), RM(min(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MIN(cons(s(z0), cons(s(z1), x2)))) MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil))) MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) MIN(cons(s(x0), cons(s(x1), x2))) -> c12(LE(s(x0), s(x1))) RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), cons(s(x1), x2)) -> c16(EQ(s(x0), s(x1))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) IF1(true, 0, x0, x1) -> c13(MIN(cons(0, x1))) IF1(true, s(0), s(x0), x1) -> c13(MIN(cons(s(0), x1))) IF1(true, s(s(x0)), s(s(x1)), x2) -> c13(MIN(cons(s(s(x0)), x2))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2)) MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2)) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2)) RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2)) RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2)) IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1))) IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2))) IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) IF2(true, 0, 0, x0) -> c17(RM(0, x0)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2)) IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0)) IF2(false, 0, s(x0), x1) -> c18(RM(0, x1)) IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2)) IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1)) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(z0, cons(z0, nil))) S tuples: MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2)) MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2)) RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2)) RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2)) RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2)) MINSORT(cons(z1, cons(x1, x2))) -> c9(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), RM(min(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), MIN(cons(z1, cons(x1, x2)))) MINSORT(cons(0, cons(z0, x2))) -> c9(MINSORT(rm(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), RM(min(cons(0, cons(z0, x2))), cons(0, cons(z0, x2))), MIN(cons(0, cons(z0, x2)))) MINSORT(cons(s(z0), cons(0, x2))) -> c9(MINSORT(rm(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), RM(min(cons(s(z0), cons(0, x2))), cons(s(z0), cons(0, x2))), MIN(cons(s(z0), cons(0, x2)))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), RM(min(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MIN(cons(s(z0), cons(s(z1), x2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) MIN(cons(s(x0), cons(s(x1), x2))) -> c12(LE(s(x0), s(x1))) RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), cons(s(x1), x2)) -> c16(EQ(s(x0), s(x1))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) IF1(true, 0, x0, x1) -> c13(MIN(cons(0, x1))) IF1(true, s(0), s(x0), x1) -> c13(MIN(cons(s(0), x1))) IF1(true, s(s(x0)), s(s(x1)), x2) -> c13(MIN(cons(s(s(x0)), x2))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2)) MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2)) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2)) RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2)) RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2)) IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1))) IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2))) IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) IF2(true, 0, 0, x0) -> c17(RM(0, x0)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2)) IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0)) IF2(false, 0, s(x0), x1) -> c18(RM(0, x1)) IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2)) IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1)) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(z0, cons(z0, nil))) K tuples: MINSORT(cons(z0, z1)) -> c8(MIN(cons(z0, z1))) MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil))) Defined Rule Symbols: rm_2, min_1, if1_4, le_2, if2_4, eq_2 Defined Pair Symbols: MINSORT_1, MIN_1, RM_2, IF1_4, LE_2, EQ_2, IF2_4 Compound Symbols: c8_1, c12_1, c16_1, c9_3, c9_1, c9_2, c12_2, c16_2, c13_1, c2_1, c6_1, c14_1, c17_1, c18_1 ---------------------------------------- (113) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace MINSORT(cons(z1, cons(x1, x2))) -> c9(MINSORT(if2(eq(if1(le(z1, x1), z1, x1, x2), z1), if1(le(z1, x1), z1, x1, x2), z1, cons(x1, x2))), RM(min(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), MIN(cons(z1, cons(x1, x2)))) by MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) ---------------------------------------- (114) Obligation: Complexity Dependency Tuples Problem Rules: rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) rm(z0, nil) -> nil min(cons(z0, nil)) -> z0 min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(cons(z0, z2)) if1(false, z0, z1, z2) -> min(cons(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) if2(true, z0, z1, z2) -> rm(z0, z2) if2(false, z0, z1, z2) -> cons(z1, rm(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)) -> c8(MIN(cons(z0, z1))) MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2)) MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2)) RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2)) RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2)) RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2)) MINSORT(cons(0, cons(z0, x2))) -> c9(MINSORT(rm(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), RM(min(cons(0, cons(z0, x2))), cons(0, cons(z0, x2))), MIN(cons(0, cons(z0, x2)))) MINSORT(cons(s(z0), cons(0, x2))) -> c9(MINSORT(rm(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), RM(min(cons(s(z0), cons(0, x2))), cons(s(z0), cons(0, x2))), MIN(cons(s(z0), cons(0, x2)))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), RM(min(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MIN(cons(s(z0), cons(s(z1), x2)))) MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil))) MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) MIN(cons(s(x0), cons(s(x1), x2))) -> c12(LE(s(x0), s(x1))) RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), cons(s(x1), x2)) -> c16(EQ(s(x0), s(x1))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) IF1(true, 0, x0, x1) -> c13(MIN(cons(0, x1))) IF1(true, s(0), s(x0), x1) -> c13(MIN(cons(s(0), x1))) IF1(true, s(s(x0)), s(s(x1)), x2) -> c13(MIN(cons(s(s(x0)), x2))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2)) MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2)) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2)) RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2)) RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2)) IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1))) IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2))) IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) IF2(true, 0, 0, x0) -> c17(RM(0, x0)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2)) IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0)) IF2(false, 0, s(x0), x1) -> c18(RM(0, x1)) IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2)) IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1)) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(z0, cons(z0, nil))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) S tuples: MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2)) MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2)) RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2)) RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2)) RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2)) MINSORT(cons(0, cons(z0, x2))) -> c9(MINSORT(rm(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), RM(min(cons(0, cons(z0, x2))), cons(0, cons(z0, x2))), MIN(cons(0, cons(z0, x2)))) MINSORT(cons(s(z0), cons(0, x2))) -> c9(MINSORT(rm(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), RM(min(cons(s(z0), cons(0, x2))), cons(s(z0), cons(0, x2))), MIN(cons(s(z0), cons(0, x2)))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), RM(min(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MIN(cons(s(z0), cons(s(z1), x2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) MIN(cons(s(x0), cons(s(x1), x2))) -> c12(LE(s(x0), s(x1))) RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), cons(s(x1), x2)) -> c16(EQ(s(x0), s(x1))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) IF1(true, 0, x0, x1) -> c13(MIN(cons(0, x1))) IF1(true, s(0), s(x0), x1) -> c13(MIN(cons(s(0), x1))) IF1(true, s(s(x0)), s(s(x1)), x2) -> c13(MIN(cons(s(s(x0)), x2))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2)) MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2)) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2)) RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2)) RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2)) IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1))) IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2))) IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) IF2(true, 0, 0, x0) -> c17(RM(0, x0)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2)) IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0)) IF2(false, 0, s(x0), x1) -> c18(RM(0, x1)) IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2)) IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1)) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(z0, cons(z0, nil))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) K tuples: MINSORT(cons(z0, z1)) -> c8(MIN(cons(z0, z1))) MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil))) Defined Rule Symbols: rm_2, min_1, if1_4, le_2, if2_4, eq_2 Defined Pair Symbols: MINSORT_1, MIN_1, RM_2, IF1_4, LE_2, EQ_2, IF2_4 Compound Symbols: c8_1, c12_1, c16_1, c9_3, c9_1, c9_2, c12_2, c16_2, c13_1, c2_1, c6_1, c14_1, c17_1, c18_1 ---------------------------------------- (115) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace MINSORT(cons(0, cons(z0, x2))) -> c9(MINSORT(rm(if1(true, 0, z0, x2), cons(0, cons(z0, x2)))), RM(min(cons(0, cons(z0, x2))), cons(0, cons(z0, x2))), MIN(cons(0, cons(z0, x2)))) by MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1)))) ---------------------------------------- (116) Obligation: Complexity Dependency Tuples Problem Rules: rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) rm(z0, nil) -> nil min(cons(z0, nil)) -> z0 min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(cons(z0, z2)) if1(false, z0, z1, z2) -> min(cons(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) if2(true, z0, z1, z2) -> rm(z0, z2) if2(false, z0, z1, z2) -> cons(z1, rm(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)) -> c8(MIN(cons(z0, z1))) MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2)) MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2)) RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2)) RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2)) RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2)) MINSORT(cons(s(z0), cons(0, x2))) -> c9(MINSORT(rm(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), RM(min(cons(s(z0), cons(0, x2))), cons(s(z0), cons(0, x2))), MIN(cons(s(z0), cons(0, x2)))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), RM(min(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MIN(cons(s(z0), cons(s(z1), x2)))) MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil))) MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) MIN(cons(s(x0), cons(s(x1), x2))) -> c12(LE(s(x0), s(x1))) RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), cons(s(x1), x2)) -> c16(EQ(s(x0), s(x1))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) IF1(true, 0, x0, x1) -> c13(MIN(cons(0, x1))) IF1(true, s(0), s(x0), x1) -> c13(MIN(cons(s(0), x1))) IF1(true, s(s(x0)), s(s(x1)), x2) -> c13(MIN(cons(s(s(x0)), x2))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2)) MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2)) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2)) RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2)) RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2)) IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1))) IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2))) IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) IF2(true, 0, 0, x0) -> c17(RM(0, x0)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2)) IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0)) IF2(false, 0, s(x0), x1) -> c18(RM(0, x1)) IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2)) IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1)) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(z0, cons(z0, nil))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1)))) S tuples: MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2)) MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2)) RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2)) RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2)) RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2)) MINSORT(cons(s(z0), cons(0, x2))) -> c9(MINSORT(rm(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), RM(min(cons(s(z0), cons(0, x2))), cons(s(z0), cons(0, x2))), MIN(cons(s(z0), cons(0, x2)))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), RM(min(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MIN(cons(s(z0), cons(s(z1), x2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) MIN(cons(s(x0), cons(s(x1), x2))) -> c12(LE(s(x0), s(x1))) RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), cons(s(x1), x2)) -> c16(EQ(s(x0), s(x1))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) IF1(true, 0, x0, x1) -> c13(MIN(cons(0, x1))) IF1(true, s(0), s(x0), x1) -> c13(MIN(cons(s(0), x1))) IF1(true, s(s(x0)), s(s(x1)), x2) -> c13(MIN(cons(s(s(x0)), x2))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2)) MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2)) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2)) RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2)) RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2)) IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1))) IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2))) IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) IF2(true, 0, 0, x0) -> c17(RM(0, x0)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2)) IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0)) IF2(false, 0, s(x0), x1) -> c18(RM(0, x1)) IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2)) IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1)) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(z0, cons(z0, nil))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1)))) K tuples: MINSORT(cons(z0, z1)) -> c8(MIN(cons(z0, z1))) MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil))) Defined Rule Symbols: rm_2, min_1, if1_4, le_2, if2_4, eq_2 Defined Pair Symbols: MINSORT_1, MIN_1, RM_2, IF1_4, LE_2, EQ_2, IF2_4 Compound Symbols: c8_1, c12_1, c16_1, c9_3, c9_1, c9_2, c12_2, c16_2, c13_1, c2_1, c6_1, c14_1, c17_1, c18_1 ---------------------------------------- (117) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace MINSORT(cons(z0, z1)) -> c8(MIN(cons(z0, z1))) by MINSORT(cons(0, cons(y0, y1))) -> c8(MIN(cons(0, cons(y0, y1)))) MINSORT(cons(s(y0), cons(0, y1))) -> c8(MIN(cons(s(y0), cons(0, y1)))) MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c8(MIN(cons(s(s(y0)), cons(s(s(y1)), y2)))) MINSORT(cons(s(y0), cons(s(y1), y2))) -> c8(MIN(cons(s(y0), cons(s(y1), y2)))) MINSORT(cons(s(0), cons(s(y0), y1))) -> c8(MIN(cons(s(0), cons(s(y0), y1)))) MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c8(MIN(cons(s(s(y0)), cons(s(0), y1)))) ---------------------------------------- (118) Obligation: Complexity Dependency Tuples Problem Rules: rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) rm(z0, nil) -> nil min(cons(z0, nil)) -> z0 min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(cons(z0, z2)) if1(false, z0, z1, z2) -> min(cons(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) if2(true, z0, z1, z2) -> rm(z0, z2) if2(false, z0, z1, z2) -> cons(z1, rm(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(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2)) MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2)) RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2)) RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2)) RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2)) MINSORT(cons(s(z0), cons(0, x2))) -> c9(MINSORT(rm(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), RM(min(cons(s(z0), cons(0, x2))), cons(s(z0), cons(0, x2))), MIN(cons(s(z0), cons(0, x2)))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), RM(min(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MIN(cons(s(z0), cons(s(z1), x2)))) MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil))) MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) MIN(cons(s(x0), cons(s(x1), x2))) -> c12(LE(s(x0), s(x1))) RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), cons(s(x1), x2)) -> c16(EQ(s(x0), s(x1))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) IF1(true, 0, x0, x1) -> c13(MIN(cons(0, x1))) IF1(true, s(0), s(x0), x1) -> c13(MIN(cons(s(0), x1))) IF1(true, s(s(x0)), s(s(x1)), x2) -> c13(MIN(cons(s(s(x0)), x2))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2)) MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2)) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2)) RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2)) RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2)) IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1))) IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2))) IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) IF2(true, 0, 0, x0) -> c17(RM(0, x0)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2)) IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0)) IF2(false, 0, s(x0), x1) -> c18(RM(0, x1)) IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2)) IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1)) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(z0, cons(z0, nil))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1)))) MINSORT(cons(0, cons(y0, y1))) -> c8(MIN(cons(0, cons(y0, y1)))) MINSORT(cons(s(y0), cons(0, y1))) -> c8(MIN(cons(s(y0), cons(0, y1)))) MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c8(MIN(cons(s(s(y0)), cons(s(s(y1)), y2)))) MINSORT(cons(s(y0), cons(s(y1), y2))) -> c8(MIN(cons(s(y0), cons(s(y1), y2)))) MINSORT(cons(s(0), cons(s(y0), y1))) -> c8(MIN(cons(s(0), cons(s(y0), y1)))) MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c8(MIN(cons(s(s(y0)), cons(s(0), y1)))) S tuples: MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2)) MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2)) RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2)) RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2)) RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2)) MINSORT(cons(s(z0), cons(0, x2))) -> c9(MINSORT(rm(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), RM(min(cons(s(z0), cons(0, x2))), cons(s(z0), cons(0, x2))), MIN(cons(s(z0), cons(0, x2)))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), RM(min(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MIN(cons(s(z0), cons(s(z1), x2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) MIN(cons(s(x0), cons(s(x1), x2))) -> c12(LE(s(x0), s(x1))) RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), cons(s(x1), x2)) -> c16(EQ(s(x0), s(x1))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) IF1(true, 0, x0, x1) -> c13(MIN(cons(0, x1))) IF1(true, s(0), s(x0), x1) -> c13(MIN(cons(s(0), x1))) IF1(true, s(s(x0)), s(s(x1)), x2) -> c13(MIN(cons(s(s(x0)), x2))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2)) MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2)) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2)) RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2)) RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2)) IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1))) IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2))) IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) IF2(true, 0, 0, x0) -> c17(RM(0, x0)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2)) IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0)) IF2(false, 0, s(x0), x1) -> c18(RM(0, x1)) IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2)) IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1)) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(z0, cons(z0, nil))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1)))) K tuples: MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil))) MINSORT(cons(0, cons(y0, y1))) -> c8(MIN(cons(0, cons(y0, y1)))) MINSORT(cons(s(y0), cons(0, y1))) -> c8(MIN(cons(s(y0), cons(0, y1)))) MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c8(MIN(cons(s(s(y0)), cons(s(s(y1)), y2)))) MINSORT(cons(s(y0), cons(s(y1), y2))) -> c8(MIN(cons(s(y0), cons(s(y1), y2)))) MINSORT(cons(s(0), cons(s(y0), y1))) -> c8(MIN(cons(s(0), cons(s(y0), y1)))) MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c8(MIN(cons(s(s(y0)), cons(s(0), y1)))) Defined Rule Symbols: rm_2, min_1, if1_4, le_2, if2_4, eq_2 Defined Pair Symbols: MIN_1, RM_2, MINSORT_1, IF1_4, LE_2, EQ_2, IF2_4 Compound Symbols: c12_1, c16_1, c9_3, c9_1, c9_2, c12_2, c16_2, c13_1, c2_1, c6_1, c14_1, c17_1, c18_1, c8_1 ---------------------------------------- (119) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace MINSORT(cons(s(z0), cons(0, x2))) -> c9(MINSORT(rm(if1(false, s(z0), 0, x2), cons(s(z0), cons(0, x2)))), RM(min(cons(s(z0), cons(0, x2))), cons(s(z0), cons(0, x2))), MIN(cons(s(z0), cons(0, x2)))) by MINSORT(cons(s(z0), cons(0, z1))) -> c9(MINSORT(rm(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), RM(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(cons(s(z0), cons(0, z1)))) ---------------------------------------- (120) Obligation: Complexity Dependency Tuples Problem Rules: rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) rm(z0, nil) -> nil min(cons(z0, nil)) -> z0 min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(cons(z0, z2)) if1(false, z0, z1, z2) -> min(cons(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) if2(true, z0, z1, z2) -> rm(z0, z2) if2(false, z0, z1, z2) -> cons(z1, rm(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(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2)) MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2)) RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2)) RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2)) RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2)) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), RM(min(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MIN(cons(s(z0), cons(s(z1), x2)))) MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil))) MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) MIN(cons(s(x0), cons(s(x1), x2))) -> c12(LE(s(x0), s(x1))) RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), cons(s(x1), x2)) -> c16(EQ(s(x0), s(x1))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) IF1(true, 0, x0, x1) -> c13(MIN(cons(0, x1))) IF1(true, s(0), s(x0), x1) -> c13(MIN(cons(s(0), x1))) IF1(true, s(s(x0)), s(s(x1)), x2) -> c13(MIN(cons(s(s(x0)), x2))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2)) MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2)) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2)) RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2)) RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2)) IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1))) IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2))) IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) IF2(true, 0, 0, x0) -> c17(RM(0, x0)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2)) IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0)) IF2(false, 0, s(x0), x1) -> c18(RM(0, x1)) IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2)) IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1)) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(z0, cons(z0, nil))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1)))) MINSORT(cons(0, cons(y0, y1))) -> c8(MIN(cons(0, cons(y0, y1)))) MINSORT(cons(s(y0), cons(0, y1))) -> c8(MIN(cons(s(y0), cons(0, y1)))) MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c8(MIN(cons(s(s(y0)), cons(s(s(y1)), y2)))) MINSORT(cons(s(y0), cons(s(y1), y2))) -> c8(MIN(cons(s(y0), cons(s(y1), y2)))) MINSORT(cons(s(0), cons(s(y0), y1))) -> c8(MIN(cons(s(0), cons(s(y0), y1)))) MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c8(MIN(cons(s(s(y0)), cons(s(0), y1)))) MINSORT(cons(s(z0), cons(0, z1))) -> c9(MINSORT(rm(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), RM(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(cons(s(z0), cons(0, z1)))) S tuples: MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2)) MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2)) RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2)) RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2)) RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2)) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), RM(min(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MIN(cons(s(z0), cons(s(z1), x2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) MIN(cons(s(x0), cons(s(x1), x2))) -> c12(LE(s(x0), s(x1))) RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), cons(s(x1), x2)) -> c16(EQ(s(x0), s(x1))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) IF1(true, 0, x0, x1) -> c13(MIN(cons(0, x1))) IF1(true, s(0), s(x0), x1) -> c13(MIN(cons(s(0), x1))) IF1(true, s(s(x0)), s(s(x1)), x2) -> c13(MIN(cons(s(s(x0)), x2))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2)) MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2)) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2)) RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2)) RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2)) IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1))) IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2))) IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) IF2(true, 0, 0, x0) -> c17(RM(0, x0)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2)) IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0)) IF2(false, 0, s(x0), x1) -> c18(RM(0, x1)) IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2)) IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1)) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(z0, cons(z0, nil))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1)))) MINSORT(cons(s(z0), cons(0, z1))) -> c9(MINSORT(rm(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), RM(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(cons(s(z0), cons(0, z1)))) K tuples: MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil))) MINSORT(cons(0, cons(y0, y1))) -> c8(MIN(cons(0, cons(y0, y1)))) MINSORT(cons(s(y0), cons(0, y1))) -> c8(MIN(cons(s(y0), cons(0, y1)))) MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c8(MIN(cons(s(s(y0)), cons(s(s(y1)), y2)))) MINSORT(cons(s(y0), cons(s(y1), y2))) -> c8(MIN(cons(s(y0), cons(s(y1), y2)))) MINSORT(cons(s(0), cons(s(y0), y1))) -> c8(MIN(cons(s(0), cons(s(y0), y1)))) MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c8(MIN(cons(s(s(y0)), cons(s(0), y1)))) Defined Rule Symbols: rm_2, min_1, if1_4, le_2, if2_4, eq_2 Defined Pair Symbols: MIN_1, RM_2, MINSORT_1, IF1_4, LE_2, EQ_2, IF2_4 Compound Symbols: c12_1, c16_1, c9_3, c9_1, c9_2, c12_2, c16_2, c13_1, c2_1, c6_1, c14_1, c17_1, c18_1, c8_1 ---------------------------------------- (121) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace MINSORT(cons(s(z0), cons(s(z1), x2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), x2), cons(s(z0), cons(s(z1), x2)))), RM(min(cons(s(z0), cons(s(z1), x2))), cons(s(z0), cons(s(z1), x2))), MIN(cons(s(z0), cons(s(z1), x2)))) by MINSORT(cons(s(z0), cons(s(z1), z2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), RM(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(cons(s(z0), cons(s(z1), z2)))) ---------------------------------------- (122) Obligation: Complexity Dependency Tuples Problem Rules: rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) rm(z0, nil) -> nil min(cons(z0, nil)) -> z0 min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(cons(z0, z2)) if1(false, z0, z1, z2) -> min(cons(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) if2(true, z0, z1, z2) -> rm(z0, z2) if2(false, z0, z1, z2) -> cons(z1, rm(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(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2)) MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2)) RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2)) RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2)) RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2)) MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil))) MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) MIN(cons(s(x0), cons(s(x1), x2))) -> c12(LE(s(x0), s(x1))) RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), cons(s(x1), x2)) -> c16(EQ(s(x0), s(x1))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) IF1(true, 0, x0, x1) -> c13(MIN(cons(0, x1))) IF1(true, s(0), s(x0), x1) -> c13(MIN(cons(s(0), x1))) IF1(true, s(s(x0)), s(s(x1)), x2) -> c13(MIN(cons(s(s(x0)), x2))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2)) MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2)) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2)) RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2)) RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2)) IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1))) IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2))) IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) IF2(true, 0, 0, x0) -> c17(RM(0, x0)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2)) IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0)) IF2(false, 0, s(x0), x1) -> c18(RM(0, x1)) IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2)) IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1)) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(z0, cons(z0, nil))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1)))) MINSORT(cons(0, cons(y0, y1))) -> c8(MIN(cons(0, cons(y0, y1)))) MINSORT(cons(s(y0), cons(0, y1))) -> c8(MIN(cons(s(y0), cons(0, y1)))) MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c8(MIN(cons(s(s(y0)), cons(s(s(y1)), y2)))) MINSORT(cons(s(y0), cons(s(y1), y2))) -> c8(MIN(cons(s(y0), cons(s(y1), y2)))) MINSORT(cons(s(0), cons(s(y0), y1))) -> c8(MIN(cons(s(0), cons(s(y0), y1)))) MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c8(MIN(cons(s(s(y0)), cons(s(0), y1)))) MINSORT(cons(s(z0), cons(0, z1))) -> c9(MINSORT(rm(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), RM(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(cons(s(z0), cons(0, z1)))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), RM(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(cons(s(z0), cons(s(z1), z2)))) S tuples: MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2)) MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2)) RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2)) RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2)) RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2)) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) MIN(cons(s(x0), cons(s(x1), x2))) -> c12(LE(s(x0), s(x1))) RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), cons(s(x1), x2)) -> c16(EQ(s(x0), s(x1))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) IF1(true, 0, x0, x1) -> c13(MIN(cons(0, x1))) IF1(true, s(0), s(x0), x1) -> c13(MIN(cons(s(0), x1))) IF1(true, s(s(x0)), s(s(x1)), x2) -> c13(MIN(cons(s(s(x0)), x2))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2)) MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2)) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2)) RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2)) RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2)) IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1))) IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2))) IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) IF2(true, 0, 0, x0) -> c17(RM(0, x0)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2)) IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0)) IF2(false, 0, s(x0), x1) -> c18(RM(0, x1)) IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2)) IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1)) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(z0, cons(z0, nil))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1)))) MINSORT(cons(s(z0), cons(0, z1))) -> c9(MINSORT(rm(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), RM(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(cons(s(z0), cons(0, z1)))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), RM(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(cons(s(z0), cons(s(z1), z2)))) K tuples: MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil))) MINSORT(cons(0, cons(y0, y1))) -> c8(MIN(cons(0, cons(y0, y1)))) MINSORT(cons(s(y0), cons(0, y1))) -> c8(MIN(cons(s(y0), cons(0, y1)))) MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c8(MIN(cons(s(s(y0)), cons(s(s(y1)), y2)))) MINSORT(cons(s(y0), cons(s(y1), y2))) -> c8(MIN(cons(s(y0), cons(s(y1), y2)))) MINSORT(cons(s(0), cons(s(y0), y1))) -> c8(MIN(cons(s(0), cons(s(y0), y1)))) MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c8(MIN(cons(s(s(y0)), cons(s(0), y1)))) Defined Rule Symbols: rm_2, min_1, if1_4, le_2, if2_4, eq_2 Defined Pair Symbols: MIN_1, RM_2, MINSORT_1, IF1_4, LE_2, EQ_2, IF2_4 Compound Symbols: c12_1, c16_1, c9_1, c9_2, c12_2, c16_2, c9_3, c13_1, c2_1, c6_1, c14_1, c17_1, c18_1, c8_1 ---------------------------------------- (123) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) by MINSORT(cons(z0, cons(z1, z2))) -> c9(RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))) ---------------------------------------- (124) Obligation: Complexity Dependency Tuples Problem Rules: rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) rm(z0, nil) -> nil min(cons(z0, nil)) -> z0 min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(cons(z0, z2)) if1(false, z0, z1, z2) -> min(cons(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) if2(true, z0, z1, z2) -> rm(z0, z2) if2(false, z0, z1, z2) -> cons(z1, rm(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(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2)) MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2)) RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2)) RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2)) RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2)) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil))) MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) MIN(cons(s(x0), cons(s(x1), x2))) -> c12(LE(s(x0), s(x1))) RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), cons(s(x1), x2)) -> c16(EQ(s(x0), s(x1))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) IF1(true, 0, x0, x1) -> c13(MIN(cons(0, x1))) IF1(true, s(0), s(x0), x1) -> c13(MIN(cons(s(0), x1))) IF1(true, s(s(x0)), s(s(x1)), x2) -> c13(MIN(cons(s(s(x0)), x2))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2)) MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2)) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2)) RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2)) RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2)) IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1))) IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2))) IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) IF2(true, 0, 0, x0) -> c17(RM(0, x0)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2)) IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0)) IF2(false, 0, s(x0), x1) -> c18(RM(0, x1)) IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2)) IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1)) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(z0, cons(z0, nil))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1)))) MINSORT(cons(0, cons(y0, y1))) -> c8(MIN(cons(0, cons(y0, y1)))) MINSORT(cons(s(y0), cons(0, y1))) -> c8(MIN(cons(s(y0), cons(0, y1)))) MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c8(MIN(cons(s(s(y0)), cons(s(s(y1)), y2)))) MINSORT(cons(s(y0), cons(s(y1), y2))) -> c8(MIN(cons(s(y0), cons(s(y1), y2)))) MINSORT(cons(s(0), cons(s(y0), y1))) -> c8(MIN(cons(s(0), cons(s(y0), y1)))) MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c8(MIN(cons(s(s(y0)), cons(s(0), y1)))) MINSORT(cons(s(z0), cons(0, z1))) -> c9(MINSORT(rm(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), RM(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(cons(s(z0), cons(0, z1)))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), RM(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(cons(s(z0), cons(s(z1), z2)))) MINSORT(cons(z0, cons(z1, z2))) -> c9(RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))) S tuples: MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2)) MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2)) RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2)) RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2)) RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2)) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) MIN(cons(s(x0), cons(s(x1), x2))) -> c12(LE(s(x0), s(x1))) RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), cons(s(x1), x2)) -> c16(EQ(s(x0), s(x1))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) IF1(true, 0, x0, x1) -> c13(MIN(cons(0, x1))) IF1(true, s(0), s(x0), x1) -> c13(MIN(cons(s(0), x1))) IF1(true, s(s(x0)), s(s(x1)), x2) -> c13(MIN(cons(s(s(x0)), x2))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2)) MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2)) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2)) RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2)) RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2)) IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1))) IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2))) IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) IF2(true, 0, 0, x0) -> c17(RM(0, x0)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2)) IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0)) IF2(false, 0, s(x0), x1) -> c18(RM(0, x1)) IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2)) IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1)) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(z0, cons(z0, nil))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1)))) MINSORT(cons(s(z0), cons(0, z1))) -> c9(MINSORT(rm(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), RM(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(cons(s(z0), cons(0, z1)))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), RM(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(cons(s(z0), cons(s(z1), z2)))) K tuples: MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil))) MINSORT(cons(0, cons(y0, y1))) -> c8(MIN(cons(0, cons(y0, y1)))) MINSORT(cons(s(y0), cons(0, y1))) -> c8(MIN(cons(s(y0), cons(0, y1)))) MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c8(MIN(cons(s(s(y0)), cons(s(s(y1)), y2)))) MINSORT(cons(s(y0), cons(s(y1), y2))) -> c8(MIN(cons(s(y0), cons(s(y1), y2)))) MINSORT(cons(s(0), cons(s(y0), y1))) -> c8(MIN(cons(s(0), cons(s(y0), y1)))) MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c8(MIN(cons(s(s(y0)), cons(s(0), y1)))) Defined Rule Symbols: rm_2, min_1, if1_4, le_2, if2_4, eq_2 Defined Pair Symbols: MIN_1, RM_2, MINSORT_1, IF1_4, LE_2, EQ_2, IF2_4 Compound Symbols: c12_1, c16_1, c9_2, c9_1, c12_2, c16_2, c9_3, c13_1, c2_1, c6_1, c14_1, c17_1, c18_1, c8_1 ---------------------------------------- (125) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(min(cons(z0, nil)), cons(z0, nil))) by MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(z0, cons(z0, nil))) ---------------------------------------- (126) Obligation: Complexity Dependency Tuples Problem Rules: rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) rm(z0, nil) -> nil min(cons(z0, nil)) -> z0 min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(cons(z0, z2)) if1(false, z0, z1, z2) -> min(cons(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) if2(true, z0, z1, z2) -> rm(z0, z2) if2(false, z0, z1, z2) -> cons(z1, rm(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(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2)) MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2)) RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2)) RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2)) RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2)) MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil))) MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) MIN(cons(s(x0), cons(s(x1), x2))) -> c12(LE(s(x0), s(x1))) RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), cons(s(x1), x2)) -> c16(EQ(s(x0), s(x1))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) IF1(true, 0, x0, x1) -> c13(MIN(cons(0, x1))) IF1(true, s(0), s(x0), x1) -> c13(MIN(cons(s(0), x1))) IF1(true, s(s(x0)), s(s(x1)), x2) -> c13(MIN(cons(s(s(x0)), x2))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2)) MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2)) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2)) RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2)) RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2)) IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1))) IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2))) IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) IF2(true, 0, 0, x0) -> c17(RM(0, x0)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2)) IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0)) IF2(false, 0, s(x0), x1) -> c18(RM(0, x1)) IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2)) IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1)) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(z0, cons(z0, nil))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1)))) MINSORT(cons(0, cons(y0, y1))) -> c8(MIN(cons(0, cons(y0, y1)))) MINSORT(cons(s(y0), cons(0, y1))) -> c8(MIN(cons(s(y0), cons(0, y1)))) MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c8(MIN(cons(s(s(y0)), cons(s(s(y1)), y2)))) MINSORT(cons(s(y0), cons(s(y1), y2))) -> c8(MIN(cons(s(y0), cons(s(y1), y2)))) MINSORT(cons(s(0), cons(s(y0), y1))) -> c8(MIN(cons(s(0), cons(s(y0), y1)))) MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c8(MIN(cons(s(s(y0)), cons(s(0), y1)))) MINSORT(cons(s(z0), cons(0, z1))) -> c9(MINSORT(rm(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), RM(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(cons(s(z0), cons(0, z1)))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), RM(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(cons(s(z0), cons(s(z1), z2)))) MINSORT(cons(z0, cons(z1, z2))) -> c9(RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(z0, cons(z0, nil))) S tuples: MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2)) MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2)) RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2)) RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2)) RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2)) MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) MIN(cons(s(x0), cons(s(x1), x2))) -> c12(LE(s(x0), s(x1))) RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), cons(s(x1), x2)) -> c16(EQ(s(x0), s(x1))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) IF1(true, 0, x0, x1) -> c13(MIN(cons(0, x1))) IF1(true, s(0), s(x0), x1) -> c13(MIN(cons(s(0), x1))) IF1(true, s(s(x0)), s(s(x1)), x2) -> c13(MIN(cons(s(s(x0)), x2))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2)) MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2)) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2)) RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2)) RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2)) IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1))) IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2))) IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) IF2(true, 0, 0, x0) -> c17(RM(0, x0)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2)) IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0)) IF2(false, 0, s(x0), x1) -> c18(RM(0, x1)) IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2)) IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1)) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(z0, cons(z0, nil))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1)))) MINSORT(cons(s(z0), cons(0, z1))) -> c9(MINSORT(rm(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), RM(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(cons(s(z0), cons(0, z1)))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), RM(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(cons(s(z0), cons(s(z1), z2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(z0, cons(z0, nil))) K tuples: MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil))) MINSORT(cons(0, cons(y0, y1))) -> c8(MIN(cons(0, cons(y0, y1)))) MINSORT(cons(s(y0), cons(0, y1))) -> c8(MIN(cons(s(y0), cons(0, y1)))) MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c8(MIN(cons(s(s(y0)), cons(s(s(y1)), y2)))) MINSORT(cons(s(y0), cons(s(y1), y2))) -> c8(MIN(cons(s(y0), cons(s(y1), y2)))) MINSORT(cons(s(0), cons(s(y0), y1))) -> c8(MIN(cons(s(0), cons(s(y0), y1)))) MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c8(MIN(cons(s(s(y0)), cons(s(0), y1)))) Defined Rule Symbols: rm_2, min_1, if1_4, le_2, if2_4, eq_2 Defined Pair Symbols: MIN_1, RM_2, MINSORT_1, IF1_4, LE_2, EQ_2, IF2_4 Compound Symbols: c12_1, c16_1, c9_1, c12_2, c16_2, c9_3, c13_1, c2_1, c6_1, c14_1, c17_1, c18_1, c9_2, c8_1 ---------------------------------------- (127) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil))) by MINSORT(cons(z0, nil)) -> c9(RM(z0, cons(z0, nil))) ---------------------------------------- (128) Obligation: Complexity Dependency Tuples Problem Rules: rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) rm(z0, nil) -> nil min(cons(z0, nil)) -> z0 min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(cons(z0, z2)) if1(false, z0, z1, z2) -> min(cons(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) if2(true, z0, z1, z2) -> rm(z0, z2) if2(false, z0, z1, z2) -> cons(z1, rm(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(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2)) MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2)) RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2)) RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2)) RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2)) MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) MIN(cons(s(x0), cons(s(x1), x2))) -> c12(LE(s(x0), s(x1))) RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), cons(s(x1), x2)) -> c16(EQ(s(x0), s(x1))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) IF1(true, 0, x0, x1) -> c13(MIN(cons(0, x1))) IF1(true, s(0), s(x0), x1) -> c13(MIN(cons(s(0), x1))) IF1(true, s(s(x0)), s(s(x1)), x2) -> c13(MIN(cons(s(s(x0)), x2))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2)) MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2)) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2)) RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2)) RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2)) IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1))) IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2))) IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) IF2(true, 0, 0, x0) -> c17(RM(0, x0)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2)) IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0)) IF2(false, 0, s(x0), x1) -> c18(RM(0, x1)) IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2)) IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1)) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(z0, cons(z0, nil))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1)))) MINSORT(cons(0, cons(y0, y1))) -> c8(MIN(cons(0, cons(y0, y1)))) MINSORT(cons(s(y0), cons(0, y1))) -> c8(MIN(cons(s(y0), cons(0, y1)))) MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c8(MIN(cons(s(s(y0)), cons(s(s(y1)), y2)))) MINSORT(cons(s(y0), cons(s(y1), y2))) -> c8(MIN(cons(s(y0), cons(s(y1), y2)))) MINSORT(cons(s(0), cons(s(y0), y1))) -> c8(MIN(cons(s(0), cons(s(y0), y1)))) MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c8(MIN(cons(s(s(y0)), cons(s(0), y1)))) MINSORT(cons(s(z0), cons(0, z1))) -> c9(MINSORT(rm(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), RM(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(cons(s(z0), cons(0, z1)))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), RM(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(cons(s(z0), cons(s(z1), z2)))) MINSORT(cons(z0, cons(z1, z2))) -> c9(RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c9(RM(z0, cons(z0, nil))) S tuples: MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2)) MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2)) RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2)) RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2)) RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2)) MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) MIN(cons(s(x0), cons(s(x1), x2))) -> c12(LE(s(x0), s(x1))) RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), cons(s(x1), x2)) -> c16(EQ(s(x0), s(x1))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) IF1(true, 0, x0, x1) -> c13(MIN(cons(0, x1))) IF1(true, s(0), s(x0), x1) -> c13(MIN(cons(s(0), x1))) IF1(true, s(s(x0)), s(s(x1)), x2) -> c13(MIN(cons(s(s(x0)), x2))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2)) MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2)) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2)) RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2)) RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2)) IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1))) IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2))) IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) IF2(true, 0, 0, x0) -> c17(RM(0, x0)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2)) IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0)) IF2(false, 0, s(x0), x1) -> c18(RM(0, x1)) IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2)) IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1)) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(z0, cons(z0, nil))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1)))) MINSORT(cons(s(z0), cons(0, z1))) -> c9(MINSORT(rm(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), RM(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(cons(s(z0), cons(0, z1)))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), RM(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(cons(s(z0), cons(s(z1), z2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(z0, cons(z0, nil))) K tuples: MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil))) MINSORT(cons(0, cons(y0, y1))) -> c8(MIN(cons(0, cons(y0, y1)))) MINSORT(cons(s(y0), cons(0, y1))) -> c8(MIN(cons(s(y0), cons(0, y1)))) MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c8(MIN(cons(s(s(y0)), cons(s(s(y1)), y2)))) MINSORT(cons(s(y0), cons(s(y1), y2))) -> c8(MIN(cons(s(y0), cons(s(y1), y2)))) MINSORT(cons(s(0), cons(s(y0), y1))) -> c8(MIN(cons(s(0), cons(s(y0), y1)))) MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c8(MIN(cons(s(s(y0)), cons(s(0), y1)))) Defined Rule Symbols: rm_2, min_1, if1_4, le_2, if2_4, eq_2 Defined Pair Symbols: MIN_1, RM_2, MINSORT_1, IF1_4, LE_2, EQ_2, IF2_4 Compound Symbols: c12_1, c16_1, c12_2, c16_2, c9_3, c13_1, c2_1, c6_1, c14_1, c17_1, c18_1, c9_2, c8_1, c9_1 ---------------------------------------- (129) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(min(cons(z0, cons(z1, z2))), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) by MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) ---------------------------------------- (130) Obligation: Complexity Dependency Tuples Problem Rules: rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) rm(z0, nil) -> nil min(cons(z0, nil)) -> z0 min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(cons(z0, z2)) if1(false, z0, z1, z2) -> min(cons(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) if2(true, z0, z1, z2) -> rm(z0, z2) if2(false, z0, z1, z2) -> cons(z1, rm(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(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2)) MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2)) RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2)) RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2)) RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2)) MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) MIN(cons(s(x0), cons(s(x1), x2))) -> c12(LE(s(x0), s(x1))) RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), cons(s(x1), x2)) -> c16(EQ(s(x0), s(x1))) IF1(true, 0, x0, x1) -> c13(MIN(cons(0, x1))) IF1(true, s(0), s(x0), x1) -> c13(MIN(cons(s(0), x1))) IF1(true, s(s(x0)), s(s(x1)), x2) -> c13(MIN(cons(s(s(x0)), x2))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2)) MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2)) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2)) RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2)) RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2)) IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1))) IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2))) IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) IF2(true, 0, 0, x0) -> c17(RM(0, x0)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2)) IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0)) IF2(false, 0, s(x0), x1) -> c18(RM(0, x1)) IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2)) IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1)) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(z0, cons(z0, nil))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1)))) MINSORT(cons(0, cons(y0, y1))) -> c8(MIN(cons(0, cons(y0, y1)))) MINSORT(cons(s(y0), cons(0, y1))) -> c8(MIN(cons(s(y0), cons(0, y1)))) MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c8(MIN(cons(s(s(y0)), cons(s(s(y1)), y2)))) MINSORT(cons(s(y0), cons(s(y1), y2))) -> c8(MIN(cons(s(y0), cons(s(y1), y2)))) MINSORT(cons(s(0), cons(s(y0), y1))) -> c8(MIN(cons(s(0), cons(s(y0), y1)))) MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c8(MIN(cons(s(s(y0)), cons(s(0), y1)))) MINSORT(cons(s(z0), cons(0, z1))) -> c9(MINSORT(rm(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), RM(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(cons(s(z0), cons(0, z1)))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), RM(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(cons(s(z0), cons(s(z1), z2)))) MINSORT(cons(z0, cons(z1, z2))) -> c9(RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c9(RM(z0, cons(z0, nil))) S tuples: MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2)) MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2)) RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2)) RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2)) RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2)) MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) MIN(cons(s(x0), cons(s(x1), x2))) -> c12(LE(s(x0), s(x1))) RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), cons(s(x1), x2)) -> c16(EQ(s(x0), s(x1))) IF1(true, 0, x0, x1) -> c13(MIN(cons(0, x1))) IF1(true, s(0), s(x0), x1) -> c13(MIN(cons(s(0), x1))) IF1(true, s(s(x0)), s(s(x1)), x2) -> c13(MIN(cons(s(s(x0)), x2))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2)) MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2)) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2)) RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2)) RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2)) IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1))) IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2))) IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) IF2(true, 0, 0, x0) -> c17(RM(0, x0)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2)) IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0)) IF2(false, 0, s(x0), x1) -> c18(RM(0, x1)) IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2)) IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1)) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(z0, cons(z0, nil))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1)))) MINSORT(cons(s(z0), cons(0, z1))) -> c9(MINSORT(rm(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), RM(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(cons(s(z0), cons(0, z1)))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), RM(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(cons(s(z0), cons(s(z1), z2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(z0, cons(z0, nil))) K tuples: MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil))) MINSORT(cons(0, cons(y0, y1))) -> c8(MIN(cons(0, cons(y0, y1)))) MINSORT(cons(s(y0), cons(0, y1))) -> c8(MIN(cons(s(y0), cons(0, y1)))) MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c8(MIN(cons(s(s(y0)), cons(s(s(y1)), y2)))) MINSORT(cons(s(y0), cons(s(y1), y2))) -> c8(MIN(cons(s(y0), cons(s(y1), y2)))) MINSORT(cons(s(0), cons(s(y0), y1))) -> c8(MIN(cons(s(0), cons(s(y0), y1)))) MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c8(MIN(cons(s(s(y0)), cons(s(0), y1)))) Defined Rule Symbols: rm_2, min_1, if1_4, le_2, if2_4, eq_2 Defined Pair Symbols: MIN_1, RM_2, IF1_4, LE_2, EQ_2, MINSORT_1, IF2_4 Compound Symbols: c12_1, c16_1, c12_2, c16_2, c13_1, c2_1, c6_1, c14_1, c9_3, c17_1, c18_1, c9_2, c8_1, c9_1 ---------------------------------------- (131) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace MIN(cons(s(x0), cons(s(x1), x2))) -> c12(LE(s(x0), s(x1))) by MIN(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c12(LE(s(s(y0)), s(s(y1)))) ---------------------------------------- (132) Obligation: Complexity Dependency Tuples Problem Rules: rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) rm(z0, nil) -> nil min(cons(z0, nil)) -> z0 min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(cons(z0, z2)) if1(false, z0, z1, z2) -> min(cons(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) if2(true, z0, z1, z2) -> rm(z0, z2) if2(false, z0, z1, z2) -> cons(z1, rm(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(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2)) MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2)) RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2)) RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2)) RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2)) MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), cons(s(x1), x2)) -> c16(EQ(s(x0), s(x1))) IF1(true, 0, x0, x1) -> c13(MIN(cons(0, x1))) IF1(true, s(0), s(x0), x1) -> c13(MIN(cons(s(0), x1))) IF1(true, s(s(x0)), s(s(x1)), x2) -> c13(MIN(cons(s(s(x0)), x2))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2)) MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2)) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2)) RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2)) RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2)) IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1))) IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2))) IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) IF2(true, 0, 0, x0) -> c17(RM(0, x0)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2)) IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0)) IF2(false, 0, s(x0), x1) -> c18(RM(0, x1)) IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2)) IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1)) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(z0, cons(z0, nil))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1)))) MINSORT(cons(0, cons(y0, y1))) -> c8(MIN(cons(0, cons(y0, y1)))) MINSORT(cons(s(y0), cons(0, y1))) -> c8(MIN(cons(s(y0), cons(0, y1)))) MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c8(MIN(cons(s(s(y0)), cons(s(s(y1)), y2)))) MINSORT(cons(s(y0), cons(s(y1), y2))) -> c8(MIN(cons(s(y0), cons(s(y1), y2)))) MINSORT(cons(s(0), cons(s(y0), y1))) -> c8(MIN(cons(s(0), cons(s(y0), y1)))) MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c8(MIN(cons(s(s(y0)), cons(s(0), y1)))) MINSORT(cons(s(z0), cons(0, z1))) -> c9(MINSORT(rm(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), RM(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(cons(s(z0), cons(0, z1)))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), RM(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(cons(s(z0), cons(s(z1), z2)))) MINSORT(cons(z0, cons(z1, z2))) -> c9(RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c9(RM(z0, cons(z0, nil))) MIN(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c12(LE(s(s(y0)), s(s(y1)))) S tuples: MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2)) MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2)) RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2)) RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2)) RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2)) MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) RM(s(x0), cons(s(x1), x2)) -> c16(EQ(s(x0), s(x1))) IF1(true, 0, x0, x1) -> c13(MIN(cons(0, x1))) IF1(true, s(0), s(x0), x1) -> c13(MIN(cons(s(0), x1))) IF1(true, s(s(x0)), s(s(x1)), x2) -> c13(MIN(cons(s(s(x0)), x2))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2)) MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2)) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2)) RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2)) RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2)) IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1))) IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2))) IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) IF2(true, 0, 0, x0) -> c17(RM(0, x0)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2)) IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0)) IF2(false, 0, s(x0), x1) -> c18(RM(0, x1)) IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2)) IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1)) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(z0, cons(z0, nil))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1)))) MINSORT(cons(s(z0), cons(0, z1))) -> c9(MINSORT(rm(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), RM(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(cons(s(z0), cons(0, z1)))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), RM(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(cons(s(z0), cons(s(z1), z2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(z0, cons(z0, nil))) MIN(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c12(LE(s(s(y0)), s(s(y1)))) K tuples: MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil))) MINSORT(cons(0, cons(y0, y1))) -> c8(MIN(cons(0, cons(y0, y1)))) MINSORT(cons(s(y0), cons(0, y1))) -> c8(MIN(cons(s(y0), cons(0, y1)))) MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c8(MIN(cons(s(s(y0)), cons(s(s(y1)), y2)))) MINSORT(cons(s(y0), cons(s(y1), y2))) -> c8(MIN(cons(s(y0), cons(s(y1), y2)))) MINSORT(cons(s(0), cons(s(y0), y1))) -> c8(MIN(cons(s(0), cons(s(y0), y1)))) MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c8(MIN(cons(s(s(y0)), cons(s(0), y1)))) Defined Rule Symbols: rm_2, min_1, if1_4, le_2, if2_4, eq_2 Defined Pair Symbols: MIN_1, RM_2, IF1_4, LE_2, EQ_2, MINSORT_1, IF2_4 Compound Symbols: c12_1, c16_1, c12_2, c16_2, c13_1, c2_1, c6_1, c14_1, c9_3, c17_1, c18_1, c9_2, c8_1, c9_1 ---------------------------------------- (133) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace RM(s(x0), cons(s(x1), x2)) -> c16(EQ(s(x0), s(x1))) by RM(s(s(y0)), cons(s(s(y1)), z2)) -> c16(EQ(s(s(y0)), s(s(y1)))) ---------------------------------------- (134) Obligation: Complexity Dependency Tuples Problem Rules: rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) rm(z0, nil) -> nil min(cons(z0, nil)) -> z0 min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(cons(z0, z2)) if1(false, z0, z1, z2) -> min(cons(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) if2(true, z0, z1, z2) -> rm(z0, z2) if2(false, z0, z1, z2) -> cons(z1, rm(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(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2)) MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2)) RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2)) RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2)) RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2)) MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) IF1(true, 0, x0, x1) -> c13(MIN(cons(0, x1))) IF1(true, s(0), s(x0), x1) -> c13(MIN(cons(s(0), x1))) IF1(true, s(s(x0)), s(s(x1)), x2) -> c13(MIN(cons(s(s(x0)), x2))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2)) MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2)) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2)) RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2)) RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2)) IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1))) IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2))) IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) IF2(true, 0, 0, x0) -> c17(RM(0, x0)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2)) IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0)) IF2(false, 0, s(x0), x1) -> c18(RM(0, x1)) IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2)) IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1)) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(z0, cons(z0, nil))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1)))) MINSORT(cons(0, cons(y0, y1))) -> c8(MIN(cons(0, cons(y0, y1)))) MINSORT(cons(s(y0), cons(0, y1))) -> c8(MIN(cons(s(y0), cons(0, y1)))) MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c8(MIN(cons(s(s(y0)), cons(s(s(y1)), y2)))) MINSORT(cons(s(y0), cons(s(y1), y2))) -> c8(MIN(cons(s(y0), cons(s(y1), y2)))) MINSORT(cons(s(0), cons(s(y0), y1))) -> c8(MIN(cons(s(0), cons(s(y0), y1)))) MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c8(MIN(cons(s(s(y0)), cons(s(0), y1)))) MINSORT(cons(s(z0), cons(0, z1))) -> c9(MINSORT(rm(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), RM(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(cons(s(z0), cons(0, z1)))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), RM(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(cons(s(z0), cons(s(z1), z2)))) MINSORT(cons(z0, cons(z1, z2))) -> c9(RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c9(RM(z0, cons(z0, nil))) MIN(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c12(LE(s(s(y0)), s(s(y1)))) RM(s(s(y0)), cons(s(s(y1)), z2)) -> c16(EQ(s(s(y0)), s(s(y1)))) S tuples: MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2)) MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2)) RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2)) RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2)) RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2)) MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) IF1(true, 0, x0, x1) -> c13(MIN(cons(0, x1))) IF1(true, s(0), s(x0), x1) -> c13(MIN(cons(s(0), x1))) IF1(true, s(s(x0)), s(s(x1)), x2) -> c13(MIN(cons(s(s(x0)), x2))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2)) MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2)) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2)) RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2)) RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2)) IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1))) IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2))) IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) IF2(true, 0, 0, x0) -> c17(RM(0, x0)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2)) IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0)) IF2(false, 0, s(x0), x1) -> c18(RM(0, x1)) IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2)) IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1)) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(z0, cons(z0, nil))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1)))) MINSORT(cons(s(z0), cons(0, z1))) -> c9(MINSORT(rm(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), RM(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(cons(s(z0), cons(0, z1)))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), RM(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(cons(s(z0), cons(s(z1), z2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(z0, cons(z0, nil))) MIN(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c12(LE(s(s(y0)), s(s(y1)))) RM(s(s(y0)), cons(s(s(y1)), z2)) -> c16(EQ(s(s(y0)), s(s(y1)))) K tuples: MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil))) MINSORT(cons(0, cons(y0, y1))) -> c8(MIN(cons(0, cons(y0, y1)))) MINSORT(cons(s(y0), cons(0, y1))) -> c8(MIN(cons(s(y0), cons(0, y1)))) MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c8(MIN(cons(s(s(y0)), cons(s(s(y1)), y2)))) MINSORT(cons(s(y0), cons(s(y1), y2))) -> c8(MIN(cons(s(y0), cons(s(y1), y2)))) MINSORT(cons(s(0), cons(s(y0), y1))) -> c8(MIN(cons(s(0), cons(s(y0), y1)))) MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c8(MIN(cons(s(s(y0)), cons(s(0), y1)))) Defined Rule Symbols: rm_2, min_1, if1_4, le_2, if2_4, eq_2 Defined Pair Symbols: MIN_1, RM_2, IF1_4, LE_2, EQ_2, MINSORT_1, IF2_4 Compound Symbols: c12_1, c16_1, c12_2, c16_2, c13_1, c2_1, c6_1, c14_1, c9_3, c17_1, c18_1, c9_2, c8_1, c9_1 ---------------------------------------- (135) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace IF1(true, 0, x0, x1) -> c13(MIN(cons(0, x1))) by IF1(true, 0, z0, cons(y0, y1)) -> c13(MIN(cons(0, cons(y0, y1)))) ---------------------------------------- (136) Obligation: Complexity Dependency Tuples Problem Rules: rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) rm(z0, nil) -> nil min(cons(z0, nil)) -> z0 min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(cons(z0, z2)) if1(false, z0, z1, z2) -> min(cons(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) if2(true, z0, z1, z2) -> rm(z0, z2) if2(false, z0, z1, z2) -> cons(z1, rm(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(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2)) MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2)) RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2)) RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2)) RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2)) MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(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) -> c13(MIN(cons(s(0), x1))) IF1(true, s(s(x0)), s(s(x1)), x2) -> c13(MIN(cons(s(s(x0)), x2))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2)) MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2)) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2)) RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2)) RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2)) IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1))) IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2))) IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) IF2(true, 0, 0, x0) -> c17(RM(0, x0)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2)) IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0)) IF2(false, 0, s(x0), x1) -> c18(RM(0, x1)) IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2)) IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1)) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(z0, cons(z0, nil))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1)))) MINSORT(cons(0, cons(y0, y1))) -> c8(MIN(cons(0, cons(y0, y1)))) MINSORT(cons(s(y0), cons(0, y1))) -> c8(MIN(cons(s(y0), cons(0, y1)))) MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c8(MIN(cons(s(s(y0)), cons(s(s(y1)), y2)))) MINSORT(cons(s(y0), cons(s(y1), y2))) -> c8(MIN(cons(s(y0), cons(s(y1), y2)))) MINSORT(cons(s(0), cons(s(y0), y1))) -> c8(MIN(cons(s(0), cons(s(y0), y1)))) MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c8(MIN(cons(s(s(y0)), cons(s(0), y1)))) MINSORT(cons(s(z0), cons(0, z1))) -> c9(MINSORT(rm(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), RM(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(cons(s(z0), cons(0, z1)))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), RM(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(cons(s(z0), cons(s(z1), z2)))) MINSORT(cons(z0, cons(z1, z2))) -> c9(RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c9(RM(z0, cons(z0, nil))) MIN(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c12(LE(s(s(y0)), s(s(y1)))) RM(s(s(y0)), cons(s(s(y1)), z2)) -> c16(EQ(s(s(y0)), s(s(y1)))) IF1(true, 0, z0, cons(y0, y1)) -> c13(MIN(cons(0, cons(y0, y1)))) S tuples: MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2)) MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2)) RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2)) RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2)) RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2)) MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(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) -> c13(MIN(cons(s(0), x1))) IF1(true, s(s(x0)), s(s(x1)), x2) -> c13(MIN(cons(s(s(x0)), x2))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2)) MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2)) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2)) RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2)) RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2)) IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1))) IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2))) IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) IF2(true, 0, 0, x0) -> c17(RM(0, x0)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2)) IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0)) IF2(false, 0, s(x0), x1) -> c18(RM(0, x1)) IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2)) IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1)) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(z0, cons(z0, nil))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1)))) MINSORT(cons(s(z0), cons(0, z1))) -> c9(MINSORT(rm(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), RM(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(cons(s(z0), cons(0, z1)))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), RM(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(cons(s(z0), cons(s(z1), z2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(z0, cons(z0, nil))) MIN(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c12(LE(s(s(y0)), s(s(y1)))) RM(s(s(y0)), cons(s(s(y1)), z2)) -> c16(EQ(s(s(y0)), s(s(y1)))) IF1(true, 0, z0, cons(y0, y1)) -> c13(MIN(cons(0, cons(y0, y1)))) K tuples: MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil))) MINSORT(cons(0, cons(y0, y1))) -> c8(MIN(cons(0, cons(y0, y1)))) MINSORT(cons(s(y0), cons(0, y1))) -> c8(MIN(cons(s(y0), cons(0, y1)))) MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c8(MIN(cons(s(s(y0)), cons(s(s(y1)), y2)))) MINSORT(cons(s(y0), cons(s(y1), y2))) -> c8(MIN(cons(s(y0), cons(s(y1), y2)))) MINSORT(cons(s(0), cons(s(y0), y1))) -> c8(MIN(cons(s(0), cons(s(y0), y1)))) MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c8(MIN(cons(s(s(y0)), cons(s(0), y1)))) Defined Rule Symbols: rm_2, min_1, if1_4, le_2, if2_4, eq_2 Defined Pair Symbols: MIN_1, RM_2, IF1_4, LE_2, EQ_2, MINSORT_1, IF2_4 Compound Symbols: c12_1, c16_1, c12_2, c16_2, c13_1, c2_1, c6_1, c14_1, c9_3, c17_1, c18_1, c9_2, c8_1, c9_1 ---------------------------------------- (137) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace MIN(cons(0, cons(z0, x2))) -> c12(IF1(true, 0, z0, x2)) by MIN(cons(0, cons(z0, cons(y1, y2)))) -> c12(IF1(true, 0, z0, cons(y1, y2))) ---------------------------------------- (138) Obligation: Complexity Dependency Tuples Problem Rules: rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) rm(z0, nil) -> nil min(cons(z0, nil)) -> z0 min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(cons(z0, z2)) if1(false, z0, z1, z2) -> min(cons(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) if2(true, z0, z1, z2) -> rm(z0, z2) if2(false, z0, z1, z2) -> cons(z1, rm(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(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2)) RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2)) RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2)) RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2)) MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(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) -> c13(MIN(cons(s(0), x1))) IF1(true, s(s(x0)), s(s(x1)), x2) -> c13(MIN(cons(s(s(x0)), x2))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2)) MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2)) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2)) RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2)) RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2)) IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1))) IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2))) IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) IF2(true, 0, 0, x0) -> c17(RM(0, x0)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2)) IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0)) IF2(false, 0, s(x0), x1) -> c18(RM(0, x1)) IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2)) IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1)) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(z0, cons(z0, nil))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1)))) MINSORT(cons(0, cons(y0, y1))) -> c8(MIN(cons(0, cons(y0, y1)))) MINSORT(cons(s(y0), cons(0, y1))) -> c8(MIN(cons(s(y0), cons(0, y1)))) MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c8(MIN(cons(s(s(y0)), cons(s(s(y1)), y2)))) MINSORT(cons(s(y0), cons(s(y1), y2))) -> c8(MIN(cons(s(y0), cons(s(y1), y2)))) MINSORT(cons(s(0), cons(s(y0), y1))) -> c8(MIN(cons(s(0), cons(s(y0), y1)))) MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c8(MIN(cons(s(s(y0)), cons(s(0), y1)))) MINSORT(cons(s(z0), cons(0, z1))) -> c9(MINSORT(rm(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), RM(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(cons(s(z0), cons(0, z1)))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), RM(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(cons(s(z0), cons(s(z1), z2)))) MINSORT(cons(z0, cons(z1, z2))) -> c9(RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c9(RM(z0, cons(z0, nil))) MIN(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c12(LE(s(s(y0)), s(s(y1)))) RM(s(s(y0)), cons(s(s(y1)), z2)) -> c16(EQ(s(s(y0)), s(s(y1)))) IF1(true, 0, z0, cons(y0, y1)) -> c13(MIN(cons(0, cons(y0, y1)))) MIN(cons(0, cons(z0, cons(y1, y2)))) -> c12(IF1(true, 0, z0, cons(y1, y2))) S tuples: MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2)) RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2)) RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2)) RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2)) MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(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) -> c13(MIN(cons(s(0), x1))) IF1(true, s(s(x0)), s(s(x1)), x2) -> c13(MIN(cons(s(s(x0)), x2))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2)) MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2)) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2)) RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2)) RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2)) IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1))) IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2))) IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) IF2(true, 0, 0, x0) -> c17(RM(0, x0)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2)) IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0)) IF2(false, 0, s(x0), x1) -> c18(RM(0, x1)) IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2)) IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1)) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(z0, cons(z0, nil))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1)))) MINSORT(cons(s(z0), cons(0, z1))) -> c9(MINSORT(rm(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), RM(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(cons(s(z0), cons(0, z1)))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), RM(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(cons(s(z0), cons(s(z1), z2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(z0, cons(z0, nil))) MIN(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c12(LE(s(s(y0)), s(s(y1)))) RM(s(s(y0)), cons(s(s(y1)), z2)) -> c16(EQ(s(s(y0)), s(s(y1)))) IF1(true, 0, z0, cons(y0, y1)) -> c13(MIN(cons(0, cons(y0, y1)))) MIN(cons(0, cons(z0, cons(y1, y2)))) -> c12(IF1(true, 0, z0, cons(y1, y2))) K tuples: MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil))) MINSORT(cons(0, cons(y0, y1))) -> c8(MIN(cons(0, cons(y0, y1)))) MINSORT(cons(s(y0), cons(0, y1))) -> c8(MIN(cons(s(y0), cons(0, y1)))) MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c8(MIN(cons(s(s(y0)), cons(s(s(y1)), y2)))) MINSORT(cons(s(y0), cons(s(y1), y2))) -> c8(MIN(cons(s(y0), cons(s(y1), y2)))) MINSORT(cons(s(0), cons(s(y0), y1))) -> c8(MIN(cons(s(0), cons(s(y0), y1)))) MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c8(MIN(cons(s(s(y0)), cons(s(0), y1)))) Defined Rule Symbols: rm_2, min_1, if1_4, le_2, if2_4, eq_2 Defined Pair Symbols: MIN_1, RM_2, IF1_4, LE_2, EQ_2, MINSORT_1, IF2_4 Compound Symbols: c12_1, c16_1, c12_2, c16_2, c13_1, c2_1, c6_1, c14_1, c9_3, c17_1, c18_1, c9_2, c8_1, c9_1 ---------------------------------------- (139) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace IF1(true, s(0), s(x0), x1) -> c13(MIN(cons(s(0), x1))) by IF1(true, s(0), s(z0), cons(0, y1)) -> c13(MIN(cons(s(0), cons(0, y1)))) IF1(true, s(0), s(z0), cons(s(y0), y1)) -> c13(MIN(cons(s(0), cons(s(y0), y1)))) ---------------------------------------- (140) Obligation: Complexity Dependency Tuples Problem Rules: rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) rm(z0, nil) -> nil min(cons(z0, nil)) -> z0 min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(cons(z0, z2)) if1(false, z0, z1, z2) -> min(cons(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) if2(true, z0, z1, z2) -> rm(z0, z2) if2(false, z0, z1, z2) -> cons(z1, rm(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(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2)) RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2)) RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2)) RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2)) MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(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) -> c13(MIN(cons(s(s(x0)), x2))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2)) MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2)) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2)) RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2)) RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2)) IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1))) IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2))) IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) IF2(true, 0, 0, x0) -> c17(RM(0, x0)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2)) IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0)) IF2(false, 0, s(x0), x1) -> c18(RM(0, x1)) IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2)) IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1)) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(z0, cons(z0, nil))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1)))) MINSORT(cons(0, cons(y0, y1))) -> c8(MIN(cons(0, cons(y0, y1)))) MINSORT(cons(s(y0), cons(0, y1))) -> c8(MIN(cons(s(y0), cons(0, y1)))) MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c8(MIN(cons(s(s(y0)), cons(s(s(y1)), y2)))) MINSORT(cons(s(y0), cons(s(y1), y2))) -> c8(MIN(cons(s(y0), cons(s(y1), y2)))) MINSORT(cons(s(0), cons(s(y0), y1))) -> c8(MIN(cons(s(0), cons(s(y0), y1)))) MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c8(MIN(cons(s(s(y0)), cons(s(0), y1)))) MINSORT(cons(s(z0), cons(0, z1))) -> c9(MINSORT(rm(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), RM(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(cons(s(z0), cons(0, z1)))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), RM(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(cons(s(z0), cons(s(z1), z2)))) MINSORT(cons(z0, cons(z1, z2))) -> c9(RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c9(RM(z0, cons(z0, nil))) MIN(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c12(LE(s(s(y0)), s(s(y1)))) RM(s(s(y0)), cons(s(s(y1)), z2)) -> c16(EQ(s(s(y0)), s(s(y1)))) IF1(true, 0, z0, cons(y0, y1)) -> c13(MIN(cons(0, cons(y0, y1)))) MIN(cons(0, cons(z0, cons(y1, y2)))) -> c12(IF1(true, 0, z0, cons(y1, y2))) IF1(true, s(0), s(z0), cons(0, y1)) -> c13(MIN(cons(s(0), cons(0, y1)))) IF1(true, s(0), s(z0), cons(s(y0), y1)) -> c13(MIN(cons(s(0), cons(s(y0), y1)))) S tuples: MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2)) RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2)) RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2)) RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2)) MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(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) -> c13(MIN(cons(s(s(x0)), x2))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2)) MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2)) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2)) RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2)) RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2)) IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1))) IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2))) IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) IF2(true, 0, 0, x0) -> c17(RM(0, x0)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2)) IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0)) IF2(false, 0, s(x0), x1) -> c18(RM(0, x1)) IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2)) IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1)) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(z0, cons(z0, nil))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1)))) MINSORT(cons(s(z0), cons(0, z1))) -> c9(MINSORT(rm(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), RM(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(cons(s(z0), cons(0, z1)))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), RM(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(cons(s(z0), cons(s(z1), z2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(z0, cons(z0, nil))) MIN(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c12(LE(s(s(y0)), s(s(y1)))) RM(s(s(y0)), cons(s(s(y1)), z2)) -> c16(EQ(s(s(y0)), s(s(y1)))) IF1(true, 0, z0, cons(y0, y1)) -> c13(MIN(cons(0, cons(y0, y1)))) MIN(cons(0, cons(z0, cons(y1, y2)))) -> c12(IF1(true, 0, z0, cons(y1, y2))) IF1(true, s(0), s(z0), cons(0, y1)) -> c13(MIN(cons(s(0), cons(0, y1)))) IF1(true, s(0), s(z0), cons(s(y0), y1)) -> c13(MIN(cons(s(0), cons(s(y0), y1)))) K tuples: MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil))) MINSORT(cons(0, cons(y0, y1))) -> c8(MIN(cons(0, cons(y0, y1)))) MINSORT(cons(s(y0), cons(0, y1))) -> c8(MIN(cons(s(y0), cons(0, y1)))) MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c8(MIN(cons(s(s(y0)), cons(s(s(y1)), y2)))) MINSORT(cons(s(y0), cons(s(y1), y2))) -> c8(MIN(cons(s(y0), cons(s(y1), y2)))) MINSORT(cons(s(0), cons(s(y0), y1))) -> c8(MIN(cons(s(0), cons(s(y0), y1)))) MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c8(MIN(cons(s(s(y0)), cons(s(0), y1)))) Defined Rule Symbols: rm_2, min_1, if1_4, le_2, if2_4, eq_2 Defined Pair Symbols: MIN_1, RM_2, IF1_4, LE_2, EQ_2, MINSORT_1, IF2_4 Compound Symbols: c12_1, c16_1, c12_2, c16_2, c13_1, c2_1, c6_1, c14_1, c9_3, c17_1, c18_1, c9_2, c8_1, c9_1 ---------------------------------------- (141) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace IF1(true, s(s(x0)), s(s(x1)), x2) -> c13(MIN(cons(s(s(x0)), x2))) by IF1(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c13(MIN(cons(s(s(z0)), cons(0, y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c13(MIN(cons(s(s(z0)), cons(s(s(y1)), y2)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c13(MIN(cons(s(s(z0)), cons(s(0), y1)))) ---------------------------------------- (142) Obligation: Complexity Dependency Tuples Problem Rules: rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) rm(z0, nil) -> nil min(cons(z0, nil)) -> z0 min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(cons(z0, z2)) if1(false, z0, z1, z2) -> min(cons(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) if2(true, z0, z1, z2) -> rm(z0, z2) if2(false, z0, z1, z2) -> cons(z1, rm(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(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2)) RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2)) RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2)) RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2)) MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2)) MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2)) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2)) RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2)) RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2)) IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1))) IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2))) IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) IF2(true, 0, 0, x0) -> c17(RM(0, x0)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2)) IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0)) IF2(false, 0, s(x0), x1) -> c18(RM(0, x1)) IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2)) IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1)) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(z0, cons(z0, nil))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1)))) MINSORT(cons(0, cons(y0, y1))) -> c8(MIN(cons(0, cons(y0, y1)))) MINSORT(cons(s(y0), cons(0, y1))) -> c8(MIN(cons(s(y0), cons(0, y1)))) MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c8(MIN(cons(s(s(y0)), cons(s(s(y1)), y2)))) MINSORT(cons(s(y0), cons(s(y1), y2))) -> c8(MIN(cons(s(y0), cons(s(y1), y2)))) MINSORT(cons(s(0), cons(s(y0), y1))) -> c8(MIN(cons(s(0), cons(s(y0), y1)))) MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c8(MIN(cons(s(s(y0)), cons(s(0), y1)))) MINSORT(cons(s(z0), cons(0, z1))) -> c9(MINSORT(rm(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), RM(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(cons(s(z0), cons(0, z1)))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), RM(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(cons(s(z0), cons(s(z1), z2)))) MINSORT(cons(z0, cons(z1, z2))) -> c9(RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c9(RM(z0, cons(z0, nil))) MIN(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c12(LE(s(s(y0)), s(s(y1)))) RM(s(s(y0)), cons(s(s(y1)), z2)) -> c16(EQ(s(s(y0)), s(s(y1)))) IF1(true, 0, z0, cons(y0, y1)) -> c13(MIN(cons(0, cons(y0, y1)))) MIN(cons(0, cons(z0, cons(y1, y2)))) -> c12(IF1(true, 0, z0, cons(y1, y2))) IF1(true, s(0), s(z0), cons(0, y1)) -> c13(MIN(cons(s(0), cons(0, y1)))) IF1(true, s(0), s(z0), cons(s(y0), y1)) -> c13(MIN(cons(s(0), cons(s(y0), y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c13(MIN(cons(s(s(z0)), cons(0, y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c13(MIN(cons(s(s(z0)), cons(s(s(y1)), y2)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c13(MIN(cons(s(s(z0)), cons(s(0), y1)))) S tuples: MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2)) RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2)) RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2)) RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2)) MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2)) MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2)) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2)) RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2)) RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2)) IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1))) IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2))) IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) IF2(true, 0, 0, x0) -> c17(RM(0, x0)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2)) IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0)) IF2(false, 0, s(x0), x1) -> c18(RM(0, x1)) IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2)) IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1)) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(z0, cons(z0, nil))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1)))) MINSORT(cons(s(z0), cons(0, z1))) -> c9(MINSORT(rm(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), RM(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(cons(s(z0), cons(0, z1)))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), RM(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(cons(s(z0), cons(s(z1), z2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(z0, cons(z0, nil))) MIN(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c12(LE(s(s(y0)), s(s(y1)))) RM(s(s(y0)), cons(s(s(y1)), z2)) -> c16(EQ(s(s(y0)), s(s(y1)))) IF1(true, 0, z0, cons(y0, y1)) -> c13(MIN(cons(0, cons(y0, y1)))) MIN(cons(0, cons(z0, cons(y1, y2)))) -> c12(IF1(true, 0, z0, cons(y1, y2))) IF1(true, s(0), s(z0), cons(0, y1)) -> c13(MIN(cons(s(0), cons(0, y1)))) IF1(true, s(0), s(z0), cons(s(y0), y1)) -> c13(MIN(cons(s(0), cons(s(y0), y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c13(MIN(cons(s(s(z0)), cons(0, y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c13(MIN(cons(s(s(z0)), cons(s(s(y1)), y2)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c13(MIN(cons(s(s(z0)), cons(s(0), y1)))) K tuples: MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil))) MINSORT(cons(0, cons(y0, y1))) -> c8(MIN(cons(0, cons(y0, y1)))) MINSORT(cons(s(y0), cons(0, y1))) -> c8(MIN(cons(s(y0), cons(0, y1)))) MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c8(MIN(cons(s(s(y0)), cons(s(s(y1)), y2)))) MINSORT(cons(s(y0), cons(s(y1), y2))) -> c8(MIN(cons(s(y0), cons(s(y1), y2)))) MINSORT(cons(s(0), cons(s(y0), y1))) -> c8(MIN(cons(s(0), cons(s(y0), y1)))) MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c8(MIN(cons(s(s(y0)), cons(s(0), y1)))) Defined Rule Symbols: rm_2, min_1, if1_4, le_2, if2_4, eq_2 Defined Pair Symbols: MIN_1, RM_2, LE_2, EQ_2, IF1_4, MINSORT_1, IF2_4 Compound Symbols: c12_1, c16_1, c12_2, c16_2, c2_1, c6_1, c14_1, c9_3, c17_1, c18_1, c9_2, c8_1, c9_1, c13_1 ---------------------------------------- (143) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) by LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1)))) ---------------------------------------- (144) Obligation: Complexity Dependency Tuples Problem Rules: rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) rm(z0, nil) -> nil min(cons(z0, nil)) -> z0 min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(cons(z0, z2)) if1(false, z0, z1, z2) -> min(cons(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) if2(true, z0, z1, z2) -> rm(z0, z2) if2(false, z0, z1, z2) -> cons(z1, rm(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(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2)) RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2)) RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2)) RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2)) MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2)) MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2)) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2)) RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2)) RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2)) IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1))) IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2))) IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) IF2(true, 0, 0, x0) -> c17(RM(0, x0)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2)) IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0)) IF2(false, 0, s(x0), x1) -> c18(RM(0, x1)) IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2)) IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1)) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(z0, cons(z0, nil))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1)))) MINSORT(cons(0, cons(y0, y1))) -> c8(MIN(cons(0, cons(y0, y1)))) MINSORT(cons(s(y0), cons(0, y1))) -> c8(MIN(cons(s(y0), cons(0, y1)))) MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c8(MIN(cons(s(s(y0)), cons(s(s(y1)), y2)))) MINSORT(cons(s(y0), cons(s(y1), y2))) -> c8(MIN(cons(s(y0), cons(s(y1), y2)))) MINSORT(cons(s(0), cons(s(y0), y1))) -> c8(MIN(cons(s(0), cons(s(y0), y1)))) MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c8(MIN(cons(s(s(y0)), cons(s(0), y1)))) MINSORT(cons(s(z0), cons(0, z1))) -> c9(MINSORT(rm(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), RM(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(cons(s(z0), cons(0, z1)))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), RM(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(cons(s(z0), cons(s(z1), z2)))) MINSORT(cons(z0, cons(z1, z2))) -> c9(RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c9(RM(z0, cons(z0, nil))) MIN(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c12(LE(s(s(y0)), s(s(y1)))) RM(s(s(y0)), cons(s(s(y1)), z2)) -> c16(EQ(s(s(y0)), s(s(y1)))) IF1(true, 0, z0, cons(y0, y1)) -> c13(MIN(cons(0, cons(y0, y1)))) MIN(cons(0, cons(z0, cons(y1, y2)))) -> c12(IF1(true, 0, z0, cons(y1, y2))) IF1(true, s(0), s(z0), cons(0, y1)) -> c13(MIN(cons(s(0), cons(0, y1)))) IF1(true, s(0), s(z0), cons(s(y0), y1)) -> c13(MIN(cons(s(0), cons(s(y0), y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c13(MIN(cons(s(s(z0)), cons(0, y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c13(MIN(cons(s(s(z0)), cons(s(s(y1)), y2)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c13(MIN(cons(s(s(z0)), cons(s(0), y1)))) LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1)))) S tuples: MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2)) RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2)) RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2)) RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2)) MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2)) MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2)) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2)) RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2)) RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2)) IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1))) IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2))) IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) IF2(true, 0, 0, x0) -> c17(RM(0, x0)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2)) IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0)) IF2(false, 0, s(x0), x1) -> c18(RM(0, x1)) IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2)) IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1)) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(z0, cons(z0, nil))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1)))) MINSORT(cons(s(z0), cons(0, z1))) -> c9(MINSORT(rm(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), RM(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(cons(s(z0), cons(0, z1)))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), RM(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(cons(s(z0), cons(s(z1), z2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(z0, cons(z0, nil))) MIN(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c12(LE(s(s(y0)), s(s(y1)))) RM(s(s(y0)), cons(s(s(y1)), z2)) -> c16(EQ(s(s(y0)), s(s(y1)))) IF1(true, 0, z0, cons(y0, y1)) -> c13(MIN(cons(0, cons(y0, y1)))) MIN(cons(0, cons(z0, cons(y1, y2)))) -> c12(IF1(true, 0, z0, cons(y1, y2))) IF1(true, s(0), s(z0), cons(0, y1)) -> c13(MIN(cons(s(0), cons(0, y1)))) IF1(true, s(0), s(z0), cons(s(y0), y1)) -> c13(MIN(cons(s(0), cons(s(y0), y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c13(MIN(cons(s(s(z0)), cons(0, y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c13(MIN(cons(s(s(z0)), cons(s(s(y1)), y2)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c13(MIN(cons(s(s(z0)), cons(s(0), y1)))) LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1)))) K tuples: MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil))) MINSORT(cons(0, cons(y0, y1))) -> c8(MIN(cons(0, cons(y0, y1)))) MINSORT(cons(s(y0), cons(0, y1))) -> c8(MIN(cons(s(y0), cons(0, y1)))) MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c8(MIN(cons(s(s(y0)), cons(s(s(y1)), y2)))) MINSORT(cons(s(y0), cons(s(y1), y2))) -> c8(MIN(cons(s(y0), cons(s(y1), y2)))) MINSORT(cons(s(0), cons(s(y0), y1))) -> c8(MIN(cons(s(0), cons(s(y0), y1)))) MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c8(MIN(cons(s(s(y0)), cons(s(0), y1)))) Defined Rule Symbols: rm_2, min_1, if1_4, le_2, if2_4, eq_2 Defined Pair Symbols: MIN_1, RM_2, EQ_2, IF1_4, MINSORT_1, IF2_4, LE_2 Compound Symbols: c12_1, c16_1, c12_2, c16_2, c6_1, c14_1, c9_3, c17_1, c18_1, c9_2, c8_1, c9_1, c13_1, c2_1 ---------------------------------------- (145) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), min(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) by MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) ---------------------------------------- (146) Obligation: Complexity Dependency Tuples Problem Rules: rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) rm(z0, nil) -> nil min(cons(z0, nil)) -> z0 min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(cons(z0, z2)) if1(false, z0, z1, z2) -> min(cons(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) if2(true, z0, z1, z2) -> rm(z0, z2) if2(false, z0, z1, z2) -> cons(z1, rm(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(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2)) RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2)) RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2)) RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2)) MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2)) MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2)) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2)) RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2)) RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2)) IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1))) IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2))) IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1))) IF2(true, 0, 0, x0) -> c17(RM(0, x0)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2)) IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0)) IF2(false, 0, s(x0), x1) -> c18(RM(0, x1)) IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2)) IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1)) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(z0, cons(z0, nil))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1)))) MINSORT(cons(0, cons(y0, y1))) -> c8(MIN(cons(0, cons(y0, y1)))) MINSORT(cons(s(y0), cons(0, y1))) -> c8(MIN(cons(s(y0), cons(0, y1)))) MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c8(MIN(cons(s(s(y0)), cons(s(s(y1)), y2)))) MINSORT(cons(s(y0), cons(s(y1), y2))) -> c8(MIN(cons(s(y0), cons(s(y1), y2)))) MINSORT(cons(s(0), cons(s(y0), y1))) -> c8(MIN(cons(s(0), cons(s(y0), y1)))) MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c8(MIN(cons(s(s(y0)), cons(s(0), y1)))) MINSORT(cons(s(z0), cons(0, z1))) -> c9(MINSORT(rm(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), RM(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(cons(s(z0), cons(0, z1)))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), RM(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(cons(s(z0), cons(s(z1), z2)))) MINSORT(cons(z0, cons(z1, z2))) -> c9(RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c9(RM(z0, cons(z0, nil))) MIN(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c12(LE(s(s(y0)), s(s(y1)))) RM(s(s(y0)), cons(s(s(y1)), z2)) -> c16(EQ(s(s(y0)), s(s(y1)))) IF1(true, 0, z0, cons(y0, y1)) -> c13(MIN(cons(0, cons(y0, y1)))) MIN(cons(0, cons(z0, cons(y1, y2)))) -> c12(IF1(true, 0, z0, cons(y1, y2))) IF1(true, s(0), s(z0), cons(0, y1)) -> c13(MIN(cons(s(0), cons(0, y1)))) IF1(true, s(0), s(z0), cons(s(y0), y1)) -> c13(MIN(cons(s(0), cons(s(y0), y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c13(MIN(cons(s(s(z0)), cons(0, y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c13(MIN(cons(s(s(z0)), cons(s(s(y1)), y2)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c13(MIN(cons(s(s(z0)), cons(s(0), y1)))) LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1)))) S tuples: MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2)) RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2)) RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2)) RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2)) MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2)) MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2)) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2)) RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2)) RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2)) IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1))) IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2))) IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1))) IF2(true, 0, 0, x0) -> c17(RM(0, x0)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2)) IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0)) IF2(false, 0, s(x0), x1) -> c18(RM(0, x1)) IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2)) IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1)) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(z0, cons(z0, nil))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1)))) MINSORT(cons(s(z0), cons(0, z1))) -> c9(MINSORT(rm(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), RM(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(cons(s(z0), cons(0, z1)))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), RM(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(cons(s(z0), cons(s(z1), z2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(z0, cons(z0, nil))) MIN(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c12(LE(s(s(y0)), s(s(y1)))) RM(s(s(y0)), cons(s(s(y1)), z2)) -> c16(EQ(s(s(y0)), s(s(y1)))) IF1(true, 0, z0, cons(y0, y1)) -> c13(MIN(cons(0, cons(y0, y1)))) MIN(cons(0, cons(z0, cons(y1, y2)))) -> c12(IF1(true, 0, z0, cons(y1, y2))) IF1(true, s(0), s(z0), cons(0, y1)) -> c13(MIN(cons(s(0), cons(0, y1)))) IF1(true, s(0), s(z0), cons(s(y0), y1)) -> c13(MIN(cons(s(0), cons(s(y0), y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c13(MIN(cons(s(s(z0)), cons(0, y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c13(MIN(cons(s(s(z0)), cons(s(s(y1)), y2)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c13(MIN(cons(s(s(z0)), cons(s(0), y1)))) LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1)))) K tuples: MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil))) MINSORT(cons(0, cons(y0, y1))) -> c8(MIN(cons(0, cons(y0, y1)))) MINSORT(cons(s(y0), cons(0, y1))) -> c8(MIN(cons(s(y0), cons(0, y1)))) MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c8(MIN(cons(s(s(y0)), cons(s(s(y1)), y2)))) MINSORT(cons(s(y0), cons(s(y1), y2))) -> c8(MIN(cons(s(y0), cons(s(y1), y2)))) MINSORT(cons(s(0), cons(s(y0), y1))) -> c8(MIN(cons(s(0), cons(s(y0), y1)))) MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c8(MIN(cons(s(s(y0)), cons(s(0), y1)))) Defined Rule Symbols: rm_2, min_1, if1_4, le_2, if2_4, eq_2 Defined Pair Symbols: MIN_1, RM_2, EQ_2, IF1_4, IF2_4, MINSORT_1, LE_2 Compound Symbols: c12_1, c16_1, c12_2, c16_2, c6_1, c14_1, c17_1, c18_1, c9_2, c9_3, c8_1, c9_1, c13_1, c2_1 ---------------------------------------- (147) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(min(cons(z0, nil)), z0), z0, z0, nil)), RM(z0, cons(z0, nil))) by MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(z0, cons(z0, nil))) ---------------------------------------- (148) Obligation: Complexity Dependency Tuples Problem Rules: rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) rm(z0, nil) -> nil min(cons(z0, nil)) -> z0 min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(cons(z0, z2)) if1(false, z0, z1, z2) -> min(cons(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) if2(true, z0, z1, z2) -> rm(z0, z2) if2(false, z0, z1, z2) -> cons(z1, rm(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(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2)) RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2)) RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2)) RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2)) MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2)) MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2)) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2)) RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2)) RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2)) IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1))) IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2))) IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1))) IF2(true, 0, 0, x0) -> c17(RM(0, x0)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2)) IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0)) IF2(false, 0, s(x0), x1) -> c18(RM(0, x1)) IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2)) IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1)) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(z0, cons(z0, nil))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1)))) MINSORT(cons(0, cons(y0, y1))) -> c8(MIN(cons(0, cons(y0, y1)))) MINSORT(cons(s(y0), cons(0, y1))) -> c8(MIN(cons(s(y0), cons(0, y1)))) MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c8(MIN(cons(s(s(y0)), cons(s(s(y1)), y2)))) MINSORT(cons(s(y0), cons(s(y1), y2))) -> c8(MIN(cons(s(y0), cons(s(y1), y2)))) MINSORT(cons(s(0), cons(s(y0), y1))) -> c8(MIN(cons(s(0), cons(s(y0), y1)))) MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c8(MIN(cons(s(s(y0)), cons(s(0), y1)))) MINSORT(cons(s(z0), cons(0, z1))) -> c9(MINSORT(rm(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), RM(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(cons(s(z0), cons(0, z1)))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), RM(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(cons(s(z0), cons(s(z1), z2)))) MINSORT(cons(z0, cons(z1, z2))) -> c9(RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c9(RM(z0, cons(z0, nil))) MIN(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c12(LE(s(s(y0)), s(s(y1)))) RM(s(s(y0)), cons(s(s(y1)), z2)) -> c16(EQ(s(s(y0)), s(s(y1)))) IF1(true, 0, z0, cons(y0, y1)) -> c13(MIN(cons(0, cons(y0, y1)))) MIN(cons(0, cons(z0, cons(y1, y2)))) -> c12(IF1(true, 0, z0, cons(y1, y2))) IF1(true, s(0), s(z0), cons(0, y1)) -> c13(MIN(cons(s(0), cons(0, y1)))) IF1(true, s(0), s(z0), cons(s(y0), y1)) -> c13(MIN(cons(s(0), cons(s(y0), y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c13(MIN(cons(s(s(z0)), cons(0, y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c13(MIN(cons(s(s(z0)), cons(s(s(y1)), y2)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c13(MIN(cons(s(s(z0)), cons(s(0), y1)))) LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1)))) S tuples: MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2)) RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2)) RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2)) RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2)) MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2)) MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2)) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2)) RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2)) RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2)) IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1))) IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2))) IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1))) IF2(true, 0, 0, x0) -> c17(RM(0, x0)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2)) IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0)) IF2(false, 0, s(x0), x1) -> c18(RM(0, x1)) IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2)) IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1)) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(z0, cons(z0, nil))) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1)))) MINSORT(cons(s(z0), cons(0, z1))) -> c9(MINSORT(rm(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), RM(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(cons(s(z0), cons(0, z1)))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), RM(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(cons(s(z0), cons(s(z1), z2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(z0, cons(z0, nil))) MIN(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c12(LE(s(s(y0)), s(s(y1)))) RM(s(s(y0)), cons(s(s(y1)), z2)) -> c16(EQ(s(s(y0)), s(s(y1)))) IF1(true, 0, z0, cons(y0, y1)) -> c13(MIN(cons(0, cons(y0, y1)))) MIN(cons(0, cons(z0, cons(y1, y2)))) -> c12(IF1(true, 0, z0, cons(y1, y2))) IF1(true, s(0), s(z0), cons(0, y1)) -> c13(MIN(cons(s(0), cons(0, y1)))) IF1(true, s(0), s(z0), cons(s(y0), y1)) -> c13(MIN(cons(s(0), cons(s(y0), y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c13(MIN(cons(s(s(z0)), cons(0, y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c13(MIN(cons(s(s(z0)), cons(s(s(y1)), y2)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c13(MIN(cons(s(s(z0)), cons(s(0), y1)))) LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1)))) K tuples: MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil))) MINSORT(cons(0, cons(y0, y1))) -> c8(MIN(cons(0, cons(y0, y1)))) MINSORT(cons(s(y0), cons(0, y1))) -> c8(MIN(cons(s(y0), cons(0, y1)))) MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c8(MIN(cons(s(s(y0)), cons(s(s(y1)), y2)))) MINSORT(cons(s(y0), cons(s(y1), y2))) -> c8(MIN(cons(s(y0), cons(s(y1), y2)))) MINSORT(cons(s(0), cons(s(y0), y1))) -> c8(MIN(cons(s(0), cons(s(y0), y1)))) MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c8(MIN(cons(s(s(y0)), cons(s(0), y1)))) Defined Rule Symbols: rm_2, min_1, if1_4, le_2, if2_4, eq_2 Defined Pair Symbols: MIN_1, RM_2, EQ_2, IF1_4, IF2_4, MINSORT_1, LE_2 Compound Symbols: c12_1, c16_1, c12_2, c16_2, c6_1, c14_1, c17_1, c18_1, c9_2, c9_3, c8_1, c9_1, c13_1, c2_1 ---------------------------------------- (149) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), min(cons(z0, nil)), z0, nil)), RM(z0, cons(z0, nil))) by MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(z0, cons(z0, nil))) ---------------------------------------- (150) Obligation: Complexity Dependency Tuples Problem Rules: rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) rm(z0, nil) -> nil min(cons(z0, nil)) -> z0 min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(cons(z0, z2)) if1(false, z0, z1, z2) -> min(cons(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) if2(true, z0, z1, z2) -> rm(z0, z2) if2(false, z0, z1, z2) -> cons(z1, rm(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(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2)) RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2)) RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2)) RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2)) MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2)) MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2)) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2)) RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2)) RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2)) IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1))) IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2))) IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1))) IF2(true, 0, 0, x0) -> c17(RM(0, x0)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2)) IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0)) IF2(false, 0, s(x0), x1) -> c18(RM(0, x1)) IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2)) IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1)) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1)))) MINSORT(cons(0, cons(y0, y1))) -> c8(MIN(cons(0, cons(y0, y1)))) MINSORT(cons(s(y0), cons(0, y1))) -> c8(MIN(cons(s(y0), cons(0, y1)))) MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c8(MIN(cons(s(s(y0)), cons(s(s(y1)), y2)))) MINSORT(cons(s(y0), cons(s(y1), y2))) -> c8(MIN(cons(s(y0), cons(s(y1), y2)))) MINSORT(cons(s(0), cons(s(y0), y1))) -> c8(MIN(cons(s(0), cons(s(y0), y1)))) MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c8(MIN(cons(s(s(y0)), cons(s(0), y1)))) MINSORT(cons(s(z0), cons(0, z1))) -> c9(MINSORT(rm(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), RM(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(cons(s(z0), cons(0, z1)))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), RM(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(cons(s(z0), cons(s(z1), z2)))) MINSORT(cons(z0, cons(z1, z2))) -> c9(RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c9(RM(z0, cons(z0, nil))) MIN(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c12(LE(s(s(y0)), s(s(y1)))) RM(s(s(y0)), cons(s(s(y1)), z2)) -> c16(EQ(s(s(y0)), s(s(y1)))) IF1(true, 0, z0, cons(y0, y1)) -> c13(MIN(cons(0, cons(y0, y1)))) MIN(cons(0, cons(z0, cons(y1, y2)))) -> c12(IF1(true, 0, z0, cons(y1, y2))) IF1(true, s(0), s(z0), cons(0, y1)) -> c13(MIN(cons(s(0), cons(0, y1)))) IF1(true, s(0), s(z0), cons(s(y0), y1)) -> c13(MIN(cons(s(0), cons(s(y0), y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c13(MIN(cons(s(s(z0)), cons(0, y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c13(MIN(cons(s(s(z0)), cons(s(s(y1)), y2)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c13(MIN(cons(s(s(z0)), cons(s(0), y1)))) LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1)))) S tuples: MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2)) RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2)) RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2)) RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2)) MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2)) MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2)) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2)) RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2)) RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2)) IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1))) IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2))) IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1))) IF2(true, 0, 0, x0) -> c17(RM(0, x0)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2)) IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0)) IF2(false, 0, s(x0), x1) -> c18(RM(0, x1)) IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2)) IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1)) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1)))) MINSORT(cons(s(z0), cons(0, z1))) -> c9(MINSORT(rm(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), RM(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(cons(s(z0), cons(0, z1)))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), RM(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(cons(s(z0), cons(s(z1), z2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(z0, cons(z0, nil))) MIN(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c12(LE(s(s(y0)), s(s(y1)))) RM(s(s(y0)), cons(s(s(y1)), z2)) -> c16(EQ(s(s(y0)), s(s(y1)))) IF1(true, 0, z0, cons(y0, y1)) -> c13(MIN(cons(0, cons(y0, y1)))) MIN(cons(0, cons(z0, cons(y1, y2)))) -> c12(IF1(true, 0, z0, cons(y1, y2))) IF1(true, s(0), s(z0), cons(0, y1)) -> c13(MIN(cons(s(0), cons(0, y1)))) IF1(true, s(0), s(z0), cons(s(y0), y1)) -> c13(MIN(cons(s(0), cons(s(y0), y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c13(MIN(cons(s(s(z0)), cons(0, y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c13(MIN(cons(s(s(z0)), cons(s(s(y1)), y2)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c13(MIN(cons(s(s(z0)), cons(s(0), y1)))) LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1)))) K tuples: MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil))) MINSORT(cons(0, cons(y0, y1))) -> c8(MIN(cons(0, cons(y0, y1)))) MINSORT(cons(s(y0), cons(0, y1))) -> c8(MIN(cons(s(y0), cons(0, y1)))) MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c8(MIN(cons(s(s(y0)), cons(s(s(y1)), y2)))) MINSORT(cons(s(y0), cons(s(y1), y2))) -> c8(MIN(cons(s(y0), cons(s(y1), y2)))) MINSORT(cons(s(0), cons(s(y0), y1))) -> c8(MIN(cons(s(0), cons(s(y0), y1)))) MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c8(MIN(cons(s(s(y0)), cons(s(0), y1)))) Defined Rule Symbols: rm_2, min_1, if1_4, le_2, if2_4, eq_2 Defined Pair Symbols: MIN_1, RM_2, EQ_2, IF1_4, IF2_4, MINSORT_1, LE_2 Compound Symbols: c12_1, c16_1, c12_2, c16_2, c6_1, c14_1, c17_1, c18_1, c9_3, c8_1, c9_1, c9_2, c13_1, c2_1 ---------------------------------------- (151) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(le(0, z0), 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1)))) by MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(true, 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1)))) ---------------------------------------- (152) Obligation: Complexity Dependency Tuples Problem Rules: rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) rm(z0, nil) -> nil min(cons(z0, nil)) -> z0 min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(cons(z0, z2)) if1(false, z0, z1, z2) -> min(cons(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) if2(true, z0, z1, z2) -> rm(z0, z2) if2(false, z0, z1, z2) -> cons(z1, rm(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(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2)) RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2)) RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2)) RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2)) MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2)) MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2)) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2)) RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2)) RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2)) IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1))) IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2))) IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1))) IF2(true, 0, 0, x0) -> c17(RM(0, x0)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2)) IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0)) IF2(false, 0, s(x0), x1) -> c18(RM(0, x1)) IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2)) IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1)) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(0, cons(y0, y1))) -> c8(MIN(cons(0, cons(y0, y1)))) MINSORT(cons(s(y0), cons(0, y1))) -> c8(MIN(cons(s(y0), cons(0, y1)))) MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c8(MIN(cons(s(s(y0)), cons(s(s(y1)), y2)))) MINSORT(cons(s(y0), cons(s(y1), y2))) -> c8(MIN(cons(s(y0), cons(s(y1), y2)))) MINSORT(cons(s(0), cons(s(y0), y1))) -> c8(MIN(cons(s(0), cons(s(y0), y1)))) MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c8(MIN(cons(s(s(y0)), cons(s(0), y1)))) MINSORT(cons(s(z0), cons(0, z1))) -> c9(MINSORT(rm(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), RM(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(cons(s(z0), cons(0, z1)))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), RM(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(cons(s(z0), cons(s(z1), z2)))) MINSORT(cons(z0, cons(z1, z2))) -> c9(RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c9(RM(z0, cons(z0, nil))) MIN(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c12(LE(s(s(y0)), s(s(y1)))) RM(s(s(y0)), cons(s(s(y1)), z2)) -> c16(EQ(s(s(y0)), s(s(y1)))) IF1(true, 0, z0, cons(y0, y1)) -> c13(MIN(cons(0, cons(y0, y1)))) MIN(cons(0, cons(z0, cons(y1, y2)))) -> c12(IF1(true, 0, z0, cons(y1, y2))) IF1(true, s(0), s(z0), cons(0, y1)) -> c13(MIN(cons(s(0), cons(0, y1)))) IF1(true, s(0), s(z0), cons(s(y0), y1)) -> c13(MIN(cons(s(0), cons(s(y0), y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c13(MIN(cons(s(s(z0)), cons(0, y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c13(MIN(cons(s(s(z0)), cons(s(s(y1)), y2)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c13(MIN(cons(s(s(z0)), cons(s(0), y1)))) LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1)))) MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(true, 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1)))) S tuples: MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2)) RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2)) RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2)) RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2)) MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2)) MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2)) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2)) RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2)) RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2)) IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1))) IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2))) IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1))) IF2(true, 0, 0, x0) -> c17(RM(0, x0)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2)) IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0)) IF2(false, 0, s(x0), x1) -> c18(RM(0, x1)) IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2)) IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1)) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(s(z0), cons(0, z1))) -> c9(MINSORT(rm(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), RM(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(cons(s(z0), cons(0, z1)))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), RM(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(cons(s(z0), cons(s(z1), z2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(z0, cons(z0, nil))) MIN(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c12(LE(s(s(y0)), s(s(y1)))) RM(s(s(y0)), cons(s(s(y1)), z2)) -> c16(EQ(s(s(y0)), s(s(y1)))) IF1(true, 0, z0, cons(y0, y1)) -> c13(MIN(cons(0, cons(y0, y1)))) MIN(cons(0, cons(z0, cons(y1, y2)))) -> c12(IF1(true, 0, z0, cons(y1, y2))) IF1(true, s(0), s(z0), cons(0, y1)) -> c13(MIN(cons(s(0), cons(0, y1)))) IF1(true, s(0), s(z0), cons(s(y0), y1)) -> c13(MIN(cons(s(0), cons(s(y0), y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c13(MIN(cons(s(s(z0)), cons(0, y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c13(MIN(cons(s(s(z0)), cons(s(s(y1)), y2)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c13(MIN(cons(s(s(z0)), cons(s(0), y1)))) LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1)))) MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(true, 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1)))) K tuples: MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil))) MINSORT(cons(0, cons(y0, y1))) -> c8(MIN(cons(0, cons(y0, y1)))) MINSORT(cons(s(y0), cons(0, y1))) -> c8(MIN(cons(s(y0), cons(0, y1)))) MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c8(MIN(cons(s(s(y0)), cons(s(s(y1)), y2)))) MINSORT(cons(s(y0), cons(s(y1), y2))) -> c8(MIN(cons(s(y0), cons(s(y1), y2)))) MINSORT(cons(s(0), cons(s(y0), y1))) -> c8(MIN(cons(s(0), cons(s(y0), y1)))) MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c8(MIN(cons(s(s(y0)), cons(s(0), y1)))) Defined Rule Symbols: rm_2, min_1, if1_4, le_2, if2_4, eq_2 Defined Pair Symbols: MIN_1, RM_2, EQ_2, IF1_4, IF2_4, MINSORT_1, LE_2 Compound Symbols: c12_1, c16_1, c12_2, c16_2, c6_1, c14_1, c17_1, c18_1, c9_3, c8_1, c9_1, c9_2, c13_1, c2_1 ---------------------------------------- (153) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace MINSORT(cons(s(z0), cons(0, z1))) -> c9(MINSORT(rm(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), RM(if1(le(s(z0), 0), s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(cons(s(z0), cons(0, z1)))) by MINSORT(cons(s(z0), cons(0, z1))) -> c9(MINSORT(rm(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), RM(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(cons(s(z0), cons(0, z1)))) ---------------------------------------- (154) Obligation: Complexity Dependency Tuples Problem Rules: rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) rm(z0, nil) -> nil min(cons(z0, nil)) -> z0 min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(cons(z0, z2)) if1(false, z0, z1, z2) -> min(cons(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) if2(true, z0, z1, z2) -> rm(z0, z2) if2(false, z0, z1, z2) -> cons(z1, rm(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(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2)) RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2)) RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2)) RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2)) MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2)) MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2)) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2)) RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2)) RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2)) IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1))) IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2))) IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1))) IF2(true, 0, 0, x0) -> c17(RM(0, x0)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2)) IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0)) IF2(false, 0, s(x0), x1) -> c18(RM(0, x1)) IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2)) IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1)) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(0, cons(y0, y1))) -> c8(MIN(cons(0, cons(y0, y1)))) MINSORT(cons(s(y0), cons(0, y1))) -> c8(MIN(cons(s(y0), cons(0, y1)))) MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c8(MIN(cons(s(s(y0)), cons(s(s(y1)), y2)))) MINSORT(cons(s(y0), cons(s(y1), y2))) -> c8(MIN(cons(s(y0), cons(s(y1), y2)))) MINSORT(cons(s(0), cons(s(y0), y1))) -> c8(MIN(cons(s(0), cons(s(y0), y1)))) MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c8(MIN(cons(s(s(y0)), cons(s(0), y1)))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), RM(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(cons(s(z0), cons(s(z1), z2)))) MINSORT(cons(z0, cons(z1, z2))) -> c9(RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c9(RM(z0, cons(z0, nil))) MIN(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c12(LE(s(s(y0)), s(s(y1)))) RM(s(s(y0)), cons(s(s(y1)), z2)) -> c16(EQ(s(s(y0)), s(s(y1)))) IF1(true, 0, z0, cons(y0, y1)) -> c13(MIN(cons(0, cons(y0, y1)))) MIN(cons(0, cons(z0, cons(y1, y2)))) -> c12(IF1(true, 0, z0, cons(y1, y2))) IF1(true, s(0), s(z0), cons(0, y1)) -> c13(MIN(cons(s(0), cons(0, y1)))) IF1(true, s(0), s(z0), cons(s(y0), y1)) -> c13(MIN(cons(s(0), cons(s(y0), y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c13(MIN(cons(s(s(z0)), cons(0, y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c13(MIN(cons(s(s(z0)), cons(s(s(y1)), y2)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c13(MIN(cons(s(s(z0)), cons(s(0), y1)))) LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1)))) MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(true, 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1)))) MINSORT(cons(s(z0), cons(0, z1))) -> c9(MINSORT(rm(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), RM(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(cons(s(z0), cons(0, z1)))) S tuples: MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2)) RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2)) RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2)) RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2)) MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2)) MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2)) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2)) RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2)) RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2)) IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1))) IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2))) IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1))) IF2(true, 0, 0, x0) -> c17(RM(0, x0)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2)) IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0)) IF2(false, 0, s(x0), x1) -> c18(RM(0, x1)) IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2)) IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1)) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), RM(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(cons(s(z0), cons(s(z1), z2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(z0, cons(z0, nil))) MIN(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c12(LE(s(s(y0)), s(s(y1)))) RM(s(s(y0)), cons(s(s(y1)), z2)) -> c16(EQ(s(s(y0)), s(s(y1)))) IF1(true, 0, z0, cons(y0, y1)) -> c13(MIN(cons(0, cons(y0, y1)))) MIN(cons(0, cons(z0, cons(y1, y2)))) -> c12(IF1(true, 0, z0, cons(y1, y2))) IF1(true, s(0), s(z0), cons(0, y1)) -> c13(MIN(cons(s(0), cons(0, y1)))) IF1(true, s(0), s(z0), cons(s(y0), y1)) -> c13(MIN(cons(s(0), cons(s(y0), y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c13(MIN(cons(s(s(z0)), cons(0, y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c13(MIN(cons(s(s(z0)), cons(s(s(y1)), y2)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c13(MIN(cons(s(s(z0)), cons(s(0), y1)))) LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1)))) MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(true, 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1)))) MINSORT(cons(s(z0), cons(0, z1))) -> c9(MINSORT(rm(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), RM(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(cons(s(z0), cons(0, z1)))) K tuples: MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil))) MINSORT(cons(0, cons(y0, y1))) -> c8(MIN(cons(0, cons(y0, y1)))) MINSORT(cons(s(y0), cons(0, y1))) -> c8(MIN(cons(s(y0), cons(0, y1)))) MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c8(MIN(cons(s(s(y0)), cons(s(s(y1)), y2)))) MINSORT(cons(s(y0), cons(s(y1), y2))) -> c8(MIN(cons(s(y0), cons(s(y1), y2)))) MINSORT(cons(s(0), cons(s(y0), y1))) -> c8(MIN(cons(s(0), cons(s(y0), y1)))) MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c8(MIN(cons(s(s(y0)), cons(s(0), y1)))) Defined Rule Symbols: rm_2, min_1, if1_4, le_2, if2_4, eq_2 Defined Pair Symbols: MIN_1, RM_2, EQ_2, IF1_4, IF2_4, MINSORT_1, LE_2 Compound Symbols: c12_1, c16_1, c12_2, c16_2, c6_1, c14_1, c17_1, c18_1, c9_3, c8_1, c9_1, c9_2, c13_1, c2_1 ---------------------------------------- (155) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace MIN(cons(s(0), cons(s(z0), x2))) -> c12(IF1(true, s(0), s(z0), x2)) by MIN(cons(s(0), cons(s(z0), cons(0, y1)))) -> c12(IF1(true, s(0), s(z0), cons(0, y1))) MIN(cons(s(0), cons(s(z0), cons(s(y1), y2)))) -> c12(IF1(true, s(0), s(z0), cons(s(y1), y2))) ---------------------------------------- (156) Obligation: Complexity Dependency Tuples Problem Rules: rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) rm(z0, nil) -> nil min(cons(z0, nil)) -> z0 min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(cons(z0, z2)) if1(false, z0, z1, z2) -> min(cons(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) if2(true, z0, z1, z2) -> rm(z0, z2) if2(false, z0, z1, z2) -> cons(z1, rm(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(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2)) RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2)) RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2)) RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2)) MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2)) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2)) RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2)) RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2)) IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1))) IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2))) IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1))) IF2(true, 0, 0, x0) -> c17(RM(0, x0)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2)) IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0)) IF2(false, 0, s(x0), x1) -> c18(RM(0, x1)) IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2)) IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1)) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(0, cons(y0, y1))) -> c8(MIN(cons(0, cons(y0, y1)))) MINSORT(cons(s(y0), cons(0, y1))) -> c8(MIN(cons(s(y0), cons(0, y1)))) MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c8(MIN(cons(s(s(y0)), cons(s(s(y1)), y2)))) MINSORT(cons(s(y0), cons(s(y1), y2))) -> c8(MIN(cons(s(y0), cons(s(y1), y2)))) MINSORT(cons(s(0), cons(s(y0), y1))) -> c8(MIN(cons(s(0), cons(s(y0), y1)))) MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c8(MIN(cons(s(s(y0)), cons(s(0), y1)))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), RM(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(cons(s(z0), cons(s(z1), z2)))) MINSORT(cons(z0, cons(z1, z2))) -> c9(RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c9(RM(z0, cons(z0, nil))) MIN(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c12(LE(s(s(y0)), s(s(y1)))) RM(s(s(y0)), cons(s(s(y1)), z2)) -> c16(EQ(s(s(y0)), s(s(y1)))) IF1(true, 0, z0, cons(y0, y1)) -> c13(MIN(cons(0, cons(y0, y1)))) MIN(cons(0, cons(z0, cons(y1, y2)))) -> c12(IF1(true, 0, z0, cons(y1, y2))) IF1(true, s(0), s(z0), cons(0, y1)) -> c13(MIN(cons(s(0), cons(0, y1)))) IF1(true, s(0), s(z0), cons(s(y0), y1)) -> c13(MIN(cons(s(0), cons(s(y0), y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c13(MIN(cons(s(s(z0)), cons(0, y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c13(MIN(cons(s(s(z0)), cons(s(s(y1)), y2)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c13(MIN(cons(s(s(z0)), cons(s(0), y1)))) LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1)))) MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(true, 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1)))) MINSORT(cons(s(z0), cons(0, z1))) -> c9(MINSORT(rm(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), RM(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(cons(s(z0), cons(0, z1)))) MIN(cons(s(0), cons(s(z0), cons(0, y1)))) -> c12(IF1(true, s(0), s(z0), cons(0, y1))) MIN(cons(s(0), cons(s(z0), cons(s(y1), y2)))) -> c12(IF1(true, s(0), s(z0), cons(s(y1), y2))) S tuples: MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2)) RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2)) RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2)) RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2)) MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2)) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2)) RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2)) RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2)) IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1))) IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2))) IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1))) IF2(true, 0, 0, x0) -> c17(RM(0, x0)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2)) IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0)) IF2(false, 0, s(x0), x1) -> c18(RM(0, x1)) IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2)) IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1)) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), RM(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(cons(s(z0), cons(s(z1), z2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(z0, cons(z0, nil))) MIN(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c12(LE(s(s(y0)), s(s(y1)))) RM(s(s(y0)), cons(s(s(y1)), z2)) -> c16(EQ(s(s(y0)), s(s(y1)))) IF1(true, 0, z0, cons(y0, y1)) -> c13(MIN(cons(0, cons(y0, y1)))) MIN(cons(0, cons(z0, cons(y1, y2)))) -> c12(IF1(true, 0, z0, cons(y1, y2))) IF1(true, s(0), s(z0), cons(0, y1)) -> c13(MIN(cons(s(0), cons(0, y1)))) IF1(true, s(0), s(z0), cons(s(y0), y1)) -> c13(MIN(cons(s(0), cons(s(y0), y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c13(MIN(cons(s(s(z0)), cons(0, y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c13(MIN(cons(s(s(z0)), cons(s(s(y1)), y2)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c13(MIN(cons(s(s(z0)), cons(s(0), y1)))) LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1)))) MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(true, 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1)))) MINSORT(cons(s(z0), cons(0, z1))) -> c9(MINSORT(rm(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), RM(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(cons(s(z0), cons(0, z1)))) MIN(cons(s(0), cons(s(z0), cons(0, y1)))) -> c12(IF1(true, s(0), s(z0), cons(0, y1))) MIN(cons(s(0), cons(s(z0), cons(s(y1), y2)))) -> c12(IF1(true, s(0), s(z0), cons(s(y1), y2))) K tuples: MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil))) MINSORT(cons(0, cons(y0, y1))) -> c8(MIN(cons(0, cons(y0, y1)))) MINSORT(cons(s(y0), cons(0, y1))) -> c8(MIN(cons(s(y0), cons(0, y1)))) MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c8(MIN(cons(s(s(y0)), cons(s(s(y1)), y2)))) MINSORT(cons(s(y0), cons(s(y1), y2))) -> c8(MIN(cons(s(y0), cons(s(y1), y2)))) MINSORT(cons(s(0), cons(s(y0), y1))) -> c8(MIN(cons(s(0), cons(s(y0), y1)))) MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c8(MIN(cons(s(s(y0)), cons(s(0), y1)))) Defined Rule Symbols: rm_2, min_1, if1_4, le_2, if2_4, eq_2 Defined Pair Symbols: MIN_1, RM_2, EQ_2, IF1_4, IF2_4, MINSORT_1, LE_2 Compound Symbols: c12_1, c16_1, c12_2, c16_2, c6_1, c14_1, c17_1, c18_1, c9_3, c8_1, c9_1, c9_2, c13_1, c2_1 ---------------------------------------- (157) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) by EQ(s(s(s(y0))), s(s(s(y1)))) -> c6(EQ(s(s(y0)), s(s(y1)))) ---------------------------------------- (158) Obligation: Complexity Dependency Tuples Problem Rules: rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) rm(z0, nil) -> nil min(cons(z0, nil)) -> z0 min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(cons(z0, z2)) if1(false, z0, z1, z2) -> min(cons(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) if2(true, z0, z1, z2) -> rm(z0, z2) if2(false, z0, z1, z2) -> cons(z1, rm(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(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2)) RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2)) RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2)) RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2)) MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2)) RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2)) RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2)) RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2)) IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1))) IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2))) IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1))) IF2(true, 0, 0, x0) -> c17(RM(0, x0)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2)) IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0)) IF2(false, 0, s(x0), x1) -> c18(RM(0, x1)) IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2)) IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1)) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(0, cons(y0, y1))) -> c8(MIN(cons(0, cons(y0, y1)))) MINSORT(cons(s(y0), cons(0, y1))) -> c8(MIN(cons(s(y0), cons(0, y1)))) MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c8(MIN(cons(s(s(y0)), cons(s(s(y1)), y2)))) MINSORT(cons(s(y0), cons(s(y1), y2))) -> c8(MIN(cons(s(y0), cons(s(y1), y2)))) MINSORT(cons(s(0), cons(s(y0), y1))) -> c8(MIN(cons(s(0), cons(s(y0), y1)))) MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c8(MIN(cons(s(s(y0)), cons(s(0), y1)))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), RM(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(cons(s(z0), cons(s(z1), z2)))) MINSORT(cons(z0, cons(z1, z2))) -> c9(RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c9(RM(z0, cons(z0, nil))) MIN(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c12(LE(s(s(y0)), s(s(y1)))) RM(s(s(y0)), cons(s(s(y1)), z2)) -> c16(EQ(s(s(y0)), s(s(y1)))) IF1(true, 0, z0, cons(y0, y1)) -> c13(MIN(cons(0, cons(y0, y1)))) MIN(cons(0, cons(z0, cons(y1, y2)))) -> c12(IF1(true, 0, z0, cons(y1, y2))) IF1(true, s(0), s(z0), cons(0, y1)) -> c13(MIN(cons(s(0), cons(0, y1)))) IF1(true, s(0), s(z0), cons(s(y0), y1)) -> c13(MIN(cons(s(0), cons(s(y0), y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c13(MIN(cons(s(s(z0)), cons(0, y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c13(MIN(cons(s(s(z0)), cons(s(s(y1)), y2)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c13(MIN(cons(s(s(z0)), cons(s(0), y1)))) LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1)))) MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(true, 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1)))) MINSORT(cons(s(z0), cons(0, z1))) -> c9(MINSORT(rm(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), RM(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(cons(s(z0), cons(0, z1)))) MIN(cons(s(0), cons(s(z0), cons(0, y1)))) -> c12(IF1(true, s(0), s(z0), cons(0, y1))) MIN(cons(s(0), cons(s(z0), cons(s(y1), y2)))) -> c12(IF1(true, s(0), s(z0), cons(s(y1), y2))) EQ(s(s(s(y0))), s(s(s(y1)))) -> c6(EQ(s(s(y0)), s(s(y1)))) S tuples: MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2)) RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2)) RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2)) RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2)) MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2)) RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2)) RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2)) RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2)) IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1))) IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2))) IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1))) IF2(true, 0, 0, x0) -> c17(RM(0, x0)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2)) IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0)) IF2(false, 0, s(x0), x1) -> c18(RM(0, x1)) IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2)) IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1)) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), RM(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(cons(s(z0), cons(s(z1), z2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(z0, cons(z0, nil))) MIN(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c12(LE(s(s(y0)), s(s(y1)))) RM(s(s(y0)), cons(s(s(y1)), z2)) -> c16(EQ(s(s(y0)), s(s(y1)))) IF1(true, 0, z0, cons(y0, y1)) -> c13(MIN(cons(0, cons(y0, y1)))) MIN(cons(0, cons(z0, cons(y1, y2)))) -> c12(IF1(true, 0, z0, cons(y1, y2))) IF1(true, s(0), s(z0), cons(0, y1)) -> c13(MIN(cons(s(0), cons(0, y1)))) IF1(true, s(0), s(z0), cons(s(y0), y1)) -> c13(MIN(cons(s(0), cons(s(y0), y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c13(MIN(cons(s(s(z0)), cons(0, y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c13(MIN(cons(s(s(z0)), cons(s(s(y1)), y2)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c13(MIN(cons(s(s(z0)), cons(s(0), y1)))) LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1)))) MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(true, 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1)))) MINSORT(cons(s(z0), cons(0, z1))) -> c9(MINSORT(rm(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), RM(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(cons(s(z0), cons(0, z1)))) MIN(cons(s(0), cons(s(z0), cons(0, y1)))) -> c12(IF1(true, s(0), s(z0), cons(0, y1))) MIN(cons(s(0), cons(s(z0), cons(s(y1), y2)))) -> c12(IF1(true, s(0), s(z0), cons(s(y1), y2))) EQ(s(s(s(y0))), s(s(s(y1)))) -> c6(EQ(s(s(y0)), s(s(y1)))) K tuples: MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil))) MINSORT(cons(0, cons(y0, y1))) -> c8(MIN(cons(0, cons(y0, y1)))) MINSORT(cons(s(y0), cons(0, y1))) -> c8(MIN(cons(s(y0), cons(0, y1)))) MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c8(MIN(cons(s(s(y0)), cons(s(s(y1)), y2)))) MINSORT(cons(s(y0), cons(s(y1), y2))) -> c8(MIN(cons(s(y0), cons(s(y1), y2)))) MINSORT(cons(s(0), cons(s(y0), y1))) -> c8(MIN(cons(s(0), cons(s(y0), y1)))) MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c8(MIN(cons(s(s(y0)), cons(s(0), y1)))) Defined Rule Symbols: rm_2, min_1, if1_4, le_2, if2_4, eq_2 Defined Pair Symbols: MIN_1, RM_2, IF1_4, IF2_4, MINSORT_1, LE_2, EQ_2 Compound Symbols: c12_1, c16_1, c12_2, c16_2, c14_1, c17_1, c18_1, c9_3, c8_1, c9_1, c9_2, c13_1, c2_1, c6_1 ---------------------------------------- (159) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace IF1(false, s(x0), 0, x1) -> c14(MIN(cons(0, x1))) by IF1(false, s(z0), 0, cons(y0, cons(y1, y2))) -> c14(MIN(cons(0, cons(y0, cons(y1, y2))))) ---------------------------------------- (160) Obligation: Complexity Dependency Tuples Problem Rules: rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) rm(z0, nil) -> nil min(cons(z0, nil)) -> z0 min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(cons(z0, z2)) if1(false, z0, z1, z2) -> min(cons(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) if2(true, z0, z1, z2) -> rm(z0, z2) if2(false, z0, z1, z2) -> cons(z1, rm(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(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2)) RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2)) RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2)) RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2)) MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2)) RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2)) RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2)) RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2))) IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1))) IF2(true, 0, 0, x0) -> c17(RM(0, x0)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2)) IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0)) IF2(false, 0, s(x0), x1) -> c18(RM(0, x1)) IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2)) IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1)) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(0, cons(y0, y1))) -> c8(MIN(cons(0, cons(y0, y1)))) MINSORT(cons(s(y0), cons(0, y1))) -> c8(MIN(cons(s(y0), cons(0, y1)))) MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c8(MIN(cons(s(s(y0)), cons(s(s(y1)), y2)))) MINSORT(cons(s(y0), cons(s(y1), y2))) -> c8(MIN(cons(s(y0), cons(s(y1), y2)))) MINSORT(cons(s(0), cons(s(y0), y1))) -> c8(MIN(cons(s(0), cons(s(y0), y1)))) MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c8(MIN(cons(s(s(y0)), cons(s(0), y1)))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), RM(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(cons(s(z0), cons(s(z1), z2)))) MINSORT(cons(z0, cons(z1, z2))) -> c9(RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c9(RM(z0, cons(z0, nil))) MIN(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c12(LE(s(s(y0)), s(s(y1)))) RM(s(s(y0)), cons(s(s(y1)), z2)) -> c16(EQ(s(s(y0)), s(s(y1)))) IF1(true, 0, z0, cons(y0, y1)) -> c13(MIN(cons(0, cons(y0, y1)))) MIN(cons(0, cons(z0, cons(y1, y2)))) -> c12(IF1(true, 0, z0, cons(y1, y2))) IF1(true, s(0), s(z0), cons(0, y1)) -> c13(MIN(cons(s(0), cons(0, y1)))) IF1(true, s(0), s(z0), cons(s(y0), y1)) -> c13(MIN(cons(s(0), cons(s(y0), y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c13(MIN(cons(s(s(z0)), cons(0, y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c13(MIN(cons(s(s(z0)), cons(s(s(y1)), y2)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c13(MIN(cons(s(s(z0)), cons(s(0), y1)))) LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1)))) MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(true, 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1)))) MINSORT(cons(s(z0), cons(0, z1))) -> c9(MINSORT(rm(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), RM(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(cons(s(z0), cons(0, z1)))) MIN(cons(s(0), cons(s(z0), cons(0, y1)))) -> c12(IF1(true, s(0), s(z0), cons(0, y1))) MIN(cons(s(0), cons(s(z0), cons(s(y1), y2)))) -> c12(IF1(true, s(0), s(z0), cons(s(y1), y2))) EQ(s(s(s(y0))), s(s(s(y1)))) -> c6(EQ(s(s(y0)), s(s(y1)))) IF1(false, s(z0), 0, cons(y0, cons(y1, y2))) -> c14(MIN(cons(0, cons(y0, cons(y1, y2))))) S tuples: MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2)) RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2)) RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2)) RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2)) MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2)) RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2)) RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2)) RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2))) IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1))) IF2(true, 0, 0, x0) -> c17(RM(0, x0)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2)) IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0)) IF2(false, 0, s(x0), x1) -> c18(RM(0, x1)) IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2)) IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1)) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), RM(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(cons(s(z0), cons(s(z1), z2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(z0, cons(z0, nil))) MIN(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c12(LE(s(s(y0)), s(s(y1)))) RM(s(s(y0)), cons(s(s(y1)), z2)) -> c16(EQ(s(s(y0)), s(s(y1)))) IF1(true, 0, z0, cons(y0, y1)) -> c13(MIN(cons(0, cons(y0, y1)))) MIN(cons(0, cons(z0, cons(y1, y2)))) -> c12(IF1(true, 0, z0, cons(y1, y2))) IF1(true, s(0), s(z0), cons(0, y1)) -> c13(MIN(cons(s(0), cons(0, y1)))) IF1(true, s(0), s(z0), cons(s(y0), y1)) -> c13(MIN(cons(s(0), cons(s(y0), y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c13(MIN(cons(s(s(z0)), cons(0, y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c13(MIN(cons(s(s(z0)), cons(s(s(y1)), y2)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c13(MIN(cons(s(s(z0)), cons(s(0), y1)))) LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1)))) MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(true, 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1)))) MINSORT(cons(s(z0), cons(0, z1))) -> c9(MINSORT(rm(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), RM(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(cons(s(z0), cons(0, z1)))) MIN(cons(s(0), cons(s(z0), cons(0, y1)))) -> c12(IF1(true, s(0), s(z0), cons(0, y1))) MIN(cons(s(0), cons(s(z0), cons(s(y1), y2)))) -> c12(IF1(true, s(0), s(z0), cons(s(y1), y2))) EQ(s(s(s(y0))), s(s(s(y1)))) -> c6(EQ(s(s(y0)), s(s(y1)))) IF1(false, s(z0), 0, cons(y0, cons(y1, y2))) -> c14(MIN(cons(0, cons(y0, cons(y1, y2))))) K tuples: MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil))) MINSORT(cons(0, cons(y0, y1))) -> c8(MIN(cons(0, cons(y0, y1)))) MINSORT(cons(s(y0), cons(0, y1))) -> c8(MIN(cons(s(y0), cons(0, y1)))) MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c8(MIN(cons(s(s(y0)), cons(s(s(y1)), y2)))) MINSORT(cons(s(y0), cons(s(y1), y2))) -> c8(MIN(cons(s(y0), cons(s(y1), y2)))) MINSORT(cons(s(0), cons(s(y0), y1))) -> c8(MIN(cons(s(0), cons(s(y0), y1)))) MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c8(MIN(cons(s(s(y0)), cons(s(0), y1)))) Defined Rule Symbols: rm_2, min_1, if1_4, le_2, if2_4, eq_2 Defined Pair Symbols: MIN_1, RM_2, IF1_4, IF2_4, MINSORT_1, LE_2, EQ_2 Compound Symbols: c12_1, c16_1, c12_2, c16_2, c14_1, c17_1, c18_1, c9_3, c8_1, c9_1, c9_2, c13_1, c2_1, c6_1 ---------------------------------------- (161) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace MINSORT(cons(s(z0), cons(s(z1), z2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), RM(if1(le(s(z0), s(z1)), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(cons(s(z0), cons(s(z1), z2)))) by MINSORT(cons(s(z0), cons(s(z1), z2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), RM(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(cons(s(z0), cons(s(z1), z2)))) ---------------------------------------- (162) Obligation: Complexity Dependency Tuples Problem Rules: rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) rm(z0, nil) -> nil min(cons(z0, nil)) -> z0 min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(cons(z0, z2)) if1(false, z0, z1, z2) -> min(cons(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) if2(true, z0, z1, z2) -> rm(z0, z2) if2(false, z0, z1, z2) -> cons(z1, rm(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(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2)) RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2)) RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2)) RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2)) MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2)) RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2)) RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2)) RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2))) IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1))) IF2(true, 0, 0, x0) -> c17(RM(0, x0)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2)) IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0)) IF2(false, 0, s(x0), x1) -> c18(RM(0, x1)) IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2)) IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1)) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(0, cons(y0, y1))) -> c8(MIN(cons(0, cons(y0, y1)))) MINSORT(cons(s(y0), cons(0, y1))) -> c8(MIN(cons(s(y0), cons(0, y1)))) MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c8(MIN(cons(s(s(y0)), cons(s(s(y1)), y2)))) MINSORT(cons(s(y0), cons(s(y1), y2))) -> c8(MIN(cons(s(y0), cons(s(y1), y2)))) MINSORT(cons(s(0), cons(s(y0), y1))) -> c8(MIN(cons(s(0), cons(s(y0), y1)))) MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c8(MIN(cons(s(s(y0)), cons(s(0), y1)))) MINSORT(cons(z0, cons(z1, z2))) -> c9(RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c9(RM(z0, cons(z0, nil))) MIN(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c12(LE(s(s(y0)), s(s(y1)))) RM(s(s(y0)), cons(s(s(y1)), z2)) -> c16(EQ(s(s(y0)), s(s(y1)))) IF1(true, 0, z0, cons(y0, y1)) -> c13(MIN(cons(0, cons(y0, y1)))) MIN(cons(0, cons(z0, cons(y1, y2)))) -> c12(IF1(true, 0, z0, cons(y1, y2))) IF1(true, s(0), s(z0), cons(0, y1)) -> c13(MIN(cons(s(0), cons(0, y1)))) IF1(true, s(0), s(z0), cons(s(y0), y1)) -> c13(MIN(cons(s(0), cons(s(y0), y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c13(MIN(cons(s(s(z0)), cons(0, y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c13(MIN(cons(s(s(z0)), cons(s(s(y1)), y2)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c13(MIN(cons(s(s(z0)), cons(s(0), y1)))) LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1)))) MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(true, 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1)))) MINSORT(cons(s(z0), cons(0, z1))) -> c9(MINSORT(rm(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), RM(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(cons(s(z0), cons(0, z1)))) MIN(cons(s(0), cons(s(z0), cons(0, y1)))) -> c12(IF1(true, s(0), s(z0), cons(0, y1))) MIN(cons(s(0), cons(s(z0), cons(s(y1), y2)))) -> c12(IF1(true, s(0), s(z0), cons(s(y1), y2))) EQ(s(s(s(y0))), s(s(s(y1)))) -> c6(EQ(s(s(y0)), s(s(y1)))) IF1(false, s(z0), 0, cons(y0, cons(y1, y2))) -> c14(MIN(cons(0, cons(y0, cons(y1, y2))))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), RM(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(cons(s(z0), cons(s(z1), z2)))) S tuples: MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2)) RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2)) RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2)) RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2)) MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2)) RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2)) RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2)) RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2))) IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1))) IF2(true, 0, 0, x0) -> c17(RM(0, x0)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2)) IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0)) IF2(false, 0, s(x0), x1) -> c18(RM(0, x1)) IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2)) IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1)) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(z0, cons(z0, nil))) MIN(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c12(LE(s(s(y0)), s(s(y1)))) RM(s(s(y0)), cons(s(s(y1)), z2)) -> c16(EQ(s(s(y0)), s(s(y1)))) IF1(true, 0, z0, cons(y0, y1)) -> c13(MIN(cons(0, cons(y0, y1)))) MIN(cons(0, cons(z0, cons(y1, y2)))) -> c12(IF1(true, 0, z0, cons(y1, y2))) IF1(true, s(0), s(z0), cons(0, y1)) -> c13(MIN(cons(s(0), cons(0, y1)))) IF1(true, s(0), s(z0), cons(s(y0), y1)) -> c13(MIN(cons(s(0), cons(s(y0), y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c13(MIN(cons(s(s(z0)), cons(0, y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c13(MIN(cons(s(s(z0)), cons(s(s(y1)), y2)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c13(MIN(cons(s(s(z0)), cons(s(0), y1)))) LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1)))) MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(true, 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1)))) MINSORT(cons(s(z0), cons(0, z1))) -> c9(MINSORT(rm(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), RM(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(cons(s(z0), cons(0, z1)))) MIN(cons(s(0), cons(s(z0), cons(0, y1)))) -> c12(IF1(true, s(0), s(z0), cons(0, y1))) MIN(cons(s(0), cons(s(z0), cons(s(y1), y2)))) -> c12(IF1(true, s(0), s(z0), cons(s(y1), y2))) EQ(s(s(s(y0))), s(s(s(y1)))) -> c6(EQ(s(s(y0)), s(s(y1)))) IF1(false, s(z0), 0, cons(y0, cons(y1, y2))) -> c14(MIN(cons(0, cons(y0, cons(y1, y2))))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), RM(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(cons(s(z0), cons(s(z1), z2)))) K tuples: MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil))) MINSORT(cons(0, cons(y0, y1))) -> c8(MIN(cons(0, cons(y0, y1)))) MINSORT(cons(s(y0), cons(0, y1))) -> c8(MIN(cons(s(y0), cons(0, y1)))) MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c8(MIN(cons(s(s(y0)), cons(s(s(y1)), y2)))) MINSORT(cons(s(y0), cons(s(y1), y2))) -> c8(MIN(cons(s(y0), cons(s(y1), y2)))) MINSORT(cons(s(0), cons(s(y0), y1))) -> c8(MIN(cons(s(0), cons(s(y0), y1)))) MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c8(MIN(cons(s(s(y0)), cons(s(0), y1)))) Defined Rule Symbols: rm_2, min_1, if1_4, le_2, if2_4, eq_2 Defined Pair Symbols: MIN_1, RM_2, IF1_4, IF2_4, MINSORT_1, LE_2, EQ_2 Compound Symbols: c12_1, c16_1, c12_2, c16_2, c14_1, c17_1, c18_1, c9_3, c8_1, c9_1, c9_2, c13_1, c2_1, c6_1 ---------------------------------------- (163) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace MIN(cons(s(z0), cons(0, x2))) -> c12(IF1(false, s(z0), 0, x2)) by MIN(cons(s(z0), cons(0, cons(y1, cons(y2, y3))))) -> c12(IF1(false, s(z0), 0, cons(y1, cons(y2, y3)))) ---------------------------------------- (164) Obligation: Complexity Dependency Tuples Problem Rules: rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) rm(z0, nil) -> nil min(cons(z0, nil)) -> z0 min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(cons(z0, z2)) if1(false, z0, z1, z2) -> min(cons(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) if2(true, z0, z1, z2) -> rm(z0, z2) if2(false, z0, z1, z2) -> cons(z1, rm(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: RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2)) RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2)) RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2)) MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2)) RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2)) RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2)) RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2))) IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1))) IF2(true, 0, 0, x0) -> c17(RM(0, x0)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2)) IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0)) IF2(false, 0, s(x0), x1) -> c18(RM(0, x1)) IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2)) IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1)) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(0, cons(y0, y1))) -> c8(MIN(cons(0, cons(y0, y1)))) MINSORT(cons(s(y0), cons(0, y1))) -> c8(MIN(cons(s(y0), cons(0, y1)))) MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c8(MIN(cons(s(s(y0)), cons(s(s(y1)), y2)))) MINSORT(cons(s(y0), cons(s(y1), y2))) -> c8(MIN(cons(s(y0), cons(s(y1), y2)))) MINSORT(cons(s(0), cons(s(y0), y1))) -> c8(MIN(cons(s(0), cons(s(y0), y1)))) MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c8(MIN(cons(s(s(y0)), cons(s(0), y1)))) MINSORT(cons(z0, cons(z1, z2))) -> c9(RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c9(RM(z0, cons(z0, nil))) MIN(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c12(LE(s(s(y0)), s(s(y1)))) RM(s(s(y0)), cons(s(s(y1)), z2)) -> c16(EQ(s(s(y0)), s(s(y1)))) IF1(true, 0, z0, cons(y0, y1)) -> c13(MIN(cons(0, cons(y0, y1)))) MIN(cons(0, cons(z0, cons(y1, y2)))) -> c12(IF1(true, 0, z0, cons(y1, y2))) IF1(true, s(0), s(z0), cons(0, y1)) -> c13(MIN(cons(s(0), cons(0, y1)))) IF1(true, s(0), s(z0), cons(s(y0), y1)) -> c13(MIN(cons(s(0), cons(s(y0), y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c13(MIN(cons(s(s(z0)), cons(0, y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c13(MIN(cons(s(s(z0)), cons(s(s(y1)), y2)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c13(MIN(cons(s(s(z0)), cons(s(0), y1)))) LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1)))) MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(true, 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1)))) MINSORT(cons(s(z0), cons(0, z1))) -> c9(MINSORT(rm(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), RM(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(cons(s(z0), cons(0, z1)))) MIN(cons(s(0), cons(s(z0), cons(0, y1)))) -> c12(IF1(true, s(0), s(z0), cons(0, y1))) MIN(cons(s(0), cons(s(z0), cons(s(y1), y2)))) -> c12(IF1(true, s(0), s(z0), cons(s(y1), y2))) EQ(s(s(s(y0))), s(s(s(y1)))) -> c6(EQ(s(s(y0)), s(s(y1)))) IF1(false, s(z0), 0, cons(y0, cons(y1, y2))) -> c14(MIN(cons(0, cons(y0, cons(y1, y2))))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), RM(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(cons(s(z0), cons(s(z1), z2)))) MIN(cons(s(z0), cons(0, cons(y1, cons(y2, y3))))) -> c12(IF1(false, s(z0), 0, cons(y1, cons(y2, y3)))) S tuples: RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2)) RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2)) RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2)) MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2)) RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2)) RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2)) RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2)) IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2))) IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1))) IF2(true, 0, 0, x0) -> c17(RM(0, x0)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2)) IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0)) IF2(false, 0, s(x0), x1) -> c18(RM(0, x1)) IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2)) IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1)) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(z0, cons(z0, nil))) MIN(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c12(LE(s(s(y0)), s(s(y1)))) RM(s(s(y0)), cons(s(s(y1)), z2)) -> c16(EQ(s(s(y0)), s(s(y1)))) IF1(true, 0, z0, cons(y0, y1)) -> c13(MIN(cons(0, cons(y0, y1)))) MIN(cons(0, cons(z0, cons(y1, y2)))) -> c12(IF1(true, 0, z0, cons(y1, y2))) IF1(true, s(0), s(z0), cons(0, y1)) -> c13(MIN(cons(s(0), cons(0, y1)))) IF1(true, s(0), s(z0), cons(s(y0), y1)) -> c13(MIN(cons(s(0), cons(s(y0), y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c13(MIN(cons(s(s(z0)), cons(0, y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c13(MIN(cons(s(s(z0)), cons(s(s(y1)), y2)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c13(MIN(cons(s(s(z0)), cons(s(0), y1)))) LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1)))) MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(true, 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1)))) MINSORT(cons(s(z0), cons(0, z1))) -> c9(MINSORT(rm(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), RM(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(cons(s(z0), cons(0, z1)))) MIN(cons(s(0), cons(s(z0), cons(0, y1)))) -> c12(IF1(true, s(0), s(z0), cons(0, y1))) MIN(cons(s(0), cons(s(z0), cons(s(y1), y2)))) -> c12(IF1(true, s(0), s(z0), cons(s(y1), y2))) EQ(s(s(s(y0))), s(s(s(y1)))) -> c6(EQ(s(s(y0)), s(s(y1)))) IF1(false, s(z0), 0, cons(y0, cons(y1, y2))) -> c14(MIN(cons(0, cons(y0, cons(y1, y2))))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), RM(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(cons(s(z0), cons(s(z1), z2)))) MIN(cons(s(z0), cons(0, cons(y1, cons(y2, y3))))) -> c12(IF1(false, s(z0), 0, cons(y1, cons(y2, y3)))) K tuples: MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil))) MINSORT(cons(0, cons(y0, y1))) -> c8(MIN(cons(0, cons(y0, y1)))) MINSORT(cons(s(y0), cons(0, y1))) -> c8(MIN(cons(s(y0), cons(0, y1)))) MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c8(MIN(cons(s(s(y0)), cons(s(s(y1)), y2)))) MINSORT(cons(s(y0), cons(s(y1), y2))) -> c8(MIN(cons(s(y0), cons(s(y1), y2)))) MINSORT(cons(s(0), cons(s(y0), y1))) -> c8(MIN(cons(s(0), cons(s(y0), y1)))) MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c8(MIN(cons(s(s(y0)), cons(s(0), y1)))) Defined Rule Symbols: rm_2, min_1, if1_4, le_2, if2_4, eq_2 Defined Pair Symbols: RM_2, MIN_1, IF1_4, IF2_4, MINSORT_1, LE_2, EQ_2 Compound Symbols: c16_1, c12_2, c16_2, c12_1, c14_1, c17_1, c18_1, c9_3, c8_1, c9_1, c9_2, c13_1, c2_1, c6_1 ---------------------------------------- (165) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace IF1(false, s(s(x0)), s(s(x1)), x2) -> c14(MIN(cons(s(s(x1)), x2))) by IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c14(MIN(cons(s(s(z1)), cons(s(s(y1)), y2)))) IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c14(MIN(cons(s(s(z1)), cons(s(0), y1)))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(y1, cons(y2, y3)))) -> c14(MIN(cons(s(s(z1)), cons(0, cons(y1, cons(y2, y3)))))) ---------------------------------------- (166) Obligation: Complexity Dependency Tuples Problem Rules: rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) rm(z0, nil) -> nil min(cons(z0, nil)) -> z0 min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(cons(z0, z2)) if1(false, z0, z1, z2) -> min(cons(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) if2(true, z0, z1, z2) -> rm(z0, z2) if2(false, z0, z1, z2) -> cons(z1, rm(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: RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2)) RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2)) RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2)) MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2)) RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2)) RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2)) RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2)) IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1))) IF2(true, 0, 0, x0) -> c17(RM(0, x0)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2)) IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0)) IF2(false, 0, s(x0), x1) -> c18(RM(0, x1)) IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2)) IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1)) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(0, cons(y0, y1))) -> c8(MIN(cons(0, cons(y0, y1)))) MINSORT(cons(s(y0), cons(0, y1))) -> c8(MIN(cons(s(y0), cons(0, y1)))) MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c8(MIN(cons(s(s(y0)), cons(s(s(y1)), y2)))) MINSORT(cons(s(y0), cons(s(y1), y2))) -> c8(MIN(cons(s(y0), cons(s(y1), y2)))) MINSORT(cons(s(0), cons(s(y0), y1))) -> c8(MIN(cons(s(0), cons(s(y0), y1)))) MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c8(MIN(cons(s(s(y0)), cons(s(0), y1)))) MINSORT(cons(z0, cons(z1, z2))) -> c9(RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c9(RM(z0, cons(z0, nil))) MIN(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c12(LE(s(s(y0)), s(s(y1)))) RM(s(s(y0)), cons(s(s(y1)), z2)) -> c16(EQ(s(s(y0)), s(s(y1)))) IF1(true, 0, z0, cons(y0, y1)) -> c13(MIN(cons(0, cons(y0, y1)))) MIN(cons(0, cons(z0, cons(y1, y2)))) -> c12(IF1(true, 0, z0, cons(y1, y2))) IF1(true, s(0), s(z0), cons(0, y1)) -> c13(MIN(cons(s(0), cons(0, y1)))) IF1(true, s(0), s(z0), cons(s(y0), y1)) -> c13(MIN(cons(s(0), cons(s(y0), y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c13(MIN(cons(s(s(z0)), cons(0, y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c13(MIN(cons(s(s(z0)), cons(s(s(y1)), y2)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c13(MIN(cons(s(s(z0)), cons(s(0), y1)))) LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1)))) MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(true, 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1)))) MINSORT(cons(s(z0), cons(0, z1))) -> c9(MINSORT(rm(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), RM(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(cons(s(z0), cons(0, z1)))) MIN(cons(s(0), cons(s(z0), cons(0, y1)))) -> c12(IF1(true, s(0), s(z0), cons(0, y1))) MIN(cons(s(0), cons(s(z0), cons(s(y1), y2)))) -> c12(IF1(true, s(0), s(z0), cons(s(y1), y2))) EQ(s(s(s(y0))), s(s(s(y1)))) -> c6(EQ(s(s(y0)), s(s(y1)))) IF1(false, s(z0), 0, cons(y0, cons(y1, y2))) -> c14(MIN(cons(0, cons(y0, cons(y1, y2))))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), RM(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(cons(s(z0), cons(s(z1), z2)))) MIN(cons(s(z0), cons(0, cons(y1, cons(y2, y3))))) -> c12(IF1(false, s(z0), 0, cons(y1, cons(y2, y3)))) IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c14(MIN(cons(s(s(z1)), cons(s(s(y1)), y2)))) IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c14(MIN(cons(s(s(z1)), cons(s(0), y1)))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(y1, cons(y2, y3)))) -> c14(MIN(cons(s(s(z1)), cons(0, cons(y1, cons(y2, y3)))))) S tuples: RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2)) RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2)) RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2)) MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2)) RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2)) RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2)) RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2)) IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1))) IF2(true, 0, 0, x0) -> c17(RM(0, x0)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2)) IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0)) IF2(false, 0, s(x0), x1) -> c18(RM(0, x1)) IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2)) IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1)) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(z0, cons(z0, nil))) MIN(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c12(LE(s(s(y0)), s(s(y1)))) RM(s(s(y0)), cons(s(s(y1)), z2)) -> c16(EQ(s(s(y0)), s(s(y1)))) IF1(true, 0, z0, cons(y0, y1)) -> c13(MIN(cons(0, cons(y0, y1)))) MIN(cons(0, cons(z0, cons(y1, y2)))) -> c12(IF1(true, 0, z0, cons(y1, y2))) IF1(true, s(0), s(z0), cons(0, y1)) -> c13(MIN(cons(s(0), cons(0, y1)))) IF1(true, s(0), s(z0), cons(s(y0), y1)) -> c13(MIN(cons(s(0), cons(s(y0), y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c13(MIN(cons(s(s(z0)), cons(0, y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c13(MIN(cons(s(s(z0)), cons(s(s(y1)), y2)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c13(MIN(cons(s(s(z0)), cons(s(0), y1)))) LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1)))) MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(true, 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1)))) MINSORT(cons(s(z0), cons(0, z1))) -> c9(MINSORT(rm(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), RM(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(cons(s(z0), cons(0, z1)))) MIN(cons(s(0), cons(s(z0), cons(0, y1)))) -> c12(IF1(true, s(0), s(z0), cons(0, y1))) MIN(cons(s(0), cons(s(z0), cons(s(y1), y2)))) -> c12(IF1(true, s(0), s(z0), cons(s(y1), y2))) EQ(s(s(s(y0))), s(s(s(y1)))) -> c6(EQ(s(s(y0)), s(s(y1)))) IF1(false, s(z0), 0, cons(y0, cons(y1, y2))) -> c14(MIN(cons(0, cons(y0, cons(y1, y2))))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), RM(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(cons(s(z0), cons(s(z1), z2)))) MIN(cons(s(z0), cons(0, cons(y1, cons(y2, y3))))) -> c12(IF1(false, s(z0), 0, cons(y1, cons(y2, y3)))) IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c14(MIN(cons(s(s(z1)), cons(s(s(y1)), y2)))) IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c14(MIN(cons(s(s(z1)), cons(s(0), y1)))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(y1, cons(y2, y3)))) -> c14(MIN(cons(s(s(z1)), cons(0, cons(y1, cons(y2, y3)))))) K tuples: MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil))) MINSORT(cons(0, cons(y0, y1))) -> c8(MIN(cons(0, cons(y0, y1)))) MINSORT(cons(s(y0), cons(0, y1))) -> c8(MIN(cons(s(y0), cons(0, y1)))) MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c8(MIN(cons(s(s(y0)), cons(s(s(y1)), y2)))) MINSORT(cons(s(y0), cons(s(y1), y2))) -> c8(MIN(cons(s(y0), cons(s(y1), y2)))) MINSORT(cons(s(0), cons(s(y0), y1))) -> c8(MIN(cons(s(0), cons(s(y0), y1)))) MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c8(MIN(cons(s(s(y0)), cons(s(0), y1)))) Defined Rule Symbols: rm_2, min_1, if1_4, le_2, if2_4, eq_2 Defined Pair Symbols: RM_2, MIN_1, IF1_4, IF2_4, MINSORT_1, LE_2, EQ_2 Compound Symbols: c16_1, c12_2, c16_2, c12_1, c14_1, c17_1, c18_1, c9_3, c8_1, c9_1, c9_2, c13_1, c2_1, c6_1 ---------------------------------------- (167) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace MIN(cons(s(s(z0)), cons(s(s(z1)), x2))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) by MIN(cons(s(s(z0)), cons(s(s(z1)), cons(0, y3)))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), cons(0, y3)), LE(s(s(z0)), s(s(z1)))) MIN(cons(s(s(z0)), cons(s(s(z1)), cons(s(s(y3)), y4)))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), cons(s(s(y3)), y4)), LE(s(s(z0)), s(s(z1)))) MIN(cons(s(s(z0)), cons(s(s(z1)), cons(s(0), y3)))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), cons(s(0), y3)), LE(s(s(z0)), s(s(z1)))) MIN(cons(s(s(s(y0))), cons(s(s(s(y1))), z2))) -> c12(IF1(le(s(y0), s(y1)), s(s(s(y0))), s(s(s(y1))), z2), LE(s(s(s(y0))), s(s(s(y1))))) MIN(cons(s(s(z0)), cons(s(s(z1)), cons(0, cons(y3, cons(y4, y5)))))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(y3, cons(y4, y5)))), LE(s(s(z0)), s(s(z1)))) ---------------------------------------- (168) Obligation: Complexity Dependency Tuples Problem Rules: rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) rm(z0, nil) -> nil min(cons(z0, nil)) -> z0 min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(cons(z0, z2)) if1(false, z0, z1, z2) -> min(cons(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) if2(true, z0, z1, z2) -> rm(z0, z2) if2(false, z0, z1, z2) -> cons(z1, rm(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: RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2)) RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2)) RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2)) RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2)) RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2)) RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2)) RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2)) IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1))) IF2(true, 0, 0, x0) -> c17(RM(0, x0)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2)) IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0)) IF2(false, 0, s(x0), x1) -> c18(RM(0, x1)) IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2)) IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1)) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(0, cons(y0, y1))) -> c8(MIN(cons(0, cons(y0, y1)))) MINSORT(cons(s(y0), cons(0, y1))) -> c8(MIN(cons(s(y0), cons(0, y1)))) MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c8(MIN(cons(s(s(y0)), cons(s(s(y1)), y2)))) MINSORT(cons(s(y0), cons(s(y1), y2))) -> c8(MIN(cons(s(y0), cons(s(y1), y2)))) MINSORT(cons(s(0), cons(s(y0), y1))) -> c8(MIN(cons(s(0), cons(s(y0), y1)))) MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c8(MIN(cons(s(s(y0)), cons(s(0), y1)))) MINSORT(cons(z0, cons(z1, z2))) -> c9(RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c9(RM(z0, cons(z0, nil))) MIN(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c12(LE(s(s(y0)), s(s(y1)))) RM(s(s(y0)), cons(s(s(y1)), z2)) -> c16(EQ(s(s(y0)), s(s(y1)))) IF1(true, 0, z0, cons(y0, y1)) -> c13(MIN(cons(0, cons(y0, y1)))) MIN(cons(0, cons(z0, cons(y1, y2)))) -> c12(IF1(true, 0, z0, cons(y1, y2))) IF1(true, s(0), s(z0), cons(0, y1)) -> c13(MIN(cons(s(0), cons(0, y1)))) IF1(true, s(0), s(z0), cons(s(y0), y1)) -> c13(MIN(cons(s(0), cons(s(y0), y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c13(MIN(cons(s(s(z0)), cons(0, y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c13(MIN(cons(s(s(z0)), cons(s(s(y1)), y2)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c13(MIN(cons(s(s(z0)), cons(s(0), y1)))) LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1)))) MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(true, 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1)))) MINSORT(cons(s(z0), cons(0, z1))) -> c9(MINSORT(rm(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), RM(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(cons(s(z0), cons(0, z1)))) MIN(cons(s(0), cons(s(z0), cons(0, y1)))) -> c12(IF1(true, s(0), s(z0), cons(0, y1))) MIN(cons(s(0), cons(s(z0), cons(s(y1), y2)))) -> c12(IF1(true, s(0), s(z0), cons(s(y1), y2))) EQ(s(s(s(y0))), s(s(s(y1)))) -> c6(EQ(s(s(y0)), s(s(y1)))) IF1(false, s(z0), 0, cons(y0, cons(y1, y2))) -> c14(MIN(cons(0, cons(y0, cons(y1, y2))))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), RM(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(cons(s(z0), cons(s(z1), z2)))) MIN(cons(s(z0), cons(0, cons(y1, cons(y2, y3))))) -> c12(IF1(false, s(z0), 0, cons(y1, cons(y2, y3)))) IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c14(MIN(cons(s(s(z1)), cons(s(s(y1)), y2)))) IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c14(MIN(cons(s(s(z1)), cons(s(0), y1)))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(y1, cons(y2, y3)))) -> c14(MIN(cons(s(s(z1)), cons(0, cons(y1, cons(y2, y3)))))) MIN(cons(s(s(z0)), cons(s(s(z1)), cons(0, y3)))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), cons(0, y3)), LE(s(s(z0)), s(s(z1)))) MIN(cons(s(s(z0)), cons(s(s(z1)), cons(s(s(y3)), y4)))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), cons(s(s(y3)), y4)), LE(s(s(z0)), s(s(z1)))) MIN(cons(s(s(z0)), cons(s(s(z1)), cons(s(0), y3)))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), cons(s(0), y3)), LE(s(s(z0)), s(s(z1)))) MIN(cons(s(s(s(y0))), cons(s(s(s(y1))), z2))) -> c12(IF1(le(s(y0), s(y1)), s(s(s(y0))), s(s(s(y1))), z2), LE(s(s(s(y0))), s(s(s(y1))))) MIN(cons(s(s(z0)), cons(s(s(z1)), cons(0, cons(y3, cons(y4, y5)))))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(y3, cons(y4, y5)))), LE(s(s(z0)), s(s(z1)))) S tuples: RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2)) RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2)) RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2)) RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2)) RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2)) RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2)) RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2)) IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1))) IF2(true, 0, 0, x0) -> c17(RM(0, x0)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2)) IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0)) IF2(false, 0, s(x0), x1) -> c18(RM(0, x1)) IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2)) IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1)) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(z0, cons(z0, nil))) MIN(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c12(LE(s(s(y0)), s(s(y1)))) RM(s(s(y0)), cons(s(s(y1)), z2)) -> c16(EQ(s(s(y0)), s(s(y1)))) IF1(true, 0, z0, cons(y0, y1)) -> c13(MIN(cons(0, cons(y0, y1)))) MIN(cons(0, cons(z0, cons(y1, y2)))) -> c12(IF1(true, 0, z0, cons(y1, y2))) IF1(true, s(0), s(z0), cons(0, y1)) -> c13(MIN(cons(s(0), cons(0, y1)))) IF1(true, s(0), s(z0), cons(s(y0), y1)) -> c13(MIN(cons(s(0), cons(s(y0), y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c13(MIN(cons(s(s(z0)), cons(0, y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c13(MIN(cons(s(s(z0)), cons(s(s(y1)), y2)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c13(MIN(cons(s(s(z0)), cons(s(0), y1)))) LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1)))) MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(true, 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1)))) MINSORT(cons(s(z0), cons(0, z1))) -> c9(MINSORT(rm(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), RM(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(cons(s(z0), cons(0, z1)))) MIN(cons(s(0), cons(s(z0), cons(0, y1)))) -> c12(IF1(true, s(0), s(z0), cons(0, y1))) MIN(cons(s(0), cons(s(z0), cons(s(y1), y2)))) -> c12(IF1(true, s(0), s(z0), cons(s(y1), y2))) EQ(s(s(s(y0))), s(s(s(y1)))) -> c6(EQ(s(s(y0)), s(s(y1)))) IF1(false, s(z0), 0, cons(y0, cons(y1, y2))) -> c14(MIN(cons(0, cons(y0, cons(y1, y2))))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), RM(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(cons(s(z0), cons(s(z1), z2)))) MIN(cons(s(z0), cons(0, cons(y1, cons(y2, y3))))) -> c12(IF1(false, s(z0), 0, cons(y1, cons(y2, y3)))) IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c14(MIN(cons(s(s(z1)), cons(s(s(y1)), y2)))) IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c14(MIN(cons(s(s(z1)), cons(s(0), y1)))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(y1, cons(y2, y3)))) -> c14(MIN(cons(s(s(z1)), cons(0, cons(y1, cons(y2, y3)))))) MIN(cons(s(s(z0)), cons(s(s(z1)), cons(0, y3)))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), cons(0, y3)), LE(s(s(z0)), s(s(z1)))) MIN(cons(s(s(z0)), cons(s(s(z1)), cons(s(s(y3)), y4)))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), cons(s(s(y3)), y4)), LE(s(s(z0)), s(s(z1)))) MIN(cons(s(s(z0)), cons(s(s(z1)), cons(s(0), y3)))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), cons(s(0), y3)), LE(s(s(z0)), s(s(z1)))) MIN(cons(s(s(s(y0))), cons(s(s(s(y1))), z2))) -> c12(IF1(le(s(y0), s(y1)), s(s(s(y0))), s(s(s(y1))), z2), LE(s(s(s(y0))), s(s(s(y1))))) MIN(cons(s(s(z0)), cons(s(s(z1)), cons(0, cons(y3, cons(y4, y5)))))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(y3, cons(y4, y5)))), LE(s(s(z0)), s(s(z1)))) K tuples: MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil))) MINSORT(cons(0, cons(y0, y1))) -> c8(MIN(cons(0, cons(y0, y1)))) MINSORT(cons(s(y0), cons(0, y1))) -> c8(MIN(cons(s(y0), cons(0, y1)))) MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c8(MIN(cons(s(s(y0)), cons(s(s(y1)), y2)))) MINSORT(cons(s(y0), cons(s(y1), y2))) -> c8(MIN(cons(s(y0), cons(s(y1), y2)))) MINSORT(cons(s(0), cons(s(y0), y1))) -> c8(MIN(cons(s(0), cons(s(y0), y1)))) MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c8(MIN(cons(s(s(y0)), cons(s(0), y1)))) Defined Rule Symbols: rm_2, min_1, if1_4, le_2, if2_4, eq_2 Defined Pair Symbols: RM_2, MIN_1, IF1_4, IF2_4, MINSORT_1, LE_2, EQ_2 Compound Symbols: c16_1, c16_2, c12_1, c14_1, c17_1, c18_1, c9_3, c8_1, c9_1, c9_2, c13_1, c2_1, c6_1, c12_2 ---------------------------------------- (169) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (170) Obligation: Complexity Dependency Tuples Problem Rules: rm(z0, cons(z1, z2)) -> if2(eq(z0, z1), z0, z1, z2) rm(z0, nil) -> nil min(cons(z0, nil)) -> z0 min(cons(z0, cons(z1, z2))) -> if1(le(z0, z1), z0, z1, z2) if1(true, z0, z1, z2) -> min(cons(z0, z2)) if1(false, z0, z1, z2) -> min(cons(z1, z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) if2(true, z0, z1, z2) -> rm(z0, z2) if2(false, z0, z1, z2) -> cons(z1, rm(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: RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2)) RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2)) RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2)) RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2)) RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2)) RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2)) RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2)) IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1))) IF2(true, 0, 0, x0) -> c17(RM(0, x0)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2)) IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0)) IF2(false, 0, s(x0), x1) -> c18(RM(0, x1)) IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2)) IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1)) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(0, cons(y0, y1))) -> c8(MIN(cons(0, cons(y0, y1)))) MINSORT(cons(s(y0), cons(0, y1))) -> c8(MIN(cons(s(y0), cons(0, y1)))) MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c8(MIN(cons(s(s(y0)), cons(s(s(y1)), y2)))) MINSORT(cons(s(y0), cons(s(y1), y2))) -> c8(MIN(cons(s(y0), cons(s(y1), y2)))) MINSORT(cons(s(0), cons(s(y0), y1))) -> c8(MIN(cons(s(0), cons(s(y0), y1)))) MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c8(MIN(cons(s(s(y0)), cons(s(0), y1)))) MINSORT(cons(z0, cons(z1, z2))) -> c9(RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c9(RM(z0, cons(z0, nil))) MIN(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c12(LE(s(s(y0)), s(s(y1)))) RM(s(s(y0)), cons(s(s(y1)), z2)) -> c16(EQ(s(s(y0)), s(s(y1)))) IF1(true, 0, z0, cons(y0, y1)) -> c13(MIN(cons(0, cons(y0, y1)))) MIN(cons(0, cons(z0, cons(y1, y2)))) -> c12(IF1(true, 0, z0, cons(y1, y2))) IF1(true, s(0), s(z0), cons(0, y1)) -> c13(MIN(cons(s(0), cons(0, y1)))) IF1(true, s(0), s(z0), cons(s(y0), y1)) -> c13(MIN(cons(s(0), cons(s(y0), y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c13(MIN(cons(s(s(z0)), cons(0, y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c13(MIN(cons(s(s(z0)), cons(s(s(y1)), y2)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c13(MIN(cons(s(s(z0)), cons(s(0), y1)))) LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1)))) MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(true, 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1)))) MINSORT(cons(s(z0), cons(0, z1))) -> c9(MINSORT(rm(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), RM(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(cons(s(z0), cons(0, z1)))) MIN(cons(s(0), cons(s(z0), cons(0, y1)))) -> c12(IF1(true, s(0), s(z0), cons(0, y1))) MIN(cons(s(0), cons(s(z0), cons(s(y1), y2)))) -> c12(IF1(true, s(0), s(z0), cons(s(y1), y2))) EQ(s(s(s(y0))), s(s(s(y1)))) -> c6(EQ(s(s(y0)), s(s(y1)))) IF1(false, s(z0), 0, cons(y0, cons(y1, y2))) -> c14(MIN(cons(0, cons(y0, cons(y1, y2))))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), RM(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(cons(s(z0), cons(s(z1), z2)))) MIN(cons(s(z0), cons(0, cons(y1, cons(y2, y3))))) -> c12(IF1(false, s(z0), 0, cons(y1, cons(y2, y3)))) IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c14(MIN(cons(s(s(z1)), cons(s(s(y1)), y2)))) IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c14(MIN(cons(s(s(z1)), cons(s(0), y1)))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(y1, cons(y2, y3)))) -> c14(MIN(cons(s(s(z1)), cons(0, cons(y1, cons(y2, y3)))))) MIN(cons(s(s(z0)), cons(s(s(z1)), cons(s(s(y3)), y4)))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), cons(s(s(y3)), y4)), LE(s(s(z0)), s(s(z1)))) MIN(cons(s(s(s(y0))), cons(s(s(s(y1))), z2))) -> c12(IF1(le(s(y0), s(y1)), s(s(s(y0))), s(s(s(y1))), z2), LE(s(s(s(y0))), s(s(s(y1))))) MIN(cons(s(s(z0)), cons(s(s(z1)), cons(0, y3)))) -> c(IF1(le(z0, z1), s(s(z0)), s(s(z1)), cons(0, y3))) MIN(cons(s(s(z0)), cons(s(s(z1)), cons(0, y3)))) -> c(LE(s(s(z0)), s(s(z1)))) MIN(cons(s(s(z0)), cons(s(s(z1)), cons(s(0), y3)))) -> c(IF1(le(z0, z1), s(s(z0)), s(s(z1)), cons(s(0), y3))) MIN(cons(s(s(z0)), cons(s(s(z1)), cons(s(0), y3)))) -> c(LE(s(s(z0)), s(s(z1)))) MIN(cons(s(s(z0)), cons(s(s(z1)), cons(0, cons(y3, cons(y4, y5)))))) -> c(IF1(le(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(y3, cons(y4, y5))))) MIN(cons(s(s(z0)), cons(s(s(z1)), cons(0, cons(y3, cons(y4, y5)))))) -> c(LE(s(s(z0)), s(s(z1)))) S tuples: RM(0, cons(0, x2)) -> c16(IF2(true, 0, 0, x2)) RM(0, cons(s(z0), x2)) -> c16(IF2(false, 0, s(z0), x2)) RM(s(z0), cons(0, x2)) -> c16(IF2(false, s(z0), 0, x2)) RM(s(s(z0)), cons(s(s(z1)), x2)) -> c16(IF2(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) MIN(cons(s(s(z0)), cons(s(0), x2))) -> c12(IF1(false, s(s(z0)), s(0), x2)) RM(s(0), cons(s(0), x2)) -> c16(IF2(true, s(0), s(0), x2)) RM(s(0), cons(s(s(z0)), x2)) -> c16(IF2(false, s(0), s(s(z0)), x2)) RM(s(s(z0)), cons(s(0), x2)) -> c16(IF2(false, s(s(z0)), s(0), x2)) IF1(false, s(s(x0)), s(0), x1) -> c14(MIN(cons(s(0), x1))) IF2(true, 0, 0, x0) -> c17(RM(0, x0)) IF2(true, s(s(x0)), s(s(x1)), x2) -> c17(RM(s(s(x0)), x2)) IF2(true, s(0), s(0), x0) -> c17(RM(s(0), x0)) IF2(false, 0, s(x0), x1) -> c18(RM(0, x1)) IF2(false, s(x0), 0, x1) -> c18(RM(s(x0), x1)) IF2(false, s(s(x0)), s(s(x1)), x2) -> c18(RM(s(s(x0)), x2)) IF2(false, s(0), s(s(x0)), x1) -> c18(RM(s(0), x1)) IF2(false, s(s(x0)), s(0), x1) -> c18(RM(s(s(x0)), x1)) MINSORT(cons(z0, cons(z1, z2))) -> c9(MINSORT(if2(eq(if1(le(z0, z1), z0, z1, z2), z0), if1(le(z0, z1), z0, z1, z2), z0, cons(z1, z2))), RM(if1(le(z0, z1), z0, z1, z2), cons(z0, cons(z1, z2))), MIN(cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c9(MINSORT(if2(eq(z0, z0), z0, z0, nil)), RM(z0, cons(z0, nil))) MIN(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c12(LE(s(s(y0)), s(s(y1)))) RM(s(s(y0)), cons(s(s(y1)), z2)) -> c16(EQ(s(s(y0)), s(s(y1)))) IF1(true, 0, z0, cons(y0, y1)) -> c13(MIN(cons(0, cons(y0, y1)))) MIN(cons(0, cons(z0, cons(y1, y2)))) -> c12(IF1(true, 0, z0, cons(y1, y2))) IF1(true, s(0), s(z0), cons(0, y1)) -> c13(MIN(cons(s(0), cons(0, y1)))) IF1(true, s(0), s(z0), cons(s(y0), y1)) -> c13(MIN(cons(s(0), cons(s(y0), y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(0, y1)) -> c13(MIN(cons(s(s(z0)), cons(0, y1)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c13(MIN(cons(s(s(z0)), cons(s(s(y1)), y2)))) IF1(true, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c13(MIN(cons(s(s(z0)), cons(s(0), y1)))) LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1)))) MINSORT(cons(0, cons(z0, z1))) -> c9(MINSORT(rm(if1(true, 0, z0, z1), cons(0, cons(z0, z1)))), RM(if1(true, 0, z0, z1), cons(0, cons(z0, z1))), MIN(cons(0, cons(z0, z1)))) MINSORT(cons(s(z0), cons(0, z1))) -> c9(MINSORT(rm(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1)))), RM(if1(false, s(z0), 0, z1), cons(s(z0), cons(0, z1))), MIN(cons(s(z0), cons(0, z1)))) MIN(cons(s(0), cons(s(z0), cons(0, y1)))) -> c12(IF1(true, s(0), s(z0), cons(0, y1))) MIN(cons(s(0), cons(s(z0), cons(s(y1), y2)))) -> c12(IF1(true, s(0), s(z0), cons(s(y1), y2))) EQ(s(s(s(y0))), s(s(s(y1)))) -> c6(EQ(s(s(y0)), s(s(y1)))) IF1(false, s(z0), 0, cons(y0, cons(y1, y2))) -> c14(MIN(cons(0, cons(y0, cons(y1, y2))))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c9(MINSORT(rm(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2)))), RM(if1(le(z0, z1), s(z0), s(z1), z2), cons(s(z0), cons(s(z1), z2))), MIN(cons(s(z0), cons(s(z1), z2)))) MIN(cons(s(z0), cons(0, cons(y1, cons(y2, y3))))) -> c12(IF1(false, s(z0), 0, cons(y1, cons(y2, y3)))) IF1(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c14(MIN(cons(s(s(z1)), cons(s(s(y1)), y2)))) IF1(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c14(MIN(cons(s(s(z1)), cons(s(0), y1)))) IF1(false, s(s(z0)), s(s(z1)), cons(0, cons(y1, cons(y2, y3)))) -> c14(MIN(cons(s(s(z1)), cons(0, cons(y1, cons(y2, y3)))))) MIN(cons(s(s(z0)), cons(s(s(z1)), cons(s(s(y3)), y4)))) -> c12(IF1(le(z0, z1), s(s(z0)), s(s(z1)), cons(s(s(y3)), y4)), LE(s(s(z0)), s(s(z1)))) MIN(cons(s(s(s(y0))), cons(s(s(s(y1))), z2))) -> c12(IF1(le(s(y0), s(y1)), s(s(s(y0))), s(s(s(y1))), z2), LE(s(s(s(y0))), s(s(s(y1))))) MIN(cons(s(s(z0)), cons(s(s(z1)), cons(0, y3)))) -> c(IF1(le(z0, z1), s(s(z0)), s(s(z1)), cons(0, y3))) MIN(cons(s(s(z0)), cons(s(s(z1)), cons(0, y3)))) -> c(LE(s(s(z0)), s(s(z1)))) MIN(cons(s(s(z0)), cons(s(s(z1)), cons(s(0), y3)))) -> c(IF1(le(z0, z1), s(s(z0)), s(s(z1)), cons(s(0), y3))) MIN(cons(s(s(z0)), cons(s(s(z1)), cons(s(0), y3)))) -> c(LE(s(s(z0)), s(s(z1)))) MIN(cons(s(s(z0)), cons(s(s(z1)), cons(0, cons(y3, cons(y4, y5)))))) -> c(IF1(le(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(y3, cons(y4, y5))))) MIN(cons(s(s(z0)), cons(s(s(z1)), cons(0, cons(y3, cons(y4, y5)))))) -> c(LE(s(s(z0)), s(s(z1)))) K tuples: MINSORT(cons(x0, cons(x1, x2))) -> c9(RM(min(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, nil)) -> c9(RM(min(cons(x0, nil)), cons(x0, nil))) MINSORT(cons(0, cons(y0, y1))) -> c8(MIN(cons(0, cons(y0, y1)))) MINSORT(cons(s(y0), cons(0, y1))) -> c8(MIN(cons(s(y0), cons(0, y1)))) MINSORT(cons(s(s(y0)), cons(s(s(y1)), y2))) -> c8(MIN(cons(s(s(y0)), cons(s(s(y1)), y2)))) MINSORT(cons(s(y0), cons(s(y1), y2))) -> c8(MIN(cons(s(y0), cons(s(y1), y2)))) MINSORT(cons(s(0), cons(s(y0), y1))) -> c8(MIN(cons(s(0), cons(s(y0), y1)))) MINSORT(cons(s(s(y0)), cons(s(0), y1))) -> c8(MIN(cons(s(s(y0)), cons(s(0), y1)))) Defined Rule Symbols: rm_2, min_1, if1_4, le_2, if2_4, eq_2 Defined Pair Symbols: RM_2, MIN_1, IF1_4, IF2_4, MINSORT_1, LE_2, EQ_2 Compound Symbols: c16_1, c16_2, c12_1, c14_1, c17_1, c18_1, c9_3, c8_1, c9_1, c9_2, c13_1, c2_1, c6_1, c12_2, c_1