KILLED proof of input_1DQUF7bWzY.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). (0) CpxTRS (1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxTRS (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (4) CpxTRS (5) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxWeightedTrs (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (8) CpxTypedWeightedTrs (9) CompletionProof [UPPER BOUND(ID), 0 ms] (10) CpxTypedWeightedCompleteTrs (11) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (12) CpxTypedWeightedCompleteTrs (13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 8 ms] (14) CpxRNTS (15) InliningProof [UPPER BOUND(ID), 668 ms] (16) CpxRNTS (17) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (18) CpxRNTS (19) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms] (20) CpxRNTS (21) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (22) CpxRNTS (23) IntTrsBoundProof [UPPER BOUND(ID), 350 ms] (24) CpxRNTS (25) IntTrsBoundProof [UPPER BOUND(ID), 140 ms] (26) CpxRNTS (27) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (28) CpxRNTS (29) IntTrsBoundProof [UPPER BOUND(ID), 182 ms] (30) CpxRNTS (31) IntTrsBoundProof [UPPER BOUND(ID), 73 ms] (32) CpxRNTS (33) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (34) CpxRNTS (35) IntTrsBoundProof [UPPER BOUND(ID), 465 ms] (36) CpxRNTS (37) IntTrsBoundProof [UPPER BOUND(ID), 145 ms] (38) CpxRNTS (39) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (40) CpxRNTS (41) IntTrsBoundProof [UPPER BOUND(ID), 1647 ms] (42) CpxRNTS (43) IntTrsBoundProof [UPPER BOUND(ID), 226 ms] (44) CpxRNTS (45) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (46) CpxRNTS (47) IntTrsBoundProof [UPPER BOUND(ID), 4093 ms] (48) CpxRNTS (49) IntTrsBoundProof [UPPER BOUND(ID), 932 ms] (50) CpxRNTS (51) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (52) CpxRNTS (53) IntTrsBoundProof [UPPER BOUND(ID), 1431 ms] (54) CpxRNTS (55) IntTrsBoundProof [UPPER BOUND(ID), 187 ms] (56) CpxRNTS (57) CompletionProof [UPPER BOUND(ID), 0 ms] (58) CpxTypedWeightedCompleteTrs (59) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (60) CpxRNTS (61) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (62) CdtProblem (63) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (64) CdtProblem (65) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (66) CdtProblem (67) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (68) CdtProblem (69) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 153 ms] (70) CdtProblem (71) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (72) CdtProblem (73) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (74) CdtProblem (75) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (76) CdtProblem (77) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (78) CdtProblem (79) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 65 ms] (80) CdtProblem (81) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (82) CdtProblem (83) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (84) CdtProblem (85) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 386 ms] (86) CdtProblem (87) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (88) CdtProblem (89) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 315 ms] (90) CdtProblem (91) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (92) CdtProblem (93) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (94) CdtProblem (95) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (96) CdtProblem (97) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (98) CdtProblem (99) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (100) CdtProblem (101) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (102) CdtProblem (103) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (104) CdtProblem (105) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (106) CdtProblem (107) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (108) CdtProblem (109) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (110) CdtProblem (111) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (112) CdtProblem (113) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (114) CdtProblem (115) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (116) CdtProblem (117) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (118) CdtProblem (119) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (120) CdtProblem (121) 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) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (132) CdtProblem (133) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (134) CdtProblem (135) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (136) CdtProblem (137) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (138) CdtProblem (139) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (140) CdtProblem (141) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (142) CdtProblem (143) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (144) CdtProblem (145) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (146) CdtProblem (147) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (148) CdtProblem (149) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (150) CdtProblem (151) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (152) CdtProblem (153) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (154) CdtProblem (155) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (156) CdtProblem (157) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (158) CdtProblem (159) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (160) CdtProblem (161) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (162) CdtProblem (163) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (164) CdtProblem (165) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (166) CdtProblem (167) CdtRewritingProof [BOTH BOUNDS(ID, ID), 2 ms] (168) CdtProblem (169) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (170) CdtProblem (171) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (172) CdtProblem ---------------------------------------- (0) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: 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) if(true, x, y) -> x if(false, x, y) -> y minsort(nil) -> nil minsort(cons(x, y)) -> cons(min(x, y), minsort(del(min(x, y), cons(x, y)))) min(x, nil) -> x min(x, cons(y, z)) -> if(le(x, y), min(x, z), min(y, z)) del(x, nil) -> nil del(x, cons(y, z)) -> if(eq(x, y), z, cons(y, del(x, z))) S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (1) RenamingProof (BOTH BOUNDS(ID, ID)) Renamed function symbols to avoid clashes with predefined symbol. ---------------------------------------- (2) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: le(0', y) -> true le(s(x), 0') -> false le(s(x), s(y)) -> le(x, y) eq(0', 0') -> true eq(0', s(y)) -> false eq(s(x), 0') -> false eq(s(x), s(y)) -> eq(x, y) if(true, x, y) -> x if(false, x, y) -> y minsort(nil) -> nil minsort(cons(x, y)) -> cons(min(x, y), minsort(del(min(x, y), cons(x, y)))) min(x, nil) -> x min(x, cons(y, z)) -> if(le(x, y), min(x, z), min(y, z)) del(x, nil) -> nil del(x, cons(y, z)) -> if(eq(x, y), z, cons(y, del(x, z))) S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (3) RelTrsToTrsProof (UPPER BOUND(ID)) transformed relative TRS to TRS ---------------------------------------- (4) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: le(0, y) -> true le(s(x), 0) -> false le(s(x), s(y)) -> le(x, y) eq(0, 0) -> true eq(0, s(y)) -> false eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) if(true, x, y) -> x if(false, x, y) -> y minsort(nil) -> nil minsort(cons(x, y)) -> cons(min(x, y), minsort(del(min(x, y), cons(x, y)))) min(x, nil) -> x min(x, cons(y, z)) -> if(le(x, y), min(x, z), min(y, z)) del(x, nil) -> nil del(x, cons(y, z)) -> if(eq(x, y), z, cons(y, del(x, z))) S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (5) RelTrsToWeightedTrsProof (UPPER BOUND(ID)) Transformed relative TRS to weighted TRS ---------------------------------------- (6) Obligation: The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: le(0, y) -> true [1] le(s(x), 0) -> false [1] le(s(x), s(y)) -> le(x, y) [1] eq(0, 0) -> true [1] eq(0, s(y)) -> false [1] eq(s(x), 0) -> false [1] eq(s(x), s(y)) -> eq(x, y) [1] if(true, x, y) -> x [1] if(false, x, y) -> y [1] minsort(nil) -> nil [1] minsort(cons(x, y)) -> cons(min(x, y), minsort(del(min(x, y), cons(x, y)))) [1] min(x, nil) -> x [1] min(x, cons(y, z)) -> if(le(x, y), min(x, z), min(y, z)) [1] del(x, nil) -> nil [1] del(x, cons(y, z)) -> if(eq(x, y), z, cons(y, del(x, z))) [1] Rewrite Strategy: INNERMOST ---------------------------------------- (7) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (8) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: le(0, y) -> true [1] le(s(x), 0) -> false [1] le(s(x), s(y)) -> le(x, y) [1] eq(0, 0) -> true [1] eq(0, s(y)) -> false [1] eq(s(x), 0) -> false [1] eq(s(x), s(y)) -> eq(x, y) [1] if(true, x, y) -> x [1] if(false, x, y) -> y [1] minsort(nil) -> nil [1] minsort(cons(x, y)) -> cons(min(x, y), minsort(del(min(x, y), cons(x, y)))) [1] min(x, nil) -> x [1] min(x, cons(y, z)) -> if(le(x, y), min(x, z), min(y, z)) [1] del(x, nil) -> nil [1] del(x, cons(y, z)) -> if(eq(x, y), z, cons(y, del(x, z))) [1] The TRS has the following type information: le :: 0:s:nil:cons -> 0:s:nil:cons -> true:false 0 :: 0:s:nil:cons true :: true:false s :: 0:s:nil:cons -> 0:s:nil:cons false :: true:false eq :: 0:s:nil:cons -> 0:s:nil:cons -> true:false if :: true:false -> 0:s:nil:cons -> 0:s:nil:cons -> 0:s:nil:cons minsort :: 0:s:nil:cons -> 0:s:nil:cons nil :: 0:s:nil:cons cons :: 0:s:nil:cons -> 0:s:nil:cons -> 0:s:nil:cons min :: 0:s:nil:cons -> 0:s:nil:cons -> 0:s:nil:cons del :: 0:s:nil:cons -> 0:s:nil:cons -> 0:s:nil:cons Rewrite Strategy: INNERMOST ---------------------------------------- (9) CompletionProof (UPPER BOUND(ID)) The transformation into a RNTS is sound, since: (a) The obligation is a constructor system where every type has a constant constructor, (b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols: minsort_1 (c) The following functions are completely defined: le_2 min_2 del_2 eq_2 if_3 Due to the following rules being added: le(v0, v1) -> null_le [0] min(v0, v1) -> null_min [0] del(v0, v1) -> null_del [0] eq(v0, v1) -> null_eq [0] if(v0, v1, v2) -> null_if [0] And the following fresh constants: null_le, null_min, null_del, null_eq, null_if ---------------------------------------- (10) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: 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] if(true, x, y) -> x [1] if(false, x, y) -> y [1] minsort(nil) -> nil [1] minsort(cons(x, y)) -> cons(min(x, y), minsort(del(min(x, y), cons(x, y)))) [1] min(x, nil) -> x [1] min(x, cons(y, z)) -> if(le(x, y), min(x, z), min(y, z)) [1] del(x, nil) -> nil [1] del(x, cons(y, z)) -> if(eq(x, y), z, cons(y, del(x, z))) [1] le(v0, v1) -> null_le [0] min(v0, v1) -> null_min [0] del(v0, v1) -> null_del [0] eq(v0, v1) -> null_eq [0] if(v0, v1, v2) -> null_if [0] The TRS has the following type information: le :: 0:s:nil:cons:null_min:null_del:null_if -> 0:s:nil:cons:null_min:null_del:null_if -> true:false:null_le:null_eq 0 :: 0:s:nil:cons:null_min:null_del:null_if true :: true:false:null_le:null_eq s :: 0:s:nil:cons:null_min:null_del:null_if -> 0:s:nil:cons:null_min:null_del:null_if false :: true:false:null_le:null_eq eq :: 0:s:nil:cons:null_min:null_del:null_if -> 0:s:nil:cons:null_min:null_del:null_if -> true:false:null_le:null_eq if :: true:false:null_le:null_eq -> 0:s:nil:cons:null_min:null_del:null_if -> 0:s:nil:cons:null_min:null_del:null_if -> 0:s:nil:cons:null_min:null_del:null_if minsort :: 0:s:nil:cons:null_min:null_del:null_if -> 0:s:nil:cons:null_min:null_del:null_if nil :: 0:s:nil:cons:null_min:null_del:null_if cons :: 0:s:nil:cons:null_min:null_del:null_if -> 0:s:nil:cons:null_min:null_del:null_if -> 0:s:nil:cons:null_min:null_del:null_if min :: 0:s:nil:cons:null_min:null_del:null_if -> 0:s:nil:cons:null_min:null_del:null_if -> 0:s:nil:cons:null_min:null_del:null_if del :: 0:s:nil:cons:null_min:null_del:null_if -> 0:s:nil:cons:null_min:null_del:null_if -> 0:s:nil:cons:null_min:null_del:null_if null_le :: true:false:null_le:null_eq null_min :: 0:s:nil:cons:null_min:null_del:null_if null_del :: 0:s:nil:cons:null_min:null_del:null_if null_eq :: true:false:null_le:null_eq null_if :: 0:s:nil:cons:null_min:null_del:null_if Rewrite Strategy: INNERMOST ---------------------------------------- (11) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (12) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: 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] if(true, x, y) -> x [1] if(false, x, y) -> y [1] minsort(nil) -> nil [1] minsort(cons(x, nil)) -> cons(min(x, nil), minsort(del(x, cons(x, nil)))) [2] minsort(cons(x, cons(y', z'))) -> cons(min(x, cons(y', z')), minsort(del(if(le(x, y'), min(x, z'), min(y', z')), cons(x, cons(y', z'))))) [2] minsort(cons(x, y)) -> cons(min(x, y), minsort(del(null_min, cons(x, y)))) [1] min(x, nil) -> x [1] min(x, cons(y, z)) -> if(le(x, y), min(x, z), min(y, z)) [1] del(x, nil) -> nil [1] del(0, cons(0, nil)) -> if(true, nil, cons(0, nil)) [3] del(0, cons(0, cons(y11, z8))) -> if(true, cons(y11, z8), cons(0, if(eq(0, y11), z8, cons(y11, del(0, z8))))) [3] del(0, cons(0, z)) -> if(true, z, cons(0, null_del)) [2] del(0, cons(s(y9), nil)) -> if(false, nil, cons(s(y9), nil)) [3] del(0, cons(s(y9), cons(y12, z9))) -> if(false, cons(y12, z9), cons(s(y9), if(eq(0, y12), z9, cons(y12, del(0, z9))))) [3] del(0, cons(s(y9), z)) -> if(false, z, cons(s(y9), null_del)) [2] del(s(x1), cons(0, nil)) -> if(false, nil, cons(0, nil)) [3] del(s(x1), cons(0, cons(y13, z10))) -> if(false, cons(y13, z10), cons(0, if(eq(s(x1), y13), z10, cons(y13, del(s(x1), z10))))) [3] del(s(x1), cons(0, z)) -> if(false, z, cons(0, null_del)) [2] del(s(x2), cons(s(y10), nil)) -> if(eq(x2, y10), nil, cons(s(y10), nil)) [3] del(s(x2), cons(s(y10), cons(y14, z11))) -> if(eq(x2, y10), cons(y14, z11), cons(s(y10), if(eq(s(x2), y14), z11, cons(y14, del(s(x2), z11))))) [3] del(s(x2), cons(s(y10), z)) -> if(eq(x2, y10), z, cons(s(y10), null_del)) [2] del(x, cons(y, nil)) -> if(null_eq, nil, cons(y, nil)) [2] del(x, cons(y, cons(y15, z12))) -> if(null_eq, cons(y15, z12), cons(y, if(eq(x, y15), z12, cons(y15, del(x, z12))))) [2] del(x, cons(y, z)) -> if(null_eq, z, cons(y, null_del)) [1] le(v0, v1) -> null_le [0] min(v0, v1) -> null_min [0] del(v0, v1) -> null_del [0] eq(v0, v1) -> null_eq [0] if(v0, v1, v2) -> null_if [0] The TRS has the following type information: le :: 0:s:nil:cons:null_min:null_del:null_if -> 0:s:nil:cons:null_min:null_del:null_if -> true:false:null_le:null_eq 0 :: 0:s:nil:cons:null_min:null_del:null_if true :: true:false:null_le:null_eq s :: 0:s:nil:cons:null_min:null_del:null_if -> 0:s:nil:cons:null_min:null_del:null_if false :: true:false:null_le:null_eq eq :: 0:s:nil:cons:null_min:null_del:null_if -> 0:s:nil:cons:null_min:null_del:null_if -> true:false:null_le:null_eq if :: true:false:null_le:null_eq -> 0:s:nil:cons:null_min:null_del:null_if -> 0:s:nil:cons:null_min:null_del:null_if -> 0:s:nil:cons:null_min:null_del:null_if minsort :: 0:s:nil:cons:null_min:null_del:null_if -> 0:s:nil:cons:null_min:null_del:null_if nil :: 0:s:nil:cons:null_min:null_del:null_if cons :: 0:s:nil:cons:null_min:null_del:null_if -> 0:s:nil:cons:null_min:null_del:null_if -> 0:s:nil:cons:null_min:null_del:null_if min :: 0:s:nil:cons:null_min:null_del:null_if -> 0:s:nil:cons:null_min:null_del:null_if -> 0:s:nil:cons:null_min:null_del:null_if del :: 0:s:nil:cons:null_min:null_del:null_if -> 0:s:nil:cons:null_min:null_del:null_if -> 0:s:nil:cons:null_min:null_del:null_if null_le :: true:false:null_le:null_eq null_min :: 0:s:nil:cons:null_min:null_del:null_if null_del :: 0:s:nil:cons:null_min:null_del:null_if null_eq :: true:false:null_le:null_eq null_if :: 0:s:nil:cons:null_min:null_del:null_if Rewrite Strategy: INNERMOST ---------------------------------------- (13) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: 0 => 0 true => 2 false => 1 nil => 1 null_le => 0 null_min => 0 null_del => 0 null_eq => 0 null_if => 0 ---------------------------------------- (14) Obligation: Complexity RNTS consisting of the following rules: del(z'', z1) -{ 2 }-> if(eq(x2, y10), z, 1 + (1 + y10) + 0) :|: z >= 0, y10 >= 0, z'' = 1 + x2, x2 >= 0, z1 = 1 + (1 + y10) + z del(z'', z1) -{ 3 }-> if(eq(x2, y10), 1, 1 + (1 + y10) + 1) :|: y10 >= 0, z'' = 1 + x2, z1 = 1 + (1 + y10) + 1, x2 >= 0 del(z'', z1) -{ 3 }-> if(eq(x2, y10), 1 + y14 + z11, 1 + (1 + y10) + if(eq(1 + x2, y14), z11, 1 + y14 + del(1 + x2, z11))) :|: z11 >= 0, y10 >= 0, z'' = 1 + x2, z1 = 1 + (1 + y10) + (1 + y14 + z11), x2 >= 0, y14 >= 0 del(z'', z1) -{ 2 }-> if(2, z, 1 + 0 + 0) :|: z'' = 0, z >= 0, z1 = 1 + 0 + z del(z'', z1) -{ 3 }-> if(2, 1, 1 + 0 + 1) :|: z'' = 0, z1 = 1 + 0 + 1 del(z'', z1) -{ 3 }-> if(2, 1 + y11 + z8, 1 + 0 + if(eq(0, y11), z8, 1 + y11 + del(0, z8))) :|: z'' = 0, z1 = 1 + 0 + (1 + y11 + z8), y11 >= 0, z8 >= 0 del(z'', z1) -{ 2 }-> if(1, z, 1 + 0 + 0) :|: x1 >= 0, z >= 0, z'' = 1 + x1, z1 = 1 + 0 + z del(z'', z1) -{ 2 }-> if(1, z, 1 + (1 + y9) + 0) :|: z'' = 0, y9 >= 0, z >= 0, z1 = 1 + (1 + y9) + z del(z'', z1) -{ 3 }-> if(1, 1, 1 + 0 + 1) :|: x1 >= 0, z'' = 1 + x1, z1 = 1 + 0 + 1 del(z'', z1) -{ 3 }-> if(1, 1, 1 + (1 + y9) + 1) :|: z'' = 0, y9 >= 0, z1 = 1 + (1 + y9) + 1 del(z'', z1) -{ 3 }-> if(1, 1 + y12 + z9, 1 + (1 + y9) + if(eq(0, y12), z9, 1 + y12 + del(0, z9))) :|: z'' = 0, y9 >= 0, z9 >= 0, y12 >= 0, z1 = 1 + (1 + y9) + (1 + y12 + z9) del(z'', z1) -{ 3 }-> if(1, 1 + y13 + z10, 1 + 0 + if(eq(1 + x1, y13), z10, 1 + y13 + del(1 + x1, z10))) :|: z1 = 1 + 0 + (1 + y13 + z10), x1 >= 0, z'' = 1 + x1, z10 >= 0, y13 >= 0 del(z'', z1) -{ 1 }-> if(0, z, 1 + y + 0) :|: z >= 0, z1 = 1 + y + z, x >= 0, y >= 0, z'' = x del(z'', z1) -{ 2 }-> if(0, 1, 1 + y + 1) :|: x >= 0, y >= 0, z'' = x, z1 = 1 + y + 1 del(z'', z1) -{ 2 }-> if(0, 1 + y15 + z12, 1 + y + if(eq(x, y15), z12, 1 + y15 + del(x, z12))) :|: z1 = 1 + y + (1 + y15 + z12), x >= 0, y >= 0, z'' = x, z12 >= 0, y15 >= 0 del(z'', z1) -{ 1 }-> 1 :|: x >= 0, z1 = 1, z'' = x del(z'', z1) -{ 0 }-> 0 :|: z'' = v0, v0 >= 0, z1 = v1, v1 >= 0 eq(z'', z1) -{ 1 }-> eq(x, y) :|: x >= 0, y >= 0, z1 = 1 + y, z'' = 1 + x eq(z'', z1) -{ 1 }-> 2 :|: z'' = 0, z1 = 0 eq(z'', z1) -{ 1 }-> 1 :|: z'' = 0, y >= 0, z1 = 1 + y eq(z'', z1) -{ 1 }-> 1 :|: z1 = 0, x >= 0, z'' = 1 + x eq(z'', z1) -{ 0 }-> 0 :|: z'' = v0, v0 >= 0, z1 = v1, v1 >= 0 if(z'', z1, z2) -{ 1 }-> x :|: z2 = y, x >= 0, y >= 0, z'' = 2, z1 = x if(z'', z1, z2) -{ 1 }-> y :|: z2 = y, x >= 0, y >= 0, z'' = 1, z1 = x if(z'', z1, z2) -{ 0 }-> 0 :|: z'' = v0, z2 = v2, v0 >= 0, z1 = v1, v1 >= 0, v2 >= 0 le(z'', z1) -{ 1 }-> le(x, y) :|: x >= 0, y >= 0, z1 = 1 + y, z'' = 1 + x le(z'', z1) -{ 1 }-> 2 :|: z'' = 0, z1 = y, y >= 0 le(z'', z1) -{ 1 }-> 1 :|: z1 = 0, x >= 0, z'' = 1 + x le(z'', z1) -{ 0 }-> 0 :|: z'' = v0, v0 >= 0, z1 = v1, v1 >= 0 min(z'', z1) -{ 1 }-> x :|: x >= 0, z1 = 1, z'' = x min(z'', z1) -{ 1 }-> if(le(x, y), min(x, z), min(y, z)) :|: z >= 0, z1 = 1 + y + z, x >= 0, y >= 0, z'' = x min(z'', z1) -{ 0 }-> 0 :|: z'' = v0, v0 >= 0, z1 = v1, v1 >= 0 minsort(z'') -{ 1 }-> 1 :|: z'' = 1 minsort(z'') -{ 1 }-> 1 + min(x, y) + minsort(del(0, 1 + x + y)) :|: z'' = 1 + x + y, x >= 0, y >= 0 minsort(z'') -{ 2 }-> 1 + min(x, 1) + minsort(del(x, 1 + x + 1)) :|: x >= 0, z'' = 1 + x + 1 minsort(z'') -{ 2 }-> 1 + min(x, 1 + y' + z') + minsort(del(if(le(x, y'), min(x, z'), min(y', z')), 1 + x + (1 + y' + z'))) :|: z'' = 1 + x + (1 + y' + z'), x >= 0, z' >= 0, y' >= 0 ---------------------------------------- (15) InliningProof (UPPER BOUND(ID)) Inlined the following terminating rules on right-hand sides where appropriate: if(z'', z1, z2) -{ 1 }-> x :|: z2 = y, x >= 0, y >= 0, z'' = 2, z1 = x if(z'', z1, z2) -{ 1 }-> y :|: z2 = y, x >= 0, y >= 0, z'' = 1, z1 = x if(z'', z1, z2) -{ 0 }-> 0 :|: z'' = v0, z2 = v2, v0 >= 0, z1 = v1, v1 >= 0, v2 >= 0 ---------------------------------------- (16) Obligation: Complexity RNTS consisting of the following rules: del(z'', z1) -{ 4 }-> x :|: z'' = 0, z1 = 1 + 0 + 1, 1 + 0 + 1 = y, x >= 0, y >= 0, 2 = 2, 1 = x del(z'', z1) -{ 3 }-> x :|: z'' = 0, z >= 0, z1 = 1 + 0 + z, 1 + 0 + 0 = y, x >= 0, y >= 0, 2 = 2, z = x del(z'', z1) -{ 4 }-> y :|: z'' = 0, y9 >= 0, z1 = 1 + (1 + y9) + 1, 1 + (1 + y9) + 1 = y, x >= 0, y >= 0, 1 = 1, 1 = x del(z'', z1) -{ 3 }-> y :|: z'' = 0, y9 >= 0, z >= 0, z1 = 1 + (1 + y9) + z, 1 + (1 + y9) + 0 = y, x >= 0, y >= 0, 1 = 1, z = x del(z'', z1) -{ 4 }-> y :|: x1 >= 0, z'' = 1 + x1, z1 = 1 + 0 + 1, 1 + 0 + 1 = y, x >= 0, y >= 0, 1 = 1, 1 = x del(z'', z1) -{ 3 }-> y :|: x1 >= 0, z >= 0, z'' = 1 + x1, z1 = 1 + 0 + z, 1 + 0 + 0 = y, x >= 0, y >= 0, 1 = 1, z = x del(z'', z1) -{ 2 }-> if(eq(x2, y10), z, 1 + (1 + y10) + 0) :|: z >= 0, y10 >= 0, z'' = 1 + x2, x2 >= 0, z1 = 1 + (1 + y10) + z del(z'', z1) -{ 3 }-> if(eq(x2, y10), 1, 1 + (1 + y10) + 1) :|: y10 >= 0, z'' = 1 + x2, z1 = 1 + (1 + y10) + 1, x2 >= 0 del(z'', z1) -{ 3 }-> if(eq(x2, y10), 1 + y14 + z11, 1 + (1 + y10) + if(eq(1 + x2, y14), z11, 1 + y14 + del(1 + x2, z11))) :|: z11 >= 0, y10 >= 0, z'' = 1 + x2, z1 = 1 + (1 + y10) + (1 + y14 + z11), x2 >= 0, y14 >= 0 del(z'', z1) -{ 3 }-> if(2, 1 + y11 + z8, 1 + 0 + if(eq(0, y11), z8, 1 + y11 + del(0, z8))) :|: z'' = 0, z1 = 1 + 0 + (1 + y11 + z8), y11 >= 0, z8 >= 0 del(z'', z1) -{ 3 }-> if(1, 1 + y12 + z9, 1 + (1 + y9) + if(eq(0, y12), z9, 1 + y12 + del(0, z9))) :|: z'' = 0, y9 >= 0, z9 >= 0, y12 >= 0, z1 = 1 + (1 + y9) + (1 + y12 + z9) del(z'', z1) -{ 3 }-> if(1, 1 + y13 + z10, 1 + 0 + if(eq(1 + x1, y13), z10, 1 + y13 + del(1 + x1, z10))) :|: z1 = 1 + 0 + (1 + y13 + z10), x1 >= 0, z'' = 1 + x1, z10 >= 0, y13 >= 0 del(z'', z1) -{ 2 }-> if(0, 1 + y15 + z12, 1 + y + if(eq(x, y15), z12, 1 + y15 + del(x, z12))) :|: z1 = 1 + y + (1 + y15 + z12), x >= 0, y >= 0, z'' = x, z12 >= 0, y15 >= 0 del(z'', z1) -{ 1 }-> 1 :|: x >= 0, z1 = 1, z'' = x del(z'', z1) -{ 0 }-> 0 :|: z'' = v0, v0 >= 0, z1 = v1, v1 >= 0 del(z'', z1) -{ 3 }-> 0 :|: z'' = 0, z1 = 1 + 0 + 1, 2 = v0, 1 + 0 + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 2 }-> 0 :|: z'' = 0, z >= 0, z1 = 1 + 0 + z, 2 = v0, 1 + 0 + 0 = v2, v0 >= 0, z = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 3 }-> 0 :|: z'' = 0, y9 >= 0, z1 = 1 + (1 + y9) + 1, 1 = v0, 1 + (1 + y9) + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 2 }-> 0 :|: z'' = 0, y9 >= 0, z >= 0, z1 = 1 + (1 + y9) + z, 1 = v0, 1 + (1 + y9) + 0 = v2, v0 >= 0, z = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 3 }-> 0 :|: x1 >= 0, z'' = 1 + x1, z1 = 1 + 0 + 1, 1 = v0, 1 + 0 + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 2 }-> 0 :|: x1 >= 0, z >= 0, z'' = 1 + x1, z1 = 1 + 0 + z, 1 = v0, 1 + 0 + 0 = v2, v0 >= 0, z = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 2 }-> 0 :|: x >= 0, y >= 0, z'' = x, z1 = 1 + y + 1, 0 = v0, 1 + y + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 1 }-> 0 :|: z >= 0, z1 = 1 + y + z, x >= 0, y >= 0, z'' = x, 0 = v0, 1 + y + 0 = v2, v0 >= 0, z = v1, v1 >= 0, v2 >= 0 eq(z'', z1) -{ 1 }-> eq(x, y) :|: x >= 0, y >= 0, z1 = 1 + y, z'' = 1 + x eq(z'', z1) -{ 1 }-> 2 :|: z'' = 0, z1 = 0 eq(z'', z1) -{ 1 }-> 1 :|: z'' = 0, y >= 0, z1 = 1 + y eq(z'', z1) -{ 1 }-> 1 :|: z1 = 0, x >= 0, z'' = 1 + x eq(z'', z1) -{ 0 }-> 0 :|: z'' = v0, v0 >= 0, z1 = v1, v1 >= 0 if(z'', z1, z2) -{ 1 }-> x :|: z2 = y, x >= 0, y >= 0, z'' = 2, z1 = x if(z'', z1, z2) -{ 1 }-> y :|: z2 = y, x >= 0, y >= 0, z'' = 1, z1 = x if(z'', z1, z2) -{ 0 }-> 0 :|: z'' = v0, z2 = v2, v0 >= 0, z1 = v1, v1 >= 0, v2 >= 0 le(z'', z1) -{ 1 }-> le(x, y) :|: x >= 0, y >= 0, z1 = 1 + y, z'' = 1 + x le(z'', z1) -{ 1 }-> 2 :|: z'' = 0, z1 = y, y >= 0 le(z'', z1) -{ 1 }-> 1 :|: z1 = 0, x >= 0, z'' = 1 + x le(z'', z1) -{ 0 }-> 0 :|: z'' = v0, v0 >= 0, z1 = v1, v1 >= 0 min(z'', z1) -{ 1 }-> x :|: x >= 0, z1 = 1, z'' = x min(z'', z1) -{ 1 }-> if(le(x, y), min(x, z), min(y, z)) :|: z >= 0, z1 = 1 + y + z, x >= 0, y >= 0, z'' = x min(z'', z1) -{ 0 }-> 0 :|: z'' = v0, v0 >= 0, z1 = v1, v1 >= 0 minsort(z'') -{ 1 }-> 1 :|: z'' = 1 minsort(z'') -{ 1 }-> 1 + min(x, y) + minsort(del(0, 1 + x + y)) :|: z'' = 1 + x + y, x >= 0, y >= 0 minsort(z'') -{ 2 }-> 1 + min(x, 1) + minsort(del(x, 1 + x + 1)) :|: x >= 0, z'' = 1 + x + 1 minsort(z'') -{ 2 }-> 1 + min(x, 1 + y' + z') + minsort(del(if(le(x, y'), min(x, z'), min(y', z')), 1 + x + (1 + y' + z'))) :|: z'' = 1 + x + (1 + y' + z'), x >= 0, z' >= 0, y' >= 0 ---------------------------------------- (17) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (18) Obligation: Complexity RNTS consisting of the following rules: del(z'', z1) -{ 4 }-> x :|: z'' = 0, z1 = 1 + 0 + 1, 1 + 0 + 1 = y, x >= 0, y >= 0, 2 = 2, 1 = x del(z'', z1) -{ 3 }-> x :|: z'' = 0, z1 - 1 >= 0, 1 + 0 + 0 = y, x >= 0, y >= 0, 2 = 2, z1 - 1 = x del(z'', z1) -{ 4 }-> y :|: z'' = 0, z1 - 3 >= 0, 1 + (1 + (z1 - 3)) + 1 = y, x >= 0, y >= 0, 1 = 1, 1 = x del(z'', z1) -{ 3 }-> y :|: z'' = 0, y9 >= 0, z >= 0, z1 = 1 + (1 + y9) + z, 1 + (1 + y9) + 0 = y, x >= 0, y >= 0, 1 = 1, z = x del(z'', z1) -{ 4 }-> y :|: z'' - 1 >= 0, z1 = 1 + 0 + 1, 1 + 0 + 1 = y, x >= 0, y >= 0, 1 = 1, 1 = x del(z'', z1) -{ 3 }-> y :|: z'' - 1 >= 0, z1 - 1 >= 0, 1 + 0 + 0 = y, x >= 0, y >= 0, 1 = 1, z1 - 1 = x del(z'', z1) -{ 2 }-> if(eq(z'' - 1, y10), z, 1 + (1 + y10) + 0) :|: z >= 0, y10 >= 0, z'' - 1 >= 0, z1 = 1 + (1 + y10) + z del(z'', z1) -{ 3 }-> if(eq(z'' - 1, y10), 1 + y14 + z11, 1 + (1 + y10) + if(eq(1 + (z'' - 1), y14), z11, 1 + y14 + del(1 + (z'' - 1), z11))) :|: z11 >= 0, y10 >= 0, z1 = 1 + (1 + y10) + (1 + y14 + z11), z'' - 1 >= 0, y14 >= 0 del(z'', z1) -{ 3 }-> if(eq(z'' - 1, z1 - 3), 1, 1 + (1 + (z1 - 3)) + 1) :|: z1 - 3 >= 0, z'' - 1 >= 0 del(z'', z1) -{ 3 }-> if(2, 1 + y11 + z8, 1 + 0 + if(eq(0, y11), z8, 1 + y11 + del(0, z8))) :|: z'' = 0, z1 = 1 + 0 + (1 + y11 + z8), y11 >= 0, z8 >= 0 del(z'', z1) -{ 3 }-> if(1, 1 + y12 + z9, 1 + (1 + y9) + if(eq(0, y12), z9, 1 + y12 + del(0, z9))) :|: z'' = 0, y9 >= 0, z9 >= 0, y12 >= 0, z1 = 1 + (1 + y9) + (1 + y12 + z9) del(z'', z1) -{ 3 }-> if(1, 1 + y13 + z10, 1 + 0 + if(eq(1 + (z'' - 1), y13), z10, 1 + y13 + del(1 + (z'' - 1), z10))) :|: z1 = 1 + 0 + (1 + y13 + z10), z'' - 1 >= 0, z10 >= 0, y13 >= 0 del(z'', z1) -{ 2 }-> if(0, 1 + y15 + z12, 1 + y + if(eq(z'', y15), z12, 1 + y15 + del(z'', z12))) :|: z1 = 1 + y + (1 + y15 + z12), z'' >= 0, y >= 0, z12 >= 0, y15 >= 0 del(z'', z1) -{ 1 }-> 1 :|: z'' >= 0, z1 = 1 del(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0 del(z'', z1) -{ 3 }-> 0 :|: z'' = 0, z1 = 1 + 0 + 1, 2 = v0, 1 + 0 + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 2 }-> 0 :|: z'' = 0, z1 - 1 >= 0, 2 = v0, 1 + 0 + 0 = v2, v0 >= 0, z1 - 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 3 }-> 0 :|: z'' = 0, z1 - 3 >= 0, 1 = v0, 1 + (1 + (z1 - 3)) + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 2 }-> 0 :|: z'' = 0, y9 >= 0, z >= 0, z1 = 1 + (1 + y9) + z, 1 = v0, 1 + (1 + y9) + 0 = v2, v0 >= 0, z = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 3 }-> 0 :|: z'' - 1 >= 0, z1 = 1 + 0 + 1, 1 = v0, 1 + 0 + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 2 }-> 0 :|: z'' - 1 >= 0, z1 - 1 >= 0, 1 = v0, 1 + 0 + 0 = v2, v0 >= 0, z1 - 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 2 }-> 0 :|: z'' >= 0, z1 - 2 >= 0, 0 = v0, 1 + (z1 - 2) + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 1 }-> 0 :|: z >= 0, z1 = 1 + y + z, z'' >= 0, y >= 0, 0 = v0, 1 + y + 0 = v2, v0 >= 0, z = v1, v1 >= 0, v2 >= 0 eq(z'', z1) -{ 1 }-> eq(z'' - 1, z1 - 1) :|: z'' - 1 >= 0, z1 - 1 >= 0 eq(z'', z1) -{ 1 }-> 2 :|: z'' = 0, z1 = 0 eq(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 - 1 >= 0 eq(z'', z1) -{ 1 }-> 1 :|: z1 = 0, z'' - 1 >= 0 eq(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0 if(z'', z1, z2) -{ 1 }-> z1 :|: z1 >= 0, z2 >= 0, z'' = 2 if(z'', z1, z2) -{ 1 }-> z2 :|: z1 >= 0, z2 >= 0, z'' = 1 if(z'', z1, z2) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0, z2 >= 0 le(z'', z1) -{ 1 }-> le(z'' - 1, z1 - 1) :|: z'' - 1 >= 0, z1 - 1 >= 0 le(z'', z1) -{ 1 }-> 2 :|: z'' = 0, z1 >= 0 le(z'', z1) -{ 1 }-> 1 :|: z1 = 0, z'' - 1 >= 0 le(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0 min(z'', z1) -{ 1 }-> z'' :|: z'' >= 0, z1 = 1 min(z'', z1) -{ 1 }-> if(le(z'', y), min(z'', z), min(y, z)) :|: z >= 0, z1 = 1 + y + z, z'' >= 0, y >= 0 min(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0 minsort(z'') -{ 1 }-> 1 :|: z'' = 1 minsort(z'') -{ 1 }-> 1 + min(x, y) + minsort(del(0, 1 + x + y)) :|: z'' = 1 + x + y, x >= 0, y >= 0 minsort(z'') -{ 2 }-> 1 + min(x, 1 + y' + z') + minsort(del(if(le(x, y'), min(x, z'), min(y', z')), 1 + x + (1 + y' + z'))) :|: z'' = 1 + x + (1 + y' + z'), x >= 0, z' >= 0, y' >= 0 minsort(z'') -{ 2 }-> 1 + min(z'' - 2, 1) + minsort(del(z'' - 2, 1 + (z'' - 2) + 1)) :|: z'' - 2 >= 0 ---------------------------------------- (19) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { le } { if } { eq } { min } { del } { minsort } ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: del(z'', z1) -{ 4 }-> x :|: z'' = 0, z1 = 1 + 0 + 1, 1 + 0 + 1 = y, x >= 0, y >= 0, 2 = 2, 1 = x del(z'', z1) -{ 3 }-> x :|: z'' = 0, z1 - 1 >= 0, 1 + 0 + 0 = y, x >= 0, y >= 0, 2 = 2, z1 - 1 = x del(z'', z1) -{ 4 }-> y :|: z'' = 0, z1 - 3 >= 0, 1 + (1 + (z1 - 3)) + 1 = y, x >= 0, y >= 0, 1 = 1, 1 = x del(z'', z1) -{ 3 }-> y :|: z'' = 0, y9 >= 0, z >= 0, z1 = 1 + (1 + y9) + z, 1 + (1 + y9) + 0 = y, x >= 0, y >= 0, 1 = 1, z = x del(z'', z1) -{ 4 }-> y :|: z'' - 1 >= 0, z1 = 1 + 0 + 1, 1 + 0 + 1 = y, x >= 0, y >= 0, 1 = 1, 1 = x del(z'', z1) -{ 3 }-> y :|: z'' - 1 >= 0, z1 - 1 >= 0, 1 + 0 + 0 = y, x >= 0, y >= 0, 1 = 1, z1 - 1 = x del(z'', z1) -{ 2 }-> if(eq(z'' - 1, y10), z, 1 + (1 + y10) + 0) :|: z >= 0, y10 >= 0, z'' - 1 >= 0, z1 = 1 + (1 + y10) + z del(z'', z1) -{ 3 }-> if(eq(z'' - 1, y10), 1 + y14 + z11, 1 + (1 + y10) + if(eq(1 + (z'' - 1), y14), z11, 1 + y14 + del(1 + (z'' - 1), z11))) :|: z11 >= 0, y10 >= 0, z1 = 1 + (1 + y10) + (1 + y14 + z11), z'' - 1 >= 0, y14 >= 0 del(z'', z1) -{ 3 }-> if(eq(z'' - 1, z1 - 3), 1, 1 + (1 + (z1 - 3)) + 1) :|: z1 - 3 >= 0, z'' - 1 >= 0 del(z'', z1) -{ 3 }-> if(2, 1 + y11 + z8, 1 + 0 + if(eq(0, y11), z8, 1 + y11 + del(0, z8))) :|: z'' = 0, z1 = 1 + 0 + (1 + y11 + z8), y11 >= 0, z8 >= 0 del(z'', z1) -{ 3 }-> if(1, 1 + y12 + z9, 1 + (1 + y9) + if(eq(0, y12), z9, 1 + y12 + del(0, z9))) :|: z'' = 0, y9 >= 0, z9 >= 0, y12 >= 0, z1 = 1 + (1 + y9) + (1 + y12 + z9) del(z'', z1) -{ 3 }-> if(1, 1 + y13 + z10, 1 + 0 + if(eq(1 + (z'' - 1), y13), z10, 1 + y13 + del(1 + (z'' - 1), z10))) :|: z1 = 1 + 0 + (1 + y13 + z10), z'' - 1 >= 0, z10 >= 0, y13 >= 0 del(z'', z1) -{ 2 }-> if(0, 1 + y15 + z12, 1 + y + if(eq(z'', y15), z12, 1 + y15 + del(z'', z12))) :|: z1 = 1 + y + (1 + y15 + z12), z'' >= 0, y >= 0, z12 >= 0, y15 >= 0 del(z'', z1) -{ 1 }-> 1 :|: z'' >= 0, z1 = 1 del(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0 del(z'', z1) -{ 3 }-> 0 :|: z'' = 0, z1 = 1 + 0 + 1, 2 = v0, 1 + 0 + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 2 }-> 0 :|: z'' = 0, z1 - 1 >= 0, 2 = v0, 1 + 0 + 0 = v2, v0 >= 0, z1 - 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 3 }-> 0 :|: z'' = 0, z1 - 3 >= 0, 1 = v0, 1 + (1 + (z1 - 3)) + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 2 }-> 0 :|: z'' = 0, y9 >= 0, z >= 0, z1 = 1 + (1 + y9) + z, 1 = v0, 1 + (1 + y9) + 0 = v2, v0 >= 0, z = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 3 }-> 0 :|: z'' - 1 >= 0, z1 = 1 + 0 + 1, 1 = v0, 1 + 0 + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 2 }-> 0 :|: z'' - 1 >= 0, z1 - 1 >= 0, 1 = v0, 1 + 0 + 0 = v2, v0 >= 0, z1 - 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 2 }-> 0 :|: z'' >= 0, z1 - 2 >= 0, 0 = v0, 1 + (z1 - 2) + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 1 }-> 0 :|: z >= 0, z1 = 1 + y + z, z'' >= 0, y >= 0, 0 = v0, 1 + y + 0 = v2, v0 >= 0, z = v1, v1 >= 0, v2 >= 0 eq(z'', z1) -{ 1 }-> eq(z'' - 1, z1 - 1) :|: z'' - 1 >= 0, z1 - 1 >= 0 eq(z'', z1) -{ 1 }-> 2 :|: z'' = 0, z1 = 0 eq(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 - 1 >= 0 eq(z'', z1) -{ 1 }-> 1 :|: z1 = 0, z'' - 1 >= 0 eq(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0 if(z'', z1, z2) -{ 1 }-> z1 :|: z1 >= 0, z2 >= 0, z'' = 2 if(z'', z1, z2) -{ 1 }-> z2 :|: z1 >= 0, z2 >= 0, z'' = 1 if(z'', z1, z2) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0, z2 >= 0 le(z'', z1) -{ 1 }-> le(z'' - 1, z1 - 1) :|: z'' - 1 >= 0, z1 - 1 >= 0 le(z'', z1) -{ 1 }-> 2 :|: z'' = 0, z1 >= 0 le(z'', z1) -{ 1 }-> 1 :|: z1 = 0, z'' - 1 >= 0 le(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0 min(z'', z1) -{ 1 }-> z'' :|: z'' >= 0, z1 = 1 min(z'', z1) -{ 1 }-> if(le(z'', y), min(z'', z), min(y, z)) :|: z >= 0, z1 = 1 + y + z, z'' >= 0, y >= 0 min(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0 minsort(z'') -{ 1 }-> 1 :|: z'' = 1 minsort(z'') -{ 1 }-> 1 + min(x, y) + minsort(del(0, 1 + x + y)) :|: z'' = 1 + x + y, x >= 0, y >= 0 minsort(z'') -{ 2 }-> 1 + min(x, 1 + y' + z') + minsort(del(if(le(x, y'), min(x, z'), min(y', z')), 1 + x + (1 + y' + z'))) :|: z'' = 1 + x + (1 + y' + z'), x >= 0, z' >= 0, y' >= 0 minsort(z'') -{ 2 }-> 1 + min(z'' - 2, 1) + minsort(del(z'' - 2, 1 + (z'' - 2) + 1)) :|: z'' - 2 >= 0 Function symbols to be analyzed: {le}, {if}, {eq}, {min}, {del}, {minsort} ---------------------------------------- (21) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (22) Obligation: Complexity RNTS consisting of the following rules: del(z'', z1) -{ 4 }-> x :|: z'' = 0, z1 = 1 + 0 + 1, 1 + 0 + 1 = y, x >= 0, y >= 0, 2 = 2, 1 = x del(z'', z1) -{ 3 }-> x :|: z'' = 0, z1 - 1 >= 0, 1 + 0 + 0 = y, x >= 0, y >= 0, 2 = 2, z1 - 1 = x del(z'', z1) -{ 4 }-> y :|: z'' = 0, z1 - 3 >= 0, 1 + (1 + (z1 - 3)) + 1 = y, x >= 0, y >= 0, 1 = 1, 1 = x del(z'', z1) -{ 3 }-> y :|: z'' = 0, y9 >= 0, z >= 0, z1 = 1 + (1 + y9) + z, 1 + (1 + y9) + 0 = y, x >= 0, y >= 0, 1 = 1, z = x del(z'', z1) -{ 4 }-> y :|: z'' - 1 >= 0, z1 = 1 + 0 + 1, 1 + 0 + 1 = y, x >= 0, y >= 0, 1 = 1, 1 = x del(z'', z1) -{ 3 }-> y :|: z'' - 1 >= 0, z1 - 1 >= 0, 1 + 0 + 0 = y, x >= 0, y >= 0, 1 = 1, z1 - 1 = x del(z'', z1) -{ 2 }-> if(eq(z'' - 1, y10), z, 1 + (1 + y10) + 0) :|: z >= 0, y10 >= 0, z'' - 1 >= 0, z1 = 1 + (1 + y10) + z del(z'', z1) -{ 3 }-> if(eq(z'' - 1, y10), 1 + y14 + z11, 1 + (1 + y10) + if(eq(1 + (z'' - 1), y14), z11, 1 + y14 + del(1 + (z'' - 1), z11))) :|: z11 >= 0, y10 >= 0, z1 = 1 + (1 + y10) + (1 + y14 + z11), z'' - 1 >= 0, y14 >= 0 del(z'', z1) -{ 3 }-> if(eq(z'' - 1, z1 - 3), 1, 1 + (1 + (z1 - 3)) + 1) :|: z1 - 3 >= 0, z'' - 1 >= 0 del(z'', z1) -{ 3 }-> if(2, 1 + y11 + z8, 1 + 0 + if(eq(0, y11), z8, 1 + y11 + del(0, z8))) :|: z'' = 0, z1 = 1 + 0 + (1 + y11 + z8), y11 >= 0, z8 >= 0 del(z'', z1) -{ 3 }-> if(1, 1 + y12 + z9, 1 + (1 + y9) + if(eq(0, y12), z9, 1 + y12 + del(0, z9))) :|: z'' = 0, y9 >= 0, z9 >= 0, y12 >= 0, z1 = 1 + (1 + y9) + (1 + y12 + z9) del(z'', z1) -{ 3 }-> if(1, 1 + y13 + z10, 1 + 0 + if(eq(1 + (z'' - 1), y13), z10, 1 + y13 + del(1 + (z'' - 1), z10))) :|: z1 = 1 + 0 + (1 + y13 + z10), z'' - 1 >= 0, z10 >= 0, y13 >= 0 del(z'', z1) -{ 2 }-> if(0, 1 + y15 + z12, 1 + y + if(eq(z'', y15), z12, 1 + y15 + del(z'', z12))) :|: z1 = 1 + y + (1 + y15 + z12), z'' >= 0, y >= 0, z12 >= 0, y15 >= 0 del(z'', z1) -{ 1 }-> 1 :|: z'' >= 0, z1 = 1 del(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0 del(z'', z1) -{ 3 }-> 0 :|: z'' = 0, z1 = 1 + 0 + 1, 2 = v0, 1 + 0 + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 2 }-> 0 :|: z'' = 0, z1 - 1 >= 0, 2 = v0, 1 + 0 + 0 = v2, v0 >= 0, z1 - 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 3 }-> 0 :|: z'' = 0, z1 - 3 >= 0, 1 = v0, 1 + (1 + (z1 - 3)) + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 2 }-> 0 :|: z'' = 0, y9 >= 0, z >= 0, z1 = 1 + (1 + y9) + z, 1 = v0, 1 + (1 + y9) + 0 = v2, v0 >= 0, z = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 3 }-> 0 :|: z'' - 1 >= 0, z1 = 1 + 0 + 1, 1 = v0, 1 + 0 + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 2 }-> 0 :|: z'' - 1 >= 0, z1 - 1 >= 0, 1 = v0, 1 + 0 + 0 = v2, v0 >= 0, z1 - 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 2 }-> 0 :|: z'' >= 0, z1 - 2 >= 0, 0 = v0, 1 + (z1 - 2) + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 1 }-> 0 :|: z >= 0, z1 = 1 + y + z, z'' >= 0, y >= 0, 0 = v0, 1 + y + 0 = v2, v0 >= 0, z = v1, v1 >= 0, v2 >= 0 eq(z'', z1) -{ 1 }-> eq(z'' - 1, z1 - 1) :|: z'' - 1 >= 0, z1 - 1 >= 0 eq(z'', z1) -{ 1 }-> 2 :|: z'' = 0, z1 = 0 eq(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 - 1 >= 0 eq(z'', z1) -{ 1 }-> 1 :|: z1 = 0, z'' - 1 >= 0 eq(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0 if(z'', z1, z2) -{ 1 }-> z1 :|: z1 >= 0, z2 >= 0, z'' = 2 if(z'', z1, z2) -{ 1 }-> z2 :|: z1 >= 0, z2 >= 0, z'' = 1 if(z'', z1, z2) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0, z2 >= 0 le(z'', z1) -{ 1 }-> le(z'' - 1, z1 - 1) :|: z'' - 1 >= 0, z1 - 1 >= 0 le(z'', z1) -{ 1 }-> 2 :|: z'' = 0, z1 >= 0 le(z'', z1) -{ 1 }-> 1 :|: z1 = 0, z'' - 1 >= 0 le(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0 min(z'', z1) -{ 1 }-> z'' :|: z'' >= 0, z1 = 1 min(z'', z1) -{ 1 }-> if(le(z'', y), min(z'', z), min(y, z)) :|: z >= 0, z1 = 1 + y + z, z'' >= 0, y >= 0 min(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0 minsort(z'') -{ 1 }-> 1 :|: z'' = 1 minsort(z'') -{ 1 }-> 1 + min(x, y) + minsort(del(0, 1 + x + y)) :|: z'' = 1 + x + y, x >= 0, y >= 0 minsort(z'') -{ 2 }-> 1 + min(x, 1 + y' + z') + minsort(del(if(le(x, y'), min(x, z'), min(y', z')), 1 + x + (1 + y' + z'))) :|: z'' = 1 + x + (1 + y' + z'), x >= 0, z' >= 0, y' >= 0 minsort(z'') -{ 2 }-> 1 + min(z'' - 2, 1) + minsort(del(z'' - 2, 1 + (z'' - 2) + 1)) :|: z'' - 2 >= 0 Function symbols to be analyzed: {le}, {if}, {eq}, {min}, {del}, {minsort} ---------------------------------------- (23) 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: 2 ---------------------------------------- (24) Obligation: Complexity RNTS consisting of the following rules: del(z'', z1) -{ 4 }-> x :|: z'' = 0, z1 = 1 + 0 + 1, 1 + 0 + 1 = y, x >= 0, y >= 0, 2 = 2, 1 = x del(z'', z1) -{ 3 }-> x :|: z'' = 0, z1 - 1 >= 0, 1 + 0 + 0 = y, x >= 0, y >= 0, 2 = 2, z1 - 1 = x del(z'', z1) -{ 4 }-> y :|: z'' = 0, z1 - 3 >= 0, 1 + (1 + (z1 - 3)) + 1 = y, x >= 0, y >= 0, 1 = 1, 1 = x del(z'', z1) -{ 3 }-> y :|: z'' = 0, y9 >= 0, z >= 0, z1 = 1 + (1 + y9) + z, 1 + (1 + y9) + 0 = y, x >= 0, y >= 0, 1 = 1, z = x del(z'', z1) -{ 4 }-> y :|: z'' - 1 >= 0, z1 = 1 + 0 + 1, 1 + 0 + 1 = y, x >= 0, y >= 0, 1 = 1, 1 = x del(z'', z1) -{ 3 }-> y :|: z'' - 1 >= 0, z1 - 1 >= 0, 1 + 0 + 0 = y, x >= 0, y >= 0, 1 = 1, z1 - 1 = x del(z'', z1) -{ 2 }-> if(eq(z'' - 1, y10), z, 1 + (1 + y10) + 0) :|: z >= 0, y10 >= 0, z'' - 1 >= 0, z1 = 1 + (1 + y10) + z del(z'', z1) -{ 3 }-> if(eq(z'' - 1, y10), 1 + y14 + z11, 1 + (1 + y10) + if(eq(1 + (z'' - 1), y14), z11, 1 + y14 + del(1 + (z'' - 1), z11))) :|: z11 >= 0, y10 >= 0, z1 = 1 + (1 + y10) + (1 + y14 + z11), z'' - 1 >= 0, y14 >= 0 del(z'', z1) -{ 3 }-> if(eq(z'' - 1, z1 - 3), 1, 1 + (1 + (z1 - 3)) + 1) :|: z1 - 3 >= 0, z'' - 1 >= 0 del(z'', z1) -{ 3 }-> if(2, 1 + y11 + z8, 1 + 0 + if(eq(0, y11), z8, 1 + y11 + del(0, z8))) :|: z'' = 0, z1 = 1 + 0 + (1 + y11 + z8), y11 >= 0, z8 >= 0 del(z'', z1) -{ 3 }-> if(1, 1 + y12 + z9, 1 + (1 + y9) + if(eq(0, y12), z9, 1 + y12 + del(0, z9))) :|: z'' = 0, y9 >= 0, z9 >= 0, y12 >= 0, z1 = 1 + (1 + y9) + (1 + y12 + z9) del(z'', z1) -{ 3 }-> if(1, 1 + y13 + z10, 1 + 0 + if(eq(1 + (z'' - 1), y13), z10, 1 + y13 + del(1 + (z'' - 1), z10))) :|: z1 = 1 + 0 + (1 + y13 + z10), z'' - 1 >= 0, z10 >= 0, y13 >= 0 del(z'', z1) -{ 2 }-> if(0, 1 + y15 + z12, 1 + y + if(eq(z'', y15), z12, 1 + y15 + del(z'', z12))) :|: z1 = 1 + y + (1 + y15 + z12), z'' >= 0, y >= 0, z12 >= 0, y15 >= 0 del(z'', z1) -{ 1 }-> 1 :|: z'' >= 0, z1 = 1 del(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0 del(z'', z1) -{ 3 }-> 0 :|: z'' = 0, z1 = 1 + 0 + 1, 2 = v0, 1 + 0 + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 2 }-> 0 :|: z'' = 0, z1 - 1 >= 0, 2 = v0, 1 + 0 + 0 = v2, v0 >= 0, z1 - 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 3 }-> 0 :|: z'' = 0, z1 - 3 >= 0, 1 = v0, 1 + (1 + (z1 - 3)) + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 2 }-> 0 :|: z'' = 0, y9 >= 0, z >= 0, z1 = 1 + (1 + y9) + z, 1 = v0, 1 + (1 + y9) + 0 = v2, v0 >= 0, z = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 3 }-> 0 :|: z'' - 1 >= 0, z1 = 1 + 0 + 1, 1 = v0, 1 + 0 + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 2 }-> 0 :|: z'' - 1 >= 0, z1 - 1 >= 0, 1 = v0, 1 + 0 + 0 = v2, v0 >= 0, z1 - 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 2 }-> 0 :|: z'' >= 0, z1 - 2 >= 0, 0 = v0, 1 + (z1 - 2) + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 1 }-> 0 :|: z >= 0, z1 = 1 + y + z, z'' >= 0, y >= 0, 0 = v0, 1 + y + 0 = v2, v0 >= 0, z = v1, v1 >= 0, v2 >= 0 eq(z'', z1) -{ 1 }-> eq(z'' - 1, z1 - 1) :|: z'' - 1 >= 0, z1 - 1 >= 0 eq(z'', z1) -{ 1 }-> 2 :|: z'' = 0, z1 = 0 eq(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 - 1 >= 0 eq(z'', z1) -{ 1 }-> 1 :|: z1 = 0, z'' - 1 >= 0 eq(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0 if(z'', z1, z2) -{ 1 }-> z1 :|: z1 >= 0, z2 >= 0, z'' = 2 if(z'', z1, z2) -{ 1 }-> z2 :|: z1 >= 0, z2 >= 0, z'' = 1 if(z'', z1, z2) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0, z2 >= 0 le(z'', z1) -{ 1 }-> le(z'' - 1, z1 - 1) :|: z'' - 1 >= 0, z1 - 1 >= 0 le(z'', z1) -{ 1 }-> 2 :|: z'' = 0, z1 >= 0 le(z'', z1) -{ 1 }-> 1 :|: z1 = 0, z'' - 1 >= 0 le(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0 min(z'', z1) -{ 1 }-> z'' :|: z'' >= 0, z1 = 1 min(z'', z1) -{ 1 }-> if(le(z'', y), min(z'', z), min(y, z)) :|: z >= 0, z1 = 1 + y + z, z'' >= 0, y >= 0 min(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0 minsort(z'') -{ 1 }-> 1 :|: z'' = 1 minsort(z'') -{ 1 }-> 1 + min(x, y) + minsort(del(0, 1 + x + y)) :|: z'' = 1 + x + y, x >= 0, y >= 0 minsort(z'') -{ 2 }-> 1 + min(x, 1 + y' + z') + minsort(del(if(le(x, y'), min(x, z'), min(y', z')), 1 + x + (1 + y' + z'))) :|: z'' = 1 + x + (1 + y' + z'), x >= 0, z' >= 0, y' >= 0 minsort(z'') -{ 2 }-> 1 + min(z'' - 2, 1) + minsort(del(z'' - 2, 1 + (z'' - 2) + 1)) :|: z'' - 2 >= 0 Function symbols to be analyzed: {le}, {if}, {eq}, {min}, {del}, {minsort} Previous analysis results are: le: runtime: ?, size: O(1) [2] ---------------------------------------- (25) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: le after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2 + z1 ---------------------------------------- (26) Obligation: Complexity RNTS consisting of the following rules: del(z'', z1) -{ 4 }-> x :|: z'' = 0, z1 = 1 + 0 + 1, 1 + 0 + 1 = y, x >= 0, y >= 0, 2 = 2, 1 = x del(z'', z1) -{ 3 }-> x :|: z'' = 0, z1 - 1 >= 0, 1 + 0 + 0 = y, x >= 0, y >= 0, 2 = 2, z1 - 1 = x del(z'', z1) -{ 4 }-> y :|: z'' = 0, z1 - 3 >= 0, 1 + (1 + (z1 - 3)) + 1 = y, x >= 0, y >= 0, 1 = 1, 1 = x del(z'', z1) -{ 3 }-> y :|: z'' = 0, y9 >= 0, z >= 0, z1 = 1 + (1 + y9) + z, 1 + (1 + y9) + 0 = y, x >= 0, y >= 0, 1 = 1, z = x del(z'', z1) -{ 4 }-> y :|: z'' - 1 >= 0, z1 = 1 + 0 + 1, 1 + 0 + 1 = y, x >= 0, y >= 0, 1 = 1, 1 = x del(z'', z1) -{ 3 }-> y :|: z'' - 1 >= 0, z1 - 1 >= 0, 1 + 0 + 0 = y, x >= 0, y >= 0, 1 = 1, z1 - 1 = x del(z'', z1) -{ 2 }-> if(eq(z'' - 1, y10), z, 1 + (1 + y10) + 0) :|: z >= 0, y10 >= 0, z'' - 1 >= 0, z1 = 1 + (1 + y10) + z del(z'', z1) -{ 3 }-> if(eq(z'' - 1, y10), 1 + y14 + z11, 1 + (1 + y10) + if(eq(1 + (z'' - 1), y14), z11, 1 + y14 + del(1 + (z'' - 1), z11))) :|: z11 >= 0, y10 >= 0, z1 = 1 + (1 + y10) + (1 + y14 + z11), z'' - 1 >= 0, y14 >= 0 del(z'', z1) -{ 3 }-> if(eq(z'' - 1, z1 - 3), 1, 1 + (1 + (z1 - 3)) + 1) :|: z1 - 3 >= 0, z'' - 1 >= 0 del(z'', z1) -{ 3 }-> if(2, 1 + y11 + z8, 1 + 0 + if(eq(0, y11), z8, 1 + y11 + del(0, z8))) :|: z'' = 0, z1 = 1 + 0 + (1 + y11 + z8), y11 >= 0, z8 >= 0 del(z'', z1) -{ 3 }-> if(1, 1 + y12 + z9, 1 + (1 + y9) + if(eq(0, y12), z9, 1 + y12 + del(0, z9))) :|: z'' = 0, y9 >= 0, z9 >= 0, y12 >= 0, z1 = 1 + (1 + y9) + (1 + y12 + z9) del(z'', z1) -{ 3 }-> if(1, 1 + y13 + z10, 1 + 0 + if(eq(1 + (z'' - 1), y13), z10, 1 + y13 + del(1 + (z'' - 1), z10))) :|: z1 = 1 + 0 + (1 + y13 + z10), z'' - 1 >= 0, z10 >= 0, y13 >= 0 del(z'', z1) -{ 2 }-> if(0, 1 + y15 + z12, 1 + y + if(eq(z'', y15), z12, 1 + y15 + del(z'', z12))) :|: z1 = 1 + y + (1 + y15 + z12), z'' >= 0, y >= 0, z12 >= 0, y15 >= 0 del(z'', z1) -{ 1 }-> 1 :|: z'' >= 0, z1 = 1 del(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0 del(z'', z1) -{ 3 }-> 0 :|: z'' = 0, z1 = 1 + 0 + 1, 2 = v0, 1 + 0 + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 2 }-> 0 :|: z'' = 0, z1 - 1 >= 0, 2 = v0, 1 + 0 + 0 = v2, v0 >= 0, z1 - 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 3 }-> 0 :|: z'' = 0, z1 - 3 >= 0, 1 = v0, 1 + (1 + (z1 - 3)) + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 2 }-> 0 :|: z'' = 0, y9 >= 0, z >= 0, z1 = 1 + (1 + y9) + z, 1 = v0, 1 + (1 + y9) + 0 = v2, v0 >= 0, z = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 3 }-> 0 :|: z'' - 1 >= 0, z1 = 1 + 0 + 1, 1 = v0, 1 + 0 + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 2 }-> 0 :|: z'' - 1 >= 0, z1 - 1 >= 0, 1 = v0, 1 + 0 + 0 = v2, v0 >= 0, z1 - 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 2 }-> 0 :|: z'' >= 0, z1 - 2 >= 0, 0 = v0, 1 + (z1 - 2) + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 1 }-> 0 :|: z >= 0, z1 = 1 + y + z, z'' >= 0, y >= 0, 0 = v0, 1 + y + 0 = v2, v0 >= 0, z = v1, v1 >= 0, v2 >= 0 eq(z'', z1) -{ 1 }-> eq(z'' - 1, z1 - 1) :|: z'' - 1 >= 0, z1 - 1 >= 0 eq(z'', z1) -{ 1 }-> 2 :|: z'' = 0, z1 = 0 eq(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 - 1 >= 0 eq(z'', z1) -{ 1 }-> 1 :|: z1 = 0, z'' - 1 >= 0 eq(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0 if(z'', z1, z2) -{ 1 }-> z1 :|: z1 >= 0, z2 >= 0, z'' = 2 if(z'', z1, z2) -{ 1 }-> z2 :|: z1 >= 0, z2 >= 0, z'' = 1 if(z'', z1, z2) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0, z2 >= 0 le(z'', z1) -{ 1 }-> le(z'' - 1, z1 - 1) :|: z'' - 1 >= 0, z1 - 1 >= 0 le(z'', z1) -{ 1 }-> 2 :|: z'' = 0, z1 >= 0 le(z'', z1) -{ 1 }-> 1 :|: z1 = 0, z'' - 1 >= 0 le(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0 min(z'', z1) -{ 1 }-> z'' :|: z'' >= 0, z1 = 1 min(z'', z1) -{ 1 }-> if(le(z'', y), min(z'', z), min(y, z)) :|: z >= 0, z1 = 1 + y + z, z'' >= 0, y >= 0 min(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0 minsort(z'') -{ 1 }-> 1 :|: z'' = 1 minsort(z'') -{ 1 }-> 1 + min(x, y) + minsort(del(0, 1 + x + y)) :|: z'' = 1 + x + y, x >= 0, y >= 0 minsort(z'') -{ 2 }-> 1 + min(x, 1 + y' + z') + minsort(del(if(le(x, y'), min(x, z'), min(y', z')), 1 + x + (1 + y' + z'))) :|: z'' = 1 + x + (1 + y' + z'), x >= 0, z' >= 0, y' >= 0 minsort(z'') -{ 2 }-> 1 + min(z'' - 2, 1) + minsort(del(z'' - 2, 1 + (z'' - 2) + 1)) :|: z'' - 2 >= 0 Function symbols to be analyzed: {if}, {eq}, {min}, {del}, {minsort} Previous analysis results are: le: runtime: O(n^1) [2 + z1], size: O(1) [2] ---------------------------------------- (27) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: del(z'', z1) -{ 4 }-> x :|: z'' = 0, z1 = 1 + 0 + 1, 1 + 0 + 1 = y, x >= 0, y >= 0, 2 = 2, 1 = x del(z'', z1) -{ 3 }-> x :|: z'' = 0, z1 - 1 >= 0, 1 + 0 + 0 = y, x >= 0, y >= 0, 2 = 2, z1 - 1 = x del(z'', z1) -{ 4 }-> y :|: z'' = 0, z1 - 3 >= 0, 1 + (1 + (z1 - 3)) + 1 = y, x >= 0, y >= 0, 1 = 1, 1 = x del(z'', z1) -{ 3 }-> y :|: z'' = 0, y9 >= 0, z >= 0, z1 = 1 + (1 + y9) + z, 1 + (1 + y9) + 0 = y, x >= 0, y >= 0, 1 = 1, z = x del(z'', z1) -{ 4 }-> y :|: z'' - 1 >= 0, z1 = 1 + 0 + 1, 1 + 0 + 1 = y, x >= 0, y >= 0, 1 = 1, 1 = x del(z'', z1) -{ 3 }-> y :|: z'' - 1 >= 0, z1 - 1 >= 0, 1 + 0 + 0 = y, x >= 0, y >= 0, 1 = 1, z1 - 1 = x del(z'', z1) -{ 2 }-> if(eq(z'' - 1, y10), z, 1 + (1 + y10) + 0) :|: z >= 0, y10 >= 0, z'' - 1 >= 0, z1 = 1 + (1 + y10) + z del(z'', z1) -{ 3 }-> if(eq(z'' - 1, y10), 1 + y14 + z11, 1 + (1 + y10) + if(eq(1 + (z'' - 1), y14), z11, 1 + y14 + del(1 + (z'' - 1), z11))) :|: z11 >= 0, y10 >= 0, z1 = 1 + (1 + y10) + (1 + y14 + z11), z'' - 1 >= 0, y14 >= 0 del(z'', z1) -{ 3 }-> if(eq(z'' - 1, z1 - 3), 1, 1 + (1 + (z1 - 3)) + 1) :|: z1 - 3 >= 0, z'' - 1 >= 0 del(z'', z1) -{ 3 }-> if(2, 1 + y11 + z8, 1 + 0 + if(eq(0, y11), z8, 1 + y11 + del(0, z8))) :|: z'' = 0, z1 = 1 + 0 + (1 + y11 + z8), y11 >= 0, z8 >= 0 del(z'', z1) -{ 3 }-> if(1, 1 + y12 + z9, 1 + (1 + y9) + if(eq(0, y12), z9, 1 + y12 + del(0, z9))) :|: z'' = 0, y9 >= 0, z9 >= 0, y12 >= 0, z1 = 1 + (1 + y9) + (1 + y12 + z9) del(z'', z1) -{ 3 }-> if(1, 1 + y13 + z10, 1 + 0 + if(eq(1 + (z'' - 1), y13), z10, 1 + y13 + del(1 + (z'' - 1), z10))) :|: z1 = 1 + 0 + (1 + y13 + z10), z'' - 1 >= 0, z10 >= 0, y13 >= 0 del(z'', z1) -{ 2 }-> if(0, 1 + y15 + z12, 1 + y + if(eq(z'', y15), z12, 1 + y15 + del(z'', z12))) :|: z1 = 1 + y + (1 + y15 + z12), z'' >= 0, y >= 0, z12 >= 0, y15 >= 0 del(z'', z1) -{ 1 }-> 1 :|: z'' >= 0, z1 = 1 del(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0 del(z'', z1) -{ 3 }-> 0 :|: z'' = 0, z1 = 1 + 0 + 1, 2 = v0, 1 + 0 + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 2 }-> 0 :|: z'' = 0, z1 - 1 >= 0, 2 = v0, 1 + 0 + 0 = v2, v0 >= 0, z1 - 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 3 }-> 0 :|: z'' = 0, z1 - 3 >= 0, 1 = v0, 1 + (1 + (z1 - 3)) + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 2 }-> 0 :|: z'' = 0, y9 >= 0, z >= 0, z1 = 1 + (1 + y9) + z, 1 = v0, 1 + (1 + y9) + 0 = v2, v0 >= 0, z = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 3 }-> 0 :|: z'' - 1 >= 0, z1 = 1 + 0 + 1, 1 = v0, 1 + 0 + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 2 }-> 0 :|: z'' - 1 >= 0, z1 - 1 >= 0, 1 = v0, 1 + 0 + 0 = v2, v0 >= 0, z1 - 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 2 }-> 0 :|: z'' >= 0, z1 - 2 >= 0, 0 = v0, 1 + (z1 - 2) + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 1 }-> 0 :|: z >= 0, z1 = 1 + y + z, z'' >= 0, y >= 0, 0 = v0, 1 + y + 0 = v2, v0 >= 0, z = v1, v1 >= 0, v2 >= 0 eq(z'', z1) -{ 1 }-> eq(z'' - 1, z1 - 1) :|: z'' - 1 >= 0, z1 - 1 >= 0 eq(z'', z1) -{ 1 }-> 2 :|: z'' = 0, z1 = 0 eq(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 - 1 >= 0 eq(z'', z1) -{ 1 }-> 1 :|: z1 = 0, z'' - 1 >= 0 eq(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0 if(z'', z1, z2) -{ 1 }-> z1 :|: z1 >= 0, z2 >= 0, z'' = 2 if(z'', z1, z2) -{ 1 }-> z2 :|: z1 >= 0, z2 >= 0, z'' = 1 if(z'', z1, z2) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0, z2 >= 0 le(z'', z1) -{ 2 + z1 }-> s :|: s >= 0, s <= 2, z'' - 1 >= 0, z1 - 1 >= 0 le(z'', z1) -{ 1 }-> 2 :|: z'' = 0, z1 >= 0 le(z'', z1) -{ 1 }-> 1 :|: z1 = 0, z'' - 1 >= 0 le(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0 min(z'', z1) -{ 1 }-> z'' :|: z'' >= 0, z1 = 1 min(z'', z1) -{ 3 + y }-> if(s'', min(z'', z), min(y, z)) :|: s'' >= 0, s'' <= 2, z >= 0, z1 = 1 + y + z, z'' >= 0, y >= 0 min(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0 minsort(z'') -{ 1 }-> 1 :|: z'' = 1 minsort(z'') -{ 1 }-> 1 + min(x, y) + minsort(del(0, 1 + x + y)) :|: z'' = 1 + x + y, x >= 0, y >= 0 minsort(z'') -{ 4 + y' }-> 1 + min(x, 1 + y' + z') + minsort(del(if(s', min(x, z'), min(y', z')), 1 + x + (1 + y' + z'))) :|: s' >= 0, s' <= 2, z'' = 1 + x + (1 + y' + z'), x >= 0, z' >= 0, y' >= 0 minsort(z'') -{ 2 }-> 1 + min(z'' - 2, 1) + minsort(del(z'' - 2, 1 + (z'' - 2) + 1)) :|: z'' - 2 >= 0 Function symbols to be analyzed: {if}, {eq}, {min}, {del}, {minsort} Previous analysis results are: le: runtime: O(n^1) [2 + z1], size: O(1) [2] ---------------------------------------- (29) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: if after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z1 + z2 ---------------------------------------- (30) Obligation: Complexity RNTS consisting of the following rules: del(z'', z1) -{ 4 }-> x :|: z'' = 0, z1 = 1 + 0 + 1, 1 + 0 + 1 = y, x >= 0, y >= 0, 2 = 2, 1 = x del(z'', z1) -{ 3 }-> x :|: z'' = 0, z1 - 1 >= 0, 1 + 0 + 0 = y, x >= 0, y >= 0, 2 = 2, z1 - 1 = x del(z'', z1) -{ 4 }-> y :|: z'' = 0, z1 - 3 >= 0, 1 + (1 + (z1 - 3)) + 1 = y, x >= 0, y >= 0, 1 = 1, 1 = x del(z'', z1) -{ 3 }-> y :|: z'' = 0, y9 >= 0, z >= 0, z1 = 1 + (1 + y9) + z, 1 + (1 + y9) + 0 = y, x >= 0, y >= 0, 1 = 1, z = x del(z'', z1) -{ 4 }-> y :|: z'' - 1 >= 0, z1 = 1 + 0 + 1, 1 + 0 + 1 = y, x >= 0, y >= 0, 1 = 1, 1 = x del(z'', z1) -{ 3 }-> y :|: z'' - 1 >= 0, z1 - 1 >= 0, 1 + 0 + 0 = y, x >= 0, y >= 0, 1 = 1, z1 - 1 = x del(z'', z1) -{ 2 }-> if(eq(z'' - 1, y10), z, 1 + (1 + y10) + 0) :|: z >= 0, y10 >= 0, z'' - 1 >= 0, z1 = 1 + (1 + y10) + z del(z'', z1) -{ 3 }-> if(eq(z'' - 1, y10), 1 + y14 + z11, 1 + (1 + y10) + if(eq(1 + (z'' - 1), y14), z11, 1 + y14 + del(1 + (z'' - 1), z11))) :|: z11 >= 0, y10 >= 0, z1 = 1 + (1 + y10) + (1 + y14 + z11), z'' - 1 >= 0, y14 >= 0 del(z'', z1) -{ 3 }-> if(eq(z'' - 1, z1 - 3), 1, 1 + (1 + (z1 - 3)) + 1) :|: z1 - 3 >= 0, z'' - 1 >= 0 del(z'', z1) -{ 3 }-> if(2, 1 + y11 + z8, 1 + 0 + if(eq(0, y11), z8, 1 + y11 + del(0, z8))) :|: z'' = 0, z1 = 1 + 0 + (1 + y11 + z8), y11 >= 0, z8 >= 0 del(z'', z1) -{ 3 }-> if(1, 1 + y12 + z9, 1 + (1 + y9) + if(eq(0, y12), z9, 1 + y12 + del(0, z9))) :|: z'' = 0, y9 >= 0, z9 >= 0, y12 >= 0, z1 = 1 + (1 + y9) + (1 + y12 + z9) del(z'', z1) -{ 3 }-> if(1, 1 + y13 + z10, 1 + 0 + if(eq(1 + (z'' - 1), y13), z10, 1 + y13 + del(1 + (z'' - 1), z10))) :|: z1 = 1 + 0 + (1 + y13 + z10), z'' - 1 >= 0, z10 >= 0, y13 >= 0 del(z'', z1) -{ 2 }-> if(0, 1 + y15 + z12, 1 + y + if(eq(z'', y15), z12, 1 + y15 + del(z'', z12))) :|: z1 = 1 + y + (1 + y15 + z12), z'' >= 0, y >= 0, z12 >= 0, y15 >= 0 del(z'', z1) -{ 1 }-> 1 :|: z'' >= 0, z1 = 1 del(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0 del(z'', z1) -{ 3 }-> 0 :|: z'' = 0, z1 = 1 + 0 + 1, 2 = v0, 1 + 0 + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 2 }-> 0 :|: z'' = 0, z1 - 1 >= 0, 2 = v0, 1 + 0 + 0 = v2, v0 >= 0, z1 - 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 3 }-> 0 :|: z'' = 0, z1 - 3 >= 0, 1 = v0, 1 + (1 + (z1 - 3)) + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 2 }-> 0 :|: z'' = 0, y9 >= 0, z >= 0, z1 = 1 + (1 + y9) + z, 1 = v0, 1 + (1 + y9) + 0 = v2, v0 >= 0, z = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 3 }-> 0 :|: z'' - 1 >= 0, z1 = 1 + 0 + 1, 1 = v0, 1 + 0 + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 2 }-> 0 :|: z'' - 1 >= 0, z1 - 1 >= 0, 1 = v0, 1 + 0 + 0 = v2, v0 >= 0, z1 - 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 2 }-> 0 :|: z'' >= 0, z1 - 2 >= 0, 0 = v0, 1 + (z1 - 2) + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 1 }-> 0 :|: z >= 0, z1 = 1 + y + z, z'' >= 0, y >= 0, 0 = v0, 1 + y + 0 = v2, v0 >= 0, z = v1, v1 >= 0, v2 >= 0 eq(z'', z1) -{ 1 }-> eq(z'' - 1, z1 - 1) :|: z'' - 1 >= 0, z1 - 1 >= 0 eq(z'', z1) -{ 1 }-> 2 :|: z'' = 0, z1 = 0 eq(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 - 1 >= 0 eq(z'', z1) -{ 1 }-> 1 :|: z1 = 0, z'' - 1 >= 0 eq(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0 if(z'', z1, z2) -{ 1 }-> z1 :|: z1 >= 0, z2 >= 0, z'' = 2 if(z'', z1, z2) -{ 1 }-> z2 :|: z1 >= 0, z2 >= 0, z'' = 1 if(z'', z1, z2) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0, z2 >= 0 le(z'', z1) -{ 2 + z1 }-> s :|: s >= 0, s <= 2, z'' - 1 >= 0, z1 - 1 >= 0 le(z'', z1) -{ 1 }-> 2 :|: z'' = 0, z1 >= 0 le(z'', z1) -{ 1 }-> 1 :|: z1 = 0, z'' - 1 >= 0 le(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0 min(z'', z1) -{ 1 }-> z'' :|: z'' >= 0, z1 = 1 min(z'', z1) -{ 3 + y }-> if(s'', min(z'', z), min(y, z)) :|: s'' >= 0, s'' <= 2, z >= 0, z1 = 1 + y + z, z'' >= 0, y >= 0 min(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0 minsort(z'') -{ 1 }-> 1 :|: z'' = 1 minsort(z'') -{ 1 }-> 1 + min(x, y) + minsort(del(0, 1 + x + y)) :|: z'' = 1 + x + y, x >= 0, y >= 0 minsort(z'') -{ 4 + y' }-> 1 + min(x, 1 + y' + z') + minsort(del(if(s', min(x, z'), min(y', z')), 1 + x + (1 + y' + z'))) :|: s' >= 0, s' <= 2, z'' = 1 + x + (1 + y' + z'), x >= 0, z' >= 0, y' >= 0 minsort(z'') -{ 2 }-> 1 + min(z'' - 2, 1) + minsort(del(z'' - 2, 1 + (z'' - 2) + 1)) :|: z'' - 2 >= 0 Function symbols to be analyzed: {if}, {eq}, {min}, {del}, {minsort} Previous analysis results are: le: runtime: O(n^1) [2 + z1], size: O(1) [2] if: runtime: ?, size: O(n^1) [z1 + z2] ---------------------------------------- (31) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: if after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (32) Obligation: Complexity RNTS consisting of the following rules: del(z'', z1) -{ 4 }-> x :|: z'' = 0, z1 = 1 + 0 + 1, 1 + 0 + 1 = y, x >= 0, y >= 0, 2 = 2, 1 = x del(z'', z1) -{ 3 }-> x :|: z'' = 0, z1 - 1 >= 0, 1 + 0 + 0 = y, x >= 0, y >= 0, 2 = 2, z1 - 1 = x del(z'', z1) -{ 4 }-> y :|: z'' = 0, z1 - 3 >= 0, 1 + (1 + (z1 - 3)) + 1 = y, x >= 0, y >= 0, 1 = 1, 1 = x del(z'', z1) -{ 3 }-> y :|: z'' = 0, y9 >= 0, z >= 0, z1 = 1 + (1 + y9) + z, 1 + (1 + y9) + 0 = y, x >= 0, y >= 0, 1 = 1, z = x del(z'', z1) -{ 4 }-> y :|: z'' - 1 >= 0, z1 = 1 + 0 + 1, 1 + 0 + 1 = y, x >= 0, y >= 0, 1 = 1, 1 = x del(z'', z1) -{ 3 }-> y :|: z'' - 1 >= 0, z1 - 1 >= 0, 1 + 0 + 0 = y, x >= 0, y >= 0, 1 = 1, z1 - 1 = x del(z'', z1) -{ 2 }-> if(eq(z'' - 1, y10), z, 1 + (1 + y10) + 0) :|: z >= 0, y10 >= 0, z'' - 1 >= 0, z1 = 1 + (1 + y10) + z del(z'', z1) -{ 3 }-> if(eq(z'' - 1, y10), 1 + y14 + z11, 1 + (1 + y10) + if(eq(1 + (z'' - 1), y14), z11, 1 + y14 + del(1 + (z'' - 1), z11))) :|: z11 >= 0, y10 >= 0, z1 = 1 + (1 + y10) + (1 + y14 + z11), z'' - 1 >= 0, y14 >= 0 del(z'', z1) -{ 3 }-> if(eq(z'' - 1, z1 - 3), 1, 1 + (1 + (z1 - 3)) + 1) :|: z1 - 3 >= 0, z'' - 1 >= 0 del(z'', z1) -{ 3 }-> if(2, 1 + y11 + z8, 1 + 0 + if(eq(0, y11), z8, 1 + y11 + del(0, z8))) :|: z'' = 0, z1 = 1 + 0 + (1 + y11 + z8), y11 >= 0, z8 >= 0 del(z'', z1) -{ 3 }-> if(1, 1 + y12 + z9, 1 + (1 + y9) + if(eq(0, y12), z9, 1 + y12 + del(0, z9))) :|: z'' = 0, y9 >= 0, z9 >= 0, y12 >= 0, z1 = 1 + (1 + y9) + (1 + y12 + z9) del(z'', z1) -{ 3 }-> if(1, 1 + y13 + z10, 1 + 0 + if(eq(1 + (z'' - 1), y13), z10, 1 + y13 + del(1 + (z'' - 1), z10))) :|: z1 = 1 + 0 + (1 + y13 + z10), z'' - 1 >= 0, z10 >= 0, y13 >= 0 del(z'', z1) -{ 2 }-> if(0, 1 + y15 + z12, 1 + y + if(eq(z'', y15), z12, 1 + y15 + del(z'', z12))) :|: z1 = 1 + y + (1 + y15 + z12), z'' >= 0, y >= 0, z12 >= 0, y15 >= 0 del(z'', z1) -{ 1 }-> 1 :|: z'' >= 0, z1 = 1 del(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0 del(z'', z1) -{ 3 }-> 0 :|: z'' = 0, z1 = 1 + 0 + 1, 2 = v0, 1 + 0 + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 2 }-> 0 :|: z'' = 0, z1 - 1 >= 0, 2 = v0, 1 + 0 + 0 = v2, v0 >= 0, z1 - 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 3 }-> 0 :|: z'' = 0, z1 - 3 >= 0, 1 = v0, 1 + (1 + (z1 - 3)) + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 2 }-> 0 :|: z'' = 0, y9 >= 0, z >= 0, z1 = 1 + (1 + y9) + z, 1 = v0, 1 + (1 + y9) + 0 = v2, v0 >= 0, z = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 3 }-> 0 :|: z'' - 1 >= 0, z1 = 1 + 0 + 1, 1 = v0, 1 + 0 + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 2 }-> 0 :|: z'' - 1 >= 0, z1 - 1 >= 0, 1 = v0, 1 + 0 + 0 = v2, v0 >= 0, z1 - 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 2 }-> 0 :|: z'' >= 0, z1 - 2 >= 0, 0 = v0, 1 + (z1 - 2) + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 1 }-> 0 :|: z >= 0, z1 = 1 + y + z, z'' >= 0, y >= 0, 0 = v0, 1 + y + 0 = v2, v0 >= 0, z = v1, v1 >= 0, v2 >= 0 eq(z'', z1) -{ 1 }-> eq(z'' - 1, z1 - 1) :|: z'' - 1 >= 0, z1 - 1 >= 0 eq(z'', z1) -{ 1 }-> 2 :|: z'' = 0, z1 = 0 eq(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 - 1 >= 0 eq(z'', z1) -{ 1 }-> 1 :|: z1 = 0, z'' - 1 >= 0 eq(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0 if(z'', z1, z2) -{ 1 }-> z1 :|: z1 >= 0, z2 >= 0, z'' = 2 if(z'', z1, z2) -{ 1 }-> z2 :|: z1 >= 0, z2 >= 0, z'' = 1 if(z'', z1, z2) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0, z2 >= 0 le(z'', z1) -{ 2 + z1 }-> s :|: s >= 0, s <= 2, z'' - 1 >= 0, z1 - 1 >= 0 le(z'', z1) -{ 1 }-> 2 :|: z'' = 0, z1 >= 0 le(z'', z1) -{ 1 }-> 1 :|: z1 = 0, z'' - 1 >= 0 le(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0 min(z'', z1) -{ 1 }-> z'' :|: z'' >= 0, z1 = 1 min(z'', z1) -{ 3 + y }-> if(s'', min(z'', z), min(y, z)) :|: s'' >= 0, s'' <= 2, z >= 0, z1 = 1 + y + z, z'' >= 0, y >= 0 min(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0 minsort(z'') -{ 1 }-> 1 :|: z'' = 1 minsort(z'') -{ 1 }-> 1 + min(x, y) + minsort(del(0, 1 + x + y)) :|: z'' = 1 + x + y, x >= 0, y >= 0 minsort(z'') -{ 4 + y' }-> 1 + min(x, 1 + y' + z') + minsort(del(if(s', min(x, z'), min(y', z')), 1 + x + (1 + y' + z'))) :|: s' >= 0, s' <= 2, z'' = 1 + x + (1 + y' + z'), x >= 0, z' >= 0, y' >= 0 minsort(z'') -{ 2 }-> 1 + min(z'' - 2, 1) + minsort(del(z'' - 2, 1 + (z'' - 2) + 1)) :|: z'' - 2 >= 0 Function symbols to be analyzed: {eq}, {min}, {del}, {minsort} Previous analysis results are: le: runtime: O(n^1) [2 + z1], size: O(1) [2] if: runtime: O(1) [1], size: O(n^1) [z1 + z2] ---------------------------------------- (33) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (34) Obligation: Complexity RNTS consisting of the following rules: del(z'', z1) -{ 4 }-> x :|: z'' = 0, z1 = 1 + 0 + 1, 1 + 0 + 1 = y, x >= 0, y >= 0, 2 = 2, 1 = x del(z'', z1) -{ 3 }-> x :|: z'' = 0, z1 - 1 >= 0, 1 + 0 + 0 = y, x >= 0, y >= 0, 2 = 2, z1 - 1 = x del(z'', z1) -{ 4 }-> y :|: z'' = 0, z1 - 3 >= 0, 1 + (1 + (z1 - 3)) + 1 = y, x >= 0, y >= 0, 1 = 1, 1 = x del(z'', z1) -{ 3 }-> y :|: z'' = 0, y9 >= 0, z >= 0, z1 = 1 + (1 + y9) + z, 1 + (1 + y9) + 0 = y, x >= 0, y >= 0, 1 = 1, z = x del(z'', z1) -{ 4 }-> y :|: z'' - 1 >= 0, z1 = 1 + 0 + 1, 1 + 0 + 1 = y, x >= 0, y >= 0, 1 = 1, 1 = x del(z'', z1) -{ 3 }-> y :|: z'' - 1 >= 0, z1 - 1 >= 0, 1 + 0 + 0 = y, x >= 0, y >= 0, 1 = 1, z1 - 1 = x del(z'', z1) -{ 2 }-> if(eq(z'' - 1, y10), z, 1 + (1 + y10) + 0) :|: z >= 0, y10 >= 0, z'' - 1 >= 0, z1 = 1 + (1 + y10) + z del(z'', z1) -{ 3 }-> if(eq(z'' - 1, y10), 1 + y14 + z11, 1 + (1 + y10) + if(eq(1 + (z'' - 1), y14), z11, 1 + y14 + del(1 + (z'' - 1), z11))) :|: z11 >= 0, y10 >= 0, z1 = 1 + (1 + y10) + (1 + y14 + z11), z'' - 1 >= 0, y14 >= 0 del(z'', z1) -{ 3 }-> if(eq(z'' - 1, z1 - 3), 1, 1 + (1 + (z1 - 3)) + 1) :|: z1 - 3 >= 0, z'' - 1 >= 0 del(z'', z1) -{ 3 }-> if(2, 1 + y11 + z8, 1 + 0 + if(eq(0, y11), z8, 1 + y11 + del(0, z8))) :|: z'' = 0, z1 = 1 + 0 + (1 + y11 + z8), y11 >= 0, z8 >= 0 del(z'', z1) -{ 3 }-> if(1, 1 + y12 + z9, 1 + (1 + y9) + if(eq(0, y12), z9, 1 + y12 + del(0, z9))) :|: z'' = 0, y9 >= 0, z9 >= 0, y12 >= 0, z1 = 1 + (1 + y9) + (1 + y12 + z9) del(z'', z1) -{ 3 }-> if(1, 1 + y13 + z10, 1 + 0 + if(eq(1 + (z'' - 1), y13), z10, 1 + y13 + del(1 + (z'' - 1), z10))) :|: z1 = 1 + 0 + (1 + y13 + z10), z'' - 1 >= 0, z10 >= 0, y13 >= 0 del(z'', z1) -{ 2 }-> if(0, 1 + y15 + z12, 1 + y + if(eq(z'', y15), z12, 1 + y15 + del(z'', z12))) :|: z1 = 1 + y + (1 + y15 + z12), z'' >= 0, y >= 0, z12 >= 0, y15 >= 0 del(z'', z1) -{ 1 }-> 1 :|: z'' >= 0, z1 = 1 del(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0 del(z'', z1) -{ 3 }-> 0 :|: z'' = 0, z1 = 1 + 0 + 1, 2 = v0, 1 + 0 + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 2 }-> 0 :|: z'' = 0, z1 - 1 >= 0, 2 = v0, 1 + 0 + 0 = v2, v0 >= 0, z1 - 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 3 }-> 0 :|: z'' = 0, z1 - 3 >= 0, 1 = v0, 1 + (1 + (z1 - 3)) + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 2 }-> 0 :|: z'' = 0, y9 >= 0, z >= 0, z1 = 1 + (1 + y9) + z, 1 = v0, 1 + (1 + y9) + 0 = v2, v0 >= 0, z = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 3 }-> 0 :|: z'' - 1 >= 0, z1 = 1 + 0 + 1, 1 = v0, 1 + 0 + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 2 }-> 0 :|: z'' - 1 >= 0, z1 - 1 >= 0, 1 = v0, 1 + 0 + 0 = v2, v0 >= 0, z1 - 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 2 }-> 0 :|: z'' >= 0, z1 - 2 >= 0, 0 = v0, 1 + (z1 - 2) + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 1 }-> 0 :|: z >= 0, z1 = 1 + y + z, z'' >= 0, y >= 0, 0 = v0, 1 + y + 0 = v2, v0 >= 0, z = v1, v1 >= 0, v2 >= 0 eq(z'', z1) -{ 1 }-> eq(z'' - 1, z1 - 1) :|: z'' - 1 >= 0, z1 - 1 >= 0 eq(z'', z1) -{ 1 }-> 2 :|: z'' = 0, z1 = 0 eq(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 - 1 >= 0 eq(z'', z1) -{ 1 }-> 1 :|: z1 = 0, z'' - 1 >= 0 eq(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0 if(z'', z1, z2) -{ 1 }-> z1 :|: z1 >= 0, z2 >= 0, z'' = 2 if(z'', z1, z2) -{ 1 }-> z2 :|: z1 >= 0, z2 >= 0, z'' = 1 if(z'', z1, z2) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0, z2 >= 0 le(z'', z1) -{ 2 + z1 }-> s :|: s >= 0, s <= 2, z'' - 1 >= 0, z1 - 1 >= 0 le(z'', z1) -{ 1 }-> 2 :|: z'' = 0, z1 >= 0 le(z'', z1) -{ 1 }-> 1 :|: z1 = 0, z'' - 1 >= 0 le(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0 min(z'', z1) -{ 1 }-> z'' :|: z'' >= 0, z1 = 1 min(z'', z1) -{ 3 + y }-> if(s'', min(z'', z), min(y, z)) :|: s'' >= 0, s'' <= 2, z >= 0, z1 = 1 + y + z, z'' >= 0, y >= 0 min(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0 minsort(z'') -{ 1 }-> 1 :|: z'' = 1 minsort(z'') -{ 1 }-> 1 + min(x, y) + minsort(del(0, 1 + x + y)) :|: z'' = 1 + x + y, x >= 0, y >= 0 minsort(z'') -{ 4 + y' }-> 1 + min(x, 1 + y' + z') + minsort(del(if(s', min(x, z'), min(y', z')), 1 + x + (1 + y' + z'))) :|: s' >= 0, s' <= 2, z'' = 1 + x + (1 + y' + z'), x >= 0, z' >= 0, y' >= 0 minsort(z'') -{ 2 }-> 1 + min(z'' - 2, 1) + minsort(del(z'' - 2, 1 + (z'' - 2) + 1)) :|: z'' - 2 >= 0 Function symbols to be analyzed: {eq}, {min}, {del}, {minsort} Previous analysis results are: le: runtime: O(n^1) [2 + z1], size: O(1) [2] if: runtime: O(1) [1], size: O(n^1) [z1 + z2] ---------------------------------------- (35) 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: 2 ---------------------------------------- (36) Obligation: Complexity RNTS consisting of the following rules: del(z'', z1) -{ 4 }-> x :|: z'' = 0, z1 = 1 + 0 + 1, 1 + 0 + 1 = y, x >= 0, y >= 0, 2 = 2, 1 = x del(z'', z1) -{ 3 }-> x :|: z'' = 0, z1 - 1 >= 0, 1 + 0 + 0 = y, x >= 0, y >= 0, 2 = 2, z1 - 1 = x del(z'', z1) -{ 4 }-> y :|: z'' = 0, z1 - 3 >= 0, 1 + (1 + (z1 - 3)) + 1 = y, x >= 0, y >= 0, 1 = 1, 1 = x del(z'', z1) -{ 3 }-> y :|: z'' = 0, y9 >= 0, z >= 0, z1 = 1 + (1 + y9) + z, 1 + (1 + y9) + 0 = y, x >= 0, y >= 0, 1 = 1, z = x del(z'', z1) -{ 4 }-> y :|: z'' - 1 >= 0, z1 = 1 + 0 + 1, 1 + 0 + 1 = y, x >= 0, y >= 0, 1 = 1, 1 = x del(z'', z1) -{ 3 }-> y :|: z'' - 1 >= 0, z1 - 1 >= 0, 1 + 0 + 0 = y, x >= 0, y >= 0, 1 = 1, z1 - 1 = x del(z'', z1) -{ 2 }-> if(eq(z'' - 1, y10), z, 1 + (1 + y10) + 0) :|: z >= 0, y10 >= 0, z'' - 1 >= 0, z1 = 1 + (1 + y10) + z del(z'', z1) -{ 3 }-> if(eq(z'' - 1, y10), 1 + y14 + z11, 1 + (1 + y10) + if(eq(1 + (z'' - 1), y14), z11, 1 + y14 + del(1 + (z'' - 1), z11))) :|: z11 >= 0, y10 >= 0, z1 = 1 + (1 + y10) + (1 + y14 + z11), z'' - 1 >= 0, y14 >= 0 del(z'', z1) -{ 3 }-> if(eq(z'' - 1, z1 - 3), 1, 1 + (1 + (z1 - 3)) + 1) :|: z1 - 3 >= 0, z'' - 1 >= 0 del(z'', z1) -{ 3 }-> if(2, 1 + y11 + z8, 1 + 0 + if(eq(0, y11), z8, 1 + y11 + del(0, z8))) :|: z'' = 0, z1 = 1 + 0 + (1 + y11 + z8), y11 >= 0, z8 >= 0 del(z'', z1) -{ 3 }-> if(1, 1 + y12 + z9, 1 + (1 + y9) + if(eq(0, y12), z9, 1 + y12 + del(0, z9))) :|: z'' = 0, y9 >= 0, z9 >= 0, y12 >= 0, z1 = 1 + (1 + y9) + (1 + y12 + z9) del(z'', z1) -{ 3 }-> if(1, 1 + y13 + z10, 1 + 0 + if(eq(1 + (z'' - 1), y13), z10, 1 + y13 + del(1 + (z'' - 1), z10))) :|: z1 = 1 + 0 + (1 + y13 + z10), z'' - 1 >= 0, z10 >= 0, y13 >= 0 del(z'', z1) -{ 2 }-> if(0, 1 + y15 + z12, 1 + y + if(eq(z'', y15), z12, 1 + y15 + del(z'', z12))) :|: z1 = 1 + y + (1 + y15 + z12), z'' >= 0, y >= 0, z12 >= 0, y15 >= 0 del(z'', z1) -{ 1 }-> 1 :|: z'' >= 0, z1 = 1 del(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0 del(z'', z1) -{ 3 }-> 0 :|: z'' = 0, z1 = 1 + 0 + 1, 2 = v0, 1 + 0 + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 2 }-> 0 :|: z'' = 0, z1 - 1 >= 0, 2 = v0, 1 + 0 + 0 = v2, v0 >= 0, z1 - 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 3 }-> 0 :|: z'' = 0, z1 - 3 >= 0, 1 = v0, 1 + (1 + (z1 - 3)) + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 2 }-> 0 :|: z'' = 0, y9 >= 0, z >= 0, z1 = 1 + (1 + y9) + z, 1 = v0, 1 + (1 + y9) + 0 = v2, v0 >= 0, z = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 3 }-> 0 :|: z'' - 1 >= 0, z1 = 1 + 0 + 1, 1 = v0, 1 + 0 + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 2 }-> 0 :|: z'' - 1 >= 0, z1 - 1 >= 0, 1 = v0, 1 + 0 + 0 = v2, v0 >= 0, z1 - 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 2 }-> 0 :|: z'' >= 0, z1 - 2 >= 0, 0 = v0, 1 + (z1 - 2) + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 1 }-> 0 :|: z >= 0, z1 = 1 + y + z, z'' >= 0, y >= 0, 0 = v0, 1 + y + 0 = v2, v0 >= 0, z = v1, v1 >= 0, v2 >= 0 eq(z'', z1) -{ 1 }-> eq(z'' - 1, z1 - 1) :|: z'' - 1 >= 0, z1 - 1 >= 0 eq(z'', z1) -{ 1 }-> 2 :|: z'' = 0, z1 = 0 eq(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 - 1 >= 0 eq(z'', z1) -{ 1 }-> 1 :|: z1 = 0, z'' - 1 >= 0 eq(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0 if(z'', z1, z2) -{ 1 }-> z1 :|: z1 >= 0, z2 >= 0, z'' = 2 if(z'', z1, z2) -{ 1 }-> z2 :|: z1 >= 0, z2 >= 0, z'' = 1 if(z'', z1, z2) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0, z2 >= 0 le(z'', z1) -{ 2 + z1 }-> s :|: s >= 0, s <= 2, z'' - 1 >= 0, z1 - 1 >= 0 le(z'', z1) -{ 1 }-> 2 :|: z'' = 0, z1 >= 0 le(z'', z1) -{ 1 }-> 1 :|: z1 = 0, z'' - 1 >= 0 le(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0 min(z'', z1) -{ 1 }-> z'' :|: z'' >= 0, z1 = 1 min(z'', z1) -{ 3 + y }-> if(s'', min(z'', z), min(y, z)) :|: s'' >= 0, s'' <= 2, z >= 0, z1 = 1 + y + z, z'' >= 0, y >= 0 min(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0 minsort(z'') -{ 1 }-> 1 :|: z'' = 1 minsort(z'') -{ 1 }-> 1 + min(x, y) + minsort(del(0, 1 + x + y)) :|: z'' = 1 + x + y, x >= 0, y >= 0 minsort(z'') -{ 4 + y' }-> 1 + min(x, 1 + y' + z') + minsort(del(if(s', min(x, z'), min(y', z')), 1 + x + (1 + y' + z'))) :|: s' >= 0, s' <= 2, z'' = 1 + x + (1 + y' + z'), x >= 0, z' >= 0, y' >= 0 minsort(z'') -{ 2 }-> 1 + min(z'' - 2, 1) + minsort(del(z'' - 2, 1 + (z'' - 2) + 1)) :|: z'' - 2 >= 0 Function symbols to be analyzed: {eq}, {min}, {del}, {minsort} Previous analysis results are: le: runtime: O(n^1) [2 + z1], size: O(1) [2] if: runtime: O(1) [1], size: O(n^1) [z1 + z2] eq: runtime: ?, size: O(1) [2] ---------------------------------------- (37) 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'' ---------------------------------------- (38) Obligation: Complexity RNTS consisting of the following rules: del(z'', z1) -{ 4 }-> x :|: z'' = 0, z1 = 1 + 0 + 1, 1 + 0 + 1 = y, x >= 0, y >= 0, 2 = 2, 1 = x del(z'', z1) -{ 3 }-> x :|: z'' = 0, z1 - 1 >= 0, 1 + 0 + 0 = y, x >= 0, y >= 0, 2 = 2, z1 - 1 = x del(z'', z1) -{ 4 }-> y :|: z'' = 0, z1 - 3 >= 0, 1 + (1 + (z1 - 3)) + 1 = y, x >= 0, y >= 0, 1 = 1, 1 = x del(z'', z1) -{ 3 }-> y :|: z'' = 0, y9 >= 0, z >= 0, z1 = 1 + (1 + y9) + z, 1 + (1 + y9) + 0 = y, x >= 0, y >= 0, 1 = 1, z = x del(z'', z1) -{ 4 }-> y :|: z'' - 1 >= 0, z1 = 1 + 0 + 1, 1 + 0 + 1 = y, x >= 0, y >= 0, 1 = 1, 1 = x del(z'', z1) -{ 3 }-> y :|: z'' - 1 >= 0, z1 - 1 >= 0, 1 + 0 + 0 = y, x >= 0, y >= 0, 1 = 1, z1 - 1 = x del(z'', z1) -{ 2 }-> if(eq(z'' - 1, y10), z, 1 + (1 + y10) + 0) :|: z >= 0, y10 >= 0, z'' - 1 >= 0, z1 = 1 + (1 + y10) + z del(z'', z1) -{ 3 }-> if(eq(z'' - 1, y10), 1 + y14 + z11, 1 + (1 + y10) + if(eq(1 + (z'' - 1), y14), z11, 1 + y14 + del(1 + (z'' - 1), z11))) :|: z11 >= 0, y10 >= 0, z1 = 1 + (1 + y10) + (1 + y14 + z11), z'' - 1 >= 0, y14 >= 0 del(z'', z1) -{ 3 }-> if(eq(z'' - 1, z1 - 3), 1, 1 + (1 + (z1 - 3)) + 1) :|: z1 - 3 >= 0, z'' - 1 >= 0 del(z'', z1) -{ 3 }-> if(2, 1 + y11 + z8, 1 + 0 + if(eq(0, y11), z8, 1 + y11 + del(0, z8))) :|: z'' = 0, z1 = 1 + 0 + (1 + y11 + z8), y11 >= 0, z8 >= 0 del(z'', z1) -{ 3 }-> if(1, 1 + y12 + z9, 1 + (1 + y9) + if(eq(0, y12), z9, 1 + y12 + del(0, z9))) :|: z'' = 0, y9 >= 0, z9 >= 0, y12 >= 0, z1 = 1 + (1 + y9) + (1 + y12 + z9) del(z'', z1) -{ 3 }-> if(1, 1 + y13 + z10, 1 + 0 + if(eq(1 + (z'' - 1), y13), z10, 1 + y13 + del(1 + (z'' - 1), z10))) :|: z1 = 1 + 0 + (1 + y13 + z10), z'' - 1 >= 0, z10 >= 0, y13 >= 0 del(z'', z1) -{ 2 }-> if(0, 1 + y15 + z12, 1 + y + if(eq(z'', y15), z12, 1 + y15 + del(z'', z12))) :|: z1 = 1 + y + (1 + y15 + z12), z'' >= 0, y >= 0, z12 >= 0, y15 >= 0 del(z'', z1) -{ 1 }-> 1 :|: z'' >= 0, z1 = 1 del(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0 del(z'', z1) -{ 3 }-> 0 :|: z'' = 0, z1 = 1 + 0 + 1, 2 = v0, 1 + 0 + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 2 }-> 0 :|: z'' = 0, z1 - 1 >= 0, 2 = v0, 1 + 0 + 0 = v2, v0 >= 0, z1 - 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 3 }-> 0 :|: z'' = 0, z1 - 3 >= 0, 1 = v0, 1 + (1 + (z1 - 3)) + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 2 }-> 0 :|: z'' = 0, y9 >= 0, z >= 0, z1 = 1 + (1 + y9) + z, 1 = v0, 1 + (1 + y9) + 0 = v2, v0 >= 0, z = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 3 }-> 0 :|: z'' - 1 >= 0, z1 = 1 + 0 + 1, 1 = v0, 1 + 0 + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 2 }-> 0 :|: z'' - 1 >= 0, z1 - 1 >= 0, 1 = v0, 1 + 0 + 0 = v2, v0 >= 0, z1 - 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 2 }-> 0 :|: z'' >= 0, z1 - 2 >= 0, 0 = v0, 1 + (z1 - 2) + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 1 }-> 0 :|: z >= 0, z1 = 1 + y + z, z'' >= 0, y >= 0, 0 = v0, 1 + y + 0 = v2, v0 >= 0, z = v1, v1 >= 0, v2 >= 0 eq(z'', z1) -{ 1 }-> eq(z'' - 1, z1 - 1) :|: z'' - 1 >= 0, z1 - 1 >= 0 eq(z'', z1) -{ 1 }-> 2 :|: z'' = 0, z1 = 0 eq(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 - 1 >= 0 eq(z'', z1) -{ 1 }-> 1 :|: z1 = 0, z'' - 1 >= 0 eq(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0 if(z'', z1, z2) -{ 1 }-> z1 :|: z1 >= 0, z2 >= 0, z'' = 2 if(z'', z1, z2) -{ 1 }-> z2 :|: z1 >= 0, z2 >= 0, z'' = 1 if(z'', z1, z2) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0, z2 >= 0 le(z'', z1) -{ 2 + z1 }-> s :|: s >= 0, s <= 2, z'' - 1 >= 0, z1 - 1 >= 0 le(z'', z1) -{ 1 }-> 2 :|: z'' = 0, z1 >= 0 le(z'', z1) -{ 1 }-> 1 :|: z1 = 0, z'' - 1 >= 0 le(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0 min(z'', z1) -{ 1 }-> z'' :|: z'' >= 0, z1 = 1 min(z'', z1) -{ 3 + y }-> if(s'', min(z'', z), min(y, z)) :|: s'' >= 0, s'' <= 2, z >= 0, z1 = 1 + y + z, z'' >= 0, y >= 0 min(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0 minsort(z'') -{ 1 }-> 1 :|: z'' = 1 minsort(z'') -{ 1 }-> 1 + min(x, y) + minsort(del(0, 1 + x + y)) :|: z'' = 1 + x + y, x >= 0, y >= 0 minsort(z'') -{ 4 + y' }-> 1 + min(x, 1 + y' + z') + minsort(del(if(s', min(x, z'), min(y', z')), 1 + x + (1 + y' + z'))) :|: s' >= 0, s' <= 2, z'' = 1 + x + (1 + y' + z'), x >= 0, z' >= 0, y' >= 0 minsort(z'') -{ 2 }-> 1 + min(z'' - 2, 1) + minsort(del(z'' - 2, 1 + (z'' - 2) + 1)) :|: z'' - 2 >= 0 Function symbols to be analyzed: {min}, {del}, {minsort} Previous analysis results are: le: runtime: O(n^1) [2 + z1], size: O(1) [2] if: runtime: O(1) [1], size: O(n^1) [z1 + z2] eq: runtime: O(n^1) [3 + z''], size: O(1) [2] ---------------------------------------- (39) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (40) Obligation: Complexity RNTS consisting of the following rules: del(z'', z1) -{ 5 + z'' }-> s10 :|: s9 >= 0, s9 <= 2, s10 >= 0, s10 <= 1 + (1 + y10) + 0 + z, z >= 0, y10 >= 0, z'' - 1 >= 0, z1 = 1 + (1 + y10) + z del(z'', z1) -{ 6 + z'' }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= 1 + (1 + (z1 - 3)) + 1 + 1, z1 - 3 >= 0, z'' - 1 >= 0 del(z'', z1) -{ 4 }-> x :|: z'' = 0, z1 = 1 + 0 + 1, 1 + 0 + 1 = y, x >= 0, y >= 0, 2 = 2, 1 = x del(z'', z1) -{ 3 }-> x :|: z'' = 0, z1 - 1 >= 0, 1 + 0 + 0 = y, x >= 0, y >= 0, 2 = 2, z1 - 1 = x del(z'', z1) -{ 4 }-> y :|: z'' = 0, z1 - 3 >= 0, 1 + (1 + (z1 - 3)) + 1 = y, x >= 0, y >= 0, 1 = 1, 1 = x del(z'', z1) -{ 3 }-> y :|: z'' = 0, y9 >= 0, z >= 0, z1 = 1 + (1 + y9) + z, 1 + (1 + y9) + 0 = y, x >= 0, y >= 0, 1 = 1, z = x del(z'', z1) -{ 4 }-> y :|: z'' - 1 >= 0, z1 = 1 + 0 + 1, 1 + 0 + 1 = y, x >= 0, y >= 0, 1 = 1, 1 = x del(z'', z1) -{ 3 }-> y :|: z'' - 1 >= 0, z1 - 1 >= 0, 1 + 0 + 0 = y, x >= 0, y >= 0, 1 = 1, z1 - 1 = x del(z'', z1) -{ 8 + 2*z'' }-> if(s7, 1 + y14 + z11, 1 + (1 + y10) + if(s8, z11, 1 + y14 + del(1 + (z'' - 1), z11))) :|: s7 >= 0, s7 <= 2, s8 >= 0, s8 <= 2, z11 >= 0, y10 >= 0, z1 = 1 + (1 + y10) + (1 + y14 + z11), z'' - 1 >= 0, y14 >= 0 del(z'', z1) -{ 6 }-> if(2, 1 + y11 + z8, 1 + 0 + if(s2, z8, 1 + y11 + del(0, z8))) :|: s2 >= 0, s2 <= 2, z'' = 0, z1 = 1 + 0 + (1 + y11 + z8), y11 >= 0, z8 >= 0 del(z'', z1) -{ 6 }-> if(1, 1 + y12 + z9, 1 + (1 + y9) + if(s3, z9, 1 + y12 + del(0, z9))) :|: s3 >= 0, s3 <= 2, z'' = 0, y9 >= 0, z9 >= 0, y12 >= 0, z1 = 1 + (1 + y9) + (1 + y12 + z9) del(z'', z1) -{ 6 + z'' }-> if(1, 1 + y13 + z10, 1 + 0 + if(s4, z10, 1 + y13 + del(1 + (z'' - 1), z10))) :|: s4 >= 0, s4 <= 2, z1 = 1 + 0 + (1 + y13 + z10), z'' - 1 >= 0, z10 >= 0, y13 >= 0 del(z'', z1) -{ 5 + z'' }-> if(0, 1 + y15 + z12, 1 + y + if(s11, z12, 1 + y15 + del(z'', z12))) :|: s11 >= 0, s11 <= 2, z1 = 1 + y + (1 + y15 + z12), z'' >= 0, y >= 0, z12 >= 0, y15 >= 0 del(z'', z1) -{ 1 }-> 1 :|: z'' >= 0, z1 = 1 del(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0 del(z'', z1) -{ 3 }-> 0 :|: z'' = 0, z1 = 1 + 0 + 1, 2 = v0, 1 + 0 + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 2 }-> 0 :|: z'' = 0, z1 - 1 >= 0, 2 = v0, 1 + 0 + 0 = v2, v0 >= 0, z1 - 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 3 }-> 0 :|: z'' = 0, z1 - 3 >= 0, 1 = v0, 1 + (1 + (z1 - 3)) + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 2 }-> 0 :|: z'' = 0, y9 >= 0, z >= 0, z1 = 1 + (1 + y9) + z, 1 = v0, 1 + (1 + y9) + 0 = v2, v0 >= 0, z = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 3 }-> 0 :|: z'' - 1 >= 0, z1 = 1 + 0 + 1, 1 = v0, 1 + 0 + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 2 }-> 0 :|: z'' - 1 >= 0, z1 - 1 >= 0, 1 = v0, 1 + 0 + 0 = v2, v0 >= 0, z1 - 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 2 }-> 0 :|: z'' >= 0, z1 - 2 >= 0, 0 = v0, 1 + (z1 - 2) + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 1 }-> 0 :|: z >= 0, z1 = 1 + y + z, z'' >= 0, y >= 0, 0 = v0, 1 + y + 0 = v2, v0 >= 0, z = v1, v1 >= 0, v2 >= 0 eq(z'', z1) -{ 3 + z'' }-> s1 :|: s1 >= 0, s1 <= 2, z'' - 1 >= 0, z1 - 1 >= 0 eq(z'', z1) -{ 1 }-> 2 :|: z'' = 0, z1 = 0 eq(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 - 1 >= 0 eq(z'', z1) -{ 1 }-> 1 :|: z1 = 0, z'' - 1 >= 0 eq(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0 if(z'', z1, z2) -{ 1 }-> z1 :|: z1 >= 0, z2 >= 0, z'' = 2 if(z'', z1, z2) -{ 1 }-> z2 :|: z1 >= 0, z2 >= 0, z'' = 1 if(z'', z1, z2) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0, z2 >= 0 le(z'', z1) -{ 2 + z1 }-> s :|: s >= 0, s <= 2, z'' - 1 >= 0, z1 - 1 >= 0 le(z'', z1) -{ 1 }-> 2 :|: z'' = 0, z1 >= 0 le(z'', z1) -{ 1 }-> 1 :|: z1 = 0, z'' - 1 >= 0 le(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0 min(z'', z1) -{ 1 }-> z'' :|: z'' >= 0, z1 = 1 min(z'', z1) -{ 3 + y }-> if(s'', min(z'', z), min(y, z)) :|: s'' >= 0, s'' <= 2, z >= 0, z1 = 1 + y + z, z'' >= 0, y >= 0 min(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0 minsort(z'') -{ 1 }-> 1 :|: z'' = 1 minsort(z'') -{ 1 }-> 1 + min(x, y) + minsort(del(0, 1 + x + y)) :|: z'' = 1 + x + y, x >= 0, y >= 0 minsort(z'') -{ 4 + y' }-> 1 + min(x, 1 + y' + z') + minsort(del(if(s', min(x, z'), min(y', z')), 1 + x + (1 + y' + z'))) :|: s' >= 0, s' <= 2, z'' = 1 + x + (1 + y' + z'), x >= 0, z' >= 0, y' >= 0 minsort(z'') -{ 2 }-> 1 + min(z'' - 2, 1) + minsort(del(z'' - 2, 1 + (z'' - 2) + 1)) :|: z'' - 2 >= 0 Function symbols to be analyzed: {min}, {del}, {minsort} Previous analysis results are: le: runtime: O(n^1) [2 + z1], size: O(1) [2] if: runtime: O(1) [1], size: O(n^1) [z1 + z2] eq: runtime: O(n^1) [3 + z''], size: O(1) [2] ---------------------------------------- (41) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: min after applying outer abstraction to obtain an ITS, resulting in: EXP with polynomial bound: ? ---------------------------------------- (42) Obligation: Complexity RNTS consisting of the following rules: del(z'', z1) -{ 5 + z'' }-> s10 :|: s9 >= 0, s9 <= 2, s10 >= 0, s10 <= 1 + (1 + y10) + 0 + z, z >= 0, y10 >= 0, z'' - 1 >= 0, z1 = 1 + (1 + y10) + z del(z'', z1) -{ 6 + z'' }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= 1 + (1 + (z1 - 3)) + 1 + 1, z1 - 3 >= 0, z'' - 1 >= 0 del(z'', z1) -{ 4 }-> x :|: z'' = 0, z1 = 1 + 0 + 1, 1 + 0 + 1 = y, x >= 0, y >= 0, 2 = 2, 1 = x del(z'', z1) -{ 3 }-> x :|: z'' = 0, z1 - 1 >= 0, 1 + 0 + 0 = y, x >= 0, y >= 0, 2 = 2, z1 - 1 = x del(z'', z1) -{ 4 }-> y :|: z'' = 0, z1 - 3 >= 0, 1 + (1 + (z1 - 3)) + 1 = y, x >= 0, y >= 0, 1 = 1, 1 = x del(z'', z1) -{ 3 }-> y :|: z'' = 0, y9 >= 0, z >= 0, z1 = 1 + (1 + y9) + z, 1 + (1 + y9) + 0 = y, x >= 0, y >= 0, 1 = 1, z = x del(z'', z1) -{ 4 }-> y :|: z'' - 1 >= 0, z1 = 1 + 0 + 1, 1 + 0 + 1 = y, x >= 0, y >= 0, 1 = 1, 1 = x del(z'', z1) -{ 3 }-> y :|: z'' - 1 >= 0, z1 - 1 >= 0, 1 + 0 + 0 = y, x >= 0, y >= 0, 1 = 1, z1 - 1 = x del(z'', z1) -{ 8 + 2*z'' }-> if(s7, 1 + y14 + z11, 1 + (1 + y10) + if(s8, z11, 1 + y14 + del(1 + (z'' - 1), z11))) :|: s7 >= 0, s7 <= 2, s8 >= 0, s8 <= 2, z11 >= 0, y10 >= 0, z1 = 1 + (1 + y10) + (1 + y14 + z11), z'' - 1 >= 0, y14 >= 0 del(z'', z1) -{ 6 }-> if(2, 1 + y11 + z8, 1 + 0 + if(s2, z8, 1 + y11 + del(0, z8))) :|: s2 >= 0, s2 <= 2, z'' = 0, z1 = 1 + 0 + (1 + y11 + z8), y11 >= 0, z8 >= 0 del(z'', z1) -{ 6 }-> if(1, 1 + y12 + z9, 1 + (1 + y9) + if(s3, z9, 1 + y12 + del(0, z9))) :|: s3 >= 0, s3 <= 2, z'' = 0, y9 >= 0, z9 >= 0, y12 >= 0, z1 = 1 + (1 + y9) + (1 + y12 + z9) del(z'', z1) -{ 6 + z'' }-> if(1, 1 + y13 + z10, 1 + 0 + if(s4, z10, 1 + y13 + del(1 + (z'' - 1), z10))) :|: s4 >= 0, s4 <= 2, z1 = 1 + 0 + (1 + y13 + z10), z'' - 1 >= 0, z10 >= 0, y13 >= 0 del(z'', z1) -{ 5 + z'' }-> if(0, 1 + y15 + z12, 1 + y + if(s11, z12, 1 + y15 + del(z'', z12))) :|: s11 >= 0, s11 <= 2, z1 = 1 + y + (1 + y15 + z12), z'' >= 0, y >= 0, z12 >= 0, y15 >= 0 del(z'', z1) -{ 1 }-> 1 :|: z'' >= 0, z1 = 1 del(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0 del(z'', z1) -{ 3 }-> 0 :|: z'' = 0, z1 = 1 + 0 + 1, 2 = v0, 1 + 0 + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 2 }-> 0 :|: z'' = 0, z1 - 1 >= 0, 2 = v0, 1 + 0 + 0 = v2, v0 >= 0, z1 - 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 3 }-> 0 :|: z'' = 0, z1 - 3 >= 0, 1 = v0, 1 + (1 + (z1 - 3)) + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 2 }-> 0 :|: z'' = 0, y9 >= 0, z >= 0, z1 = 1 + (1 + y9) + z, 1 = v0, 1 + (1 + y9) + 0 = v2, v0 >= 0, z = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 3 }-> 0 :|: z'' - 1 >= 0, z1 = 1 + 0 + 1, 1 = v0, 1 + 0 + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 2 }-> 0 :|: z'' - 1 >= 0, z1 - 1 >= 0, 1 = v0, 1 + 0 + 0 = v2, v0 >= 0, z1 - 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 2 }-> 0 :|: z'' >= 0, z1 - 2 >= 0, 0 = v0, 1 + (z1 - 2) + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 1 }-> 0 :|: z >= 0, z1 = 1 + y + z, z'' >= 0, y >= 0, 0 = v0, 1 + y + 0 = v2, v0 >= 0, z = v1, v1 >= 0, v2 >= 0 eq(z'', z1) -{ 3 + z'' }-> s1 :|: s1 >= 0, s1 <= 2, z'' - 1 >= 0, z1 - 1 >= 0 eq(z'', z1) -{ 1 }-> 2 :|: z'' = 0, z1 = 0 eq(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 - 1 >= 0 eq(z'', z1) -{ 1 }-> 1 :|: z1 = 0, z'' - 1 >= 0 eq(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0 if(z'', z1, z2) -{ 1 }-> z1 :|: z1 >= 0, z2 >= 0, z'' = 2 if(z'', z1, z2) -{ 1 }-> z2 :|: z1 >= 0, z2 >= 0, z'' = 1 if(z'', z1, z2) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0, z2 >= 0 le(z'', z1) -{ 2 + z1 }-> s :|: s >= 0, s <= 2, z'' - 1 >= 0, z1 - 1 >= 0 le(z'', z1) -{ 1 }-> 2 :|: z'' = 0, z1 >= 0 le(z'', z1) -{ 1 }-> 1 :|: z1 = 0, z'' - 1 >= 0 le(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0 min(z'', z1) -{ 1 }-> z'' :|: z'' >= 0, z1 = 1 min(z'', z1) -{ 3 + y }-> if(s'', min(z'', z), min(y, z)) :|: s'' >= 0, s'' <= 2, z >= 0, z1 = 1 + y + z, z'' >= 0, y >= 0 min(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0 minsort(z'') -{ 1 }-> 1 :|: z'' = 1 minsort(z'') -{ 1 }-> 1 + min(x, y) + minsort(del(0, 1 + x + y)) :|: z'' = 1 + x + y, x >= 0, y >= 0 minsort(z'') -{ 4 + y' }-> 1 + min(x, 1 + y' + z') + minsort(del(if(s', min(x, z'), min(y', z')), 1 + x + (1 + y' + z'))) :|: s' >= 0, s' <= 2, z'' = 1 + x + (1 + y' + z'), x >= 0, z' >= 0, y' >= 0 minsort(z'') -{ 2 }-> 1 + min(z'' - 2, 1) + minsort(del(z'' - 2, 1 + (z'' - 2) + 1)) :|: z'' - 2 >= 0 Function symbols to be analyzed: {min}, {del}, {minsort} Previous analysis results are: le: runtime: O(n^1) [2 + z1], size: O(1) [2] if: runtime: O(1) [1], size: O(n^1) [z1 + z2] eq: runtime: O(n^1) [3 + z''], size: O(1) [2] min: runtime: ?, size: EXP ---------------------------------------- (43) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: min after applying outer abstraction to obtain an ITS, resulting in: EXP with polynomial bound: ? ---------------------------------------- (44) Obligation: Complexity RNTS consisting of the following rules: del(z'', z1) -{ 5 + z'' }-> s10 :|: s9 >= 0, s9 <= 2, s10 >= 0, s10 <= 1 + (1 + y10) + 0 + z, z >= 0, y10 >= 0, z'' - 1 >= 0, z1 = 1 + (1 + y10) + z del(z'', z1) -{ 6 + z'' }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= 1 + (1 + (z1 - 3)) + 1 + 1, z1 - 3 >= 0, z'' - 1 >= 0 del(z'', z1) -{ 4 }-> x :|: z'' = 0, z1 = 1 + 0 + 1, 1 + 0 + 1 = y, x >= 0, y >= 0, 2 = 2, 1 = x del(z'', z1) -{ 3 }-> x :|: z'' = 0, z1 - 1 >= 0, 1 + 0 + 0 = y, x >= 0, y >= 0, 2 = 2, z1 - 1 = x del(z'', z1) -{ 4 }-> y :|: z'' = 0, z1 - 3 >= 0, 1 + (1 + (z1 - 3)) + 1 = y, x >= 0, y >= 0, 1 = 1, 1 = x del(z'', z1) -{ 3 }-> y :|: z'' = 0, y9 >= 0, z >= 0, z1 = 1 + (1 + y9) + z, 1 + (1 + y9) + 0 = y, x >= 0, y >= 0, 1 = 1, z = x del(z'', z1) -{ 4 }-> y :|: z'' - 1 >= 0, z1 = 1 + 0 + 1, 1 + 0 + 1 = y, x >= 0, y >= 0, 1 = 1, 1 = x del(z'', z1) -{ 3 }-> y :|: z'' - 1 >= 0, z1 - 1 >= 0, 1 + 0 + 0 = y, x >= 0, y >= 0, 1 = 1, z1 - 1 = x del(z'', z1) -{ 8 + 2*z'' }-> if(s7, 1 + y14 + z11, 1 + (1 + y10) + if(s8, z11, 1 + y14 + del(1 + (z'' - 1), z11))) :|: s7 >= 0, s7 <= 2, s8 >= 0, s8 <= 2, z11 >= 0, y10 >= 0, z1 = 1 + (1 + y10) + (1 + y14 + z11), z'' - 1 >= 0, y14 >= 0 del(z'', z1) -{ 6 }-> if(2, 1 + y11 + z8, 1 + 0 + if(s2, z8, 1 + y11 + del(0, z8))) :|: s2 >= 0, s2 <= 2, z'' = 0, z1 = 1 + 0 + (1 + y11 + z8), y11 >= 0, z8 >= 0 del(z'', z1) -{ 6 }-> if(1, 1 + y12 + z9, 1 + (1 + y9) + if(s3, z9, 1 + y12 + del(0, z9))) :|: s3 >= 0, s3 <= 2, z'' = 0, y9 >= 0, z9 >= 0, y12 >= 0, z1 = 1 + (1 + y9) + (1 + y12 + z9) del(z'', z1) -{ 6 + z'' }-> if(1, 1 + y13 + z10, 1 + 0 + if(s4, z10, 1 + y13 + del(1 + (z'' - 1), z10))) :|: s4 >= 0, s4 <= 2, z1 = 1 + 0 + (1 + y13 + z10), z'' - 1 >= 0, z10 >= 0, y13 >= 0 del(z'', z1) -{ 5 + z'' }-> if(0, 1 + y15 + z12, 1 + y + if(s11, z12, 1 + y15 + del(z'', z12))) :|: s11 >= 0, s11 <= 2, z1 = 1 + y + (1 + y15 + z12), z'' >= 0, y >= 0, z12 >= 0, y15 >= 0 del(z'', z1) -{ 1 }-> 1 :|: z'' >= 0, z1 = 1 del(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0 del(z'', z1) -{ 3 }-> 0 :|: z'' = 0, z1 = 1 + 0 + 1, 2 = v0, 1 + 0 + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 2 }-> 0 :|: z'' = 0, z1 - 1 >= 0, 2 = v0, 1 + 0 + 0 = v2, v0 >= 0, z1 - 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 3 }-> 0 :|: z'' = 0, z1 - 3 >= 0, 1 = v0, 1 + (1 + (z1 - 3)) + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 2 }-> 0 :|: z'' = 0, y9 >= 0, z >= 0, z1 = 1 + (1 + y9) + z, 1 = v0, 1 + (1 + y9) + 0 = v2, v0 >= 0, z = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 3 }-> 0 :|: z'' - 1 >= 0, z1 = 1 + 0 + 1, 1 = v0, 1 + 0 + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 2 }-> 0 :|: z'' - 1 >= 0, z1 - 1 >= 0, 1 = v0, 1 + 0 + 0 = v2, v0 >= 0, z1 - 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 2 }-> 0 :|: z'' >= 0, z1 - 2 >= 0, 0 = v0, 1 + (z1 - 2) + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 1 }-> 0 :|: z >= 0, z1 = 1 + y + z, z'' >= 0, y >= 0, 0 = v0, 1 + y + 0 = v2, v0 >= 0, z = v1, v1 >= 0, v2 >= 0 eq(z'', z1) -{ 3 + z'' }-> s1 :|: s1 >= 0, s1 <= 2, z'' - 1 >= 0, z1 - 1 >= 0 eq(z'', z1) -{ 1 }-> 2 :|: z'' = 0, z1 = 0 eq(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 - 1 >= 0 eq(z'', z1) -{ 1 }-> 1 :|: z1 = 0, z'' - 1 >= 0 eq(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0 if(z'', z1, z2) -{ 1 }-> z1 :|: z1 >= 0, z2 >= 0, z'' = 2 if(z'', z1, z2) -{ 1 }-> z2 :|: z1 >= 0, z2 >= 0, z'' = 1 if(z'', z1, z2) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0, z2 >= 0 le(z'', z1) -{ 2 + z1 }-> s :|: s >= 0, s <= 2, z'' - 1 >= 0, z1 - 1 >= 0 le(z'', z1) -{ 1 }-> 2 :|: z'' = 0, z1 >= 0 le(z'', z1) -{ 1 }-> 1 :|: z1 = 0, z'' - 1 >= 0 le(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0 min(z'', z1) -{ 1 }-> z'' :|: z'' >= 0, z1 = 1 min(z'', z1) -{ 3 + y }-> if(s'', min(z'', z), min(y, z)) :|: s'' >= 0, s'' <= 2, z >= 0, z1 = 1 + y + z, z'' >= 0, y >= 0 min(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0 minsort(z'') -{ 1 }-> 1 :|: z'' = 1 minsort(z'') -{ 1 }-> 1 + min(x, y) + minsort(del(0, 1 + x + y)) :|: z'' = 1 + x + y, x >= 0, y >= 0 minsort(z'') -{ 4 + y' }-> 1 + min(x, 1 + y' + z') + minsort(del(if(s', min(x, z'), min(y', z')), 1 + x + (1 + y' + z'))) :|: s' >= 0, s' <= 2, z'' = 1 + x + (1 + y' + z'), x >= 0, z' >= 0, y' >= 0 minsort(z'') -{ 2 }-> 1 + min(z'' - 2, 1) + minsort(del(z'' - 2, 1 + (z'' - 2) + 1)) :|: z'' - 2 >= 0 Function symbols to be analyzed: {del}, {minsort} Previous analysis results are: le: runtime: O(n^1) [2 + z1], size: O(1) [2] if: runtime: O(1) [1], size: O(n^1) [z1 + z2] eq: runtime: O(n^1) [3 + z''], size: O(1) [2] min: runtime: EXP, size: EXP ---------------------------------------- (45) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (46) Obligation: Complexity RNTS consisting of the following rules: del(z'', z1) -{ 5 + z'' }-> s10 :|: s9 >= 0, s9 <= 2, s10 >= 0, s10 <= 1 + (1 + y10) + 0 + z, z >= 0, y10 >= 0, z'' - 1 >= 0, z1 = 1 + (1 + y10) + z del(z'', z1) -{ 6 + z'' }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= 1 + (1 + (z1 - 3)) + 1 + 1, z1 - 3 >= 0, z'' - 1 >= 0 del(z'', z1) -{ 4 }-> x :|: z'' = 0, z1 = 1 + 0 + 1, 1 + 0 + 1 = y, x >= 0, y >= 0, 2 = 2, 1 = x del(z'', z1) -{ 3 }-> x :|: z'' = 0, z1 - 1 >= 0, 1 + 0 + 0 = y, x >= 0, y >= 0, 2 = 2, z1 - 1 = x del(z'', z1) -{ 4 }-> y :|: z'' = 0, z1 - 3 >= 0, 1 + (1 + (z1 - 3)) + 1 = y, x >= 0, y >= 0, 1 = 1, 1 = x del(z'', z1) -{ 3 }-> y :|: z'' = 0, y9 >= 0, z >= 0, z1 = 1 + (1 + y9) + z, 1 + (1 + y9) + 0 = y, x >= 0, y >= 0, 1 = 1, z = x del(z'', z1) -{ 4 }-> y :|: z'' - 1 >= 0, z1 = 1 + 0 + 1, 1 + 0 + 1 = y, x >= 0, y >= 0, 1 = 1, 1 = x del(z'', z1) -{ 3 }-> y :|: z'' - 1 >= 0, z1 - 1 >= 0, 1 + 0 + 0 = y, x >= 0, y >= 0, 1 = 1, z1 - 1 = x del(z'', z1) -{ 8 + 2*z'' }-> if(s7, 1 + y14 + z11, 1 + (1 + y10) + if(s8, z11, 1 + y14 + del(1 + (z'' - 1), z11))) :|: s7 >= 0, s7 <= 2, s8 >= 0, s8 <= 2, z11 >= 0, y10 >= 0, z1 = 1 + (1 + y10) + (1 + y14 + z11), z'' - 1 >= 0, y14 >= 0 del(z'', z1) -{ 6 }-> if(2, 1 + y11 + z8, 1 + 0 + if(s2, z8, 1 + y11 + del(0, z8))) :|: s2 >= 0, s2 <= 2, z'' = 0, z1 = 1 + 0 + (1 + y11 + z8), y11 >= 0, z8 >= 0 del(z'', z1) -{ 6 }-> if(1, 1 + y12 + z9, 1 + (1 + y9) + if(s3, z9, 1 + y12 + del(0, z9))) :|: s3 >= 0, s3 <= 2, z'' = 0, y9 >= 0, z9 >= 0, y12 >= 0, z1 = 1 + (1 + y9) + (1 + y12 + z9) del(z'', z1) -{ 6 + z'' }-> if(1, 1 + y13 + z10, 1 + 0 + if(s4, z10, 1 + y13 + del(1 + (z'' - 1), z10))) :|: s4 >= 0, s4 <= 2, z1 = 1 + 0 + (1 + y13 + z10), z'' - 1 >= 0, z10 >= 0, y13 >= 0 del(z'', z1) -{ 5 + z'' }-> if(0, 1 + y15 + z12, 1 + y + if(s11, z12, 1 + y15 + del(z'', z12))) :|: s11 >= 0, s11 <= 2, z1 = 1 + y + (1 + y15 + z12), z'' >= 0, y >= 0, z12 >= 0, y15 >= 0 del(z'', z1) -{ 1 }-> 1 :|: z'' >= 0, z1 = 1 del(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0 del(z'', z1) -{ 3 }-> 0 :|: z'' = 0, z1 = 1 + 0 + 1, 2 = v0, 1 + 0 + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 2 }-> 0 :|: z'' = 0, z1 - 1 >= 0, 2 = v0, 1 + 0 + 0 = v2, v0 >= 0, z1 - 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 3 }-> 0 :|: z'' = 0, z1 - 3 >= 0, 1 = v0, 1 + (1 + (z1 - 3)) + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 2 }-> 0 :|: z'' = 0, y9 >= 0, z >= 0, z1 = 1 + (1 + y9) + z, 1 = v0, 1 + (1 + y9) + 0 = v2, v0 >= 0, z = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 3 }-> 0 :|: z'' - 1 >= 0, z1 = 1 + 0 + 1, 1 = v0, 1 + 0 + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 2 }-> 0 :|: z'' - 1 >= 0, z1 - 1 >= 0, 1 = v0, 1 + 0 + 0 = v2, v0 >= 0, z1 - 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 2 }-> 0 :|: z'' >= 0, z1 - 2 >= 0, 0 = v0, 1 + (z1 - 2) + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 1 }-> 0 :|: z >= 0, z1 = 1 + y + z, z'' >= 0, y >= 0, 0 = v0, 1 + y + 0 = v2, v0 >= 0, z = v1, v1 >= 0, v2 >= 0 eq(z'', z1) -{ 3 + z'' }-> s1 :|: s1 >= 0, s1 <= 2, z'' - 1 >= 0, z1 - 1 >= 0 eq(z'', z1) -{ 1 }-> 2 :|: z'' = 0, z1 = 0 eq(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 - 1 >= 0 eq(z'', z1) -{ 1 }-> 1 :|: z1 = 0, z'' - 1 >= 0 eq(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0 if(z'', z1, z2) -{ 1 }-> z1 :|: z1 >= 0, z2 >= 0, z'' = 2 if(z'', z1, z2) -{ 1 }-> z2 :|: z1 >= 0, z2 >= 0, z'' = 1 if(z'', z1, z2) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0, z2 >= 0 le(z'', z1) -{ 2 + z1 }-> s :|: s >= 0, s <= 2, z'' - 1 >= 0, z1 - 1 >= 0 le(z'', z1) -{ 1 }-> 2 :|: z'' = 0, z1 >= 0 le(z'', z1) -{ 1 }-> 1 :|: z1 = 0, z'' - 1 >= 0 le(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0 min(z'', z1) -{ 4 + inf10 + inf8 + y }-> s20 :|: s18 >= 0, s18 <= inf9, s19 >= 0, s19 <= inf11, s20 >= 0, s20 <= s19 + s18, s'' >= 0, s'' <= 2, z >= 0, z1 = 1 + y + z, z'' >= 0, y >= 0 min(z'', z1) -{ 1 }-> z'' :|: z'' >= 0, z1 = 1 min(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0 minsort(z'') -{ 1 }-> 1 :|: z'' = 1 minsort(z'') -{ 2 + inf }-> 1 + s12 + minsort(del(z'' - 2, 1 + (z'' - 2) + 1)) :|: s12 >= 0, s12 <= inf', z'' - 2 >= 0 minsort(z'') -{ 1 + inf'' }-> 1 + s13 + minsort(del(0, 1 + x + y)) :|: s13 >= 0, s13 <= inf1, z'' = 1 + x + y, x >= 0, y >= 0 minsort(z'') -{ 5 + inf2 + inf4 + inf6 + y' }-> 1 + s14 + minsort(del(s17, 1 + x + (1 + y' + z'))) :|: s14 >= 0, s14 <= inf3, s15 >= 0, s15 <= inf5, s16 >= 0, s16 <= inf7, s17 >= 0, s17 <= s16 + s15, s' >= 0, s' <= 2, z'' = 1 + x + (1 + y' + z'), x >= 0, z' >= 0, y' >= 0 Function symbols to be analyzed: {del}, {minsort} Previous analysis results are: le: runtime: O(n^1) [2 + z1], size: O(1) [2] if: runtime: O(1) [1], size: O(n^1) [z1 + z2] eq: runtime: O(n^1) [3 + z''], size: O(1) [2] min: runtime: EXP, size: EXP ---------------------------------------- (47) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: del after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 8 + 11*z1 + 4*z1^2 ---------------------------------------- (48) Obligation: Complexity RNTS consisting of the following rules: del(z'', z1) -{ 5 + z'' }-> s10 :|: s9 >= 0, s9 <= 2, s10 >= 0, s10 <= 1 + (1 + y10) + 0 + z, z >= 0, y10 >= 0, z'' - 1 >= 0, z1 = 1 + (1 + y10) + z del(z'', z1) -{ 6 + z'' }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= 1 + (1 + (z1 - 3)) + 1 + 1, z1 - 3 >= 0, z'' - 1 >= 0 del(z'', z1) -{ 4 }-> x :|: z'' = 0, z1 = 1 + 0 + 1, 1 + 0 + 1 = y, x >= 0, y >= 0, 2 = 2, 1 = x del(z'', z1) -{ 3 }-> x :|: z'' = 0, z1 - 1 >= 0, 1 + 0 + 0 = y, x >= 0, y >= 0, 2 = 2, z1 - 1 = x del(z'', z1) -{ 4 }-> y :|: z'' = 0, z1 - 3 >= 0, 1 + (1 + (z1 - 3)) + 1 = y, x >= 0, y >= 0, 1 = 1, 1 = x del(z'', z1) -{ 3 }-> y :|: z'' = 0, y9 >= 0, z >= 0, z1 = 1 + (1 + y9) + z, 1 + (1 + y9) + 0 = y, x >= 0, y >= 0, 1 = 1, z = x del(z'', z1) -{ 4 }-> y :|: z'' - 1 >= 0, z1 = 1 + 0 + 1, 1 + 0 + 1 = y, x >= 0, y >= 0, 1 = 1, 1 = x del(z'', z1) -{ 3 }-> y :|: z'' - 1 >= 0, z1 - 1 >= 0, 1 + 0 + 0 = y, x >= 0, y >= 0, 1 = 1, z1 - 1 = x del(z'', z1) -{ 8 + 2*z'' }-> if(s7, 1 + y14 + z11, 1 + (1 + y10) + if(s8, z11, 1 + y14 + del(1 + (z'' - 1), z11))) :|: s7 >= 0, s7 <= 2, s8 >= 0, s8 <= 2, z11 >= 0, y10 >= 0, z1 = 1 + (1 + y10) + (1 + y14 + z11), z'' - 1 >= 0, y14 >= 0 del(z'', z1) -{ 6 }-> if(2, 1 + y11 + z8, 1 + 0 + if(s2, z8, 1 + y11 + del(0, z8))) :|: s2 >= 0, s2 <= 2, z'' = 0, z1 = 1 + 0 + (1 + y11 + z8), y11 >= 0, z8 >= 0 del(z'', z1) -{ 6 }-> if(1, 1 + y12 + z9, 1 + (1 + y9) + if(s3, z9, 1 + y12 + del(0, z9))) :|: s3 >= 0, s3 <= 2, z'' = 0, y9 >= 0, z9 >= 0, y12 >= 0, z1 = 1 + (1 + y9) + (1 + y12 + z9) del(z'', z1) -{ 6 + z'' }-> if(1, 1 + y13 + z10, 1 + 0 + if(s4, z10, 1 + y13 + del(1 + (z'' - 1), z10))) :|: s4 >= 0, s4 <= 2, z1 = 1 + 0 + (1 + y13 + z10), z'' - 1 >= 0, z10 >= 0, y13 >= 0 del(z'', z1) -{ 5 + z'' }-> if(0, 1 + y15 + z12, 1 + y + if(s11, z12, 1 + y15 + del(z'', z12))) :|: s11 >= 0, s11 <= 2, z1 = 1 + y + (1 + y15 + z12), z'' >= 0, y >= 0, z12 >= 0, y15 >= 0 del(z'', z1) -{ 1 }-> 1 :|: z'' >= 0, z1 = 1 del(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0 del(z'', z1) -{ 3 }-> 0 :|: z'' = 0, z1 = 1 + 0 + 1, 2 = v0, 1 + 0 + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 2 }-> 0 :|: z'' = 0, z1 - 1 >= 0, 2 = v0, 1 + 0 + 0 = v2, v0 >= 0, z1 - 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 3 }-> 0 :|: z'' = 0, z1 - 3 >= 0, 1 = v0, 1 + (1 + (z1 - 3)) + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 2 }-> 0 :|: z'' = 0, y9 >= 0, z >= 0, z1 = 1 + (1 + y9) + z, 1 = v0, 1 + (1 + y9) + 0 = v2, v0 >= 0, z = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 3 }-> 0 :|: z'' - 1 >= 0, z1 = 1 + 0 + 1, 1 = v0, 1 + 0 + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 2 }-> 0 :|: z'' - 1 >= 0, z1 - 1 >= 0, 1 = v0, 1 + 0 + 0 = v2, v0 >= 0, z1 - 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 2 }-> 0 :|: z'' >= 0, z1 - 2 >= 0, 0 = v0, 1 + (z1 - 2) + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 1 }-> 0 :|: z >= 0, z1 = 1 + y + z, z'' >= 0, y >= 0, 0 = v0, 1 + y + 0 = v2, v0 >= 0, z = v1, v1 >= 0, v2 >= 0 eq(z'', z1) -{ 3 + z'' }-> s1 :|: s1 >= 0, s1 <= 2, z'' - 1 >= 0, z1 - 1 >= 0 eq(z'', z1) -{ 1 }-> 2 :|: z'' = 0, z1 = 0 eq(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 - 1 >= 0 eq(z'', z1) -{ 1 }-> 1 :|: z1 = 0, z'' - 1 >= 0 eq(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0 if(z'', z1, z2) -{ 1 }-> z1 :|: z1 >= 0, z2 >= 0, z'' = 2 if(z'', z1, z2) -{ 1 }-> z2 :|: z1 >= 0, z2 >= 0, z'' = 1 if(z'', z1, z2) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0, z2 >= 0 le(z'', z1) -{ 2 + z1 }-> s :|: s >= 0, s <= 2, z'' - 1 >= 0, z1 - 1 >= 0 le(z'', z1) -{ 1 }-> 2 :|: z'' = 0, z1 >= 0 le(z'', z1) -{ 1 }-> 1 :|: z1 = 0, z'' - 1 >= 0 le(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0 min(z'', z1) -{ 4 + inf10 + inf8 + y }-> s20 :|: s18 >= 0, s18 <= inf9, s19 >= 0, s19 <= inf11, s20 >= 0, s20 <= s19 + s18, s'' >= 0, s'' <= 2, z >= 0, z1 = 1 + y + z, z'' >= 0, y >= 0 min(z'', z1) -{ 1 }-> z'' :|: z'' >= 0, z1 = 1 min(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0 minsort(z'') -{ 1 }-> 1 :|: z'' = 1 minsort(z'') -{ 2 + inf }-> 1 + s12 + minsort(del(z'' - 2, 1 + (z'' - 2) + 1)) :|: s12 >= 0, s12 <= inf', z'' - 2 >= 0 minsort(z'') -{ 1 + inf'' }-> 1 + s13 + minsort(del(0, 1 + x + y)) :|: s13 >= 0, s13 <= inf1, z'' = 1 + x + y, x >= 0, y >= 0 minsort(z'') -{ 5 + inf2 + inf4 + inf6 + y' }-> 1 + s14 + minsort(del(s17, 1 + x + (1 + y' + z'))) :|: s14 >= 0, s14 <= inf3, s15 >= 0, s15 <= inf5, s16 >= 0, s16 <= inf7, s17 >= 0, s17 <= s16 + s15, s' >= 0, s' <= 2, z'' = 1 + x + (1 + y' + z'), x >= 0, z' >= 0, y' >= 0 Function symbols to be analyzed: {del}, {minsort} Previous analysis results are: le: runtime: O(n^1) [2 + z1], size: O(1) [2] if: runtime: O(1) [1], size: O(n^1) [z1 + z2] eq: runtime: O(n^1) [3 + z''], size: O(1) [2] min: runtime: EXP, size: EXP del: runtime: ?, size: O(n^2) [8 + 11*z1 + 4*z1^2] ---------------------------------------- (49) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: del after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 45 + 7*z'' + 2*z''*z1 + 13*z1 ---------------------------------------- (50) Obligation: Complexity RNTS consisting of the following rules: del(z'', z1) -{ 5 + z'' }-> s10 :|: s9 >= 0, s9 <= 2, s10 >= 0, s10 <= 1 + (1 + y10) + 0 + z, z >= 0, y10 >= 0, z'' - 1 >= 0, z1 = 1 + (1 + y10) + z del(z'', z1) -{ 6 + z'' }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= 1 + (1 + (z1 - 3)) + 1 + 1, z1 - 3 >= 0, z'' - 1 >= 0 del(z'', z1) -{ 4 }-> x :|: z'' = 0, z1 = 1 + 0 + 1, 1 + 0 + 1 = y, x >= 0, y >= 0, 2 = 2, 1 = x del(z'', z1) -{ 3 }-> x :|: z'' = 0, z1 - 1 >= 0, 1 + 0 + 0 = y, x >= 0, y >= 0, 2 = 2, z1 - 1 = x del(z'', z1) -{ 4 }-> y :|: z'' = 0, z1 - 3 >= 0, 1 + (1 + (z1 - 3)) + 1 = y, x >= 0, y >= 0, 1 = 1, 1 = x del(z'', z1) -{ 3 }-> y :|: z'' = 0, y9 >= 0, z >= 0, z1 = 1 + (1 + y9) + z, 1 + (1 + y9) + 0 = y, x >= 0, y >= 0, 1 = 1, z = x del(z'', z1) -{ 4 }-> y :|: z'' - 1 >= 0, z1 = 1 + 0 + 1, 1 + 0 + 1 = y, x >= 0, y >= 0, 1 = 1, 1 = x del(z'', z1) -{ 3 }-> y :|: z'' - 1 >= 0, z1 - 1 >= 0, 1 + 0 + 0 = y, x >= 0, y >= 0, 1 = 1, z1 - 1 = x del(z'', z1) -{ 8 + 2*z'' }-> if(s7, 1 + y14 + z11, 1 + (1 + y10) + if(s8, z11, 1 + y14 + del(1 + (z'' - 1), z11))) :|: s7 >= 0, s7 <= 2, s8 >= 0, s8 <= 2, z11 >= 0, y10 >= 0, z1 = 1 + (1 + y10) + (1 + y14 + z11), z'' - 1 >= 0, y14 >= 0 del(z'', z1) -{ 6 }-> if(2, 1 + y11 + z8, 1 + 0 + if(s2, z8, 1 + y11 + del(0, z8))) :|: s2 >= 0, s2 <= 2, z'' = 0, z1 = 1 + 0 + (1 + y11 + z8), y11 >= 0, z8 >= 0 del(z'', z1) -{ 6 }-> if(1, 1 + y12 + z9, 1 + (1 + y9) + if(s3, z9, 1 + y12 + del(0, z9))) :|: s3 >= 0, s3 <= 2, z'' = 0, y9 >= 0, z9 >= 0, y12 >= 0, z1 = 1 + (1 + y9) + (1 + y12 + z9) del(z'', z1) -{ 6 + z'' }-> if(1, 1 + y13 + z10, 1 + 0 + if(s4, z10, 1 + y13 + del(1 + (z'' - 1), z10))) :|: s4 >= 0, s4 <= 2, z1 = 1 + 0 + (1 + y13 + z10), z'' - 1 >= 0, z10 >= 0, y13 >= 0 del(z'', z1) -{ 5 + z'' }-> if(0, 1 + y15 + z12, 1 + y + if(s11, z12, 1 + y15 + del(z'', z12))) :|: s11 >= 0, s11 <= 2, z1 = 1 + y + (1 + y15 + z12), z'' >= 0, y >= 0, z12 >= 0, y15 >= 0 del(z'', z1) -{ 1 }-> 1 :|: z'' >= 0, z1 = 1 del(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0 del(z'', z1) -{ 3 }-> 0 :|: z'' = 0, z1 = 1 + 0 + 1, 2 = v0, 1 + 0 + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 2 }-> 0 :|: z'' = 0, z1 - 1 >= 0, 2 = v0, 1 + 0 + 0 = v2, v0 >= 0, z1 - 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 3 }-> 0 :|: z'' = 0, z1 - 3 >= 0, 1 = v0, 1 + (1 + (z1 - 3)) + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 2 }-> 0 :|: z'' = 0, y9 >= 0, z >= 0, z1 = 1 + (1 + y9) + z, 1 = v0, 1 + (1 + y9) + 0 = v2, v0 >= 0, z = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 3 }-> 0 :|: z'' - 1 >= 0, z1 = 1 + 0 + 1, 1 = v0, 1 + 0 + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 2 }-> 0 :|: z'' - 1 >= 0, z1 - 1 >= 0, 1 = v0, 1 + 0 + 0 = v2, v0 >= 0, z1 - 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 2 }-> 0 :|: z'' >= 0, z1 - 2 >= 0, 0 = v0, 1 + (z1 - 2) + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 1 }-> 0 :|: z >= 0, z1 = 1 + y + z, z'' >= 0, y >= 0, 0 = v0, 1 + y + 0 = v2, v0 >= 0, z = v1, v1 >= 0, v2 >= 0 eq(z'', z1) -{ 3 + z'' }-> s1 :|: s1 >= 0, s1 <= 2, z'' - 1 >= 0, z1 - 1 >= 0 eq(z'', z1) -{ 1 }-> 2 :|: z'' = 0, z1 = 0 eq(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 - 1 >= 0 eq(z'', z1) -{ 1 }-> 1 :|: z1 = 0, z'' - 1 >= 0 eq(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0 if(z'', z1, z2) -{ 1 }-> z1 :|: z1 >= 0, z2 >= 0, z'' = 2 if(z'', z1, z2) -{ 1 }-> z2 :|: z1 >= 0, z2 >= 0, z'' = 1 if(z'', z1, z2) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0, z2 >= 0 le(z'', z1) -{ 2 + z1 }-> s :|: s >= 0, s <= 2, z'' - 1 >= 0, z1 - 1 >= 0 le(z'', z1) -{ 1 }-> 2 :|: z'' = 0, z1 >= 0 le(z'', z1) -{ 1 }-> 1 :|: z1 = 0, z'' - 1 >= 0 le(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0 min(z'', z1) -{ 4 + inf10 + inf8 + y }-> s20 :|: s18 >= 0, s18 <= inf9, s19 >= 0, s19 <= inf11, s20 >= 0, s20 <= s19 + s18, s'' >= 0, s'' <= 2, z >= 0, z1 = 1 + y + z, z'' >= 0, y >= 0 min(z'', z1) -{ 1 }-> z'' :|: z'' >= 0, z1 = 1 min(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0 minsort(z'') -{ 1 }-> 1 :|: z'' = 1 minsort(z'') -{ 2 + inf }-> 1 + s12 + minsort(del(z'' - 2, 1 + (z'' - 2) + 1)) :|: s12 >= 0, s12 <= inf', z'' - 2 >= 0 minsort(z'') -{ 1 + inf'' }-> 1 + s13 + minsort(del(0, 1 + x + y)) :|: s13 >= 0, s13 <= inf1, z'' = 1 + x + y, x >= 0, y >= 0 minsort(z'') -{ 5 + inf2 + inf4 + inf6 + y' }-> 1 + s14 + minsort(del(s17, 1 + x + (1 + y' + z'))) :|: s14 >= 0, s14 <= inf3, s15 >= 0, s15 <= inf5, s16 >= 0, s16 <= inf7, s17 >= 0, s17 <= s16 + s15, s' >= 0, s' <= 2, z'' = 1 + x + (1 + y' + z'), x >= 0, z' >= 0, y' >= 0 Function symbols to be analyzed: {minsort} Previous analysis results are: le: runtime: O(n^1) [2 + z1], size: O(1) [2] if: runtime: O(1) [1], size: O(n^1) [z1 + z2] eq: runtime: O(n^1) [3 + z''], size: O(1) [2] min: runtime: EXP, size: EXP del: runtime: O(n^2) [45 + 7*z'' + 2*z''*z1 + 13*z1], size: O(n^2) [8 + 11*z1 + 4*z1^2] ---------------------------------------- (51) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (52) Obligation: Complexity RNTS consisting of the following rules: del(z'', z1) -{ 5 + z'' }-> s10 :|: s9 >= 0, s9 <= 2, s10 >= 0, s10 <= 1 + (1 + y10) + 0 + z, z >= 0, y10 >= 0, z'' - 1 >= 0, z1 = 1 + (1 + y10) + z del(z'', z1) -{ 53 + 13*z8 }-> s26 :|: s24 >= 0, s24 <= 8 + 11 * z8 + 4 * (z8 * z8), s25 >= 0, s25 <= 1 + y11 + s24 + z8, s26 >= 0, s26 <= 1 + 0 + s25 + (1 + y11 + z8), s2 >= 0, s2 <= 2, z'' = 0, z1 = 1 + 0 + (1 + y11 + z8), y11 >= 0, z8 >= 0 del(z'', z1) -{ 53 + 13*z9 }-> s29 :|: s27 >= 0, s27 <= 8 + 11 * z9 + 4 * (z9 * z9), s28 >= 0, s28 <= 1 + y12 + s27 + z9, s29 >= 0, s29 <= 1 + (1 + y9) + s28 + (1 + y12 + z9), s3 >= 0, s3 <= 2, z'' = 0, y9 >= 0, z9 >= 0, y12 >= 0, z1 = 1 + (1 + y9) + (1 + y12 + z9) del(z'', z1) -{ 53 + 8*z'' + 2*z''*z10 + 13*z10 }-> s32 :|: s30 >= 0, s30 <= 8 + 11 * z10 + 4 * (z10 * z10), s31 >= 0, s31 <= 1 + y13 + s30 + z10, s32 >= 0, s32 <= 1 + 0 + s31 + (1 + y13 + z10), s4 >= 0, s4 <= 2, z1 = 1 + 0 + (1 + y13 + z10), z'' - 1 >= 0, z10 >= 0, y13 >= 0 del(z'', z1) -{ 55 + 9*z'' + 2*z''*z11 + 13*z11 }-> s35 :|: s33 >= 0, s33 <= 8 + 11 * z11 + 4 * (z11 * z11), s34 >= 0, s34 <= 1 + y14 + s33 + z11, s35 >= 0, s35 <= 1 + (1 + y10) + s34 + (1 + y14 + z11), s7 >= 0, s7 <= 2, s8 >= 0, s8 <= 2, z11 >= 0, y10 >= 0, z1 = 1 + (1 + y10) + (1 + y14 + z11), z'' - 1 >= 0, y14 >= 0 del(z'', z1) -{ 52 + 8*z'' + 2*z''*z12 + 13*z12 }-> s38 :|: s36 >= 0, s36 <= 8 + 11 * z12 + 4 * (z12 * z12), s37 >= 0, s37 <= 1 + y15 + s36 + z12, s38 >= 0, s38 <= 1 + y + s37 + (1 + y15 + z12), s11 >= 0, s11 <= 2, z1 = 1 + y + (1 + y15 + z12), z'' >= 0, y >= 0, z12 >= 0, y15 >= 0 del(z'', z1) -{ 6 + z'' }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= 1 + (1 + (z1 - 3)) + 1 + 1, z1 - 3 >= 0, z'' - 1 >= 0 del(z'', z1) -{ 4 }-> x :|: z'' = 0, z1 = 1 + 0 + 1, 1 + 0 + 1 = y, x >= 0, y >= 0, 2 = 2, 1 = x del(z'', z1) -{ 3 }-> x :|: z'' = 0, z1 - 1 >= 0, 1 + 0 + 0 = y, x >= 0, y >= 0, 2 = 2, z1 - 1 = x del(z'', z1) -{ 4 }-> y :|: z'' = 0, z1 - 3 >= 0, 1 + (1 + (z1 - 3)) + 1 = y, x >= 0, y >= 0, 1 = 1, 1 = x del(z'', z1) -{ 3 }-> y :|: z'' = 0, y9 >= 0, z >= 0, z1 = 1 + (1 + y9) + z, 1 + (1 + y9) + 0 = y, x >= 0, y >= 0, 1 = 1, z = x del(z'', z1) -{ 4 }-> y :|: z'' - 1 >= 0, z1 = 1 + 0 + 1, 1 + 0 + 1 = y, x >= 0, y >= 0, 1 = 1, 1 = x del(z'', z1) -{ 3 }-> y :|: z'' - 1 >= 0, z1 - 1 >= 0, 1 + 0 + 0 = y, x >= 0, y >= 0, 1 = 1, z1 - 1 = x del(z'', z1) -{ 1 }-> 1 :|: z'' >= 0, z1 = 1 del(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0 del(z'', z1) -{ 3 }-> 0 :|: z'' = 0, z1 = 1 + 0 + 1, 2 = v0, 1 + 0 + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 2 }-> 0 :|: z'' = 0, z1 - 1 >= 0, 2 = v0, 1 + 0 + 0 = v2, v0 >= 0, z1 - 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 3 }-> 0 :|: z'' = 0, z1 - 3 >= 0, 1 = v0, 1 + (1 + (z1 - 3)) + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 2 }-> 0 :|: z'' = 0, y9 >= 0, z >= 0, z1 = 1 + (1 + y9) + z, 1 = v0, 1 + (1 + y9) + 0 = v2, v0 >= 0, z = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 3 }-> 0 :|: z'' - 1 >= 0, z1 = 1 + 0 + 1, 1 = v0, 1 + 0 + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 2 }-> 0 :|: z'' - 1 >= 0, z1 - 1 >= 0, 1 = v0, 1 + 0 + 0 = v2, v0 >= 0, z1 - 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 2 }-> 0 :|: z'' >= 0, z1 - 2 >= 0, 0 = v0, 1 + (z1 - 2) + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 1 }-> 0 :|: z >= 0, z1 = 1 + y + z, z'' >= 0, y >= 0, 0 = v0, 1 + y + 0 = v2, v0 >= 0, z = v1, v1 >= 0, v2 >= 0 eq(z'', z1) -{ 3 + z'' }-> s1 :|: s1 >= 0, s1 <= 2, z'' - 1 >= 0, z1 - 1 >= 0 eq(z'', z1) -{ 1 }-> 2 :|: z'' = 0, z1 = 0 eq(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 - 1 >= 0 eq(z'', z1) -{ 1 }-> 1 :|: z1 = 0, z'' - 1 >= 0 eq(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0 if(z'', z1, z2) -{ 1 }-> z1 :|: z1 >= 0, z2 >= 0, z'' = 2 if(z'', z1, z2) -{ 1 }-> z2 :|: z1 >= 0, z2 >= 0, z'' = 1 if(z'', z1, z2) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0, z2 >= 0 le(z'', z1) -{ 2 + z1 }-> s :|: s >= 0, s <= 2, z'' - 1 >= 0, z1 - 1 >= 0 le(z'', z1) -{ 1 }-> 2 :|: z'' = 0, z1 >= 0 le(z'', z1) -{ 1 }-> 1 :|: z1 = 0, z'' - 1 >= 0 le(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0 min(z'', z1) -{ 4 + inf10 + inf8 + y }-> s20 :|: s18 >= 0, s18 <= inf9, s19 >= 0, s19 <= inf11, s20 >= 0, s20 <= s19 + s18, s'' >= 0, s'' <= 2, z >= 0, z1 = 1 + y + z, z'' >= 0, y >= 0 min(z'', z1) -{ 1 }-> z'' :|: z'' >= 0, z1 = 1 min(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0 minsort(z'') -{ 1 }-> 1 :|: z'' = 1 minsort(z'') -{ 33 + inf + 16*z'' + 2*z''^2 }-> 1 + s12 + minsort(s21) :|: s21 >= 0, s21 <= 8 + 11 * (1 + (z'' - 2) + 1) + 4 * ((1 + (z'' - 2) + 1) * (1 + (z'' - 2) + 1)), s12 >= 0, s12 <= inf', z'' - 2 >= 0 minsort(z'') -{ 59 + inf'' + 13*x + 13*y }-> 1 + s13 + minsort(s22) :|: s22 >= 0, s22 <= 8 + 11 * (1 + x + y) + 4 * ((1 + x + y) * (1 + x + y)), s13 >= 0, s13 <= inf1, z'' = 1 + x + y, x >= 0, y >= 0 minsort(z'') -{ 76 + inf2 + inf4 + inf6 + 11*s17 + 2*s17*x + 2*s17*y' + 2*s17*z' + 13*x + 14*y' + 13*z' }-> 1 + s14 + minsort(s23) :|: s23 >= 0, s23 <= 8 + 11 * (1 + x + (1 + y' + z')) + 4 * ((1 + x + (1 + y' + z')) * (1 + x + (1 + y' + z'))), s14 >= 0, s14 <= inf3, s15 >= 0, s15 <= inf5, s16 >= 0, s16 <= inf7, s17 >= 0, s17 <= s16 + s15, s' >= 0, s' <= 2, z'' = 1 + x + (1 + y' + z'), x >= 0, z' >= 0, y' >= 0 Function symbols to be analyzed: {minsort} Previous analysis results are: le: runtime: O(n^1) [2 + z1], size: O(1) [2] if: runtime: O(1) [1], size: O(n^1) [z1 + z2] eq: runtime: O(n^1) [3 + z''], size: O(1) [2] min: runtime: EXP, size: EXP del: runtime: O(n^2) [45 + 7*z'' + 2*z''*z1 + 13*z1], size: O(n^2) [8 + 11*z1 + 4*z1^2] ---------------------------------------- (53) 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: ? ---------------------------------------- (54) Obligation: Complexity RNTS consisting of the following rules: del(z'', z1) -{ 5 + z'' }-> s10 :|: s9 >= 0, s9 <= 2, s10 >= 0, s10 <= 1 + (1 + y10) + 0 + z, z >= 0, y10 >= 0, z'' - 1 >= 0, z1 = 1 + (1 + y10) + z del(z'', z1) -{ 53 + 13*z8 }-> s26 :|: s24 >= 0, s24 <= 8 + 11 * z8 + 4 * (z8 * z8), s25 >= 0, s25 <= 1 + y11 + s24 + z8, s26 >= 0, s26 <= 1 + 0 + s25 + (1 + y11 + z8), s2 >= 0, s2 <= 2, z'' = 0, z1 = 1 + 0 + (1 + y11 + z8), y11 >= 0, z8 >= 0 del(z'', z1) -{ 53 + 13*z9 }-> s29 :|: s27 >= 0, s27 <= 8 + 11 * z9 + 4 * (z9 * z9), s28 >= 0, s28 <= 1 + y12 + s27 + z9, s29 >= 0, s29 <= 1 + (1 + y9) + s28 + (1 + y12 + z9), s3 >= 0, s3 <= 2, z'' = 0, y9 >= 0, z9 >= 0, y12 >= 0, z1 = 1 + (1 + y9) + (1 + y12 + z9) del(z'', z1) -{ 53 + 8*z'' + 2*z''*z10 + 13*z10 }-> s32 :|: s30 >= 0, s30 <= 8 + 11 * z10 + 4 * (z10 * z10), s31 >= 0, s31 <= 1 + y13 + s30 + z10, s32 >= 0, s32 <= 1 + 0 + s31 + (1 + y13 + z10), s4 >= 0, s4 <= 2, z1 = 1 + 0 + (1 + y13 + z10), z'' - 1 >= 0, z10 >= 0, y13 >= 0 del(z'', z1) -{ 55 + 9*z'' + 2*z''*z11 + 13*z11 }-> s35 :|: s33 >= 0, s33 <= 8 + 11 * z11 + 4 * (z11 * z11), s34 >= 0, s34 <= 1 + y14 + s33 + z11, s35 >= 0, s35 <= 1 + (1 + y10) + s34 + (1 + y14 + z11), s7 >= 0, s7 <= 2, s8 >= 0, s8 <= 2, z11 >= 0, y10 >= 0, z1 = 1 + (1 + y10) + (1 + y14 + z11), z'' - 1 >= 0, y14 >= 0 del(z'', z1) -{ 52 + 8*z'' + 2*z''*z12 + 13*z12 }-> s38 :|: s36 >= 0, s36 <= 8 + 11 * z12 + 4 * (z12 * z12), s37 >= 0, s37 <= 1 + y15 + s36 + z12, s38 >= 0, s38 <= 1 + y + s37 + (1 + y15 + z12), s11 >= 0, s11 <= 2, z1 = 1 + y + (1 + y15 + z12), z'' >= 0, y >= 0, z12 >= 0, y15 >= 0 del(z'', z1) -{ 6 + z'' }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= 1 + (1 + (z1 - 3)) + 1 + 1, z1 - 3 >= 0, z'' - 1 >= 0 del(z'', z1) -{ 4 }-> x :|: z'' = 0, z1 = 1 + 0 + 1, 1 + 0 + 1 = y, x >= 0, y >= 0, 2 = 2, 1 = x del(z'', z1) -{ 3 }-> x :|: z'' = 0, z1 - 1 >= 0, 1 + 0 + 0 = y, x >= 0, y >= 0, 2 = 2, z1 - 1 = x del(z'', z1) -{ 4 }-> y :|: z'' = 0, z1 - 3 >= 0, 1 + (1 + (z1 - 3)) + 1 = y, x >= 0, y >= 0, 1 = 1, 1 = x del(z'', z1) -{ 3 }-> y :|: z'' = 0, y9 >= 0, z >= 0, z1 = 1 + (1 + y9) + z, 1 + (1 + y9) + 0 = y, x >= 0, y >= 0, 1 = 1, z = x del(z'', z1) -{ 4 }-> y :|: z'' - 1 >= 0, z1 = 1 + 0 + 1, 1 + 0 + 1 = y, x >= 0, y >= 0, 1 = 1, 1 = x del(z'', z1) -{ 3 }-> y :|: z'' - 1 >= 0, z1 - 1 >= 0, 1 + 0 + 0 = y, x >= 0, y >= 0, 1 = 1, z1 - 1 = x del(z'', z1) -{ 1 }-> 1 :|: z'' >= 0, z1 = 1 del(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0 del(z'', z1) -{ 3 }-> 0 :|: z'' = 0, z1 = 1 + 0 + 1, 2 = v0, 1 + 0 + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 2 }-> 0 :|: z'' = 0, z1 - 1 >= 0, 2 = v0, 1 + 0 + 0 = v2, v0 >= 0, z1 - 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 3 }-> 0 :|: z'' = 0, z1 - 3 >= 0, 1 = v0, 1 + (1 + (z1 - 3)) + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 2 }-> 0 :|: z'' = 0, y9 >= 0, z >= 0, z1 = 1 + (1 + y9) + z, 1 = v0, 1 + (1 + y9) + 0 = v2, v0 >= 0, z = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 3 }-> 0 :|: z'' - 1 >= 0, z1 = 1 + 0 + 1, 1 = v0, 1 + 0 + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 2 }-> 0 :|: z'' - 1 >= 0, z1 - 1 >= 0, 1 = v0, 1 + 0 + 0 = v2, v0 >= 0, z1 - 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 2 }-> 0 :|: z'' >= 0, z1 - 2 >= 0, 0 = v0, 1 + (z1 - 2) + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 1 }-> 0 :|: z >= 0, z1 = 1 + y + z, z'' >= 0, y >= 0, 0 = v0, 1 + y + 0 = v2, v0 >= 0, z = v1, v1 >= 0, v2 >= 0 eq(z'', z1) -{ 3 + z'' }-> s1 :|: s1 >= 0, s1 <= 2, z'' - 1 >= 0, z1 - 1 >= 0 eq(z'', z1) -{ 1 }-> 2 :|: z'' = 0, z1 = 0 eq(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 - 1 >= 0 eq(z'', z1) -{ 1 }-> 1 :|: z1 = 0, z'' - 1 >= 0 eq(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0 if(z'', z1, z2) -{ 1 }-> z1 :|: z1 >= 0, z2 >= 0, z'' = 2 if(z'', z1, z2) -{ 1 }-> z2 :|: z1 >= 0, z2 >= 0, z'' = 1 if(z'', z1, z2) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0, z2 >= 0 le(z'', z1) -{ 2 + z1 }-> s :|: s >= 0, s <= 2, z'' - 1 >= 0, z1 - 1 >= 0 le(z'', z1) -{ 1 }-> 2 :|: z'' = 0, z1 >= 0 le(z'', z1) -{ 1 }-> 1 :|: z1 = 0, z'' - 1 >= 0 le(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0 min(z'', z1) -{ 4 + inf10 + inf8 + y }-> s20 :|: s18 >= 0, s18 <= inf9, s19 >= 0, s19 <= inf11, s20 >= 0, s20 <= s19 + s18, s'' >= 0, s'' <= 2, z >= 0, z1 = 1 + y + z, z'' >= 0, y >= 0 min(z'', z1) -{ 1 }-> z'' :|: z'' >= 0, z1 = 1 min(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0 minsort(z'') -{ 1 }-> 1 :|: z'' = 1 minsort(z'') -{ 33 + inf + 16*z'' + 2*z''^2 }-> 1 + s12 + minsort(s21) :|: s21 >= 0, s21 <= 8 + 11 * (1 + (z'' - 2) + 1) + 4 * ((1 + (z'' - 2) + 1) * (1 + (z'' - 2) + 1)), s12 >= 0, s12 <= inf', z'' - 2 >= 0 minsort(z'') -{ 59 + inf'' + 13*x + 13*y }-> 1 + s13 + minsort(s22) :|: s22 >= 0, s22 <= 8 + 11 * (1 + x + y) + 4 * ((1 + x + y) * (1 + x + y)), s13 >= 0, s13 <= inf1, z'' = 1 + x + y, x >= 0, y >= 0 minsort(z'') -{ 76 + inf2 + inf4 + inf6 + 11*s17 + 2*s17*x + 2*s17*y' + 2*s17*z' + 13*x + 14*y' + 13*z' }-> 1 + s14 + minsort(s23) :|: s23 >= 0, s23 <= 8 + 11 * (1 + x + (1 + y' + z')) + 4 * ((1 + x + (1 + y' + z')) * (1 + x + (1 + y' + z'))), s14 >= 0, s14 <= inf3, s15 >= 0, s15 <= inf5, s16 >= 0, s16 <= inf7, s17 >= 0, s17 <= s16 + s15, s' >= 0, s' <= 2, z'' = 1 + x + (1 + y' + z'), x >= 0, z' >= 0, y' >= 0 Function symbols to be analyzed: {minsort} Previous analysis results are: le: runtime: O(n^1) [2 + z1], size: O(1) [2] if: runtime: O(1) [1], size: O(n^1) [z1 + z2] eq: runtime: O(n^1) [3 + z''], size: O(1) [2] min: runtime: EXP, size: EXP del: runtime: O(n^2) [45 + 7*z'' + 2*z''*z1 + 13*z1], size: O(n^2) [8 + 11*z1 + 4*z1^2] minsort: runtime: ?, size: INF ---------------------------------------- (55) 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: ? ---------------------------------------- (56) Obligation: Complexity RNTS consisting of the following rules: del(z'', z1) -{ 5 + z'' }-> s10 :|: s9 >= 0, s9 <= 2, s10 >= 0, s10 <= 1 + (1 + y10) + 0 + z, z >= 0, y10 >= 0, z'' - 1 >= 0, z1 = 1 + (1 + y10) + z del(z'', z1) -{ 53 + 13*z8 }-> s26 :|: s24 >= 0, s24 <= 8 + 11 * z8 + 4 * (z8 * z8), s25 >= 0, s25 <= 1 + y11 + s24 + z8, s26 >= 0, s26 <= 1 + 0 + s25 + (1 + y11 + z8), s2 >= 0, s2 <= 2, z'' = 0, z1 = 1 + 0 + (1 + y11 + z8), y11 >= 0, z8 >= 0 del(z'', z1) -{ 53 + 13*z9 }-> s29 :|: s27 >= 0, s27 <= 8 + 11 * z9 + 4 * (z9 * z9), s28 >= 0, s28 <= 1 + y12 + s27 + z9, s29 >= 0, s29 <= 1 + (1 + y9) + s28 + (1 + y12 + z9), s3 >= 0, s3 <= 2, z'' = 0, y9 >= 0, z9 >= 0, y12 >= 0, z1 = 1 + (1 + y9) + (1 + y12 + z9) del(z'', z1) -{ 53 + 8*z'' + 2*z''*z10 + 13*z10 }-> s32 :|: s30 >= 0, s30 <= 8 + 11 * z10 + 4 * (z10 * z10), s31 >= 0, s31 <= 1 + y13 + s30 + z10, s32 >= 0, s32 <= 1 + 0 + s31 + (1 + y13 + z10), s4 >= 0, s4 <= 2, z1 = 1 + 0 + (1 + y13 + z10), z'' - 1 >= 0, z10 >= 0, y13 >= 0 del(z'', z1) -{ 55 + 9*z'' + 2*z''*z11 + 13*z11 }-> s35 :|: s33 >= 0, s33 <= 8 + 11 * z11 + 4 * (z11 * z11), s34 >= 0, s34 <= 1 + y14 + s33 + z11, s35 >= 0, s35 <= 1 + (1 + y10) + s34 + (1 + y14 + z11), s7 >= 0, s7 <= 2, s8 >= 0, s8 <= 2, z11 >= 0, y10 >= 0, z1 = 1 + (1 + y10) + (1 + y14 + z11), z'' - 1 >= 0, y14 >= 0 del(z'', z1) -{ 52 + 8*z'' + 2*z''*z12 + 13*z12 }-> s38 :|: s36 >= 0, s36 <= 8 + 11 * z12 + 4 * (z12 * z12), s37 >= 0, s37 <= 1 + y15 + s36 + z12, s38 >= 0, s38 <= 1 + y + s37 + (1 + y15 + z12), s11 >= 0, s11 <= 2, z1 = 1 + y + (1 + y15 + z12), z'' >= 0, y >= 0, z12 >= 0, y15 >= 0 del(z'', z1) -{ 6 + z'' }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= 1 + (1 + (z1 - 3)) + 1 + 1, z1 - 3 >= 0, z'' - 1 >= 0 del(z'', z1) -{ 4 }-> x :|: z'' = 0, z1 = 1 + 0 + 1, 1 + 0 + 1 = y, x >= 0, y >= 0, 2 = 2, 1 = x del(z'', z1) -{ 3 }-> x :|: z'' = 0, z1 - 1 >= 0, 1 + 0 + 0 = y, x >= 0, y >= 0, 2 = 2, z1 - 1 = x del(z'', z1) -{ 4 }-> y :|: z'' = 0, z1 - 3 >= 0, 1 + (1 + (z1 - 3)) + 1 = y, x >= 0, y >= 0, 1 = 1, 1 = x del(z'', z1) -{ 3 }-> y :|: z'' = 0, y9 >= 0, z >= 0, z1 = 1 + (1 + y9) + z, 1 + (1 + y9) + 0 = y, x >= 0, y >= 0, 1 = 1, z = x del(z'', z1) -{ 4 }-> y :|: z'' - 1 >= 0, z1 = 1 + 0 + 1, 1 + 0 + 1 = y, x >= 0, y >= 0, 1 = 1, 1 = x del(z'', z1) -{ 3 }-> y :|: z'' - 1 >= 0, z1 - 1 >= 0, 1 + 0 + 0 = y, x >= 0, y >= 0, 1 = 1, z1 - 1 = x del(z'', z1) -{ 1 }-> 1 :|: z'' >= 0, z1 = 1 del(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0 del(z'', z1) -{ 3 }-> 0 :|: z'' = 0, z1 = 1 + 0 + 1, 2 = v0, 1 + 0 + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 2 }-> 0 :|: z'' = 0, z1 - 1 >= 0, 2 = v0, 1 + 0 + 0 = v2, v0 >= 0, z1 - 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 3 }-> 0 :|: z'' = 0, z1 - 3 >= 0, 1 = v0, 1 + (1 + (z1 - 3)) + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 2 }-> 0 :|: z'' = 0, y9 >= 0, z >= 0, z1 = 1 + (1 + y9) + z, 1 = v0, 1 + (1 + y9) + 0 = v2, v0 >= 0, z = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 3 }-> 0 :|: z'' - 1 >= 0, z1 = 1 + 0 + 1, 1 = v0, 1 + 0 + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 2 }-> 0 :|: z'' - 1 >= 0, z1 - 1 >= 0, 1 = v0, 1 + 0 + 0 = v2, v0 >= 0, z1 - 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 2 }-> 0 :|: z'' >= 0, z1 - 2 >= 0, 0 = v0, 1 + (z1 - 2) + 1 = v2, v0 >= 0, 1 = v1, v1 >= 0, v2 >= 0 del(z'', z1) -{ 1 }-> 0 :|: z >= 0, z1 = 1 + y + z, z'' >= 0, y >= 0, 0 = v0, 1 + y + 0 = v2, v0 >= 0, z = v1, v1 >= 0, v2 >= 0 eq(z'', z1) -{ 3 + z'' }-> s1 :|: s1 >= 0, s1 <= 2, z'' - 1 >= 0, z1 - 1 >= 0 eq(z'', z1) -{ 1 }-> 2 :|: z'' = 0, z1 = 0 eq(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 - 1 >= 0 eq(z'', z1) -{ 1 }-> 1 :|: z1 = 0, z'' - 1 >= 0 eq(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0 if(z'', z1, z2) -{ 1 }-> z1 :|: z1 >= 0, z2 >= 0, z'' = 2 if(z'', z1, z2) -{ 1 }-> z2 :|: z1 >= 0, z2 >= 0, z'' = 1 if(z'', z1, z2) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0, z2 >= 0 le(z'', z1) -{ 2 + z1 }-> s :|: s >= 0, s <= 2, z'' - 1 >= 0, z1 - 1 >= 0 le(z'', z1) -{ 1 }-> 2 :|: z'' = 0, z1 >= 0 le(z'', z1) -{ 1 }-> 1 :|: z1 = 0, z'' - 1 >= 0 le(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0 min(z'', z1) -{ 4 + inf10 + inf8 + y }-> s20 :|: s18 >= 0, s18 <= inf9, s19 >= 0, s19 <= inf11, s20 >= 0, s20 <= s19 + s18, s'' >= 0, s'' <= 2, z >= 0, z1 = 1 + y + z, z'' >= 0, y >= 0 min(z'', z1) -{ 1 }-> z'' :|: z'' >= 0, z1 = 1 min(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0 minsort(z'') -{ 1 }-> 1 :|: z'' = 1 minsort(z'') -{ 33 + inf + 16*z'' + 2*z''^2 }-> 1 + s12 + minsort(s21) :|: s21 >= 0, s21 <= 8 + 11 * (1 + (z'' - 2) + 1) + 4 * ((1 + (z'' - 2) + 1) * (1 + (z'' - 2) + 1)), s12 >= 0, s12 <= inf', z'' - 2 >= 0 minsort(z'') -{ 59 + inf'' + 13*x + 13*y }-> 1 + s13 + minsort(s22) :|: s22 >= 0, s22 <= 8 + 11 * (1 + x + y) + 4 * ((1 + x + y) * (1 + x + y)), s13 >= 0, s13 <= inf1, z'' = 1 + x + y, x >= 0, y >= 0 minsort(z'') -{ 76 + inf2 + inf4 + inf6 + 11*s17 + 2*s17*x + 2*s17*y' + 2*s17*z' + 13*x + 14*y' + 13*z' }-> 1 + s14 + minsort(s23) :|: s23 >= 0, s23 <= 8 + 11 * (1 + x + (1 + y' + z')) + 4 * ((1 + x + (1 + y' + z')) * (1 + x + (1 + y' + z'))), s14 >= 0, s14 <= inf3, s15 >= 0, s15 <= inf5, s16 >= 0, s16 <= inf7, s17 >= 0, s17 <= s16 + s15, s' >= 0, s' <= 2, z'' = 1 + x + (1 + y' + z'), x >= 0, z' >= 0, y' >= 0 Function symbols to be analyzed: {minsort} Previous analysis results are: le: runtime: O(n^1) [2 + z1], size: O(1) [2] if: runtime: O(1) [1], size: O(n^1) [z1 + z2] eq: runtime: O(n^1) [3 + z''], size: O(1) [2] min: runtime: EXP, size: EXP del: runtime: O(n^2) [45 + 7*z'' + 2*z''*z1 + 13*z1], size: O(n^2) [8 + 11*z1 + 4*z1^2] minsort: runtime: INF, size: INF ---------------------------------------- (57) 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: le(v0, v1) -> null_le [0] eq(v0, v1) -> null_eq [0] minsort(v0) -> null_minsort [0] min(v0, v1) -> null_min [0] del(v0, v1) -> null_del [0] if(v0, v1, v2) -> null_if [0] And the following fresh constants: null_le, null_eq, null_minsort, null_min, null_del, null_if ---------------------------------------- (58) 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] if(true, x, y) -> x [1] if(false, x, y) -> y [1] minsort(nil) -> nil [1] minsort(cons(x, y)) -> cons(min(x, y), minsort(del(min(x, y), cons(x, y)))) [1] min(x, nil) -> x [1] min(x, cons(y, z)) -> if(le(x, y), min(x, z), min(y, z)) [1] del(x, nil) -> nil [1] del(x, cons(y, z)) -> if(eq(x, y), z, cons(y, del(x, z))) [1] le(v0, v1) -> null_le [0] eq(v0, v1) -> null_eq [0] minsort(v0) -> null_minsort [0] min(v0, v1) -> null_min [0] del(v0, v1) -> null_del [0] if(v0, v1, v2) -> null_if [0] The TRS has the following type information: le :: 0:s:nil:cons:null_minsort:null_min:null_del:null_if -> 0:s:nil:cons:null_minsort:null_min:null_del:null_if -> true:false:null_le:null_eq 0 :: 0:s:nil:cons:null_minsort:null_min:null_del:null_if true :: true:false:null_le:null_eq s :: 0:s:nil:cons:null_minsort:null_min:null_del:null_if -> 0:s:nil:cons:null_minsort:null_min:null_del:null_if false :: true:false:null_le:null_eq eq :: 0:s:nil:cons:null_minsort:null_min:null_del:null_if -> 0:s:nil:cons:null_minsort:null_min:null_del:null_if -> true:false:null_le:null_eq if :: true:false:null_le:null_eq -> 0:s:nil:cons:null_minsort:null_min:null_del:null_if -> 0:s:nil:cons:null_minsort:null_min:null_del:null_if -> 0:s:nil:cons:null_minsort:null_min:null_del:null_if minsort :: 0:s:nil:cons:null_minsort:null_min:null_del:null_if -> 0:s:nil:cons:null_minsort:null_min:null_del:null_if nil :: 0:s:nil:cons:null_minsort:null_min:null_del:null_if cons :: 0:s:nil:cons:null_minsort:null_min:null_del:null_if -> 0:s:nil:cons:null_minsort:null_min:null_del:null_if -> 0:s:nil:cons:null_minsort:null_min:null_del:null_if min :: 0:s:nil:cons:null_minsort:null_min:null_del:null_if -> 0:s:nil:cons:null_minsort:null_min:null_del:null_if -> 0:s:nil:cons:null_minsort:null_min:null_del:null_if del :: 0:s:nil:cons:null_minsort:null_min:null_del:null_if -> 0:s:nil:cons:null_minsort:null_min:null_del:null_if -> 0:s:nil:cons:null_minsort:null_min:null_del:null_if null_le :: true:false:null_le:null_eq null_eq :: true:false:null_le:null_eq null_minsort :: 0:s:nil:cons:null_minsort:null_min:null_del:null_if null_min :: 0:s:nil:cons:null_minsort:null_min:null_del:null_if null_del :: 0:s:nil:cons:null_minsort:null_min:null_del:null_if null_if :: 0:s:nil:cons:null_minsort:null_min:null_del:null_if Rewrite Strategy: INNERMOST ---------------------------------------- (59) 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 => 2 false => 1 nil => 1 null_le => 0 null_eq => 0 null_minsort => 0 null_min => 0 null_del => 0 null_if => 0 ---------------------------------------- (60) Obligation: Complexity RNTS consisting of the following rules: del(z', z'') -{ 1 }-> if(eq(x, y), z, 1 + y + del(x, z)) :|: z >= 0, z' = x, x >= 0, y >= 0, z'' = 1 + y + z del(z', z'') -{ 1 }-> 1 :|: z' = x, x >= 0, z'' = 1 del(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0 eq(z', z'') -{ 1 }-> eq(x, y) :|: z' = 1 + x, x >= 0, y >= 0, z'' = 1 + y eq(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' = 0 eq(z', z'') -{ 1 }-> 1 :|: y >= 0, z'' = 1 + y, z' = 0 eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' = 1 + x, x >= 0 eq(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0 if(z', z'', z1) -{ 1 }-> x :|: z1 = y, z' = 2, x >= 0, y >= 0, z'' = x if(z', z'', z1) -{ 1 }-> y :|: z1 = y, x >= 0, y >= 0, z'' = x, z' = 1 if(z', z'', z1) -{ 0 }-> 0 :|: v0 >= 0, z1 = v2, v1 >= 0, z'' = v1, v2 >= 0, z' = v0 le(z', z'') -{ 1 }-> le(x, y) :|: z' = 1 + x, x >= 0, y >= 0, z'' = 1 + y le(z', z'') -{ 1 }-> 2 :|: z'' = y, y >= 0, z' = 0 le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' = 1 + x, x >= 0 le(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0 min(z', z'') -{ 1 }-> x :|: z' = x, x >= 0, z'' = 1 min(z', z'') -{ 1 }-> if(le(x, y), min(x, z), min(y, z)) :|: z >= 0, z' = x, x >= 0, y >= 0, z'' = 1 + y + z min(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0 minsort(z') -{ 1 }-> 1 :|: z' = 1 minsort(z') -{ 0 }-> 0 :|: v0 >= 0, z' = v0 minsort(z') -{ 1 }-> 1 + min(x, y) + minsort(del(min(x, y), 1 + x + y)) :|: z' = 1 + x + y, x >= 0, y >= 0 Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (61) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (62) 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) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 minsort(nil) -> nil minsort(cons(z0, z1)) -> cons(min(z0, z1), minsort(del(min(z0, z1), cons(z0, z1)))) min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if(le(z0, z1), min(z0, z2), min(z1, z2)) del(z0, nil) -> nil del(z0, cons(z1, z2)) -> if(eq(z0, z1), z2, cons(z1, del(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)) IF(true, z0, z1) -> c7 IF(false, z0, z1) -> c8 MINSORT(nil) -> c9 MINSORT(cons(z0, z1)) -> c10(MIN(z0, z1)) MINSORT(cons(z0, z1)) -> c11(MINSORT(del(min(z0, z1), cons(z0, z1))), DEL(min(z0, z1), cons(z0, z1)), MIN(z0, z1)) MIN(z0, nil) -> c12 MIN(z0, cons(z1, z2)) -> c13(IF(le(z0, z1), min(z0, z2), min(z1, z2)), LE(z0, z1)) MIN(z0, cons(z1, z2)) -> c14(IF(le(z0, z1), min(z0, z2), min(z1, z2)), MIN(z0, z2)) MIN(z0, cons(z1, z2)) -> c15(IF(le(z0, z1), min(z0, z2), min(z1, z2)), MIN(z1, z2)) DEL(z0, nil) -> c16 DEL(z0, cons(z1, z2)) -> c17(IF(eq(z0, z1), z2, cons(z1, del(z0, z2))), EQ(z0, z1)) DEL(z0, cons(z1, z2)) -> c18(IF(eq(z0, z1), z2, cons(z1, del(z0, z2))), DEL(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)) IF(true, z0, z1) -> c7 IF(false, z0, z1) -> c8 MINSORT(nil) -> c9 MINSORT(cons(z0, z1)) -> c10(MIN(z0, z1)) MINSORT(cons(z0, z1)) -> c11(MINSORT(del(min(z0, z1), cons(z0, z1))), DEL(min(z0, z1), cons(z0, z1)), MIN(z0, z1)) MIN(z0, nil) -> c12 MIN(z0, cons(z1, z2)) -> c13(IF(le(z0, z1), min(z0, z2), min(z1, z2)), LE(z0, z1)) MIN(z0, cons(z1, z2)) -> c14(IF(le(z0, z1), min(z0, z2), min(z1, z2)), MIN(z0, z2)) MIN(z0, cons(z1, z2)) -> c15(IF(le(z0, z1), min(z0, z2), min(z1, z2)), MIN(z1, z2)) DEL(z0, nil) -> c16 DEL(z0, cons(z1, z2)) -> c17(IF(eq(z0, z1), z2, cons(z1, del(z0, z2))), EQ(z0, z1)) DEL(z0, cons(z1, z2)) -> c18(IF(eq(z0, z1), z2, cons(z1, del(z0, z2))), DEL(z0, z2)) K tuples:none Defined Rule Symbols: le_2, eq_2, if_3, minsort_1, min_2, del_2 Defined Pair Symbols: LE_2, EQ_2, IF_3, MINSORT_1, MIN_2, DEL_2 Compound Symbols: c, c1, c2_1, c3, c4, c5, c6_1, c7, c8, c9, c10_1, c11_3, c12, c13_2, c14_2, c15_2, c16, c17_2, c18_2 ---------------------------------------- (63) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 10 trailing nodes: LE(s(z0), 0) -> c1 DEL(z0, nil) -> c16 IF(true, z0, z1) -> c7 MIN(z0, nil) -> c12 IF(false, z0, z1) -> c8 EQ(0, s(z0)) -> c4 MINSORT(nil) -> c9 EQ(s(z0), 0) -> c5 LE(0, z0) -> c EQ(0, 0) -> c3 ---------------------------------------- (64) 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) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 minsort(nil) -> nil minsort(cons(z0, z1)) -> cons(min(z0, z1), minsort(del(min(z0, z1), cons(z0, z1)))) min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if(le(z0, z1), min(z0, z2), min(z1, z2)) del(z0, nil) -> nil del(z0, cons(z1, z2)) -> if(eq(z0, z1), z2, cons(z1, del(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)) -> c10(MIN(z0, z1)) MINSORT(cons(z0, z1)) -> c11(MINSORT(del(min(z0, z1), cons(z0, z1))), DEL(min(z0, z1), cons(z0, z1)), MIN(z0, z1)) MIN(z0, cons(z1, z2)) -> c13(IF(le(z0, z1), min(z0, z2), min(z1, z2)), LE(z0, z1)) MIN(z0, cons(z1, z2)) -> c14(IF(le(z0, z1), min(z0, z2), min(z1, z2)), MIN(z0, z2)) MIN(z0, cons(z1, z2)) -> c15(IF(le(z0, z1), min(z0, z2), min(z1, z2)), MIN(z1, z2)) DEL(z0, cons(z1, z2)) -> c17(IF(eq(z0, z1), z2, cons(z1, del(z0, z2))), EQ(z0, z1)) DEL(z0, cons(z1, z2)) -> c18(IF(eq(z0, z1), z2, cons(z1, del(z0, z2))), DEL(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)) -> c10(MIN(z0, z1)) MINSORT(cons(z0, z1)) -> c11(MINSORT(del(min(z0, z1), cons(z0, z1))), DEL(min(z0, z1), cons(z0, z1)), MIN(z0, z1)) MIN(z0, cons(z1, z2)) -> c13(IF(le(z0, z1), min(z0, z2), min(z1, z2)), LE(z0, z1)) MIN(z0, cons(z1, z2)) -> c14(IF(le(z0, z1), min(z0, z2), min(z1, z2)), MIN(z0, z2)) MIN(z0, cons(z1, z2)) -> c15(IF(le(z0, z1), min(z0, z2), min(z1, z2)), MIN(z1, z2)) DEL(z0, cons(z1, z2)) -> c17(IF(eq(z0, z1), z2, cons(z1, del(z0, z2))), EQ(z0, z1)) DEL(z0, cons(z1, z2)) -> c18(IF(eq(z0, z1), z2, cons(z1, del(z0, z2))), DEL(z0, z2)) K tuples:none Defined Rule Symbols: le_2, eq_2, if_3, minsort_1, min_2, del_2 Defined Pair Symbols: LE_2, EQ_2, MINSORT_1, MIN_2, DEL_2 Compound Symbols: c2_1, c6_1, c10_1, c11_3, c13_2, c14_2, c15_2, c17_2, c18_2 ---------------------------------------- (65) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 5 trailing tuple parts ---------------------------------------- (66) 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) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 minsort(nil) -> nil minsort(cons(z0, z1)) -> cons(min(z0, z1), minsort(del(min(z0, z1), cons(z0, z1)))) min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if(le(z0, z1), min(z0, z2), min(z1, z2)) del(z0, nil) -> nil del(z0, cons(z1, z2)) -> if(eq(z0, z1), z2, cons(z1, del(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)) -> c10(MIN(z0, z1)) MINSORT(cons(z0, z1)) -> c11(MINSORT(del(min(z0, z1), cons(z0, z1))), DEL(min(z0, z1), cons(z0, z1)), MIN(z0, z1)) MIN(z0, cons(z1, z2)) -> c13(LE(z0, z1)) MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2)) MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2)) DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1)) DEL(z0, cons(z1, z2)) -> c18(DEL(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)) -> c10(MIN(z0, z1)) MINSORT(cons(z0, z1)) -> c11(MINSORT(del(min(z0, z1), cons(z0, z1))), DEL(min(z0, z1), cons(z0, z1)), MIN(z0, z1)) MIN(z0, cons(z1, z2)) -> c13(LE(z0, z1)) MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2)) MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2)) DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1)) DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2)) K tuples:none Defined Rule Symbols: le_2, eq_2, if_3, minsort_1, min_2, del_2 Defined Pair Symbols: LE_2, EQ_2, MINSORT_1, MIN_2, DEL_2 Compound Symbols: c2_1, c6_1, c10_1, c11_3, c13_1, c14_1, c15_1, c17_1, c18_1 ---------------------------------------- (67) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: minsort(nil) -> nil minsort(cons(z0, z1)) -> cons(min(z0, z1), minsort(del(min(z0, z1), cons(z0, z1)))) ---------------------------------------- (68) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if(eq(z0, z1), z2, cons(z1, del(z0, z2))) del(z0, nil) -> nil min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if(le(z0, z1), min(z0, z2), min(z1, z2)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 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) Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) MINSORT(cons(z0, z1)) -> c10(MIN(z0, z1)) MINSORT(cons(z0, z1)) -> c11(MINSORT(del(min(z0, z1), cons(z0, z1))), DEL(min(z0, z1), cons(z0, z1)), MIN(z0, z1)) MIN(z0, cons(z1, z2)) -> c13(LE(z0, z1)) MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2)) MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2)) DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1)) DEL(z0, cons(z1, z2)) -> c18(DEL(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)) -> c10(MIN(z0, z1)) MINSORT(cons(z0, z1)) -> c11(MINSORT(del(min(z0, z1), cons(z0, z1))), DEL(min(z0, z1), cons(z0, z1)), MIN(z0, z1)) MIN(z0, cons(z1, z2)) -> c13(LE(z0, z1)) MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2)) MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2)) DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1)) DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2)) K tuples:none Defined Rule Symbols: del_2, min_2, if_3, le_2, eq_2 Defined Pair Symbols: LE_2, EQ_2, MINSORT_1, MIN_2, DEL_2 Compound Symbols: c2_1, c6_1, c10_1, c11_3, c13_1, c14_1, c15_1, c17_1, c18_1 ---------------------------------------- (69) 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)) -> c10(MIN(z0, z1)) We considered the (Usable) Rules:none And the Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) MINSORT(cons(z0, z1)) -> c10(MIN(z0, z1)) MINSORT(cons(z0, z1)) -> c11(MINSORT(del(min(z0, z1), cons(z0, z1))), DEL(min(z0, z1), cons(z0, z1)), MIN(z0, z1)) MIN(z0, cons(z1, z2)) -> c13(LE(z0, z1)) MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2)) MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2)) DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1)) DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2)) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = [1] POL(DEL(x_1, x_2)) = 0 POL(EQ(x_1, x_2)) = 0 POL(LE(x_1, x_2)) = 0 POL(MIN(x_1, x_2)) = 0 POL(MINSORT(x_1)) = [1] POL(c10(x_1)) = x_1 POL(c11(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c13(x_1)) = x_1 POL(c14(x_1)) = x_1 POL(c15(x_1)) = x_1 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(cons(x_1, x_2)) = [1] + x_1 + x_2 POL(del(x_1, x_2)) = [1] + x_1 + x_2 POL(eq(x_1, x_2)) = 0 POL(false) = 0 POL(if(x_1, x_2, x_3)) = x_1 + x_3 POL(le(x_1, x_2)) = x_1 POL(min(x_1, x_2)) = [1] + x_1 + x_2 POL(nil) = [1] POL(s(x_1)) = [1] + x_1 POL(true) = 0 ---------------------------------------- (70) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if(eq(z0, z1), z2, cons(z1, del(z0, z2))) del(z0, nil) -> nil min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if(le(z0, z1), min(z0, z2), min(z1, z2)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 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) Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) MINSORT(cons(z0, z1)) -> c10(MIN(z0, z1)) MINSORT(cons(z0, z1)) -> c11(MINSORT(del(min(z0, z1), cons(z0, z1))), DEL(min(z0, z1), cons(z0, z1)), MIN(z0, z1)) MIN(z0, cons(z1, z2)) -> c13(LE(z0, z1)) MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2)) MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2)) DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1)) DEL(z0, cons(z1, z2)) -> c18(DEL(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)) -> c11(MINSORT(del(min(z0, z1), cons(z0, z1))), DEL(min(z0, z1), cons(z0, z1)), MIN(z0, z1)) MIN(z0, cons(z1, z2)) -> c13(LE(z0, z1)) MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2)) MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2)) DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1)) DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2)) K tuples: MINSORT(cons(z0, z1)) -> c10(MIN(z0, z1)) Defined Rule Symbols: del_2, min_2, if_3, le_2, eq_2 Defined Pair Symbols: LE_2, EQ_2, MINSORT_1, MIN_2, DEL_2 Compound Symbols: c2_1, c6_1, c10_1, c11_3, c13_1, c14_1, c15_1, c17_1, c18_1 ---------------------------------------- (71) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MINSORT(cons(z0, z1)) -> c11(MINSORT(del(min(z0, z1), cons(z0, z1))), DEL(min(z0, z1), cons(z0, z1)), MIN(z0, z1)) by MINSORT(cons(z1, z2)) -> c11(MINSORT(if(eq(min(z1, z2), z1), z2, cons(z1, del(min(z1, z2), z2)))), DEL(min(z1, z2), cons(z1, z2)), MIN(z1, z2)) MINSORT(cons(z0, nil)) -> c11(MINSORT(del(z0, cons(z0, nil))), DEL(min(z0, nil), cons(z0, nil)), MIN(z0, nil)) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2)))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) ---------------------------------------- (72) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if(eq(z0, z1), z2, cons(z1, del(z0, z2))) del(z0, nil) -> nil min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if(le(z0, z1), min(z0, z2), min(z1, z2)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 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) Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) MINSORT(cons(z0, z1)) -> c10(MIN(z0, z1)) MIN(z0, cons(z1, z2)) -> c13(LE(z0, z1)) MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2)) MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2)) DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1)) DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2)) MINSORT(cons(z1, z2)) -> c11(MINSORT(if(eq(min(z1, z2), z1), z2, cons(z1, del(min(z1, z2), z2)))), DEL(min(z1, z2), cons(z1, z2)), MIN(z1, z2)) MINSORT(cons(z0, nil)) -> c11(MINSORT(del(z0, cons(z0, nil))), DEL(min(z0, nil), cons(z0, nil)), MIN(z0, nil)) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2)))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) S tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) MIN(z0, cons(z1, z2)) -> c13(LE(z0, z1)) MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2)) MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2)) DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1)) DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2)) MINSORT(cons(z1, z2)) -> c11(MINSORT(if(eq(min(z1, z2), z1), z2, cons(z1, del(min(z1, z2), z2)))), DEL(min(z1, z2), cons(z1, z2)), MIN(z1, z2)) MINSORT(cons(z0, nil)) -> c11(MINSORT(del(z0, cons(z0, nil))), DEL(min(z0, nil), cons(z0, nil)), MIN(z0, nil)) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2)))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) K tuples: MINSORT(cons(z0, z1)) -> c10(MIN(z0, z1)) Defined Rule Symbols: del_2, min_2, if_3, le_2, eq_2 Defined Pair Symbols: LE_2, EQ_2, MINSORT_1, MIN_2, DEL_2 Compound Symbols: c2_1, c6_1, c10_1, c13_1, c14_1, c15_1, c17_1, c18_1, c11_3 ---------------------------------------- (73) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (74) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if(eq(z0, z1), z2, cons(z1, del(z0, z2))) del(z0, nil) -> nil min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if(le(z0, z1), min(z0, z2), min(z1, z2)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 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) Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) MINSORT(cons(z0, z1)) -> c10(MIN(z0, z1)) MIN(z0, cons(z1, z2)) -> c13(LE(z0, z1)) MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2)) MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2)) DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1)) DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2)) MINSORT(cons(z1, z2)) -> c11(MINSORT(if(eq(min(z1, z2), z1), z2, cons(z1, del(min(z1, z2), z2)))), DEL(min(z1, z2), cons(z1, z2)), MIN(z1, z2)) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2)))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(del(z0, cons(z0, nil))), DEL(min(z0, nil), cons(z0, nil))) S tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) MIN(z0, cons(z1, z2)) -> c13(LE(z0, z1)) MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2)) MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2)) DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1)) DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2)) MINSORT(cons(z1, z2)) -> c11(MINSORT(if(eq(min(z1, z2), z1), z2, cons(z1, del(min(z1, z2), z2)))), DEL(min(z1, z2), cons(z1, z2)), MIN(z1, z2)) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2)))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(del(z0, cons(z0, nil))), DEL(min(z0, nil), cons(z0, nil))) K tuples: MINSORT(cons(z0, z1)) -> c10(MIN(z0, z1)) Defined Rule Symbols: del_2, min_2, if_3, le_2, eq_2 Defined Pair Symbols: LE_2, EQ_2, MINSORT_1, MIN_2, DEL_2 Compound Symbols: c2_1, c6_1, c10_1, c13_1, c14_1, c15_1, c17_1, c18_1, c11_3, c11_2 ---------------------------------------- (75) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MINSORT(cons(z1, z2)) -> c11(MINSORT(if(eq(min(z1, z2), z1), z2, cons(z1, del(min(z1, z2), z2)))), DEL(min(z1, z2), cons(z1, z2)), MIN(z1, z2)) by MINSORT(cons(x0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(x0, cons(z1, z2)), x0), cons(z1, z2), cons(x0, if(eq(min(x0, cons(z1, z2)), z1), z2, cons(z1, del(min(x0, cons(z1, z2)), z2)))))), DEL(min(x0, cons(z1, z2)), cons(x0, cons(z1, z2))), MIN(x0, cons(z1, z2))) MINSORT(cons(x0, nil)) -> c11(MINSORT(if(eq(min(x0, nil), x0), nil, cons(x0, nil))), DEL(min(x0, nil), cons(x0, nil)), MIN(x0, nil)) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil)), MIN(z0, nil)) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(min(z0, nil), cons(z0, nil)), MIN(z0, nil)) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1))) ---------------------------------------- (76) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if(eq(z0, z1), z2, cons(z1, del(z0, z2))) del(z0, nil) -> nil min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if(le(z0, z1), min(z0, z2), min(z1, z2)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 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) Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) MINSORT(cons(z0, z1)) -> c10(MIN(z0, z1)) MIN(z0, cons(z1, z2)) -> c13(LE(z0, z1)) MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2)) MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2)) DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1)) DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2)) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2)))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(del(z0, cons(z0, nil))), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(x0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(x0, cons(z1, z2)), x0), cons(z1, z2), cons(x0, if(eq(min(x0, cons(z1, z2)), z1), z2, cons(z1, del(min(x0, cons(z1, z2)), z2)))))), DEL(min(x0, cons(z1, z2)), cons(x0, cons(z1, z2))), MIN(x0, cons(z1, z2))) MINSORT(cons(x0, nil)) -> c11(MINSORT(if(eq(min(x0, nil), x0), nil, cons(x0, nil))), DEL(min(x0, nil), cons(x0, nil)), MIN(x0, nil)) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil)), MIN(z0, nil)) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(min(z0, nil), cons(z0, nil)), MIN(z0, nil)) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1))) S tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) MIN(z0, cons(z1, z2)) -> c13(LE(z0, z1)) MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2)) MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2)) DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1)) DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2)) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2)))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(del(z0, cons(z0, nil))), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(x0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(x0, cons(z1, z2)), x0), cons(z1, z2), cons(x0, if(eq(min(x0, cons(z1, z2)), z1), z2, cons(z1, del(min(x0, cons(z1, z2)), z2)))))), DEL(min(x0, cons(z1, z2)), cons(x0, cons(z1, z2))), MIN(x0, cons(z1, z2))) MINSORT(cons(x0, nil)) -> c11(MINSORT(if(eq(min(x0, nil), x0), nil, cons(x0, nil))), DEL(min(x0, nil), cons(x0, nil)), MIN(x0, nil)) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil)), MIN(z0, nil)) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(min(z0, nil), cons(z0, nil)), MIN(z0, nil)) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1))) K tuples: MINSORT(cons(z0, z1)) -> c10(MIN(z0, z1)) Defined Rule Symbols: del_2, min_2, if_3, le_2, eq_2 Defined Pair Symbols: LE_2, EQ_2, MINSORT_1, MIN_2, DEL_2 Compound Symbols: c2_1, c6_1, c10_1, c13_1, c14_1, c15_1, c17_1, c18_1, c11_3, c11_2, c11_1 ---------------------------------------- (77) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (78) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if(eq(z0, z1), z2, cons(z1, del(z0, z2))) del(z0, nil) -> nil min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if(le(z0, z1), min(z0, z2), min(z1, z2)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 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) Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) MINSORT(cons(z0, z1)) -> c10(MIN(z0, z1)) MIN(z0, cons(z1, z2)) -> c13(LE(z0, z1)) MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2)) MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2)) DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1)) DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2)) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2)))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(del(z0, cons(z0, nil))), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(x0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(x0, cons(z1, z2)), x0), cons(z1, z2), cons(x0, if(eq(min(x0, cons(z1, z2)), z1), z2, cons(z1, del(min(x0, cons(z1, z2)), z2)))))), DEL(min(x0, cons(z1, z2)), cons(x0, cons(z1, z2))), MIN(x0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1))) MINSORT(cons(x0, nil)) -> c11(MINSORT(if(eq(min(x0, nil), x0), nil, cons(x0, nil))), DEL(min(x0, nil), cons(x0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(min(z0, nil), cons(z0, nil))) S tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) MIN(z0, cons(z1, z2)) -> c13(LE(z0, z1)) MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2)) MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2)) DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1)) DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2)) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2)))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(del(z0, cons(z0, nil))), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(x0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(x0, cons(z1, z2)), x0), cons(z1, z2), cons(x0, if(eq(min(x0, cons(z1, z2)), z1), z2, cons(z1, del(min(x0, cons(z1, z2)), z2)))))), DEL(min(x0, cons(z1, z2)), cons(x0, cons(z1, z2))), MIN(x0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1))) MINSORT(cons(x0, nil)) -> c11(MINSORT(if(eq(min(x0, nil), x0), nil, cons(x0, nil))), DEL(min(x0, nil), cons(x0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(min(z0, nil), cons(z0, nil))) K tuples: MINSORT(cons(z0, z1)) -> c10(MIN(z0, z1)) Defined Rule Symbols: del_2, min_2, if_3, le_2, eq_2 Defined Pair Symbols: LE_2, EQ_2, MINSORT_1, MIN_2, DEL_2 Compound Symbols: c2_1, c6_1, c10_1, c13_1, c14_1, c15_1, c17_1, c18_1, c11_3, c11_2, c11_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, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1))) 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)) -> c10(MIN(z0, z1)) MIN(z0, cons(z1, z2)) -> c13(LE(z0, z1)) MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2)) MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2)) DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1)) DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2)) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2)))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(del(z0, cons(z0, nil))), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(x0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(x0, cons(z1, z2)), x0), cons(z1, z2), cons(x0, if(eq(min(x0, cons(z1, z2)), z1), z2, cons(z1, del(min(x0, cons(z1, z2)), z2)))))), DEL(min(x0, cons(z1, z2)), cons(x0, cons(z1, z2))), MIN(x0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1))) MINSORT(cons(x0, nil)) -> c11(MINSORT(if(eq(min(x0, nil), x0), nil, cons(x0, nil))), DEL(min(x0, nil), cons(x0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(min(z0, nil), cons(z0, nil))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = [1] POL(DEL(x_1, x_2)) = 0 POL(EQ(x_1, x_2)) = 0 POL(LE(x_1, x_2)) = 0 POL(MIN(x_1, x_2)) = 0 POL(MINSORT(x_1)) = [1] POL(c10(x_1)) = x_1 POL(c11(x_1)) = x_1 POL(c11(x_1, x_2)) = x_1 + x_2 POL(c11(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c13(x_1)) = x_1 POL(c14(x_1)) = x_1 POL(c15(x_1)) = x_1 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(cons(x_1, x_2)) = 0 POL(del(x_1, x_2)) = 0 POL(eq(x_1, x_2)) = [1] POL(false) = [1] POL(if(x_1, x_2, x_3)) = 0 POL(le(x_1, x_2)) = x_1 + x_2 POL(min(x_1, x_2)) = 0 POL(nil) = 0 POL(s(x_1)) = [1] + x_1 POL(true) = [1] ---------------------------------------- (80) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if(eq(z0, z1), z2, cons(z1, del(z0, z2))) del(z0, nil) -> nil min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if(le(z0, z1), min(z0, z2), min(z1, z2)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 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) Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) MINSORT(cons(z0, z1)) -> c10(MIN(z0, z1)) MIN(z0, cons(z1, z2)) -> c13(LE(z0, z1)) MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2)) MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2)) DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1)) DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2)) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2)))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(del(z0, cons(z0, nil))), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(x0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(x0, cons(z1, z2)), x0), cons(z1, z2), cons(x0, if(eq(min(x0, cons(z1, z2)), z1), z2, cons(z1, del(min(x0, cons(z1, z2)), z2)))))), DEL(min(x0, cons(z1, z2)), cons(x0, cons(z1, z2))), MIN(x0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1))) MINSORT(cons(x0, nil)) -> c11(MINSORT(if(eq(min(x0, nil), x0), nil, cons(x0, nil))), DEL(min(x0, nil), cons(x0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(min(z0, nil), cons(z0, nil))) S tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) MIN(z0, cons(z1, z2)) -> c13(LE(z0, z1)) MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2)) MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2)) DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1)) DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2)) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2)))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(del(z0, cons(z0, nil))), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(x0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(x0, cons(z1, z2)), x0), cons(z1, z2), cons(x0, if(eq(min(x0, cons(z1, z2)), z1), z2, cons(z1, del(min(x0, cons(z1, z2)), z2)))))), DEL(min(x0, cons(z1, z2)), cons(x0, cons(z1, z2))), MIN(x0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(x0, nil)) -> c11(MINSORT(if(eq(min(x0, nil), x0), nil, cons(x0, nil))), DEL(min(x0, nil), cons(x0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(min(z0, nil), cons(z0, nil))) K tuples: MINSORT(cons(z0, z1)) -> c10(MIN(z0, z1)) MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1))) Defined Rule Symbols: del_2, min_2, if_3, le_2, eq_2 Defined Pair Symbols: LE_2, EQ_2, MINSORT_1, MIN_2, DEL_2 Compound Symbols: c2_1, c6_1, c10_1, c13_1, c14_1, c15_1, c17_1, c18_1, c11_3, c11_2, c11_1 ---------------------------------------- (81) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2)))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) by MINSORT(cons(z1, cons(x1, x2))) -> c11(MINSORT(if(eq(if(le(z1, x1), min(z1, x2), min(x1, x2)), z1), cons(x1, x2), cons(z1, del(if(le(z1, x1), min(z1, x2), min(x1, x2)), cons(x1, x2))))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2))) MINSORT(cons(x0, cons(z0, nil))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, nil), z0), cons(x0, cons(z0, nil)))), DEL(min(x0, cons(z0, nil)), cons(x0, cons(z0, nil))), MIN(x0, cons(z0, nil))) MINSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, cons(z1, z2)), if(le(z0, z1), min(z0, z2), min(z1, z2))), cons(x0, cons(z0, cons(z1, z2))))), DEL(min(x0, cons(z0, cons(z1, z2))), cons(x0, cons(z0, cons(z1, z2)))), MIN(x0, cons(z0, cons(z1, z2)))) MINSORT(cons(z0, cons(x1, nil))) -> c11(MINSORT(del(if(le(z0, x1), z0, min(x1, nil)), cons(z0, cons(x1, nil)))), DEL(min(z0, cons(x1, nil)), cons(z0, cons(x1, nil))), MIN(z0, cons(x1, nil))) MINSORT(cons(z0, cons(x1, cons(z1, z2)))) -> c11(MINSORT(del(if(le(z0, x1), if(le(z0, z1), min(z0, z2), min(z1, z2)), min(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2))))), DEL(min(z0, cons(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2)))), MIN(z0, cons(x1, cons(z1, z2)))) MINSORT(cons(0, cons(z0, x2))) -> c11(MINSORT(del(if(true, min(0, x2), min(z0, x2)), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2))) MINSORT(cons(s(z0), cons(0, x2))) -> c11(MINSORT(del(if(false, min(s(z0), x2), min(0, x2)), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), x2), min(s(z1), x2)), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2))) MINSORT(cons(x0, cons(x1, x2))) -> c11(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))), MIN(x0, cons(x1, x2))) ---------------------------------------- (82) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if(eq(z0, z1), z2, cons(z1, del(z0, z2))) del(z0, nil) -> nil min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if(le(z0, z1), min(z0, z2), min(z1, z2)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 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) Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) MINSORT(cons(z0, z1)) -> c10(MIN(z0, z1)) MIN(z0, cons(z1, z2)) -> c13(LE(z0, z1)) MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2)) MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2)) DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1)) DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2)) MINSORT(cons(z0, nil)) -> c11(MINSORT(del(z0, cons(z0, nil))), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(x0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(x0, cons(z1, z2)), x0), cons(z1, z2), cons(x0, if(eq(min(x0, cons(z1, z2)), z1), z2, cons(z1, del(min(x0, cons(z1, z2)), z2)))))), DEL(min(x0, cons(z1, z2)), cons(x0, cons(z1, z2))), MIN(x0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1))) MINSORT(cons(x0, nil)) -> c11(MINSORT(if(eq(min(x0, nil), x0), nil, cons(x0, nil))), DEL(min(x0, nil), cons(x0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z1, cons(x1, x2))) -> c11(MINSORT(if(eq(if(le(z1, x1), min(z1, x2), min(x1, x2)), z1), cons(x1, x2), cons(z1, del(if(le(z1, x1), min(z1, x2), min(x1, x2)), cons(x1, x2))))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2))) MINSORT(cons(x0, cons(z0, nil))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, nil), z0), cons(x0, cons(z0, nil)))), DEL(min(x0, cons(z0, nil)), cons(x0, cons(z0, nil))), MIN(x0, cons(z0, nil))) MINSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, cons(z1, z2)), if(le(z0, z1), min(z0, z2), min(z1, z2))), cons(x0, cons(z0, cons(z1, z2))))), DEL(min(x0, cons(z0, cons(z1, z2))), cons(x0, cons(z0, cons(z1, z2)))), MIN(x0, cons(z0, cons(z1, z2)))) MINSORT(cons(z0, cons(x1, nil))) -> c11(MINSORT(del(if(le(z0, x1), z0, min(x1, nil)), cons(z0, cons(x1, nil)))), DEL(min(z0, cons(x1, nil)), cons(z0, cons(x1, nil))), MIN(z0, cons(x1, nil))) MINSORT(cons(z0, cons(x1, cons(z1, z2)))) -> c11(MINSORT(del(if(le(z0, x1), if(le(z0, z1), min(z0, z2), min(z1, z2)), min(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2))))), DEL(min(z0, cons(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2)))), MIN(z0, cons(x1, cons(z1, z2)))) MINSORT(cons(0, cons(z0, x2))) -> c11(MINSORT(del(if(true, min(0, x2), min(z0, x2)), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2))) MINSORT(cons(s(z0), cons(0, x2))) -> c11(MINSORT(del(if(false, min(s(z0), x2), min(0, x2)), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), x2), min(s(z1), x2)), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2))) MINSORT(cons(x0, cons(x1, x2))) -> c11(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))), MIN(x0, cons(x1, x2))) S tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) MIN(z0, cons(z1, z2)) -> c13(LE(z0, z1)) MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2)) MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2)) DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1)) DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2)) MINSORT(cons(z0, nil)) -> c11(MINSORT(del(z0, cons(z0, nil))), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(x0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(x0, cons(z1, z2)), x0), cons(z1, z2), cons(x0, if(eq(min(x0, cons(z1, z2)), z1), z2, cons(z1, del(min(x0, cons(z1, z2)), z2)))))), DEL(min(x0, cons(z1, z2)), cons(x0, cons(z1, z2))), MIN(x0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(x0, nil)) -> c11(MINSORT(if(eq(min(x0, nil), x0), nil, cons(x0, nil))), DEL(min(x0, nil), cons(x0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z1, cons(x1, x2))) -> c11(MINSORT(if(eq(if(le(z1, x1), min(z1, x2), min(x1, x2)), z1), cons(x1, x2), cons(z1, del(if(le(z1, x1), min(z1, x2), min(x1, x2)), cons(x1, x2))))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2))) MINSORT(cons(x0, cons(z0, nil))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, nil), z0), cons(x0, cons(z0, nil)))), DEL(min(x0, cons(z0, nil)), cons(x0, cons(z0, nil))), MIN(x0, cons(z0, nil))) MINSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, cons(z1, z2)), if(le(z0, z1), min(z0, z2), min(z1, z2))), cons(x0, cons(z0, cons(z1, z2))))), DEL(min(x0, cons(z0, cons(z1, z2))), cons(x0, cons(z0, cons(z1, z2)))), MIN(x0, cons(z0, cons(z1, z2)))) MINSORT(cons(z0, cons(x1, nil))) -> c11(MINSORT(del(if(le(z0, x1), z0, min(x1, nil)), cons(z0, cons(x1, nil)))), DEL(min(z0, cons(x1, nil)), cons(z0, cons(x1, nil))), MIN(z0, cons(x1, nil))) MINSORT(cons(z0, cons(x1, cons(z1, z2)))) -> c11(MINSORT(del(if(le(z0, x1), if(le(z0, z1), min(z0, z2), min(z1, z2)), min(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2))))), DEL(min(z0, cons(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2)))), MIN(z0, cons(x1, cons(z1, z2)))) MINSORT(cons(0, cons(z0, x2))) -> c11(MINSORT(del(if(true, min(0, x2), min(z0, x2)), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2))) MINSORT(cons(s(z0), cons(0, x2))) -> c11(MINSORT(del(if(false, min(s(z0), x2), min(0, x2)), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), x2), min(s(z1), x2)), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2))) MINSORT(cons(x0, cons(x1, x2))) -> c11(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2))), MIN(x0, cons(x1, x2))) K tuples: MINSORT(cons(z0, z1)) -> c10(MIN(z0, z1)) MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1))) Defined Rule Symbols: del_2, min_2, if_3, le_2, eq_2 Defined Pair Symbols: LE_2, EQ_2, MINSORT_1, MIN_2, DEL_2 Compound Symbols: c2_1, c6_1, c10_1, c13_1, c14_1, c15_1, c17_1, c18_1, c11_2, c11_3, c11_1 ---------------------------------------- (83) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (84) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if(eq(z0, z1), z2, cons(z1, del(z0, z2))) del(z0, nil) -> nil min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if(le(z0, z1), min(z0, z2), min(z1, z2)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 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) Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) MINSORT(cons(z0, z1)) -> c10(MIN(z0, z1)) MIN(z0, cons(z1, z2)) -> c13(LE(z0, z1)) MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2)) MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2)) DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1)) DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2)) MINSORT(cons(z0, nil)) -> c11(MINSORT(del(z0, cons(z0, nil))), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(x0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(x0, cons(z1, z2)), x0), cons(z1, z2), cons(x0, if(eq(min(x0, cons(z1, z2)), z1), z2, cons(z1, del(min(x0, cons(z1, z2)), z2)))))), DEL(min(x0, cons(z1, z2)), cons(x0, cons(z1, z2))), MIN(x0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1))) MINSORT(cons(x0, nil)) -> c11(MINSORT(if(eq(min(x0, nil), x0), nil, cons(x0, nil))), DEL(min(x0, nil), cons(x0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z1, cons(x1, x2))) -> c11(MINSORT(if(eq(if(le(z1, x1), min(z1, x2), min(x1, x2)), z1), cons(x1, x2), cons(z1, del(if(le(z1, x1), min(z1, x2), min(x1, x2)), cons(x1, x2))))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2))) MINSORT(cons(x0, cons(z0, nil))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, nil), z0), cons(x0, cons(z0, nil)))), DEL(min(x0, cons(z0, nil)), cons(x0, cons(z0, nil))), MIN(x0, cons(z0, nil))) MINSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, cons(z1, z2)), if(le(z0, z1), min(z0, z2), min(z1, z2))), cons(x0, cons(z0, cons(z1, z2))))), DEL(min(x0, cons(z0, cons(z1, z2))), cons(x0, cons(z0, cons(z1, z2)))), MIN(x0, cons(z0, cons(z1, z2)))) MINSORT(cons(z0, cons(x1, nil))) -> c11(MINSORT(del(if(le(z0, x1), z0, min(x1, nil)), cons(z0, cons(x1, nil)))), DEL(min(z0, cons(x1, nil)), cons(z0, cons(x1, nil))), MIN(z0, cons(x1, nil))) MINSORT(cons(z0, cons(x1, cons(z1, z2)))) -> c11(MINSORT(del(if(le(z0, x1), if(le(z0, z1), min(z0, z2), min(z1, z2)), min(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2))))), DEL(min(z0, cons(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2)))), MIN(z0, cons(x1, cons(z1, z2)))) MINSORT(cons(0, cons(z0, x2))) -> c11(MINSORT(del(if(true, min(0, x2), min(z0, x2)), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2))) MINSORT(cons(s(z0), cons(0, x2))) -> c11(MINSORT(del(if(false, min(s(z0), x2), min(0, x2)), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), x2), min(s(z1), x2)), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2))) MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2))) S tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) MIN(z0, cons(z1, z2)) -> c13(LE(z0, z1)) MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2)) MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2)) DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1)) DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2)) MINSORT(cons(z0, nil)) -> c11(MINSORT(del(z0, cons(z0, nil))), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(x0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(x0, cons(z1, z2)), x0), cons(z1, z2), cons(x0, if(eq(min(x0, cons(z1, z2)), z1), z2, cons(z1, del(min(x0, cons(z1, z2)), z2)))))), DEL(min(x0, cons(z1, z2)), cons(x0, cons(z1, z2))), MIN(x0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(x0, nil)) -> c11(MINSORT(if(eq(min(x0, nil), x0), nil, cons(x0, nil))), DEL(min(x0, nil), cons(x0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z1, cons(x1, x2))) -> c11(MINSORT(if(eq(if(le(z1, x1), min(z1, x2), min(x1, x2)), z1), cons(x1, x2), cons(z1, del(if(le(z1, x1), min(z1, x2), min(x1, x2)), cons(x1, x2))))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2))) MINSORT(cons(x0, cons(z0, nil))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, nil), z0), cons(x0, cons(z0, nil)))), DEL(min(x0, cons(z0, nil)), cons(x0, cons(z0, nil))), MIN(x0, cons(z0, nil))) MINSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, cons(z1, z2)), if(le(z0, z1), min(z0, z2), min(z1, z2))), cons(x0, cons(z0, cons(z1, z2))))), DEL(min(x0, cons(z0, cons(z1, z2))), cons(x0, cons(z0, cons(z1, z2)))), MIN(x0, cons(z0, cons(z1, z2)))) MINSORT(cons(z0, cons(x1, nil))) -> c11(MINSORT(del(if(le(z0, x1), z0, min(x1, nil)), cons(z0, cons(x1, nil)))), DEL(min(z0, cons(x1, nil)), cons(z0, cons(x1, nil))), MIN(z0, cons(x1, nil))) MINSORT(cons(z0, cons(x1, cons(z1, z2)))) -> c11(MINSORT(del(if(le(z0, x1), if(le(z0, z1), min(z0, z2), min(z1, z2)), min(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2))))), DEL(min(z0, cons(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2)))), MIN(z0, cons(x1, cons(z1, z2)))) MINSORT(cons(0, cons(z0, x2))) -> c11(MINSORT(del(if(true, min(0, x2), min(z0, x2)), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2))) MINSORT(cons(s(z0), cons(0, x2))) -> c11(MINSORT(del(if(false, min(s(z0), x2), min(0, x2)), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), x2), min(s(z1), x2)), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2))) MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2))) K tuples: MINSORT(cons(z0, z1)) -> c10(MIN(z0, z1)) MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1))) Defined Rule Symbols: del_2, min_2, if_3, le_2, eq_2 Defined Pair Symbols: LE_2, EQ_2, MINSORT_1, MIN_2, DEL_2 Compound Symbols: c2_1, c6_1, c10_1, c13_1, c14_1, c15_1, c17_1, c18_1, c11_2, c11_3, c11_1, c_1 ---------------------------------------- (85) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(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)) -> c10(MIN(z0, z1)) MIN(z0, cons(z1, z2)) -> c13(LE(z0, z1)) MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2)) MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2)) DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1)) DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2)) MINSORT(cons(z0, nil)) -> c11(MINSORT(del(z0, cons(z0, nil))), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(x0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(x0, cons(z1, z2)), x0), cons(z1, z2), cons(x0, if(eq(min(x0, cons(z1, z2)), z1), z2, cons(z1, del(min(x0, cons(z1, z2)), z2)))))), DEL(min(x0, cons(z1, z2)), cons(x0, cons(z1, z2))), MIN(x0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1))) MINSORT(cons(x0, nil)) -> c11(MINSORT(if(eq(min(x0, nil), x0), nil, cons(x0, nil))), DEL(min(x0, nil), cons(x0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z1, cons(x1, x2))) -> c11(MINSORT(if(eq(if(le(z1, x1), min(z1, x2), min(x1, x2)), z1), cons(x1, x2), cons(z1, del(if(le(z1, x1), min(z1, x2), min(x1, x2)), cons(x1, x2))))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2))) MINSORT(cons(x0, cons(z0, nil))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, nil), z0), cons(x0, cons(z0, nil)))), DEL(min(x0, cons(z0, nil)), cons(x0, cons(z0, nil))), MIN(x0, cons(z0, nil))) MINSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, cons(z1, z2)), if(le(z0, z1), min(z0, z2), min(z1, z2))), cons(x0, cons(z0, cons(z1, z2))))), DEL(min(x0, cons(z0, cons(z1, z2))), cons(x0, cons(z0, cons(z1, z2)))), MIN(x0, cons(z0, cons(z1, z2)))) MINSORT(cons(z0, cons(x1, nil))) -> c11(MINSORT(del(if(le(z0, x1), z0, min(x1, nil)), cons(z0, cons(x1, nil)))), DEL(min(z0, cons(x1, nil)), cons(z0, cons(x1, nil))), MIN(z0, cons(x1, nil))) MINSORT(cons(z0, cons(x1, cons(z1, z2)))) -> c11(MINSORT(del(if(le(z0, x1), if(le(z0, z1), min(z0, z2), min(z1, z2)), min(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2))))), DEL(min(z0, cons(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2)))), MIN(z0, cons(x1, cons(z1, z2)))) MINSORT(cons(0, cons(z0, x2))) -> c11(MINSORT(del(if(true, min(0, x2), min(z0, x2)), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2))) MINSORT(cons(s(z0), cons(0, x2))) -> c11(MINSORT(del(if(false, min(s(z0), x2), min(0, x2)), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), x2), min(s(z1), x2)), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2))) MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(DEL(x_1, x_2)) = 0 POL(EQ(x_1, x_2)) = 0 POL(LE(x_1, x_2)) = 0 POL(MIN(x_1, x_2)) = 0 POL(MINSORT(x_1)) = [1] POL(c(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c11(x_1)) = x_1 POL(c11(x_1, x_2)) = x_1 + x_2 POL(c11(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c13(x_1)) = x_1 POL(c14(x_1)) = x_1 POL(c15(x_1)) = x_1 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(cons(x_1, x_2)) = [3] POL(del(x_1, x_2)) = 0 POL(eq(x_1, x_2)) = 0 POL(false) = 0 POL(if(x_1, x_2, x_3)) = 0 POL(le(x_1, x_2)) = 0 POL(min(x_1, x_2)) = [1] + [3]x_2 POL(nil) = 0 POL(s(x_1)) = 0 POL(true) = 0 ---------------------------------------- (86) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if(eq(z0, z1), z2, cons(z1, del(z0, z2))) del(z0, nil) -> nil min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if(le(z0, z1), min(z0, z2), min(z1, z2)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 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) Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) MINSORT(cons(z0, z1)) -> c10(MIN(z0, z1)) MIN(z0, cons(z1, z2)) -> c13(LE(z0, z1)) MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2)) MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2)) DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1)) DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2)) MINSORT(cons(z0, nil)) -> c11(MINSORT(del(z0, cons(z0, nil))), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(x0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(x0, cons(z1, z2)), x0), cons(z1, z2), cons(x0, if(eq(min(x0, cons(z1, z2)), z1), z2, cons(z1, del(min(x0, cons(z1, z2)), z2)))))), DEL(min(x0, cons(z1, z2)), cons(x0, cons(z1, z2))), MIN(x0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1))) MINSORT(cons(x0, nil)) -> c11(MINSORT(if(eq(min(x0, nil), x0), nil, cons(x0, nil))), DEL(min(x0, nil), cons(x0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z1, cons(x1, x2))) -> c11(MINSORT(if(eq(if(le(z1, x1), min(z1, x2), min(x1, x2)), z1), cons(x1, x2), cons(z1, del(if(le(z1, x1), min(z1, x2), min(x1, x2)), cons(x1, x2))))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2))) MINSORT(cons(x0, cons(z0, nil))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, nil), z0), cons(x0, cons(z0, nil)))), DEL(min(x0, cons(z0, nil)), cons(x0, cons(z0, nil))), MIN(x0, cons(z0, nil))) MINSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, cons(z1, z2)), if(le(z0, z1), min(z0, z2), min(z1, z2))), cons(x0, cons(z0, cons(z1, z2))))), DEL(min(x0, cons(z0, cons(z1, z2))), cons(x0, cons(z0, cons(z1, z2)))), MIN(x0, cons(z0, cons(z1, z2)))) MINSORT(cons(z0, cons(x1, nil))) -> c11(MINSORT(del(if(le(z0, x1), z0, min(x1, nil)), cons(z0, cons(x1, nil)))), DEL(min(z0, cons(x1, nil)), cons(z0, cons(x1, nil))), MIN(z0, cons(x1, nil))) MINSORT(cons(z0, cons(x1, cons(z1, z2)))) -> c11(MINSORT(del(if(le(z0, x1), if(le(z0, z1), min(z0, z2), min(z1, z2)), min(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2))))), DEL(min(z0, cons(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2)))), MIN(z0, cons(x1, cons(z1, z2)))) MINSORT(cons(0, cons(z0, x2))) -> c11(MINSORT(del(if(true, min(0, x2), min(z0, x2)), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2))) MINSORT(cons(s(z0), cons(0, x2))) -> c11(MINSORT(del(if(false, min(s(z0), x2), min(0, x2)), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), x2), min(s(z1), x2)), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2))) MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2))) S tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) MIN(z0, cons(z1, z2)) -> c13(LE(z0, z1)) MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2)) MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2)) DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1)) DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2)) MINSORT(cons(z0, nil)) -> c11(MINSORT(del(z0, cons(z0, nil))), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(x0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(x0, cons(z1, z2)), x0), cons(z1, z2), cons(x0, if(eq(min(x0, cons(z1, z2)), z1), z2, cons(z1, del(min(x0, cons(z1, z2)), z2)))))), DEL(min(x0, cons(z1, z2)), cons(x0, cons(z1, z2))), MIN(x0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(x0, nil)) -> c11(MINSORT(if(eq(min(x0, nil), x0), nil, cons(x0, nil))), DEL(min(x0, nil), cons(x0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z1, cons(x1, x2))) -> c11(MINSORT(if(eq(if(le(z1, x1), min(z1, x2), min(x1, x2)), z1), cons(x1, x2), cons(z1, del(if(le(z1, x1), min(z1, x2), min(x1, x2)), cons(x1, x2))))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2))) MINSORT(cons(x0, cons(z0, nil))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, nil), z0), cons(x0, cons(z0, nil)))), DEL(min(x0, cons(z0, nil)), cons(x0, cons(z0, nil))), MIN(x0, cons(z0, nil))) MINSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, cons(z1, z2)), if(le(z0, z1), min(z0, z2), min(z1, z2))), cons(x0, cons(z0, cons(z1, z2))))), DEL(min(x0, cons(z0, cons(z1, z2))), cons(x0, cons(z0, cons(z1, z2)))), MIN(x0, cons(z0, cons(z1, z2)))) MINSORT(cons(z0, cons(x1, nil))) -> c11(MINSORT(del(if(le(z0, x1), z0, min(x1, nil)), cons(z0, cons(x1, nil)))), DEL(min(z0, cons(x1, nil)), cons(z0, cons(x1, nil))), MIN(z0, cons(x1, nil))) MINSORT(cons(z0, cons(x1, cons(z1, z2)))) -> c11(MINSORT(del(if(le(z0, x1), if(le(z0, z1), min(z0, z2), min(z1, z2)), min(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2))))), DEL(min(z0, cons(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2)))), MIN(z0, cons(x1, cons(z1, z2)))) MINSORT(cons(0, cons(z0, x2))) -> c11(MINSORT(del(if(true, min(0, x2), min(z0, x2)), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2))) MINSORT(cons(s(z0), cons(0, x2))) -> c11(MINSORT(del(if(false, min(s(z0), x2), min(0, x2)), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), x2), min(s(z1), x2)), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2))) K tuples: MINSORT(cons(z0, z1)) -> c10(MIN(z0, z1)) MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1))) MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2))) Defined Rule Symbols: del_2, min_2, if_3, le_2, eq_2 Defined Pair Symbols: LE_2, EQ_2, MINSORT_1, MIN_2, DEL_2 Compound Symbols: c2_1, c6_1, c10_1, c13_1, c14_1, c15_1, c17_1, c18_1, c11_2, c11_3, c11_1, c_1 ---------------------------------------- (87) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MINSORT(cons(z0, nil)) -> c11(MINSORT(del(z0, cons(z0, nil))), DEL(min(z0, nil), cons(z0, nil))) by MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil))) ---------------------------------------- (88) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if(eq(z0, z1), z2, cons(z1, del(z0, z2))) del(z0, nil) -> nil min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if(le(z0, z1), min(z0, z2), min(z1, z2)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 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) Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) MINSORT(cons(z0, z1)) -> c10(MIN(z0, z1)) MIN(z0, cons(z1, z2)) -> c13(LE(z0, z1)) MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2)) MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2)) DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1)) DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2)) MINSORT(cons(x0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(x0, cons(z1, z2)), x0), cons(z1, z2), cons(x0, if(eq(min(x0, cons(z1, z2)), z1), z2, cons(z1, del(min(x0, cons(z1, z2)), z2)))))), DEL(min(x0, cons(z1, z2)), cons(x0, cons(z1, z2))), MIN(x0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1))) MINSORT(cons(x0, nil)) -> c11(MINSORT(if(eq(min(x0, nil), x0), nil, cons(x0, nil))), DEL(min(x0, nil), cons(x0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z1, cons(x1, x2))) -> c11(MINSORT(if(eq(if(le(z1, x1), min(z1, x2), min(x1, x2)), z1), cons(x1, x2), cons(z1, del(if(le(z1, x1), min(z1, x2), min(x1, x2)), cons(x1, x2))))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2))) MINSORT(cons(x0, cons(z0, nil))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, nil), z0), cons(x0, cons(z0, nil)))), DEL(min(x0, cons(z0, nil)), cons(x0, cons(z0, nil))), MIN(x0, cons(z0, nil))) MINSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, cons(z1, z2)), if(le(z0, z1), min(z0, z2), min(z1, z2))), cons(x0, cons(z0, cons(z1, z2))))), DEL(min(x0, cons(z0, cons(z1, z2))), cons(x0, cons(z0, cons(z1, z2)))), MIN(x0, cons(z0, cons(z1, z2)))) MINSORT(cons(z0, cons(x1, nil))) -> c11(MINSORT(del(if(le(z0, x1), z0, min(x1, nil)), cons(z0, cons(x1, nil)))), DEL(min(z0, cons(x1, nil)), cons(z0, cons(x1, nil))), MIN(z0, cons(x1, nil))) MINSORT(cons(z0, cons(x1, cons(z1, z2)))) -> c11(MINSORT(del(if(le(z0, x1), if(le(z0, z1), min(z0, z2), min(z1, z2)), min(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2))))), DEL(min(z0, cons(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2)))), MIN(z0, cons(x1, cons(z1, z2)))) MINSORT(cons(0, cons(z0, x2))) -> c11(MINSORT(del(if(true, min(0, x2), min(z0, x2)), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2))) MINSORT(cons(s(z0), cons(0, x2))) -> c11(MINSORT(del(if(false, min(s(z0), x2), min(0, x2)), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), x2), min(s(z1), x2)), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2))) MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil))) S tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) MIN(z0, cons(z1, z2)) -> c13(LE(z0, z1)) MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2)) MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2)) DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1)) DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2)) MINSORT(cons(x0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(x0, cons(z1, z2)), x0), cons(z1, z2), cons(x0, if(eq(min(x0, cons(z1, z2)), z1), z2, cons(z1, del(min(x0, cons(z1, z2)), z2)))))), DEL(min(x0, cons(z1, z2)), cons(x0, cons(z1, z2))), MIN(x0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(x0, nil)) -> c11(MINSORT(if(eq(min(x0, nil), x0), nil, cons(x0, nil))), DEL(min(x0, nil), cons(x0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z1, cons(x1, x2))) -> c11(MINSORT(if(eq(if(le(z1, x1), min(z1, x2), min(x1, x2)), z1), cons(x1, x2), cons(z1, del(if(le(z1, x1), min(z1, x2), min(x1, x2)), cons(x1, x2))))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2))) MINSORT(cons(x0, cons(z0, nil))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, nil), z0), cons(x0, cons(z0, nil)))), DEL(min(x0, cons(z0, nil)), cons(x0, cons(z0, nil))), MIN(x0, cons(z0, nil))) MINSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, cons(z1, z2)), if(le(z0, z1), min(z0, z2), min(z1, z2))), cons(x0, cons(z0, cons(z1, z2))))), DEL(min(x0, cons(z0, cons(z1, z2))), cons(x0, cons(z0, cons(z1, z2)))), MIN(x0, cons(z0, cons(z1, z2)))) MINSORT(cons(z0, cons(x1, nil))) -> c11(MINSORT(del(if(le(z0, x1), z0, min(x1, nil)), cons(z0, cons(x1, nil)))), DEL(min(z0, cons(x1, nil)), cons(z0, cons(x1, nil))), MIN(z0, cons(x1, nil))) MINSORT(cons(z0, cons(x1, cons(z1, z2)))) -> c11(MINSORT(del(if(le(z0, x1), if(le(z0, z1), min(z0, z2), min(z1, z2)), min(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2))))), DEL(min(z0, cons(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2)))), MIN(z0, cons(x1, cons(z1, z2)))) MINSORT(cons(0, cons(z0, x2))) -> c11(MINSORT(del(if(true, min(0, x2), min(z0, x2)), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2))) MINSORT(cons(s(z0), cons(0, x2))) -> c11(MINSORT(del(if(false, min(s(z0), x2), min(0, x2)), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), x2), min(s(z1), x2)), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil))) K tuples: MINSORT(cons(z0, z1)) -> c10(MIN(z0, z1)) MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1))) MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2))) Defined Rule Symbols: del_2, min_2, if_3, le_2, eq_2 Defined Pair Symbols: LE_2, EQ_2, MINSORT_1, MIN_2, DEL_2 Compound Symbols: c2_1, c6_1, c10_1, c13_1, c14_1, c15_1, c17_1, c18_1, c11_3, c11_1, c11_2, c_1 ---------------------------------------- (89) 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)) -> c11(DEL(min(x0, nil), cons(x0, nil))) We considered the (Usable) Rules:none And the Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) MINSORT(cons(z0, z1)) -> c10(MIN(z0, z1)) MIN(z0, cons(z1, z2)) -> c13(LE(z0, z1)) MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2)) MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2)) DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1)) DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2)) MINSORT(cons(x0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(x0, cons(z1, z2)), x0), cons(z1, z2), cons(x0, if(eq(min(x0, cons(z1, z2)), z1), z2, cons(z1, del(min(x0, cons(z1, z2)), z2)))))), DEL(min(x0, cons(z1, z2)), cons(x0, cons(z1, z2))), MIN(x0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1))) MINSORT(cons(x0, nil)) -> c11(MINSORT(if(eq(min(x0, nil), x0), nil, cons(x0, nil))), DEL(min(x0, nil), cons(x0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z1, cons(x1, x2))) -> c11(MINSORT(if(eq(if(le(z1, x1), min(z1, x2), min(x1, x2)), z1), cons(x1, x2), cons(z1, del(if(le(z1, x1), min(z1, x2), min(x1, x2)), cons(x1, x2))))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2))) MINSORT(cons(x0, cons(z0, nil))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, nil), z0), cons(x0, cons(z0, nil)))), DEL(min(x0, cons(z0, nil)), cons(x0, cons(z0, nil))), MIN(x0, cons(z0, nil))) MINSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, cons(z1, z2)), if(le(z0, z1), min(z0, z2), min(z1, z2))), cons(x0, cons(z0, cons(z1, z2))))), DEL(min(x0, cons(z0, cons(z1, z2))), cons(x0, cons(z0, cons(z1, z2)))), MIN(x0, cons(z0, cons(z1, z2)))) MINSORT(cons(z0, cons(x1, nil))) -> c11(MINSORT(del(if(le(z0, x1), z0, min(x1, nil)), cons(z0, cons(x1, nil)))), DEL(min(z0, cons(x1, nil)), cons(z0, cons(x1, nil))), MIN(z0, cons(x1, nil))) MINSORT(cons(z0, cons(x1, cons(z1, z2)))) -> c11(MINSORT(del(if(le(z0, x1), if(le(z0, z1), min(z0, z2), min(z1, z2)), min(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2))))), DEL(min(z0, cons(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2)))), MIN(z0, cons(x1, cons(z1, z2)))) MINSORT(cons(0, cons(z0, x2))) -> c11(MINSORT(del(if(true, min(0, x2), min(z0, x2)), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2))) MINSORT(cons(s(z0), cons(0, x2))) -> c11(MINSORT(del(if(false, min(s(z0), x2), min(0, x2)), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), x2), min(s(z1), x2)), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2))) MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(DEL(x_1, x_2)) = 0 POL(EQ(x_1, x_2)) = 0 POL(LE(x_1, x_2)) = 0 POL(MIN(x_1, x_2)) = 0 POL(MINSORT(x_1)) = [2] POL(c(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c11(x_1)) = x_1 POL(c11(x_1, x_2)) = x_1 + x_2 POL(c11(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c13(x_1)) = x_1 POL(c14(x_1)) = x_1 POL(c15(x_1)) = x_1 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(cons(x_1, x_2)) = [3] POL(del(x_1, x_2)) = [1] + [3]x_2 POL(eq(x_1, x_2)) = 0 POL(false) = 0 POL(if(x_1, x_2, x_3)) = [1] + x_3 POL(le(x_1, x_2)) = 0 POL(min(x_1, x_2)) = 0 POL(nil) = 0 POL(s(x_1)) = 0 POL(true) = 0 ---------------------------------------- (90) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if(eq(z0, z1), z2, cons(z1, del(z0, z2))) del(z0, nil) -> nil min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if(le(z0, z1), min(z0, z2), min(z1, z2)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 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) Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) MINSORT(cons(z0, z1)) -> c10(MIN(z0, z1)) MIN(z0, cons(z1, z2)) -> c13(LE(z0, z1)) MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2)) MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2)) DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1)) DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2)) MINSORT(cons(x0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(x0, cons(z1, z2)), x0), cons(z1, z2), cons(x0, if(eq(min(x0, cons(z1, z2)), z1), z2, cons(z1, del(min(x0, cons(z1, z2)), z2)))))), DEL(min(x0, cons(z1, z2)), cons(x0, cons(z1, z2))), MIN(x0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1))) MINSORT(cons(x0, nil)) -> c11(MINSORT(if(eq(min(x0, nil), x0), nil, cons(x0, nil))), DEL(min(x0, nil), cons(x0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z1, cons(x1, x2))) -> c11(MINSORT(if(eq(if(le(z1, x1), min(z1, x2), min(x1, x2)), z1), cons(x1, x2), cons(z1, del(if(le(z1, x1), min(z1, x2), min(x1, x2)), cons(x1, x2))))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2))) MINSORT(cons(x0, cons(z0, nil))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, nil), z0), cons(x0, cons(z0, nil)))), DEL(min(x0, cons(z0, nil)), cons(x0, cons(z0, nil))), MIN(x0, cons(z0, nil))) MINSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, cons(z1, z2)), if(le(z0, z1), min(z0, z2), min(z1, z2))), cons(x0, cons(z0, cons(z1, z2))))), DEL(min(x0, cons(z0, cons(z1, z2))), cons(x0, cons(z0, cons(z1, z2)))), MIN(x0, cons(z0, cons(z1, z2)))) MINSORT(cons(z0, cons(x1, nil))) -> c11(MINSORT(del(if(le(z0, x1), z0, min(x1, nil)), cons(z0, cons(x1, nil)))), DEL(min(z0, cons(x1, nil)), cons(z0, cons(x1, nil))), MIN(z0, cons(x1, nil))) MINSORT(cons(z0, cons(x1, cons(z1, z2)))) -> c11(MINSORT(del(if(le(z0, x1), if(le(z0, z1), min(z0, z2), min(z1, z2)), min(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2))))), DEL(min(z0, cons(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2)))), MIN(z0, cons(x1, cons(z1, z2)))) MINSORT(cons(0, cons(z0, x2))) -> c11(MINSORT(del(if(true, min(0, x2), min(z0, x2)), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2))) MINSORT(cons(s(z0), cons(0, x2))) -> c11(MINSORT(del(if(false, min(s(z0), x2), min(0, x2)), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), x2), min(s(z1), x2)), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2))) MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil))) S tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) MIN(z0, cons(z1, z2)) -> c13(LE(z0, z1)) MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2)) MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2)) DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1)) DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2)) MINSORT(cons(x0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(x0, cons(z1, z2)), x0), cons(z1, z2), cons(x0, if(eq(min(x0, cons(z1, z2)), z1), z2, cons(z1, del(min(x0, cons(z1, z2)), z2)))))), DEL(min(x0, cons(z1, z2)), cons(x0, cons(z1, z2))), MIN(x0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(x0, nil)) -> c11(MINSORT(if(eq(min(x0, nil), x0), nil, cons(x0, nil))), DEL(min(x0, nil), cons(x0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z1, cons(x1, x2))) -> c11(MINSORT(if(eq(if(le(z1, x1), min(z1, x2), min(x1, x2)), z1), cons(x1, x2), cons(z1, del(if(le(z1, x1), min(z1, x2), min(x1, x2)), cons(x1, x2))))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2))) MINSORT(cons(x0, cons(z0, nil))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, nil), z0), cons(x0, cons(z0, nil)))), DEL(min(x0, cons(z0, nil)), cons(x0, cons(z0, nil))), MIN(x0, cons(z0, nil))) MINSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, cons(z1, z2)), if(le(z0, z1), min(z0, z2), min(z1, z2))), cons(x0, cons(z0, cons(z1, z2))))), DEL(min(x0, cons(z0, cons(z1, z2))), cons(x0, cons(z0, cons(z1, z2)))), MIN(x0, cons(z0, cons(z1, z2)))) MINSORT(cons(z0, cons(x1, nil))) -> c11(MINSORT(del(if(le(z0, x1), z0, min(x1, nil)), cons(z0, cons(x1, nil)))), DEL(min(z0, cons(x1, nil)), cons(z0, cons(x1, nil))), MIN(z0, cons(x1, nil))) MINSORT(cons(z0, cons(x1, cons(z1, z2)))) -> c11(MINSORT(del(if(le(z0, x1), if(le(z0, z1), min(z0, z2), min(z1, z2)), min(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2))))), DEL(min(z0, cons(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2)))), MIN(z0, cons(x1, cons(z1, z2)))) MINSORT(cons(0, cons(z0, x2))) -> c11(MINSORT(del(if(true, min(0, x2), min(z0, x2)), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2))) MINSORT(cons(s(z0), cons(0, x2))) -> c11(MINSORT(del(if(false, min(s(z0), x2), min(0, x2)), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), x2), min(s(z1), x2)), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil))) K tuples: MINSORT(cons(z0, z1)) -> c10(MIN(z0, z1)) MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1))) MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2))) MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil))) Defined Rule Symbols: del_2, min_2, if_3, le_2, eq_2 Defined Pair Symbols: LE_2, EQ_2, MINSORT_1, MIN_2, DEL_2 Compound Symbols: c2_1, c6_1, c10_1, c13_1, c14_1, c15_1, c17_1, c18_1, c11_3, c11_1, c11_2, c_1 ---------------------------------------- (91) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace MINSORT(cons(x0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(x0, cons(z1, z2)), x0), cons(z1, z2), cons(x0, if(eq(min(x0, cons(z1, z2)), z1), z2, cons(z1, del(min(x0, cons(z1, z2)), z2)))))), DEL(min(x0, cons(z1, z2)), cons(x0, cons(z1, z2))), MIN(x0, cons(z1, z2))) by MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) ---------------------------------------- (92) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if(eq(z0, z1), z2, cons(z1, del(z0, z2))) del(z0, nil) -> nil min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if(le(z0, z1), min(z0, z2), min(z1, z2)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 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) Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) MINSORT(cons(z0, z1)) -> c10(MIN(z0, z1)) MIN(z0, cons(z1, z2)) -> c13(LE(z0, z1)) MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2)) MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2)) DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1)) DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2)) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1))) MINSORT(cons(x0, nil)) -> c11(MINSORT(if(eq(min(x0, nil), x0), nil, cons(x0, nil))), DEL(min(x0, nil), cons(x0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z1, cons(x1, x2))) -> c11(MINSORT(if(eq(if(le(z1, x1), min(z1, x2), min(x1, x2)), z1), cons(x1, x2), cons(z1, del(if(le(z1, x1), min(z1, x2), min(x1, x2)), cons(x1, x2))))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2))) MINSORT(cons(x0, cons(z0, nil))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, nil), z0), cons(x0, cons(z0, nil)))), DEL(min(x0, cons(z0, nil)), cons(x0, cons(z0, nil))), MIN(x0, cons(z0, nil))) MINSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, cons(z1, z2)), if(le(z0, z1), min(z0, z2), min(z1, z2))), cons(x0, cons(z0, cons(z1, z2))))), DEL(min(x0, cons(z0, cons(z1, z2))), cons(x0, cons(z0, cons(z1, z2)))), MIN(x0, cons(z0, cons(z1, z2)))) MINSORT(cons(z0, cons(x1, nil))) -> c11(MINSORT(del(if(le(z0, x1), z0, min(x1, nil)), cons(z0, cons(x1, nil)))), DEL(min(z0, cons(x1, nil)), cons(z0, cons(x1, nil))), MIN(z0, cons(x1, nil))) MINSORT(cons(z0, cons(x1, cons(z1, z2)))) -> c11(MINSORT(del(if(le(z0, x1), if(le(z0, z1), min(z0, z2), min(z1, z2)), min(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2))))), DEL(min(z0, cons(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2)))), MIN(z0, cons(x1, cons(z1, z2)))) MINSORT(cons(0, cons(z0, x2))) -> c11(MINSORT(del(if(true, min(0, x2), min(z0, x2)), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2))) MINSORT(cons(s(z0), cons(0, x2))) -> c11(MINSORT(del(if(false, min(s(z0), x2), min(0, x2)), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), x2), min(s(z1), x2)), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2))) MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) S tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) MIN(z0, cons(z1, z2)) -> c13(LE(z0, z1)) MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2)) MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2)) DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1)) DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2)) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(x0, nil)) -> c11(MINSORT(if(eq(min(x0, nil), x0), nil, cons(x0, nil))), DEL(min(x0, nil), cons(x0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z1, cons(x1, x2))) -> c11(MINSORT(if(eq(if(le(z1, x1), min(z1, x2), min(x1, x2)), z1), cons(x1, x2), cons(z1, del(if(le(z1, x1), min(z1, x2), min(x1, x2)), cons(x1, x2))))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2))) MINSORT(cons(x0, cons(z0, nil))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, nil), z0), cons(x0, cons(z0, nil)))), DEL(min(x0, cons(z0, nil)), cons(x0, cons(z0, nil))), MIN(x0, cons(z0, nil))) MINSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, cons(z1, z2)), if(le(z0, z1), min(z0, z2), min(z1, z2))), cons(x0, cons(z0, cons(z1, z2))))), DEL(min(x0, cons(z0, cons(z1, z2))), cons(x0, cons(z0, cons(z1, z2)))), MIN(x0, cons(z0, cons(z1, z2)))) MINSORT(cons(z0, cons(x1, nil))) -> c11(MINSORT(del(if(le(z0, x1), z0, min(x1, nil)), cons(z0, cons(x1, nil)))), DEL(min(z0, cons(x1, nil)), cons(z0, cons(x1, nil))), MIN(z0, cons(x1, nil))) MINSORT(cons(z0, cons(x1, cons(z1, z2)))) -> c11(MINSORT(del(if(le(z0, x1), if(le(z0, z1), min(z0, z2), min(z1, z2)), min(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2))))), DEL(min(z0, cons(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2)))), MIN(z0, cons(x1, cons(z1, z2)))) MINSORT(cons(0, cons(z0, x2))) -> c11(MINSORT(del(if(true, min(0, x2), min(z0, x2)), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2))) MINSORT(cons(s(z0), cons(0, x2))) -> c11(MINSORT(del(if(false, min(s(z0), x2), min(0, x2)), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), x2), min(s(z1), x2)), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) K tuples: MINSORT(cons(z0, z1)) -> c10(MIN(z0, z1)) MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1))) MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2))) MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil))) Defined Rule Symbols: del_2, min_2, if_3, le_2, eq_2 Defined Pair Symbols: LE_2, EQ_2, MINSORT_1, MIN_2, DEL_2 Compound Symbols: c2_1, c6_1, c10_1, c13_1, c14_1, c15_1, c17_1, c18_1, c11_3, c11_1, c11_2, c_1 ---------------------------------------- (93) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) by MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) ---------------------------------------- (94) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if(eq(z0, z1), z2, cons(z1, del(z0, z2))) del(z0, nil) -> nil min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if(le(z0, z1), min(z0, z2), min(z1, z2)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 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) Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) MINSORT(cons(z0, z1)) -> c10(MIN(z0, z1)) MIN(z0, cons(z1, z2)) -> c13(LE(z0, z1)) MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2)) MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2)) DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1)) DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2)) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1))) MINSORT(cons(x0, nil)) -> c11(MINSORT(if(eq(min(x0, nil), x0), nil, cons(x0, nil))), DEL(min(x0, nil), cons(x0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z1, cons(x1, x2))) -> c11(MINSORT(if(eq(if(le(z1, x1), min(z1, x2), min(x1, x2)), z1), cons(x1, x2), cons(z1, del(if(le(z1, x1), min(z1, x2), min(x1, x2)), cons(x1, x2))))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2))) MINSORT(cons(x0, cons(z0, nil))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, nil), z0), cons(x0, cons(z0, nil)))), DEL(min(x0, cons(z0, nil)), cons(x0, cons(z0, nil))), MIN(x0, cons(z0, nil))) MINSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, cons(z1, z2)), if(le(z0, z1), min(z0, z2), min(z1, z2))), cons(x0, cons(z0, cons(z1, z2))))), DEL(min(x0, cons(z0, cons(z1, z2))), cons(x0, cons(z0, cons(z1, z2)))), MIN(x0, cons(z0, cons(z1, z2)))) MINSORT(cons(z0, cons(x1, nil))) -> c11(MINSORT(del(if(le(z0, x1), z0, min(x1, nil)), cons(z0, cons(x1, nil)))), DEL(min(z0, cons(x1, nil)), cons(z0, cons(x1, nil))), MIN(z0, cons(x1, nil))) MINSORT(cons(z0, cons(x1, cons(z1, z2)))) -> c11(MINSORT(del(if(le(z0, x1), if(le(z0, z1), min(z0, z2), min(z1, z2)), min(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2))))), DEL(min(z0, cons(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2)))), MIN(z0, cons(x1, cons(z1, z2)))) MINSORT(cons(0, cons(z0, x2))) -> c11(MINSORT(del(if(true, min(0, x2), min(z0, x2)), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2))) MINSORT(cons(s(z0), cons(0, x2))) -> c11(MINSORT(del(if(false, min(s(z0), x2), min(0, x2)), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), x2), min(s(z1), x2)), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2))) MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) S tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) MIN(z0, cons(z1, z2)) -> c13(LE(z0, z1)) MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2)) MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2)) DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1)) DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2)) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(x0, nil)) -> c11(MINSORT(if(eq(min(x0, nil), x0), nil, cons(x0, nil))), DEL(min(x0, nil), cons(x0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z1, cons(x1, x2))) -> c11(MINSORT(if(eq(if(le(z1, x1), min(z1, x2), min(x1, x2)), z1), cons(x1, x2), cons(z1, del(if(le(z1, x1), min(z1, x2), min(x1, x2)), cons(x1, x2))))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2))) MINSORT(cons(x0, cons(z0, nil))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, nil), z0), cons(x0, cons(z0, nil)))), DEL(min(x0, cons(z0, nil)), cons(x0, cons(z0, nil))), MIN(x0, cons(z0, nil))) MINSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, cons(z1, z2)), if(le(z0, z1), min(z0, z2), min(z1, z2))), cons(x0, cons(z0, cons(z1, z2))))), DEL(min(x0, cons(z0, cons(z1, z2))), cons(x0, cons(z0, cons(z1, z2)))), MIN(x0, cons(z0, cons(z1, z2)))) MINSORT(cons(z0, cons(x1, nil))) -> c11(MINSORT(del(if(le(z0, x1), z0, min(x1, nil)), cons(z0, cons(x1, nil)))), DEL(min(z0, cons(x1, nil)), cons(z0, cons(x1, nil))), MIN(z0, cons(x1, nil))) MINSORT(cons(z0, cons(x1, cons(z1, z2)))) -> c11(MINSORT(del(if(le(z0, x1), if(le(z0, z1), min(z0, z2), min(z1, z2)), min(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2))))), DEL(min(z0, cons(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2)))), MIN(z0, cons(x1, cons(z1, z2)))) MINSORT(cons(0, cons(z0, x2))) -> c11(MINSORT(del(if(true, min(0, x2), min(z0, x2)), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2))) MINSORT(cons(s(z0), cons(0, x2))) -> c11(MINSORT(del(if(false, min(s(z0), x2), min(0, x2)), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), x2), min(s(z1), x2)), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) K tuples: MINSORT(cons(z0, z1)) -> c10(MIN(z0, z1)) MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1))) MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2))) MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil))) Defined Rule Symbols: del_2, min_2, if_3, le_2, eq_2 Defined Pair Symbols: LE_2, EQ_2, MINSORT_1, MIN_2, DEL_2 Compound Symbols: c2_1, c6_1, c10_1, c13_1, c14_1, c15_1, c17_1, c18_1, c11_3, c11_1, c11_2, c_1 ---------------------------------------- (95) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(min(z0, cons(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) by MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) ---------------------------------------- (96) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if(eq(z0, z1), z2, cons(z1, del(z0, z2))) del(z0, nil) -> nil min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if(le(z0, z1), min(z0, z2), min(z1, z2)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 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) Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) MINSORT(cons(z0, z1)) -> c10(MIN(z0, z1)) MIN(z0, cons(z1, z2)) -> c13(LE(z0, z1)) MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2)) MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2)) DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1)) DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2)) MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1))) MINSORT(cons(x0, nil)) -> c11(MINSORT(if(eq(min(x0, nil), x0), nil, cons(x0, nil))), DEL(min(x0, nil), cons(x0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z1, cons(x1, x2))) -> c11(MINSORT(if(eq(if(le(z1, x1), min(z1, x2), min(x1, x2)), z1), cons(x1, x2), cons(z1, del(if(le(z1, x1), min(z1, x2), min(x1, x2)), cons(x1, x2))))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2))) MINSORT(cons(x0, cons(z0, nil))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, nil), z0), cons(x0, cons(z0, nil)))), DEL(min(x0, cons(z0, nil)), cons(x0, cons(z0, nil))), MIN(x0, cons(z0, nil))) MINSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, cons(z1, z2)), if(le(z0, z1), min(z0, z2), min(z1, z2))), cons(x0, cons(z0, cons(z1, z2))))), DEL(min(x0, cons(z0, cons(z1, z2))), cons(x0, cons(z0, cons(z1, z2)))), MIN(x0, cons(z0, cons(z1, z2)))) MINSORT(cons(z0, cons(x1, nil))) -> c11(MINSORT(del(if(le(z0, x1), z0, min(x1, nil)), cons(z0, cons(x1, nil)))), DEL(min(z0, cons(x1, nil)), cons(z0, cons(x1, nil))), MIN(z0, cons(x1, nil))) MINSORT(cons(z0, cons(x1, cons(z1, z2)))) -> c11(MINSORT(del(if(le(z0, x1), if(le(z0, z1), min(z0, z2), min(z1, z2)), min(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2))))), DEL(min(z0, cons(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2)))), MIN(z0, cons(x1, cons(z1, z2)))) MINSORT(cons(0, cons(z0, x2))) -> c11(MINSORT(del(if(true, min(0, x2), min(z0, x2)), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2))) MINSORT(cons(s(z0), cons(0, x2))) -> c11(MINSORT(del(if(false, min(s(z0), x2), min(0, x2)), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), x2), min(s(z1), x2)), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2))) MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) S tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) MIN(z0, cons(z1, z2)) -> c13(LE(z0, z1)) MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2)) MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2)) DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1)) DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2)) MINSORT(cons(x0, nil)) -> c11(MINSORT(if(eq(min(x0, nil), x0), nil, cons(x0, nil))), DEL(min(x0, nil), cons(x0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z1, cons(x1, x2))) -> c11(MINSORT(if(eq(if(le(z1, x1), min(z1, x2), min(x1, x2)), z1), cons(x1, x2), cons(z1, del(if(le(z1, x1), min(z1, x2), min(x1, x2)), cons(x1, x2))))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2))) MINSORT(cons(x0, cons(z0, nil))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, nil), z0), cons(x0, cons(z0, nil)))), DEL(min(x0, cons(z0, nil)), cons(x0, cons(z0, nil))), MIN(x0, cons(z0, nil))) MINSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, cons(z1, z2)), if(le(z0, z1), min(z0, z2), min(z1, z2))), cons(x0, cons(z0, cons(z1, z2))))), DEL(min(x0, cons(z0, cons(z1, z2))), cons(x0, cons(z0, cons(z1, z2)))), MIN(x0, cons(z0, cons(z1, z2)))) MINSORT(cons(z0, cons(x1, nil))) -> c11(MINSORT(del(if(le(z0, x1), z0, min(x1, nil)), cons(z0, cons(x1, nil)))), DEL(min(z0, cons(x1, nil)), cons(z0, cons(x1, nil))), MIN(z0, cons(x1, nil))) MINSORT(cons(z0, cons(x1, cons(z1, z2)))) -> c11(MINSORT(del(if(le(z0, x1), if(le(z0, z1), min(z0, z2), min(z1, z2)), min(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2))))), DEL(min(z0, cons(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2)))), MIN(z0, cons(x1, cons(z1, z2)))) MINSORT(cons(0, cons(z0, x2))) -> c11(MINSORT(del(if(true, min(0, x2), min(z0, x2)), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2))) MINSORT(cons(s(z0), cons(0, x2))) -> c11(MINSORT(del(if(false, min(s(z0), x2), min(0, x2)), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), x2), min(s(z1), x2)), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) K tuples: MINSORT(cons(z0, z1)) -> c10(MIN(z0, z1)) MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1))) MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2))) MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil))) Defined Rule Symbols: del_2, min_2, if_3, le_2, eq_2 Defined Pair Symbols: LE_2, EQ_2, MINSORT_1, MIN_2, DEL_2 Compound Symbols: c2_1, c6_1, c10_1, c13_1, c14_1, c15_1, c17_1, c18_1, c11_1, c11_2, c11_3, c_1 ---------------------------------------- (97) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace MINSORT(cons(x0, nil)) -> c11(MINSORT(if(eq(min(x0, nil), x0), nil, cons(x0, nil))), DEL(min(x0, nil), cons(x0, nil))) by MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil))) ---------------------------------------- (98) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if(eq(z0, z1), z2, cons(z1, del(z0, z2))) del(z0, nil) -> nil min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if(le(z0, z1), min(z0, z2), min(z1, z2)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 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) Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) MINSORT(cons(z0, z1)) -> c10(MIN(z0, z1)) MIN(z0, cons(z1, z2)) -> c13(LE(z0, z1)) MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2)) MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2)) DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1)) DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2)) MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z1, cons(x1, x2))) -> c11(MINSORT(if(eq(if(le(z1, x1), min(z1, x2), min(x1, x2)), z1), cons(x1, x2), cons(z1, del(if(le(z1, x1), min(z1, x2), min(x1, x2)), cons(x1, x2))))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2))) MINSORT(cons(x0, cons(z0, nil))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, nil), z0), cons(x0, cons(z0, nil)))), DEL(min(x0, cons(z0, nil)), cons(x0, cons(z0, nil))), MIN(x0, cons(z0, nil))) MINSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, cons(z1, z2)), if(le(z0, z1), min(z0, z2), min(z1, z2))), cons(x0, cons(z0, cons(z1, z2))))), DEL(min(x0, cons(z0, cons(z1, z2))), cons(x0, cons(z0, cons(z1, z2)))), MIN(x0, cons(z0, cons(z1, z2)))) MINSORT(cons(z0, cons(x1, nil))) -> c11(MINSORT(del(if(le(z0, x1), z0, min(x1, nil)), cons(z0, cons(x1, nil)))), DEL(min(z0, cons(x1, nil)), cons(z0, cons(x1, nil))), MIN(z0, cons(x1, nil))) MINSORT(cons(z0, cons(x1, cons(z1, z2)))) -> c11(MINSORT(del(if(le(z0, x1), if(le(z0, z1), min(z0, z2), min(z1, z2)), min(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2))))), DEL(min(z0, cons(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2)))), MIN(z0, cons(x1, cons(z1, z2)))) MINSORT(cons(0, cons(z0, x2))) -> c11(MINSORT(del(if(true, min(0, x2), min(z0, x2)), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2))) MINSORT(cons(s(z0), cons(0, x2))) -> c11(MINSORT(del(if(false, min(s(z0), x2), min(0, x2)), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), x2), min(s(z1), x2)), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2))) MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil))) S tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) MIN(z0, cons(z1, z2)) -> c13(LE(z0, z1)) MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2)) MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2)) DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1)) DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2)) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z1, cons(x1, x2))) -> c11(MINSORT(if(eq(if(le(z1, x1), min(z1, x2), min(x1, x2)), z1), cons(x1, x2), cons(z1, del(if(le(z1, x1), min(z1, x2), min(x1, x2)), cons(x1, x2))))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2))) MINSORT(cons(x0, cons(z0, nil))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, nil), z0), cons(x0, cons(z0, nil)))), DEL(min(x0, cons(z0, nil)), cons(x0, cons(z0, nil))), MIN(x0, cons(z0, nil))) MINSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, cons(z1, z2)), if(le(z0, z1), min(z0, z2), min(z1, z2))), cons(x0, cons(z0, cons(z1, z2))))), DEL(min(x0, cons(z0, cons(z1, z2))), cons(x0, cons(z0, cons(z1, z2)))), MIN(x0, cons(z0, cons(z1, z2)))) MINSORT(cons(z0, cons(x1, nil))) -> c11(MINSORT(del(if(le(z0, x1), z0, min(x1, nil)), cons(z0, cons(x1, nil)))), DEL(min(z0, cons(x1, nil)), cons(z0, cons(x1, nil))), MIN(z0, cons(x1, nil))) MINSORT(cons(z0, cons(x1, cons(z1, z2)))) -> c11(MINSORT(del(if(le(z0, x1), if(le(z0, z1), min(z0, z2), min(z1, z2)), min(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2))))), DEL(min(z0, cons(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2)))), MIN(z0, cons(x1, cons(z1, z2)))) MINSORT(cons(0, cons(z0, x2))) -> c11(MINSORT(del(if(true, min(0, x2), min(z0, x2)), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2))) MINSORT(cons(s(z0), cons(0, x2))) -> c11(MINSORT(del(if(false, min(s(z0), x2), min(0, x2)), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), x2), min(s(z1), x2)), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil))) K tuples: MINSORT(cons(z0, z1)) -> c10(MIN(z0, z1)) MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1))) MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2))) MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil))) Defined Rule Symbols: del_2, min_2, if_3, le_2, eq_2 Defined Pair Symbols: LE_2, EQ_2, MINSORT_1, MIN_2, DEL_2 Compound Symbols: c2_1, c6_1, c10_1, c13_1, c14_1, c15_1, c17_1, c18_1, c11_1, c11_2, c11_3, c_1 ---------------------------------------- (99) 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))) ---------------------------------------- (100) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if(eq(z0, z1), z2, cons(z1, del(z0, z2))) del(z0, nil) -> nil min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if(le(z0, z1), min(z0, z2), min(z1, z2)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 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) Tuples: EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) MINSORT(cons(z0, z1)) -> c10(MIN(z0, z1)) MIN(z0, cons(z1, z2)) -> c13(LE(z0, z1)) MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2)) MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2)) DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1)) DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2)) MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z1, cons(x1, x2))) -> c11(MINSORT(if(eq(if(le(z1, x1), min(z1, x2), min(x1, x2)), z1), cons(x1, x2), cons(z1, del(if(le(z1, x1), min(z1, x2), min(x1, x2)), cons(x1, x2))))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2))) MINSORT(cons(x0, cons(z0, nil))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, nil), z0), cons(x0, cons(z0, nil)))), DEL(min(x0, cons(z0, nil)), cons(x0, cons(z0, nil))), MIN(x0, cons(z0, nil))) MINSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, cons(z1, z2)), if(le(z0, z1), min(z0, z2), min(z1, z2))), cons(x0, cons(z0, cons(z1, z2))))), DEL(min(x0, cons(z0, cons(z1, z2))), cons(x0, cons(z0, cons(z1, z2)))), MIN(x0, cons(z0, cons(z1, z2)))) MINSORT(cons(z0, cons(x1, nil))) -> c11(MINSORT(del(if(le(z0, x1), z0, min(x1, nil)), cons(z0, cons(x1, nil)))), DEL(min(z0, cons(x1, nil)), cons(z0, cons(x1, nil))), MIN(z0, cons(x1, nil))) MINSORT(cons(z0, cons(x1, cons(z1, z2)))) -> c11(MINSORT(del(if(le(z0, x1), if(le(z0, z1), min(z0, z2), min(z1, z2)), min(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2))))), DEL(min(z0, cons(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2)))), MIN(z0, cons(x1, cons(z1, z2)))) MINSORT(cons(0, cons(z0, x2))) -> c11(MINSORT(del(if(true, min(0, x2), min(z0, x2)), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2))) MINSORT(cons(s(z0), cons(0, x2))) -> c11(MINSORT(del(if(false, min(s(z0), x2), min(0, x2)), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), x2), min(s(z1), x2)), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2))) MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) S tuples: EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) MIN(z0, cons(z1, z2)) -> c13(LE(z0, z1)) MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2)) MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2)) DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1)) DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2)) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z1, cons(x1, x2))) -> c11(MINSORT(if(eq(if(le(z1, x1), min(z1, x2), min(x1, x2)), z1), cons(x1, x2), cons(z1, del(if(le(z1, x1), min(z1, x2), min(x1, x2)), cons(x1, x2))))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2))) MINSORT(cons(x0, cons(z0, nil))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, nil), z0), cons(x0, cons(z0, nil)))), DEL(min(x0, cons(z0, nil)), cons(x0, cons(z0, nil))), MIN(x0, cons(z0, nil))) MINSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, cons(z1, z2)), if(le(z0, z1), min(z0, z2), min(z1, z2))), cons(x0, cons(z0, cons(z1, z2))))), DEL(min(x0, cons(z0, cons(z1, z2))), cons(x0, cons(z0, cons(z1, z2)))), MIN(x0, cons(z0, cons(z1, z2)))) MINSORT(cons(z0, cons(x1, nil))) -> c11(MINSORT(del(if(le(z0, x1), z0, min(x1, nil)), cons(z0, cons(x1, nil)))), DEL(min(z0, cons(x1, nil)), cons(z0, cons(x1, nil))), MIN(z0, cons(x1, nil))) MINSORT(cons(z0, cons(x1, cons(z1, z2)))) -> c11(MINSORT(del(if(le(z0, x1), if(le(z0, z1), min(z0, z2), min(z1, z2)), min(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2))))), DEL(min(z0, cons(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2)))), MIN(z0, cons(x1, cons(z1, z2)))) MINSORT(cons(0, cons(z0, x2))) -> c11(MINSORT(del(if(true, min(0, x2), min(z0, x2)), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2))) MINSORT(cons(s(z0), cons(0, x2))) -> c11(MINSORT(del(if(false, min(s(z0), x2), min(0, x2)), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), x2), min(s(z1), x2)), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) K tuples: MINSORT(cons(z0, z1)) -> c10(MIN(z0, z1)) MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1))) MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2))) MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil))) Defined Rule Symbols: del_2, min_2, if_3, le_2, eq_2 Defined Pair Symbols: EQ_2, MINSORT_1, MIN_2, DEL_2, LE_2 Compound Symbols: c6_1, c10_1, c13_1, c14_1, c15_1, c17_1, c18_1, c11_1, c11_2, c11_3, c_1, c2_1 ---------------------------------------- (101) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil))) by MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil))) ---------------------------------------- (102) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if(eq(z0, z1), z2, cons(z1, del(z0, z2))) del(z0, nil) -> nil min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if(le(z0, z1), min(z0, z2), min(z1, z2)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 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) Tuples: EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) MINSORT(cons(z0, z1)) -> c10(MIN(z0, z1)) MIN(z0, cons(z1, z2)) -> c13(LE(z0, z1)) MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2)) MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2)) DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1)) DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2)) MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z1, cons(x1, x2))) -> c11(MINSORT(if(eq(if(le(z1, x1), min(z1, x2), min(x1, x2)), z1), cons(x1, x2), cons(z1, del(if(le(z1, x1), min(z1, x2), min(x1, x2)), cons(x1, x2))))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2))) MINSORT(cons(x0, cons(z0, nil))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, nil), z0), cons(x0, cons(z0, nil)))), DEL(min(x0, cons(z0, nil)), cons(x0, cons(z0, nil))), MIN(x0, cons(z0, nil))) MINSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, cons(z1, z2)), if(le(z0, z1), min(z0, z2), min(z1, z2))), cons(x0, cons(z0, cons(z1, z2))))), DEL(min(x0, cons(z0, cons(z1, z2))), cons(x0, cons(z0, cons(z1, z2)))), MIN(x0, cons(z0, cons(z1, z2)))) MINSORT(cons(z0, cons(x1, nil))) -> c11(MINSORT(del(if(le(z0, x1), z0, min(x1, nil)), cons(z0, cons(x1, nil)))), DEL(min(z0, cons(x1, nil)), cons(z0, cons(x1, nil))), MIN(z0, cons(x1, nil))) MINSORT(cons(z0, cons(x1, cons(z1, z2)))) -> c11(MINSORT(del(if(le(z0, x1), if(le(z0, z1), min(z0, z2), min(z1, z2)), min(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2))))), DEL(min(z0, cons(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2)))), MIN(z0, cons(x1, cons(z1, z2)))) MINSORT(cons(0, cons(z0, x2))) -> c11(MINSORT(del(if(true, min(0, x2), min(z0, x2)), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2))) MINSORT(cons(s(z0), cons(0, x2))) -> c11(MINSORT(del(if(false, min(s(z0), x2), min(0, x2)), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), x2), min(s(z1), x2)), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2))) MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil))) S tuples: EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) MIN(z0, cons(z1, z2)) -> c13(LE(z0, z1)) MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2)) MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2)) DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1)) DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2)) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z1, cons(x1, x2))) -> c11(MINSORT(if(eq(if(le(z1, x1), min(z1, x2), min(x1, x2)), z1), cons(x1, x2), cons(z1, del(if(le(z1, x1), min(z1, x2), min(x1, x2)), cons(x1, x2))))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2))) MINSORT(cons(x0, cons(z0, nil))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, nil), z0), cons(x0, cons(z0, nil)))), DEL(min(x0, cons(z0, nil)), cons(x0, cons(z0, nil))), MIN(x0, cons(z0, nil))) MINSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, cons(z1, z2)), if(le(z0, z1), min(z0, z2), min(z1, z2))), cons(x0, cons(z0, cons(z1, z2))))), DEL(min(x0, cons(z0, cons(z1, z2))), cons(x0, cons(z0, cons(z1, z2)))), MIN(x0, cons(z0, cons(z1, z2)))) MINSORT(cons(z0, cons(x1, nil))) -> c11(MINSORT(del(if(le(z0, x1), z0, min(x1, nil)), cons(z0, cons(x1, nil)))), DEL(min(z0, cons(x1, nil)), cons(z0, cons(x1, nil))), MIN(z0, cons(x1, nil))) MINSORT(cons(z0, cons(x1, cons(z1, z2)))) -> c11(MINSORT(del(if(le(z0, x1), if(le(z0, z1), min(z0, z2), min(z1, z2)), min(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2))))), DEL(min(z0, cons(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2)))), MIN(z0, cons(x1, cons(z1, z2)))) MINSORT(cons(0, cons(z0, x2))) -> c11(MINSORT(del(if(true, min(0, x2), min(z0, x2)), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2))) MINSORT(cons(s(z0), cons(0, x2))) -> c11(MINSORT(del(if(false, min(s(z0), x2), min(0, x2)), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), x2), min(s(z1), x2)), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil))) K tuples: MINSORT(cons(z0, z1)) -> c10(MIN(z0, z1)) MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1))) MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2))) MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil))) Defined Rule Symbols: del_2, min_2, if_3, le_2, eq_2 Defined Pair Symbols: EQ_2, MINSORT_1, MIN_2, DEL_2, LE_2 Compound Symbols: c6_1, c10_1, c13_1, c14_1, c15_1, c17_1, c18_1, c11_1, c11_2, c11_3, c_1, c2_1 ---------------------------------------- (103) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(min(z0, nil), cons(z0, nil))) by MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(z0, cons(z0, nil))) ---------------------------------------- (104) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if(eq(z0, z1), z2, cons(z1, del(z0, z2))) del(z0, nil) -> nil min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if(le(z0, z1), min(z0, z2), min(z1, z2)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 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) Tuples: EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) MINSORT(cons(z0, z1)) -> c10(MIN(z0, z1)) MIN(z0, cons(z1, z2)) -> c13(LE(z0, z1)) MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2)) MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2)) DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1)) DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2)) MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1))) MINSORT(cons(z1, cons(x1, x2))) -> c11(MINSORT(if(eq(if(le(z1, x1), min(z1, x2), min(x1, x2)), z1), cons(x1, x2), cons(z1, del(if(le(z1, x1), min(z1, x2), min(x1, x2)), cons(x1, x2))))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2))) MINSORT(cons(x0, cons(z0, nil))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, nil), z0), cons(x0, cons(z0, nil)))), DEL(min(x0, cons(z0, nil)), cons(x0, cons(z0, nil))), MIN(x0, cons(z0, nil))) MINSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, cons(z1, z2)), if(le(z0, z1), min(z0, z2), min(z1, z2))), cons(x0, cons(z0, cons(z1, z2))))), DEL(min(x0, cons(z0, cons(z1, z2))), cons(x0, cons(z0, cons(z1, z2)))), MIN(x0, cons(z0, cons(z1, z2)))) MINSORT(cons(z0, cons(x1, nil))) -> c11(MINSORT(del(if(le(z0, x1), z0, min(x1, nil)), cons(z0, cons(x1, nil)))), DEL(min(z0, cons(x1, nil)), cons(z0, cons(x1, nil))), MIN(z0, cons(x1, nil))) MINSORT(cons(z0, cons(x1, cons(z1, z2)))) -> c11(MINSORT(del(if(le(z0, x1), if(le(z0, z1), min(z0, z2), min(z1, z2)), min(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2))))), DEL(min(z0, cons(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2)))), MIN(z0, cons(x1, cons(z1, z2)))) MINSORT(cons(0, cons(z0, x2))) -> c11(MINSORT(del(if(true, min(0, x2), min(z0, x2)), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2))) MINSORT(cons(s(z0), cons(0, x2))) -> c11(MINSORT(del(if(false, min(s(z0), x2), min(0, x2)), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), x2), min(s(z1), x2)), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2))) MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(z0, cons(z0, nil))) S tuples: EQ(s(z0), s(z1)) -> c6(EQ(z0, z1)) MIN(z0, cons(z1, z2)) -> c13(LE(z0, z1)) MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2)) MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2)) DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1)) DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2)) MINSORT(cons(z1, cons(x1, x2))) -> c11(MINSORT(if(eq(if(le(z1, x1), min(z1, x2), min(x1, x2)), z1), cons(x1, x2), cons(z1, del(if(le(z1, x1), min(z1, x2), min(x1, x2)), cons(x1, x2))))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2))) MINSORT(cons(x0, cons(z0, nil))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, nil), z0), cons(x0, cons(z0, nil)))), DEL(min(x0, cons(z0, nil)), cons(x0, cons(z0, nil))), MIN(x0, cons(z0, nil))) MINSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, cons(z1, z2)), if(le(z0, z1), min(z0, z2), min(z1, z2))), cons(x0, cons(z0, cons(z1, z2))))), DEL(min(x0, cons(z0, cons(z1, z2))), cons(x0, cons(z0, cons(z1, z2)))), MIN(x0, cons(z0, cons(z1, z2)))) MINSORT(cons(z0, cons(x1, nil))) -> c11(MINSORT(del(if(le(z0, x1), z0, min(x1, nil)), cons(z0, cons(x1, nil)))), DEL(min(z0, cons(x1, nil)), cons(z0, cons(x1, nil))), MIN(z0, cons(x1, nil))) MINSORT(cons(z0, cons(x1, cons(z1, z2)))) -> c11(MINSORT(del(if(le(z0, x1), if(le(z0, z1), min(z0, z2), min(z1, z2)), min(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2))))), DEL(min(z0, cons(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2)))), MIN(z0, cons(x1, cons(z1, z2)))) MINSORT(cons(0, cons(z0, x2))) -> c11(MINSORT(del(if(true, min(0, x2), min(z0, x2)), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2))) MINSORT(cons(s(z0), cons(0, x2))) -> c11(MINSORT(del(if(false, min(s(z0), x2), min(0, x2)), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), x2), min(s(z1), x2)), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(z0, cons(z0, nil))) K tuples: MINSORT(cons(z0, z1)) -> c10(MIN(z0, z1)) MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1))) MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2))) MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil))) Defined Rule Symbols: del_2, min_2, if_3, le_2, eq_2 Defined Pair Symbols: EQ_2, MINSORT_1, MIN_2, DEL_2, LE_2 Compound Symbols: c6_1, c10_1, c13_1, c14_1, c15_1, c17_1, c18_1, c11_1, c11_3, c_1, c11_2, c2_1 ---------------------------------------- (105) 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))) ---------------------------------------- (106) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if(eq(z0, z1), z2, cons(z1, del(z0, z2))) del(z0, nil) -> nil min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if(le(z0, z1), min(z0, z2), min(z1, z2)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 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) Tuples: MINSORT(cons(z0, z1)) -> c10(MIN(z0, z1)) MIN(z0, cons(z1, z2)) -> c13(LE(z0, z1)) MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2)) MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2)) DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1)) DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2)) MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1))) MINSORT(cons(z1, cons(x1, x2))) -> c11(MINSORT(if(eq(if(le(z1, x1), min(z1, x2), min(x1, x2)), z1), cons(x1, x2), cons(z1, del(if(le(z1, x1), min(z1, x2), min(x1, x2)), cons(x1, x2))))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2))) MINSORT(cons(x0, cons(z0, nil))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, nil), z0), cons(x0, cons(z0, nil)))), DEL(min(x0, cons(z0, nil)), cons(x0, cons(z0, nil))), MIN(x0, cons(z0, nil))) MINSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, cons(z1, z2)), if(le(z0, z1), min(z0, z2), min(z1, z2))), cons(x0, cons(z0, cons(z1, z2))))), DEL(min(x0, cons(z0, cons(z1, z2))), cons(x0, cons(z0, cons(z1, z2)))), MIN(x0, cons(z0, cons(z1, z2)))) MINSORT(cons(z0, cons(x1, nil))) -> c11(MINSORT(del(if(le(z0, x1), z0, min(x1, nil)), cons(z0, cons(x1, nil)))), DEL(min(z0, cons(x1, nil)), cons(z0, cons(x1, nil))), MIN(z0, cons(x1, nil))) MINSORT(cons(z0, cons(x1, cons(z1, z2)))) -> c11(MINSORT(del(if(le(z0, x1), if(le(z0, z1), min(z0, z2), min(z1, z2)), min(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2))))), DEL(min(z0, cons(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2)))), MIN(z0, cons(x1, cons(z1, z2)))) MINSORT(cons(0, cons(z0, x2))) -> c11(MINSORT(del(if(true, min(0, x2), min(z0, x2)), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2))) MINSORT(cons(s(z0), cons(0, x2))) -> c11(MINSORT(del(if(false, min(s(z0), x2), min(0, x2)), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), x2), min(s(z1), x2)), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2))) MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(z0, cons(z0, nil))) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) S tuples: MIN(z0, cons(z1, z2)) -> c13(LE(z0, z1)) MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2)) MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2)) DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1)) DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2)) MINSORT(cons(z1, cons(x1, x2))) -> c11(MINSORT(if(eq(if(le(z1, x1), min(z1, x2), min(x1, x2)), z1), cons(x1, x2), cons(z1, del(if(le(z1, x1), min(z1, x2), min(x1, x2)), cons(x1, x2))))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2))) MINSORT(cons(x0, cons(z0, nil))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, nil), z0), cons(x0, cons(z0, nil)))), DEL(min(x0, cons(z0, nil)), cons(x0, cons(z0, nil))), MIN(x0, cons(z0, nil))) MINSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, cons(z1, z2)), if(le(z0, z1), min(z0, z2), min(z1, z2))), cons(x0, cons(z0, cons(z1, z2))))), DEL(min(x0, cons(z0, cons(z1, z2))), cons(x0, cons(z0, cons(z1, z2)))), MIN(x0, cons(z0, cons(z1, z2)))) MINSORT(cons(z0, cons(x1, nil))) -> c11(MINSORT(del(if(le(z0, x1), z0, min(x1, nil)), cons(z0, cons(x1, nil)))), DEL(min(z0, cons(x1, nil)), cons(z0, cons(x1, nil))), MIN(z0, cons(x1, nil))) MINSORT(cons(z0, cons(x1, cons(z1, z2)))) -> c11(MINSORT(del(if(le(z0, x1), if(le(z0, z1), min(z0, z2), min(z1, z2)), min(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2))))), DEL(min(z0, cons(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2)))), MIN(z0, cons(x1, cons(z1, z2)))) MINSORT(cons(0, cons(z0, x2))) -> c11(MINSORT(del(if(true, min(0, x2), min(z0, x2)), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2))) MINSORT(cons(s(z0), cons(0, x2))) -> c11(MINSORT(del(if(false, min(s(z0), x2), min(0, x2)), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), x2), min(s(z1), x2)), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(z0, cons(z0, nil))) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) K tuples: MINSORT(cons(z0, z1)) -> c10(MIN(z0, z1)) MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1))) MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2))) MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil))) Defined Rule Symbols: del_2, min_2, if_3, le_2, eq_2 Defined Pair Symbols: MINSORT_1, MIN_2, DEL_2, LE_2, EQ_2 Compound Symbols: c10_1, c13_1, c14_1, c15_1, c17_1, c18_1, c11_1, c11_3, c_1, c11_2, c2_1, c6_1 ---------------------------------------- (107) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace MINSORT(cons(z0, z1)) -> c10(MIN(z0, z1)) by MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2))) ---------------------------------------- (108) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if(eq(z0, z1), z2, cons(z1, del(z0, z2))) del(z0, nil) -> nil min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if(le(z0, z1), min(z0, z2), min(z1, z2)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 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) Tuples: MIN(z0, cons(z1, z2)) -> c13(LE(z0, z1)) MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2)) MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2)) DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1)) DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2)) MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1))) MINSORT(cons(z1, cons(x1, x2))) -> c11(MINSORT(if(eq(if(le(z1, x1), min(z1, x2), min(x1, x2)), z1), cons(x1, x2), cons(z1, del(if(le(z1, x1), min(z1, x2), min(x1, x2)), cons(x1, x2))))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2))) MINSORT(cons(x0, cons(z0, nil))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, nil), z0), cons(x0, cons(z0, nil)))), DEL(min(x0, cons(z0, nil)), cons(x0, cons(z0, nil))), MIN(x0, cons(z0, nil))) MINSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, cons(z1, z2)), if(le(z0, z1), min(z0, z2), min(z1, z2))), cons(x0, cons(z0, cons(z1, z2))))), DEL(min(x0, cons(z0, cons(z1, z2))), cons(x0, cons(z0, cons(z1, z2)))), MIN(x0, cons(z0, cons(z1, z2)))) MINSORT(cons(z0, cons(x1, nil))) -> c11(MINSORT(del(if(le(z0, x1), z0, min(x1, nil)), cons(z0, cons(x1, nil)))), DEL(min(z0, cons(x1, nil)), cons(z0, cons(x1, nil))), MIN(z0, cons(x1, nil))) MINSORT(cons(z0, cons(x1, cons(z1, z2)))) -> c11(MINSORT(del(if(le(z0, x1), if(le(z0, z1), min(z0, z2), min(z1, z2)), min(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2))))), DEL(min(z0, cons(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2)))), MIN(z0, cons(x1, cons(z1, z2)))) MINSORT(cons(0, cons(z0, x2))) -> c11(MINSORT(del(if(true, min(0, x2), min(z0, x2)), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2))) MINSORT(cons(s(z0), cons(0, x2))) -> c11(MINSORT(del(if(false, min(s(z0), x2), min(0, x2)), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), x2), min(s(z1), x2)), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2))) MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(z0, cons(z0, nil))) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2))) S tuples: MIN(z0, cons(z1, z2)) -> c13(LE(z0, z1)) MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2)) MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2)) DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1)) DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2)) MINSORT(cons(z1, cons(x1, x2))) -> c11(MINSORT(if(eq(if(le(z1, x1), min(z1, x2), min(x1, x2)), z1), cons(x1, x2), cons(z1, del(if(le(z1, x1), min(z1, x2), min(x1, x2)), cons(x1, x2))))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2))) MINSORT(cons(x0, cons(z0, nil))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, nil), z0), cons(x0, cons(z0, nil)))), DEL(min(x0, cons(z0, nil)), cons(x0, cons(z0, nil))), MIN(x0, cons(z0, nil))) MINSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, cons(z1, z2)), if(le(z0, z1), min(z0, z2), min(z1, z2))), cons(x0, cons(z0, cons(z1, z2))))), DEL(min(x0, cons(z0, cons(z1, z2))), cons(x0, cons(z0, cons(z1, z2)))), MIN(x0, cons(z0, cons(z1, z2)))) MINSORT(cons(z0, cons(x1, nil))) -> c11(MINSORT(del(if(le(z0, x1), z0, min(x1, nil)), cons(z0, cons(x1, nil)))), DEL(min(z0, cons(x1, nil)), cons(z0, cons(x1, nil))), MIN(z0, cons(x1, nil))) MINSORT(cons(z0, cons(x1, cons(z1, z2)))) -> c11(MINSORT(del(if(le(z0, x1), if(le(z0, z1), min(z0, z2), min(z1, z2)), min(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2))))), DEL(min(z0, cons(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2)))), MIN(z0, cons(x1, cons(z1, z2)))) MINSORT(cons(0, cons(z0, x2))) -> c11(MINSORT(del(if(true, min(0, x2), min(z0, x2)), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2))) MINSORT(cons(s(z0), cons(0, x2))) -> c11(MINSORT(del(if(false, min(s(z0), x2), min(0, x2)), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), x2), min(s(z1), x2)), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(z0, cons(z0, nil))) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) K tuples: MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1))) MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2))) MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil))) MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2))) Defined Rule Symbols: del_2, min_2, if_3, le_2, eq_2 Defined Pair Symbols: MIN_2, DEL_2, MINSORT_1, LE_2, EQ_2 Compound Symbols: c13_1, c14_1, c15_1, c17_1, c18_1, c11_1, c11_3, c_1, c11_2, c2_1, c6_1, c10_1 ---------------------------------------- (109) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace MINSORT(cons(z1, cons(x1, x2))) -> c11(MINSORT(if(eq(if(le(z1, x1), min(z1, x2), min(x1, x2)), z1), cons(x1, x2), cons(z1, del(if(le(z1, x1), min(z1, x2), min(x1, x2)), cons(x1, x2))))), DEL(min(z1, cons(x1, x2)), cons(z1, cons(x1, x2))), MIN(z1, cons(x1, x2))) by MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) ---------------------------------------- (110) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if(eq(z0, z1), z2, cons(z1, del(z0, z2))) del(z0, nil) -> nil min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if(le(z0, z1), min(z0, z2), min(z1, z2)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 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) Tuples: MIN(z0, cons(z1, z2)) -> c13(LE(z0, z1)) MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2)) MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2)) DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1)) DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2)) MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1))) MINSORT(cons(x0, cons(z0, nil))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, nil), z0), cons(x0, cons(z0, nil)))), DEL(min(x0, cons(z0, nil)), cons(x0, cons(z0, nil))), MIN(x0, cons(z0, nil))) MINSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, cons(z1, z2)), if(le(z0, z1), min(z0, z2), min(z1, z2))), cons(x0, cons(z0, cons(z1, z2))))), DEL(min(x0, cons(z0, cons(z1, z2))), cons(x0, cons(z0, cons(z1, z2)))), MIN(x0, cons(z0, cons(z1, z2)))) MINSORT(cons(z0, cons(x1, nil))) -> c11(MINSORT(del(if(le(z0, x1), z0, min(x1, nil)), cons(z0, cons(x1, nil)))), DEL(min(z0, cons(x1, nil)), cons(z0, cons(x1, nil))), MIN(z0, cons(x1, nil))) MINSORT(cons(z0, cons(x1, cons(z1, z2)))) -> c11(MINSORT(del(if(le(z0, x1), if(le(z0, z1), min(z0, z2), min(z1, z2)), min(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2))))), DEL(min(z0, cons(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2)))), MIN(z0, cons(x1, cons(z1, z2)))) MINSORT(cons(0, cons(z0, x2))) -> c11(MINSORT(del(if(true, min(0, x2), min(z0, x2)), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2))) MINSORT(cons(s(z0), cons(0, x2))) -> c11(MINSORT(del(if(false, min(s(z0), x2), min(0, x2)), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), x2), min(s(z1), x2)), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2))) MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(z0, cons(z0, nil))) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) S tuples: MIN(z0, cons(z1, z2)) -> c13(LE(z0, z1)) MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2)) MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2)) DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1)) DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2)) MINSORT(cons(x0, cons(z0, nil))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, nil), z0), cons(x0, cons(z0, nil)))), DEL(min(x0, cons(z0, nil)), cons(x0, cons(z0, nil))), MIN(x0, cons(z0, nil))) MINSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, cons(z1, z2)), if(le(z0, z1), min(z0, z2), min(z1, z2))), cons(x0, cons(z0, cons(z1, z2))))), DEL(min(x0, cons(z0, cons(z1, z2))), cons(x0, cons(z0, cons(z1, z2)))), MIN(x0, cons(z0, cons(z1, z2)))) MINSORT(cons(z0, cons(x1, nil))) -> c11(MINSORT(del(if(le(z0, x1), z0, min(x1, nil)), cons(z0, cons(x1, nil)))), DEL(min(z0, cons(x1, nil)), cons(z0, cons(x1, nil))), MIN(z0, cons(x1, nil))) MINSORT(cons(z0, cons(x1, cons(z1, z2)))) -> c11(MINSORT(del(if(le(z0, x1), if(le(z0, z1), min(z0, z2), min(z1, z2)), min(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2))))), DEL(min(z0, cons(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2)))), MIN(z0, cons(x1, cons(z1, z2)))) MINSORT(cons(0, cons(z0, x2))) -> c11(MINSORT(del(if(true, min(0, x2), min(z0, x2)), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2))) MINSORT(cons(s(z0), cons(0, x2))) -> c11(MINSORT(del(if(false, min(s(z0), x2), min(0, x2)), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), x2), min(s(z1), x2)), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(z0, cons(z0, nil))) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) K tuples: MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1))) MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2))) MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil))) MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2))) Defined Rule Symbols: del_2, min_2, if_3, le_2, eq_2 Defined Pair Symbols: MIN_2, DEL_2, MINSORT_1, LE_2, EQ_2 Compound Symbols: c13_1, c14_1, c15_1, c17_1, c18_1, c11_1, c11_3, c_1, c11_2, c2_1, c6_1, c10_1 ---------------------------------------- (111) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace MIN(z0, cons(z1, z2)) -> c13(LE(z0, z1)) by MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1)))) ---------------------------------------- (112) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if(eq(z0, z1), z2, cons(z1, del(z0, z2))) del(z0, nil) -> nil min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if(le(z0, z1), min(z0, z2), min(z1, z2)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 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) Tuples: MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2)) MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2)) DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1)) DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2)) MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1))) MINSORT(cons(x0, cons(z0, nil))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, nil), z0), cons(x0, cons(z0, nil)))), DEL(min(x0, cons(z0, nil)), cons(x0, cons(z0, nil))), MIN(x0, cons(z0, nil))) MINSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, cons(z1, z2)), if(le(z0, z1), min(z0, z2), min(z1, z2))), cons(x0, cons(z0, cons(z1, z2))))), DEL(min(x0, cons(z0, cons(z1, z2))), cons(x0, cons(z0, cons(z1, z2)))), MIN(x0, cons(z0, cons(z1, z2)))) MINSORT(cons(z0, cons(x1, nil))) -> c11(MINSORT(del(if(le(z0, x1), z0, min(x1, nil)), cons(z0, cons(x1, nil)))), DEL(min(z0, cons(x1, nil)), cons(z0, cons(x1, nil))), MIN(z0, cons(x1, nil))) MINSORT(cons(z0, cons(x1, cons(z1, z2)))) -> c11(MINSORT(del(if(le(z0, x1), if(le(z0, z1), min(z0, z2), min(z1, z2)), min(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2))))), DEL(min(z0, cons(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2)))), MIN(z0, cons(x1, cons(z1, z2)))) MINSORT(cons(0, cons(z0, x2))) -> c11(MINSORT(del(if(true, min(0, x2), min(z0, x2)), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2))) MINSORT(cons(s(z0), cons(0, x2))) -> c11(MINSORT(del(if(false, min(s(z0), x2), min(0, x2)), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), x2), min(s(z1), x2)), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2))) MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(z0, cons(z0, nil))) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1)))) S tuples: MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2)) MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2)) DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1)) DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2)) MINSORT(cons(x0, cons(z0, nil))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, nil), z0), cons(x0, cons(z0, nil)))), DEL(min(x0, cons(z0, nil)), cons(x0, cons(z0, nil))), MIN(x0, cons(z0, nil))) MINSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, cons(z1, z2)), if(le(z0, z1), min(z0, z2), min(z1, z2))), cons(x0, cons(z0, cons(z1, z2))))), DEL(min(x0, cons(z0, cons(z1, z2))), cons(x0, cons(z0, cons(z1, z2)))), MIN(x0, cons(z0, cons(z1, z2)))) MINSORT(cons(z0, cons(x1, nil))) -> c11(MINSORT(del(if(le(z0, x1), z0, min(x1, nil)), cons(z0, cons(x1, nil)))), DEL(min(z0, cons(x1, nil)), cons(z0, cons(x1, nil))), MIN(z0, cons(x1, nil))) MINSORT(cons(z0, cons(x1, cons(z1, z2)))) -> c11(MINSORT(del(if(le(z0, x1), if(le(z0, z1), min(z0, z2), min(z1, z2)), min(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2))))), DEL(min(z0, cons(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2)))), MIN(z0, cons(x1, cons(z1, z2)))) MINSORT(cons(0, cons(z0, x2))) -> c11(MINSORT(del(if(true, min(0, x2), min(z0, x2)), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2))) MINSORT(cons(s(z0), cons(0, x2))) -> c11(MINSORT(del(if(false, min(s(z0), x2), min(0, x2)), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), x2), min(s(z1), x2)), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(z0, cons(z0, nil))) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1)))) K tuples: MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1))) MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2))) MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil))) MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2))) Defined Rule Symbols: del_2, min_2, if_3, le_2, eq_2 Defined Pair Symbols: MIN_2, DEL_2, MINSORT_1, LE_2, EQ_2 Compound Symbols: c14_1, c15_1, c17_1, c18_1, c11_1, c11_3, c_1, c11_2, c2_1, c6_1, c10_1, c13_1 ---------------------------------------- (113) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace MINSORT(cons(x0, cons(z0, nil))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, nil), z0), cons(x0, cons(z0, nil)))), DEL(min(x0, cons(z0, nil)), cons(x0, cons(z0, nil))), MIN(x0, cons(z0, nil))) by MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) ---------------------------------------- (114) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if(eq(z0, z1), z2, cons(z1, del(z0, z2))) del(z0, nil) -> nil min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if(le(z0, z1), min(z0, z2), min(z1, z2)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 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) Tuples: MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2)) MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2)) DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1)) DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2)) MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1))) MINSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, cons(z1, z2)), if(le(z0, z1), min(z0, z2), min(z1, z2))), cons(x0, cons(z0, cons(z1, z2))))), DEL(min(x0, cons(z0, cons(z1, z2))), cons(x0, cons(z0, cons(z1, z2)))), MIN(x0, cons(z0, cons(z1, z2)))) MINSORT(cons(z0, cons(x1, nil))) -> c11(MINSORT(del(if(le(z0, x1), z0, min(x1, nil)), cons(z0, cons(x1, nil)))), DEL(min(z0, cons(x1, nil)), cons(z0, cons(x1, nil))), MIN(z0, cons(x1, nil))) MINSORT(cons(z0, cons(x1, cons(z1, z2)))) -> c11(MINSORT(del(if(le(z0, x1), if(le(z0, z1), min(z0, z2), min(z1, z2)), min(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2))))), DEL(min(z0, cons(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2)))), MIN(z0, cons(x1, cons(z1, z2)))) MINSORT(cons(0, cons(z0, x2))) -> c11(MINSORT(del(if(true, min(0, x2), min(z0, x2)), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2))) MINSORT(cons(s(z0), cons(0, x2))) -> c11(MINSORT(del(if(false, min(s(z0), x2), min(0, x2)), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), x2), min(s(z1), x2)), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2))) MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(z0, cons(z0, nil))) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1)))) MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) S tuples: MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2)) MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2)) DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1)) DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2)) MINSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, cons(z1, z2)), if(le(z0, z1), min(z0, z2), min(z1, z2))), cons(x0, cons(z0, cons(z1, z2))))), DEL(min(x0, cons(z0, cons(z1, z2))), cons(x0, cons(z0, cons(z1, z2)))), MIN(x0, cons(z0, cons(z1, z2)))) MINSORT(cons(z0, cons(x1, nil))) -> c11(MINSORT(del(if(le(z0, x1), z0, min(x1, nil)), cons(z0, cons(x1, nil)))), DEL(min(z0, cons(x1, nil)), cons(z0, cons(x1, nil))), MIN(z0, cons(x1, nil))) MINSORT(cons(z0, cons(x1, cons(z1, z2)))) -> c11(MINSORT(del(if(le(z0, x1), if(le(z0, z1), min(z0, z2), min(z1, z2)), min(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2))))), DEL(min(z0, cons(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2)))), MIN(z0, cons(x1, cons(z1, z2)))) MINSORT(cons(0, cons(z0, x2))) -> c11(MINSORT(del(if(true, min(0, x2), min(z0, x2)), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2))) MINSORT(cons(s(z0), cons(0, x2))) -> c11(MINSORT(del(if(false, min(s(z0), x2), min(0, x2)), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), x2), min(s(z1), x2)), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(z0, cons(z0, nil))) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1)))) MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) K tuples: MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1))) MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2))) MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil))) MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2))) Defined Rule Symbols: del_2, min_2, if_3, le_2, eq_2 Defined Pair Symbols: MIN_2, DEL_2, MINSORT_1, LE_2, EQ_2 Compound Symbols: c14_1, c15_1, c17_1, c18_1, c11_1, c11_3, c_1, c11_2, c2_1, c6_1, c10_1, c13_1 ---------------------------------------- (115) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace MINSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c11(MINSORT(del(if(le(x0, z0), min(x0, cons(z1, z2)), if(le(z0, z1), min(z0, z2), min(z1, z2))), cons(x0, cons(z0, cons(z1, z2))))), DEL(min(x0, cons(z0, cons(z1, z2))), cons(x0, cons(z0, cons(z1, z2)))), MIN(x0, cons(z0, cons(z1, z2)))) by MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) ---------------------------------------- (116) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if(eq(z0, z1), z2, cons(z1, del(z0, z2))) del(z0, nil) -> nil min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if(le(z0, z1), min(z0, z2), min(z1, z2)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 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) Tuples: MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2)) MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2)) DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1)) DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2)) MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1))) MINSORT(cons(z0, cons(x1, nil))) -> c11(MINSORT(del(if(le(z0, x1), z0, min(x1, nil)), cons(z0, cons(x1, nil)))), DEL(min(z0, cons(x1, nil)), cons(z0, cons(x1, nil))), MIN(z0, cons(x1, nil))) MINSORT(cons(z0, cons(x1, cons(z1, z2)))) -> c11(MINSORT(del(if(le(z0, x1), if(le(z0, z1), min(z0, z2), min(z1, z2)), min(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2))))), DEL(min(z0, cons(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2)))), MIN(z0, cons(x1, cons(z1, z2)))) MINSORT(cons(0, cons(z0, x2))) -> c11(MINSORT(del(if(true, min(0, x2), min(z0, x2)), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2))) MINSORT(cons(s(z0), cons(0, x2))) -> c11(MINSORT(del(if(false, min(s(z0), x2), min(0, x2)), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), x2), min(s(z1), x2)), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2))) MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(z0, cons(z0, nil))) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1)))) MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) S tuples: MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2)) MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2)) DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1)) DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2)) MINSORT(cons(z0, cons(x1, nil))) -> c11(MINSORT(del(if(le(z0, x1), z0, min(x1, nil)), cons(z0, cons(x1, nil)))), DEL(min(z0, cons(x1, nil)), cons(z0, cons(x1, nil))), MIN(z0, cons(x1, nil))) MINSORT(cons(z0, cons(x1, cons(z1, z2)))) -> c11(MINSORT(del(if(le(z0, x1), if(le(z0, z1), min(z0, z2), min(z1, z2)), min(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2))))), DEL(min(z0, cons(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2)))), MIN(z0, cons(x1, cons(z1, z2)))) MINSORT(cons(0, cons(z0, x2))) -> c11(MINSORT(del(if(true, min(0, x2), min(z0, x2)), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2))) MINSORT(cons(s(z0), cons(0, x2))) -> c11(MINSORT(del(if(false, min(s(z0), x2), min(0, x2)), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), x2), min(s(z1), x2)), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(z0, cons(z0, nil))) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1)))) MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) K tuples: MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1))) MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2))) MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil))) MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2))) Defined Rule Symbols: del_2, min_2, if_3, le_2, eq_2 Defined Pair Symbols: MIN_2, DEL_2, MINSORT_1, LE_2, EQ_2 Compound Symbols: c14_1, c15_1, c17_1, c18_1, c11_1, c11_3, c_1, c11_2, c2_1, c6_1, c10_1, c13_1 ---------------------------------------- (117) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace MINSORT(cons(z0, cons(x1, nil))) -> c11(MINSORT(del(if(le(z0, x1), z0, min(x1, nil)), cons(z0, cons(x1, nil)))), DEL(min(z0, cons(x1, nil)), cons(z0, cons(x1, nil))), MIN(z0, cons(x1, nil))) by MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) ---------------------------------------- (118) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if(eq(z0, z1), z2, cons(z1, del(z0, z2))) del(z0, nil) -> nil min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if(le(z0, z1), min(z0, z2), min(z1, z2)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 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) Tuples: MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2)) MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2)) DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1)) DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2)) MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1))) MINSORT(cons(z0, cons(x1, cons(z1, z2)))) -> c11(MINSORT(del(if(le(z0, x1), if(le(z0, z1), min(z0, z2), min(z1, z2)), min(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2))))), DEL(min(z0, cons(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2)))), MIN(z0, cons(x1, cons(z1, z2)))) MINSORT(cons(0, cons(z0, x2))) -> c11(MINSORT(del(if(true, min(0, x2), min(z0, x2)), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2))) MINSORT(cons(s(z0), cons(0, x2))) -> c11(MINSORT(del(if(false, min(s(z0), x2), min(0, x2)), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), x2), min(s(z1), x2)), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2))) MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(z0, cons(z0, nil))) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1)))) MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) S tuples: MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2)) MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2)) DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1)) DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2)) MINSORT(cons(z0, cons(x1, cons(z1, z2)))) -> c11(MINSORT(del(if(le(z0, x1), if(le(z0, z1), min(z0, z2), min(z1, z2)), min(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2))))), DEL(min(z0, cons(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2)))), MIN(z0, cons(x1, cons(z1, z2)))) MINSORT(cons(0, cons(z0, x2))) -> c11(MINSORT(del(if(true, min(0, x2), min(z0, x2)), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2))) MINSORT(cons(s(z0), cons(0, x2))) -> c11(MINSORT(del(if(false, min(s(z0), x2), min(0, x2)), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), x2), min(s(z1), x2)), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(z0, cons(z0, nil))) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1)))) MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) K tuples: MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1))) MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2))) MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil))) MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2))) Defined Rule Symbols: del_2, min_2, if_3, le_2, eq_2 Defined Pair Symbols: MIN_2, DEL_2, MINSORT_1, LE_2, EQ_2 Compound Symbols: c14_1, c15_1, c17_1, c18_1, c11_1, c11_3, c_1, c11_2, c2_1, c6_1, c10_1, c13_1 ---------------------------------------- (119) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace MINSORT(cons(z0, cons(x1, cons(z1, z2)))) -> c11(MINSORT(del(if(le(z0, x1), if(le(z0, z1), min(z0, z2), min(z1, z2)), min(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2))))), DEL(min(z0, cons(x1, cons(z1, z2))), cons(z0, cons(x1, cons(z1, z2)))), MIN(z0, cons(x1, cons(z1, z2)))) by MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), if(le(z0, z2), min(z0, z3), min(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) ---------------------------------------- (120) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if(eq(z0, z1), z2, cons(z1, del(z0, z2))) del(z0, nil) -> nil min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if(le(z0, z1), min(z0, z2), min(z1, z2)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 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) Tuples: MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2)) MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2)) DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1)) DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2)) MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1))) MINSORT(cons(0, cons(z0, x2))) -> c11(MINSORT(del(if(true, min(0, x2), min(z0, x2)), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2))) MINSORT(cons(s(z0), cons(0, x2))) -> c11(MINSORT(del(if(false, min(s(z0), x2), min(0, x2)), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), x2), min(s(z1), x2)), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2))) MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(z0, cons(z0, nil))) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1)))) MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), if(le(z0, z2), min(z0, z3), min(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) S tuples: MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2)) MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2)) DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1)) DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2)) MINSORT(cons(0, cons(z0, x2))) -> c11(MINSORT(del(if(true, min(0, x2), min(z0, x2)), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2))) MINSORT(cons(s(z0), cons(0, x2))) -> c11(MINSORT(del(if(false, min(s(z0), x2), min(0, x2)), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), x2), min(s(z1), x2)), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(z0, cons(z0, nil))) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1)))) MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), if(le(z0, z2), min(z0, z3), min(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) K tuples: MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1))) MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2))) MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil))) MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2))) Defined Rule Symbols: del_2, min_2, if_3, le_2, eq_2 Defined Pair Symbols: MIN_2, DEL_2, MINSORT_1, LE_2, EQ_2 Compound Symbols: c14_1, c15_1, c17_1, c18_1, c11_1, c11_3, c_1, c11_2, c2_1, c6_1, c10_1, c13_1 ---------------------------------------- (121) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace MINSORT(cons(0, cons(z0, x2))) -> c11(MINSORT(del(if(true, min(0, x2), min(z0, x2)), cons(0, cons(z0, x2)))), DEL(min(0, cons(z0, x2)), cons(0, cons(z0, x2))), MIN(0, cons(z0, x2))) by MINSORT(cons(0, cons(z0, z1))) -> c11(MINSORT(del(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1)))), DEL(if(le(0, z0), min(0, z1), min(z0, z1)), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) ---------------------------------------- (122) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if(eq(z0, z1), z2, cons(z1, del(z0, z2))) del(z0, nil) -> nil min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if(le(z0, z1), min(z0, z2), min(z1, z2)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 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) Tuples: MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2)) MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2)) DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1)) DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2)) MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1))) MINSORT(cons(s(z0), cons(0, x2))) -> c11(MINSORT(del(if(false, min(s(z0), x2), min(0, x2)), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), x2), min(s(z1), x2)), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2))) MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(z0, cons(z0, nil))) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1)))) MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), if(le(z0, z2), min(z0, z3), min(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) MINSORT(cons(0, cons(z0, z1))) -> c11(MINSORT(del(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1)))), DEL(if(le(0, z0), min(0, z1), min(z0, z1)), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) S tuples: MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2)) MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2)) DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1)) DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2)) MINSORT(cons(s(z0), cons(0, x2))) -> c11(MINSORT(del(if(false, min(s(z0), x2), min(0, x2)), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), x2), min(s(z1), x2)), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(z0, cons(z0, nil))) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1)))) MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), if(le(z0, z2), min(z0, z3), min(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) MINSORT(cons(0, cons(z0, z1))) -> c11(MINSORT(del(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1)))), DEL(if(le(0, z0), min(0, z1), min(z0, z1)), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) K tuples: MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1))) MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2))) MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil))) MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2))) Defined Rule Symbols: del_2, min_2, if_3, le_2, eq_2 Defined Pair Symbols: MIN_2, DEL_2, MINSORT_1, LE_2, EQ_2 Compound Symbols: c14_1, c15_1, c17_1, c18_1, c11_1, c11_3, c_1, c11_2, c2_1, c6_1, c10_1, c13_1 ---------------------------------------- (123) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace MINSORT(cons(s(z0), cons(0, x2))) -> c11(MINSORT(del(if(false, min(s(z0), x2), min(0, x2)), cons(s(z0), cons(0, x2)))), DEL(min(s(z0), cons(0, x2)), cons(s(z0), cons(0, x2))), MIN(s(z0), cons(0, x2))) by MINSORT(cons(s(z0), cons(0, z1))) -> c11(MINSORT(del(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1)))), DEL(if(le(s(z0), 0), min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) ---------------------------------------- (124) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if(eq(z0, z1), z2, cons(z1, del(z0, z2))) del(z0, nil) -> nil min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if(le(z0, z1), min(z0, z2), min(z1, z2)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 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) Tuples: MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2)) MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2)) DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1)) DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2)) MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1))) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), x2), min(s(z1), x2)), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2))) MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(z0, cons(z0, nil))) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1)))) MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), if(le(z0, z2), min(z0, z3), min(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) MINSORT(cons(0, cons(z0, z1))) -> c11(MINSORT(del(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1)))), DEL(if(le(0, z0), min(0, z1), min(z0, z1)), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) MINSORT(cons(s(z0), cons(0, z1))) -> c11(MINSORT(del(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1)))), DEL(if(le(s(z0), 0), min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) S tuples: MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2)) MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2)) DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1)) DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2)) MINSORT(cons(s(z0), cons(s(z1), x2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), x2), min(s(z1), x2)), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(z0, cons(z0, nil))) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1)))) MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), if(le(z0, z2), min(z0, z3), min(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) MINSORT(cons(0, cons(z0, z1))) -> c11(MINSORT(del(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1)))), DEL(if(le(0, z0), min(0, z1), min(z0, z1)), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) MINSORT(cons(s(z0), cons(0, z1))) -> c11(MINSORT(del(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1)))), DEL(if(le(s(z0), 0), min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) K tuples: MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1))) MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2))) MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil))) MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2))) Defined Rule Symbols: del_2, min_2, if_3, le_2, eq_2 Defined Pair Symbols: MIN_2, DEL_2, MINSORT_1, LE_2, EQ_2 Compound Symbols: c14_1, c15_1, c17_1, c18_1, c11_1, c11_3, c_1, c11_2, c2_1, c6_1, c10_1, c13_1 ---------------------------------------- (125) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace MINSORT(cons(s(z0), cons(s(z1), x2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), x2), min(s(z1), x2)), cons(s(z0), cons(s(z1), x2)))), DEL(min(s(z0), cons(s(z1), x2)), cons(s(z0), cons(s(z1), x2))), MIN(s(z0), cons(s(z1), x2))) by MINSORT(cons(s(z0), cons(s(z1), z2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2)))), DEL(if(le(s(z0), s(z1)), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) ---------------------------------------- (126) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if(eq(z0, z1), z2, cons(z1, del(z0, z2))) del(z0, nil) -> nil min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if(le(z0, z1), min(z0, z2), min(z1, z2)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 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) Tuples: MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2)) MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2)) DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1)) DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2)) MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1))) MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(z0, cons(z0, nil))) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1)))) MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), if(le(z0, z2), min(z0, z3), min(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) MINSORT(cons(0, cons(z0, z1))) -> c11(MINSORT(del(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1)))), DEL(if(le(0, z0), min(0, z1), min(z0, z1)), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) MINSORT(cons(s(z0), cons(0, z1))) -> c11(MINSORT(del(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1)))), DEL(if(le(s(z0), 0), min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2)))), DEL(if(le(s(z0), s(z1)), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) S tuples: MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2)) MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2)) DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1)) DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2)) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(z0, cons(z0, nil))) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1)))) MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), if(le(z0, z2), min(z0, z3), min(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) MINSORT(cons(0, cons(z0, z1))) -> c11(MINSORT(del(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1)))), DEL(if(le(0, z0), min(0, z1), min(z0, z1)), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) MINSORT(cons(s(z0), cons(0, z1))) -> c11(MINSORT(del(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1)))), DEL(if(le(s(z0), 0), min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2)))), DEL(if(le(s(z0), s(z1)), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) K tuples: MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1))) MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2))) MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil))) MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2))) Defined Rule Symbols: del_2, min_2, if_3, le_2, eq_2 Defined Pair Symbols: MIN_2, DEL_2, MINSORT_1, LE_2, EQ_2 Compound Symbols: c14_1, c15_1, c17_1, c18_1, c11_1, c_1, c11_2, c11_3, c2_1, c6_1, c10_1, c13_1 ---------------------------------------- (127) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) by MINSORT(cons(z0, cons(z1, z2))) -> c(DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2)))) ---------------------------------------- (128) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if(eq(z0, z1), z2, cons(z1, del(z0, z2))) del(z0, nil) -> nil min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if(le(z0, z1), min(z0, z2), min(z1, z2)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 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) Tuples: MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2)) MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2)) DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1)) DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2)) MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1))) MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(z0, cons(z0, nil))) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1)))) MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), if(le(z0, z2), min(z0, z3), min(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) MINSORT(cons(0, cons(z0, z1))) -> c11(MINSORT(del(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1)))), DEL(if(le(0, z0), min(0, z1), min(z0, z1)), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) MINSORT(cons(s(z0), cons(0, z1))) -> c11(MINSORT(del(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1)))), DEL(if(le(s(z0), 0), min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2)))), DEL(if(le(s(z0), s(z1)), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) MINSORT(cons(z0, cons(z1, z2))) -> c(DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2)))) S tuples: MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2)) MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2)) DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1)) DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2)) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(z0, cons(z0, nil))) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1)))) MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), if(le(z0, z2), min(z0, z3), min(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) MINSORT(cons(0, cons(z0, z1))) -> c11(MINSORT(del(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1)))), DEL(if(le(0, z0), min(0, z1), min(z0, z1)), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) MINSORT(cons(s(z0), cons(0, z1))) -> c11(MINSORT(del(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1)))), DEL(if(le(s(z0), 0), min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2)))), DEL(if(le(s(z0), s(z1)), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) K tuples: MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1))) MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2))) MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil))) MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2))) Defined Rule Symbols: del_2, min_2, if_3, le_2, eq_2 Defined Pair Symbols: MIN_2, DEL_2, MINSORT_1, LE_2, EQ_2 Compound Symbols: c14_1, c15_1, c17_1, c18_1, c11_1, c_1, c11_2, c11_3, c2_1, c6_1, c10_1, c13_1 ---------------------------------------- (129) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(min(z0, nil), cons(z0, nil))) by MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil))) ---------------------------------------- (130) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if(eq(z0, z1), z2, cons(z1, del(z0, z2))) del(z0, nil) -> nil min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if(le(z0, z1), min(z0, z2), min(z1, z2)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 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) Tuples: MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2)) MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2)) DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1)) DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2)) MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1))) MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2))) MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(z0, cons(z0, nil))) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1)))) MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), if(le(z0, z2), min(z0, z3), min(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) MINSORT(cons(0, cons(z0, z1))) -> c11(MINSORT(del(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1)))), DEL(if(le(0, z0), min(0, z1), min(z0, z1)), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) MINSORT(cons(s(z0), cons(0, z1))) -> c11(MINSORT(del(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1)))), DEL(if(le(s(z0), 0), min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2)))), DEL(if(le(s(z0), s(z1)), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) MINSORT(cons(z0, cons(z1, z2))) -> c(DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil))) S tuples: MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2)) MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2)) DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1)) DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2)) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(z0, cons(z0, nil))) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1)))) MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), if(le(z0, z2), min(z0, z3), min(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) MINSORT(cons(0, cons(z0, z1))) -> c11(MINSORT(del(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1)))), DEL(if(le(0, z0), min(0, z1), min(z0, z1)), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) MINSORT(cons(s(z0), cons(0, z1))) -> c11(MINSORT(del(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1)))), DEL(if(le(s(z0), 0), min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2)))), DEL(if(le(s(z0), s(z1)), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil))) K tuples: MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1))) MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2))) MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil))) MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2))) Defined Rule Symbols: del_2, min_2, if_3, le_2, eq_2 Defined Pair Symbols: MIN_2, DEL_2, MINSORT_1, LE_2, EQ_2 Compound Symbols: c14_1, c15_1, c17_1, c18_1, c11_1, c_1, c11_3, c11_2, c2_1, c6_1, c10_1, c13_1 ---------------------------------------- (131) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil))) by MINSORT(cons(z0, nil)) -> c11(DEL(z0, cons(z0, nil))) ---------------------------------------- (132) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if(eq(z0, z1), z2, cons(z1, del(z0, z2))) del(z0, nil) -> nil min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if(le(z0, z1), min(z0, z2), min(z1, z2)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 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) Tuples: MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2)) MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2)) DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1)) DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2)) MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1))) MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(z0, cons(z0, nil))) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1)))) MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), if(le(z0, z2), min(z0, z3), min(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) MINSORT(cons(0, cons(z0, z1))) -> c11(MINSORT(del(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1)))), DEL(if(le(0, z0), min(0, z1), min(z0, z1)), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) MINSORT(cons(s(z0), cons(0, z1))) -> c11(MINSORT(del(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1)))), DEL(if(le(s(z0), 0), min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2)))), DEL(if(le(s(z0), s(z1)), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) MINSORT(cons(z0, cons(z1, z2))) -> c(DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c11(DEL(z0, cons(z0, nil))) S tuples: MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2)) MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2)) DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1)) DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2)) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(z0, cons(z0, nil))) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1)))) MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), if(le(z0, z2), min(z0, z3), min(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) MINSORT(cons(0, cons(z0, z1))) -> c11(MINSORT(del(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1)))), DEL(if(le(0, z0), min(0, z1), min(z0, z1)), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) MINSORT(cons(s(z0), cons(0, z1))) -> c11(MINSORT(del(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1)))), DEL(if(le(s(z0), 0), min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2)))), DEL(if(le(s(z0), s(z1)), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil))) K tuples: MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1))) MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2))) MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil))) MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2))) Defined Rule Symbols: del_2, min_2, if_3, le_2, eq_2 Defined Pair Symbols: MIN_2, DEL_2, MINSORT_1, LE_2, EQ_2 Compound Symbols: c14_1, c15_1, c17_1, c18_1, c11_1, c_1, c11_3, c11_2, c2_1, c6_1, c10_1, c13_1 ---------------------------------------- (133) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace MIN(z0, cons(z1, z2)) -> c14(MIN(z0, z2)) by MIN(z0, cons(z1, cons(y1, y2))) -> c14(MIN(z0, cons(y1, y2))) MIN(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c14(MIN(s(s(y0)), cons(s(s(y1)), y2))) ---------------------------------------- (134) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if(eq(z0, z1), z2, cons(z1, del(z0, z2))) del(z0, nil) -> nil min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if(le(z0, z1), min(z0, z2), min(z1, z2)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 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) Tuples: MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2)) DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1)) DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2)) MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1))) MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(z0, cons(z0, nil))) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1)))) MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), if(le(z0, z2), min(z0, z3), min(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) MINSORT(cons(0, cons(z0, z1))) -> c11(MINSORT(del(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1)))), DEL(if(le(0, z0), min(0, z1), min(z0, z1)), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) MINSORT(cons(s(z0), cons(0, z1))) -> c11(MINSORT(del(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1)))), DEL(if(le(s(z0), 0), min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2)))), DEL(if(le(s(z0), s(z1)), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) MINSORT(cons(z0, cons(z1, z2))) -> c(DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c11(DEL(z0, cons(z0, nil))) MIN(z0, cons(z1, cons(y1, y2))) -> c14(MIN(z0, cons(y1, y2))) MIN(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c14(MIN(s(s(y0)), cons(s(s(y1)), y2))) S tuples: MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2)) DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1)) DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2)) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(z0, cons(z0, nil))) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1)))) MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), if(le(z0, z2), min(z0, z3), min(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) MINSORT(cons(0, cons(z0, z1))) -> c11(MINSORT(del(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1)))), DEL(if(le(0, z0), min(0, z1), min(z0, z1)), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) MINSORT(cons(s(z0), cons(0, z1))) -> c11(MINSORT(del(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1)))), DEL(if(le(s(z0), 0), min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2)))), DEL(if(le(s(z0), s(z1)), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil))) MIN(z0, cons(z1, cons(y1, y2))) -> c14(MIN(z0, cons(y1, y2))) MIN(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c14(MIN(s(s(y0)), cons(s(s(y1)), y2))) K tuples: MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1))) MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2))) MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil))) MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2))) Defined Rule Symbols: del_2, min_2, if_3, le_2, eq_2 Defined Pair Symbols: MIN_2, DEL_2, MINSORT_1, LE_2, EQ_2 Compound Symbols: c15_1, c17_1, c18_1, c11_1, c_1, c11_3, c11_2, c2_1, c6_1, c10_1, c13_1, c14_1 ---------------------------------------- (135) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) by MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) ---------------------------------------- (136) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if(eq(z0, z1), z2, cons(z1, del(z0, z2))) del(z0, nil) -> nil min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if(le(z0, z1), min(z0, z2), min(z1, z2)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 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) Tuples: MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2)) DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1)) DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2)) MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1))) MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(z0, cons(z0, nil))) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1)))) MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), if(le(z0, z2), min(z0, z3), min(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) MINSORT(cons(0, cons(z0, z1))) -> c11(MINSORT(del(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1)))), DEL(if(le(0, z0), min(0, z1), min(z0, z1)), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) MINSORT(cons(s(z0), cons(0, z1))) -> c11(MINSORT(del(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1)))), DEL(if(le(s(z0), 0), min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2)))), DEL(if(le(s(z0), s(z1)), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) MINSORT(cons(z0, cons(z1, z2))) -> c(DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c11(DEL(z0, cons(z0, nil))) MIN(z0, cons(z1, cons(y1, y2))) -> c14(MIN(z0, cons(y1, y2))) MIN(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c14(MIN(s(s(y0)), cons(s(s(y1)), y2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) S tuples: MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2)) DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1)) DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2)) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(z0, cons(z0, nil))) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1)))) MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), if(le(z0, z2), min(z0, z3), min(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) MINSORT(cons(0, cons(z0, z1))) -> c11(MINSORT(del(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1)))), DEL(if(le(0, z0), min(0, z1), min(z0, z1)), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) MINSORT(cons(s(z0), cons(0, z1))) -> c11(MINSORT(del(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1)))), DEL(if(le(s(z0), 0), min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2)))), DEL(if(le(s(z0), s(z1)), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil))) MIN(z0, cons(z1, cons(y1, y2))) -> c14(MIN(z0, cons(y1, y2))) MIN(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c14(MIN(s(s(y0)), cons(s(s(y1)), y2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) K tuples: MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1))) MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2))) MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil))) MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2))) Defined Rule Symbols: del_2, min_2, if_3, le_2, eq_2 Defined Pair Symbols: MIN_2, DEL_2, MINSORT_1, LE_2, EQ_2 Compound Symbols: c15_1, c17_1, c18_1, c11_1, c_1, c11_3, c11_2, c2_1, c6_1, c10_1, c13_1, c14_1 ---------------------------------------- (137) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) by MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) ---------------------------------------- (138) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if(eq(z0, z1), z2, cons(z1, del(z0, z2))) del(z0, nil) -> nil min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if(le(z0, z1), min(z0, z2), min(z1, z2)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 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) Tuples: MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2)) DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1)) DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2)) MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1))) MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(z0, cons(z0, nil))) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1)))) MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), if(le(z0, z2), min(z0, z3), min(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) MINSORT(cons(0, cons(z0, z1))) -> c11(MINSORT(del(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1)))), DEL(if(le(0, z0), min(0, z1), min(z0, z1)), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) MINSORT(cons(s(z0), cons(0, z1))) -> c11(MINSORT(del(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1)))), DEL(if(le(s(z0), 0), min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2)))), DEL(if(le(s(z0), s(z1)), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) MINSORT(cons(z0, cons(z1, z2))) -> c(DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c11(DEL(z0, cons(z0, nil))) MIN(z0, cons(z1, cons(y1, y2))) -> c14(MIN(z0, cons(y1, y2))) MIN(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c14(MIN(s(s(y0)), cons(s(s(y1)), y2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) S tuples: MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2)) DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1)) DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2)) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(z0, cons(z0, nil))) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1)))) MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), if(le(z0, z2), min(z0, z3), min(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) MINSORT(cons(0, cons(z0, z1))) -> c11(MINSORT(del(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1)))), DEL(if(le(0, z0), min(0, z1), min(z0, z1)), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) MINSORT(cons(s(z0), cons(0, z1))) -> c11(MINSORT(del(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1)))), DEL(if(le(s(z0), 0), min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2)))), DEL(if(le(s(z0), s(z1)), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil))) MIN(z0, cons(z1, cons(y1, y2))) -> c14(MIN(z0, cons(y1, y2))) MIN(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c14(MIN(s(s(y0)), cons(s(s(y1)), y2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) K tuples: MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1))) MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2))) MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil))) MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2))) Defined Rule Symbols: del_2, min_2, if_3, le_2, eq_2 Defined Pair Symbols: MIN_2, DEL_2, MINSORT_1, LE_2, EQ_2 Compound Symbols: c15_1, c17_1, c18_1, c11_1, c_1, c11_3, c11_2, c2_1, c6_1, c10_1, c13_1, c14_1 ---------------------------------------- (139) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(min(z0, cons(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) by MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) ---------------------------------------- (140) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if(eq(z0, z1), z2, cons(z1, del(z0, z2))) del(z0, nil) -> nil min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if(le(z0, z1), min(z0, z2), min(z1, z2)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 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) Tuples: MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2)) DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1)) DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2)) MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1))) MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(z0, cons(z0, nil))) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1)))) MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), if(le(z0, z2), min(z0, z3), min(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) MINSORT(cons(0, cons(z0, z1))) -> c11(MINSORT(del(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1)))), DEL(if(le(0, z0), min(0, z1), min(z0, z1)), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) MINSORT(cons(s(z0), cons(0, z1))) -> c11(MINSORT(del(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1)))), DEL(if(le(s(z0), 0), min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2)))), DEL(if(le(s(z0), s(z1)), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) MINSORT(cons(z0, cons(z1, z2))) -> c(DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c11(DEL(z0, cons(z0, nil))) MIN(z0, cons(z1, cons(y1, y2))) -> c14(MIN(z0, cons(y1, y2))) MIN(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c14(MIN(s(s(y0)), cons(s(s(y1)), y2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) S tuples: MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2)) DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1)) DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2)) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(z0, cons(z0, nil))) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1)))) MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), if(le(z0, z2), min(z0, z3), min(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) MINSORT(cons(0, cons(z0, z1))) -> c11(MINSORT(del(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1)))), DEL(if(le(0, z0), min(0, z1), min(z0, z1)), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) MINSORT(cons(s(z0), cons(0, z1))) -> c11(MINSORT(del(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1)))), DEL(if(le(s(z0), 0), min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2)))), DEL(if(le(s(z0), s(z1)), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil))) MIN(z0, cons(z1, cons(y1, y2))) -> c14(MIN(z0, cons(y1, y2))) MIN(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c14(MIN(s(s(y0)), cons(s(s(y1)), y2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) K tuples: MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1))) MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2))) MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil))) MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2))) Defined Rule Symbols: del_2, min_2, if_3, le_2, eq_2 Defined Pair Symbols: MIN_2, DEL_2, MINSORT_1, LE_2, EQ_2 Compound Symbols: c15_1, c17_1, c18_1, c11_1, c_1, c11_2, c2_1, c6_1, c10_1, c11_3, c13_1, c14_1 ---------------------------------------- (141) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil))) by MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil))) ---------------------------------------- (142) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if(eq(z0, z1), z2, cons(z1, del(z0, z2))) del(z0, nil) -> nil min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if(le(z0, z1), min(z0, z2), min(z1, z2)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 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) Tuples: MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2)) DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1)) DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2)) MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1))) MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(z0, cons(z0, nil))) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1)))) MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), if(le(z0, z2), min(z0, z3), min(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) MINSORT(cons(0, cons(z0, z1))) -> c11(MINSORT(del(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1)))), DEL(if(le(0, z0), min(0, z1), min(z0, z1)), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) MINSORT(cons(s(z0), cons(0, z1))) -> c11(MINSORT(del(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1)))), DEL(if(le(s(z0), 0), min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2)))), DEL(if(le(s(z0), s(z1)), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) MINSORT(cons(z0, cons(z1, z2))) -> c(DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c11(DEL(z0, cons(z0, nil))) MIN(z0, cons(z1, cons(y1, y2))) -> c14(MIN(z0, cons(y1, y2))) MIN(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c14(MIN(s(s(y0)), cons(s(s(y1)), y2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil))) S tuples: MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2)) DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1)) DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2)) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(z0, cons(z0, nil))) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1)))) MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), if(le(z0, z2), min(z0, z3), min(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) MINSORT(cons(0, cons(z0, z1))) -> c11(MINSORT(del(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1)))), DEL(if(le(0, z0), min(0, z1), min(z0, z1)), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) MINSORT(cons(s(z0), cons(0, z1))) -> c11(MINSORT(del(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1)))), DEL(if(le(s(z0), 0), min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2)))), DEL(if(le(s(z0), s(z1)), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil))) MIN(z0, cons(z1, cons(y1, y2))) -> c14(MIN(z0, cons(y1, y2))) MIN(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c14(MIN(s(s(y0)), cons(s(s(y1)), y2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil))) K tuples: MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1))) MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2))) MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil))) MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2))) Defined Rule Symbols: del_2, min_2, if_3, le_2, eq_2 Defined Pair Symbols: MIN_2, DEL_2, MINSORT_1, LE_2, EQ_2 Compound Symbols: c15_1, c17_1, c18_1, c11_1, c_1, c2_1, c11_2, c6_1, c10_1, c11_3, c13_1, c14_1 ---------------------------------------- (143) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil))) by MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil))) ---------------------------------------- (144) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if(eq(z0, z1), z2, cons(z1, del(z0, z2))) del(z0, nil) -> nil min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if(le(z0, z1), min(z0, z2), min(z1, z2)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 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) Tuples: MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2)) DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1)) DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2)) MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1))) MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(z0, cons(z0, nil))) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1)))) MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), if(le(z0, z2), min(z0, z3), min(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) MINSORT(cons(0, cons(z0, z1))) -> c11(MINSORT(del(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1)))), DEL(if(le(0, z0), min(0, z1), min(z0, z1)), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) MINSORT(cons(s(z0), cons(0, z1))) -> c11(MINSORT(del(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1)))), DEL(if(le(s(z0), 0), min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2)))), DEL(if(le(s(z0), s(z1)), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) MINSORT(cons(z0, cons(z1, z2))) -> c(DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c11(DEL(z0, cons(z0, nil))) MIN(z0, cons(z1, cons(y1, y2))) -> c14(MIN(z0, cons(y1, y2))) MIN(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c14(MIN(s(s(y0)), cons(s(s(y1)), y2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil))) S tuples: MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2)) DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1)) DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2)) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(z0, cons(z0, nil))) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1)))) MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), if(le(z0, z2), min(z0, z3), min(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) MINSORT(cons(0, cons(z0, z1))) -> c11(MINSORT(del(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1)))), DEL(if(le(0, z0), min(0, z1), min(z0, z1)), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) MINSORT(cons(s(z0), cons(0, z1))) -> c11(MINSORT(del(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1)))), DEL(if(le(s(z0), 0), min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2)))), DEL(if(le(s(z0), s(z1)), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil))) MIN(z0, cons(z1, cons(y1, y2))) -> c14(MIN(z0, cons(y1, y2))) MIN(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c14(MIN(s(s(y0)), cons(s(s(y1)), y2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil))) K tuples: MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1))) MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2))) MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil))) MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2))) Defined Rule Symbols: del_2, min_2, if_3, le_2, eq_2 Defined Pair Symbols: MIN_2, DEL_2, MINSORT_1, LE_2, EQ_2 Compound Symbols: c15_1, c17_1, c18_1, c11_1, c_1, c2_1, c11_2, c6_1, c10_1, c11_3, c13_1, c14_1 ---------------------------------------- (145) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(min(z0, nil), nil)))), DEL(z0, cons(z0, nil))) by MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil))) ---------------------------------------- (146) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if(eq(z0, z1), z2, cons(z1, del(z0, z2))) del(z0, nil) -> nil min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if(le(z0, z1), min(z0, z2), min(z1, z2)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 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) Tuples: MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2)) DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1)) DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2)) MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1))) MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1)))) MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), if(le(z0, z2), min(z0, z3), min(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) MINSORT(cons(0, cons(z0, z1))) -> c11(MINSORT(del(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1)))), DEL(if(le(0, z0), min(0, z1), min(z0, z1)), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) MINSORT(cons(s(z0), cons(0, z1))) -> c11(MINSORT(del(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1)))), DEL(if(le(s(z0), 0), min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2)))), DEL(if(le(s(z0), s(z1)), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) MINSORT(cons(z0, cons(z1, z2))) -> c(DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c11(DEL(z0, cons(z0, nil))) MIN(z0, cons(z1, cons(y1, y2))) -> c14(MIN(z0, cons(y1, y2))) MIN(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c14(MIN(s(s(y0)), cons(s(s(y1)), y2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil))) S tuples: MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2)) DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1)) DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2)) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1)))) MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), if(le(z0, z2), min(z0, z3), min(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) MINSORT(cons(0, cons(z0, z1))) -> c11(MINSORT(del(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1)))), DEL(if(le(0, z0), min(0, z1), min(z0, z1)), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) MINSORT(cons(s(z0), cons(0, z1))) -> c11(MINSORT(del(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1)))), DEL(if(le(s(z0), 0), min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2)))), DEL(if(le(s(z0), s(z1)), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil))) MIN(z0, cons(z1, cons(y1, y2))) -> c14(MIN(z0, cons(y1, y2))) MIN(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c14(MIN(s(s(y0)), cons(s(s(y1)), y2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil))) K tuples: MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1))) MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2))) MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil))) MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2))) Defined Rule Symbols: del_2, min_2, if_3, le_2, eq_2 Defined Pair Symbols: MIN_2, DEL_2, MINSORT_1, LE_2, EQ_2 Compound Symbols: c15_1, c17_1, c18_1, c11_1, c_1, c2_1, c6_1, c10_1, c11_3, c13_1, c11_2, c14_1 ---------------------------------------- (147) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace MIN(z0, cons(z1, z2)) -> c15(MIN(z1, z2)) by MIN(z0, cons(z1, cons(y1, y2))) -> c15(MIN(z1, cons(y1, y2))) MIN(z0, cons(s(s(y0)), cons(s(s(y1)), y2))) -> c15(MIN(s(s(y0)), cons(s(s(y1)), y2))) MIN(z0, cons(z1, cons(y1, cons(y2, y3)))) -> c15(MIN(z1, cons(y1, cons(y2, y3)))) MIN(z0, cons(s(s(y0)), cons(y1, cons(s(s(y2)), y3)))) -> c15(MIN(s(s(y0)), cons(y1, cons(s(s(y2)), y3)))) ---------------------------------------- (148) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if(eq(z0, z1), z2, cons(z1, del(z0, z2))) del(z0, nil) -> nil min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if(le(z0, z1), min(z0, z2), min(z1, z2)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 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) Tuples: DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1)) DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2)) MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1))) MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1)))) MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), if(le(z0, z2), min(z0, z3), min(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) MINSORT(cons(0, cons(z0, z1))) -> c11(MINSORT(del(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1)))), DEL(if(le(0, z0), min(0, z1), min(z0, z1)), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) MINSORT(cons(s(z0), cons(0, z1))) -> c11(MINSORT(del(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1)))), DEL(if(le(s(z0), 0), min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2)))), DEL(if(le(s(z0), s(z1)), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) MINSORT(cons(z0, cons(z1, z2))) -> c(DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c11(DEL(z0, cons(z0, nil))) MIN(z0, cons(z1, cons(y1, y2))) -> c14(MIN(z0, cons(y1, y2))) MIN(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c14(MIN(s(s(y0)), cons(s(s(y1)), y2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil))) MIN(z0, cons(z1, cons(y1, y2))) -> c15(MIN(z1, cons(y1, y2))) MIN(z0, cons(s(s(y0)), cons(s(s(y1)), y2))) -> c15(MIN(s(s(y0)), cons(s(s(y1)), y2))) MIN(z0, cons(z1, cons(y1, cons(y2, y3)))) -> c15(MIN(z1, cons(y1, cons(y2, y3)))) MIN(z0, cons(s(s(y0)), cons(y1, cons(s(s(y2)), y3)))) -> c15(MIN(s(s(y0)), cons(y1, cons(s(s(y2)), y3)))) S tuples: DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1)) DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2)) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1)))) MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), if(le(z0, z2), min(z0, z3), min(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) MINSORT(cons(0, cons(z0, z1))) -> c11(MINSORT(del(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1)))), DEL(if(le(0, z0), min(0, z1), min(z0, z1)), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) MINSORT(cons(s(z0), cons(0, z1))) -> c11(MINSORT(del(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1)))), DEL(if(le(s(z0), 0), min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2)))), DEL(if(le(s(z0), s(z1)), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil))) MIN(z0, cons(z1, cons(y1, y2))) -> c14(MIN(z0, cons(y1, y2))) MIN(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c14(MIN(s(s(y0)), cons(s(s(y1)), y2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil))) MIN(z0, cons(z1, cons(y1, y2))) -> c15(MIN(z1, cons(y1, y2))) MIN(z0, cons(s(s(y0)), cons(s(s(y1)), y2))) -> c15(MIN(s(s(y0)), cons(s(s(y1)), y2))) MIN(z0, cons(z1, cons(y1, cons(y2, y3)))) -> c15(MIN(z1, cons(y1, cons(y2, y3)))) MIN(z0, cons(s(s(y0)), cons(y1, cons(s(s(y2)), y3)))) -> c15(MIN(s(s(y0)), cons(y1, cons(s(s(y2)), y3)))) K tuples: MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1))) MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2))) MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil))) MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2))) Defined Rule Symbols: del_2, min_2, if_3, le_2, eq_2 Defined Pair Symbols: DEL_2, MINSORT_1, LE_2, EQ_2, MIN_2 Compound Symbols: c17_1, c18_1, c11_1, c_1, c2_1, c6_1, c10_1, c11_3, c13_1, c11_2, c14_1, c15_1 ---------------------------------------- (149) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace DEL(z0, cons(z1, z2)) -> c17(EQ(z0, z1)) by DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c17(EQ(s(s(y0)), s(s(y1)))) ---------------------------------------- (150) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if(eq(z0, z1), z2, cons(z1, del(z0, z2))) del(z0, nil) -> nil min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if(le(z0, z1), min(z0, z2), min(z1, z2)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 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) Tuples: DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2)) MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1))) MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1)))) MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), if(le(z0, z2), min(z0, z3), min(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) MINSORT(cons(0, cons(z0, z1))) -> c11(MINSORT(del(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1)))), DEL(if(le(0, z0), min(0, z1), min(z0, z1)), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) MINSORT(cons(s(z0), cons(0, z1))) -> c11(MINSORT(del(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1)))), DEL(if(le(s(z0), 0), min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2)))), DEL(if(le(s(z0), s(z1)), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) MINSORT(cons(z0, cons(z1, z2))) -> c(DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c11(DEL(z0, cons(z0, nil))) MIN(z0, cons(z1, cons(y1, y2))) -> c14(MIN(z0, cons(y1, y2))) MIN(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c14(MIN(s(s(y0)), cons(s(s(y1)), y2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil))) MIN(z0, cons(z1, cons(y1, y2))) -> c15(MIN(z1, cons(y1, y2))) MIN(z0, cons(s(s(y0)), cons(s(s(y1)), y2))) -> c15(MIN(s(s(y0)), cons(s(s(y1)), y2))) MIN(z0, cons(z1, cons(y1, cons(y2, y3)))) -> c15(MIN(z1, cons(y1, cons(y2, y3)))) MIN(z0, cons(s(s(y0)), cons(y1, cons(s(s(y2)), y3)))) -> c15(MIN(s(s(y0)), cons(y1, cons(s(s(y2)), y3)))) DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c17(EQ(s(s(y0)), s(s(y1)))) S tuples: DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2)) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1)))) MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), if(le(z0, z2), min(z0, z3), min(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) MINSORT(cons(0, cons(z0, z1))) -> c11(MINSORT(del(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1)))), DEL(if(le(0, z0), min(0, z1), min(z0, z1)), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) MINSORT(cons(s(z0), cons(0, z1))) -> c11(MINSORT(del(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1)))), DEL(if(le(s(z0), 0), min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2)))), DEL(if(le(s(z0), s(z1)), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil))) MIN(z0, cons(z1, cons(y1, y2))) -> c14(MIN(z0, cons(y1, y2))) MIN(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c14(MIN(s(s(y0)), cons(s(s(y1)), y2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil))) MIN(z0, cons(z1, cons(y1, y2))) -> c15(MIN(z1, cons(y1, y2))) MIN(z0, cons(s(s(y0)), cons(s(s(y1)), y2))) -> c15(MIN(s(s(y0)), cons(s(s(y1)), y2))) MIN(z0, cons(z1, cons(y1, cons(y2, y3)))) -> c15(MIN(z1, cons(y1, cons(y2, y3)))) MIN(z0, cons(s(s(y0)), cons(y1, cons(s(s(y2)), y3)))) -> c15(MIN(s(s(y0)), cons(y1, cons(s(s(y2)), y3)))) DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c17(EQ(s(s(y0)), s(s(y1)))) K tuples: MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1))) MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2))) MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil))) MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2))) Defined Rule Symbols: del_2, min_2, if_3, le_2, eq_2 Defined Pair Symbols: DEL_2, MINSORT_1, LE_2, EQ_2, MIN_2 Compound Symbols: c18_1, c11_1, c_1, c2_1, c6_1, c10_1, c11_3, c13_1, c11_2, c14_1, c15_1, c17_1 ---------------------------------------- (151) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z1, z2))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) by MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) ---------------------------------------- (152) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if(eq(z0, z1), z2, cons(z1, del(z0, z2))) del(z0, nil) -> nil min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if(le(z0, z1), min(z0, z2), min(z1, z2)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 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) Tuples: DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2)) MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1))) MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2))) MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1)))) MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), if(le(z0, z2), min(z0, z3), min(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) MINSORT(cons(0, cons(z0, z1))) -> c11(MINSORT(del(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1)))), DEL(if(le(0, z0), min(0, z1), min(z0, z1)), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) MINSORT(cons(s(z0), cons(0, z1))) -> c11(MINSORT(del(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1)))), DEL(if(le(s(z0), 0), min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2)))), DEL(if(le(s(z0), s(z1)), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) MINSORT(cons(z0, cons(z1, z2))) -> c(DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c11(DEL(z0, cons(z0, nil))) MIN(z0, cons(z1, cons(y1, y2))) -> c14(MIN(z0, cons(y1, y2))) MIN(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c14(MIN(s(s(y0)), cons(s(s(y1)), y2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil))) MIN(z0, cons(z1, cons(y1, y2))) -> c15(MIN(z1, cons(y1, y2))) MIN(z0, cons(s(s(y0)), cons(s(s(y1)), y2))) -> c15(MIN(s(s(y0)), cons(s(s(y1)), y2))) MIN(z0, cons(z1, cons(y1, cons(y2, y3)))) -> c15(MIN(z1, cons(y1, cons(y2, y3)))) MIN(z0, cons(s(s(y0)), cons(y1, cons(s(s(y2)), y3)))) -> c15(MIN(s(s(y0)), cons(y1, cons(s(s(y2)), y3)))) DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c17(EQ(s(s(y0)), s(s(y1)))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) S tuples: DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2)) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1)))) MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), if(le(z0, z2), min(z0, z3), min(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) MINSORT(cons(0, cons(z0, z1))) -> c11(MINSORT(del(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1)))), DEL(if(le(0, z0), min(0, z1), min(z0, z1)), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) MINSORT(cons(s(z0), cons(0, z1))) -> c11(MINSORT(del(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1)))), DEL(if(le(s(z0), 0), min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2)))), DEL(if(le(s(z0), s(z1)), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil))) MIN(z0, cons(z1, cons(y1, y2))) -> c14(MIN(z0, cons(y1, y2))) MIN(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c14(MIN(s(s(y0)), cons(s(s(y1)), y2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil))) MIN(z0, cons(z1, cons(y1, y2))) -> c15(MIN(z1, cons(y1, y2))) MIN(z0, cons(s(s(y0)), cons(s(s(y1)), y2))) -> c15(MIN(s(s(y0)), cons(s(s(y1)), y2))) MIN(z0, cons(z1, cons(y1, cons(y2, y3)))) -> c15(MIN(z1, cons(y1, cons(y2, y3)))) MIN(z0, cons(s(s(y0)), cons(y1, cons(s(s(y2)), y3)))) -> c15(MIN(s(s(y0)), cons(y1, cons(s(s(y2)), y3)))) DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c17(EQ(s(s(y0)), s(s(y1)))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) K tuples: MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1))) MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2))) MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil))) MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2))) Defined Rule Symbols: del_2, min_2, if_3, le_2, eq_2 Defined Pair Symbols: DEL_2, MINSORT_1, LE_2, EQ_2, MIN_2 Compound Symbols: c18_1, c11_1, c_1, c2_1, c6_1, c10_1, c13_1, c11_3, c11_2, c14_1, c15_1, c17_1 ---------------------------------------- (153) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace DEL(z0, cons(z1, z2)) -> c18(DEL(z0, z2)) by DEL(z0, cons(z1, cons(y1, y2))) -> c18(DEL(z0, cons(y1, y2))) DEL(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c18(DEL(s(s(y0)), cons(s(s(y1)), y2))) ---------------------------------------- (154) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if(eq(z0, z1), z2, cons(z1, del(z0, z2))) del(z0, nil) -> nil min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if(le(z0, z1), min(z0, z2), min(z1, z2)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 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) Tuples: MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1))) MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2))) MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1)))) MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), if(le(z0, z2), min(z0, z3), min(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) MINSORT(cons(0, cons(z0, z1))) -> c11(MINSORT(del(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1)))), DEL(if(le(0, z0), min(0, z1), min(z0, z1)), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) MINSORT(cons(s(z0), cons(0, z1))) -> c11(MINSORT(del(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1)))), DEL(if(le(s(z0), 0), min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2)))), DEL(if(le(s(z0), s(z1)), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) MINSORT(cons(z0, cons(z1, z2))) -> c(DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c11(DEL(z0, cons(z0, nil))) MIN(z0, cons(z1, cons(y1, y2))) -> c14(MIN(z0, cons(y1, y2))) MIN(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c14(MIN(s(s(y0)), cons(s(s(y1)), y2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil))) MIN(z0, cons(z1, cons(y1, y2))) -> c15(MIN(z1, cons(y1, y2))) MIN(z0, cons(s(s(y0)), cons(s(s(y1)), y2))) -> c15(MIN(s(s(y0)), cons(s(s(y1)), y2))) MIN(z0, cons(z1, cons(y1, cons(y2, y3)))) -> c15(MIN(z1, cons(y1, cons(y2, y3)))) MIN(z0, cons(s(s(y0)), cons(y1, cons(s(s(y2)), y3)))) -> c15(MIN(s(s(y0)), cons(y1, cons(s(s(y2)), y3)))) DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c17(EQ(s(s(y0)), s(s(y1)))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) DEL(z0, cons(z1, cons(y1, y2))) -> c18(DEL(z0, cons(y1, y2))) DEL(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c18(DEL(s(s(y0)), cons(s(s(y1)), y2))) S tuples: LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1)))) MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), if(le(z0, z2), min(z0, z3), min(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) MINSORT(cons(0, cons(z0, z1))) -> c11(MINSORT(del(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1)))), DEL(if(le(0, z0), min(0, z1), min(z0, z1)), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) MINSORT(cons(s(z0), cons(0, z1))) -> c11(MINSORT(del(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1)))), DEL(if(le(s(z0), 0), min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2)))), DEL(if(le(s(z0), s(z1)), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil))) MIN(z0, cons(z1, cons(y1, y2))) -> c14(MIN(z0, cons(y1, y2))) MIN(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c14(MIN(s(s(y0)), cons(s(s(y1)), y2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil))) MIN(z0, cons(z1, cons(y1, y2))) -> c15(MIN(z1, cons(y1, y2))) MIN(z0, cons(s(s(y0)), cons(s(s(y1)), y2))) -> c15(MIN(s(s(y0)), cons(s(s(y1)), y2))) MIN(z0, cons(z1, cons(y1, cons(y2, y3)))) -> c15(MIN(z1, cons(y1, cons(y2, y3)))) MIN(z0, cons(s(s(y0)), cons(y1, cons(s(s(y2)), y3)))) -> c15(MIN(s(s(y0)), cons(y1, cons(s(s(y2)), y3)))) DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c17(EQ(s(s(y0)), s(s(y1)))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) DEL(z0, cons(z1, cons(y1, y2))) -> c18(DEL(z0, cons(y1, y2))) DEL(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c18(DEL(s(s(y0)), cons(s(s(y1)), y2))) K tuples: MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1))) MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2))) MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil))) MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2))) Defined Rule Symbols: del_2, min_2, if_3, le_2, eq_2 Defined Pair Symbols: MINSORT_1, LE_2, EQ_2, MIN_2, DEL_2 Compound Symbols: c11_1, c_1, c2_1, c6_1, c10_1, c13_1, c11_3, c11_2, c14_1, c15_1, c17_1, c18_1 ---------------------------------------- (155) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) by MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) ---------------------------------------- (156) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if(eq(z0, z1), z2, cons(z1, del(z0, z2))) del(z0, nil) -> nil min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if(le(z0, z1), min(z0, z2), min(z1, z2)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 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) Tuples: MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1))) MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2))) MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1)))) MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), if(le(z0, z2), min(z0, z3), min(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) MINSORT(cons(0, cons(z0, z1))) -> c11(MINSORT(del(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1)))), DEL(if(le(0, z0), min(0, z1), min(z0, z1)), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) MINSORT(cons(s(z0), cons(0, z1))) -> c11(MINSORT(del(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1)))), DEL(if(le(s(z0), 0), min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2)))), DEL(if(le(s(z0), s(z1)), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) MINSORT(cons(z0, cons(z1, z2))) -> c(DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c11(DEL(z0, cons(z0, nil))) MIN(z0, cons(z1, cons(y1, y2))) -> c14(MIN(z0, cons(y1, y2))) MIN(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c14(MIN(s(s(y0)), cons(s(s(y1)), y2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil))) MIN(z0, cons(z1, cons(y1, y2))) -> c15(MIN(z1, cons(y1, y2))) MIN(z0, cons(s(s(y0)), cons(s(s(y1)), y2))) -> c15(MIN(s(s(y0)), cons(s(s(y1)), y2))) MIN(z0, cons(z1, cons(y1, cons(y2, y3)))) -> c15(MIN(z1, cons(y1, cons(y2, y3)))) MIN(z0, cons(s(s(y0)), cons(y1, cons(s(s(y2)), y3)))) -> c15(MIN(s(s(y0)), cons(y1, cons(s(s(y2)), y3)))) DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c17(EQ(s(s(y0)), s(s(y1)))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) DEL(z0, cons(z1, cons(y1, y2))) -> c18(DEL(z0, cons(y1, y2))) DEL(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c18(DEL(s(s(y0)), cons(s(s(y1)), y2))) MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) S tuples: LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1)))) MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), if(le(z0, z2), min(z0, z3), min(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) MINSORT(cons(0, cons(z0, z1))) -> c11(MINSORT(del(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1)))), DEL(if(le(0, z0), min(0, z1), min(z0, z1)), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) MINSORT(cons(s(z0), cons(0, z1))) -> c11(MINSORT(del(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1)))), DEL(if(le(s(z0), 0), min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2)))), DEL(if(le(s(z0), s(z1)), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil))) MIN(z0, cons(z1, cons(y1, y2))) -> c14(MIN(z0, cons(y1, y2))) MIN(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c14(MIN(s(s(y0)), cons(s(s(y1)), y2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil))) MIN(z0, cons(z1, cons(y1, y2))) -> c15(MIN(z1, cons(y1, y2))) MIN(z0, cons(s(s(y0)), cons(s(s(y1)), y2))) -> c15(MIN(s(s(y0)), cons(s(s(y1)), y2))) MIN(z0, cons(z1, cons(y1, cons(y2, y3)))) -> c15(MIN(z1, cons(y1, cons(y2, y3)))) MIN(z0, cons(s(s(y0)), cons(y1, cons(s(s(y2)), y3)))) -> c15(MIN(s(s(y0)), cons(y1, cons(s(s(y2)), y3)))) DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c17(EQ(s(s(y0)), s(s(y1)))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) DEL(z0, cons(z1, cons(y1, y2))) -> c18(DEL(z0, cons(y1, y2))) DEL(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c18(DEL(s(s(y0)), cons(s(s(y1)), y2))) MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) K tuples: MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1))) MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2))) MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil))) MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2))) Defined Rule Symbols: del_2, min_2, if_3, le_2, eq_2 Defined Pair Symbols: MINSORT_1, LE_2, EQ_2, MIN_2, DEL_2 Compound Symbols: c11_1, c_1, c2_1, c6_1, c10_1, c13_1, c11_3, c11_2, c14_1, c15_1, c17_1, c18_1 ---------------------------------------- (157) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) by MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) ---------------------------------------- (158) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if(eq(z0, z1), z2, cons(z1, del(z0, z2))) del(z0, nil) -> nil min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if(le(z0, z1), min(z0, z2), min(z1, z2)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 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) Tuples: MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1))) MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2))) MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1)))) MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), if(le(z0, z2), min(z0, z3), min(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) MINSORT(cons(0, cons(z0, z1))) -> c11(MINSORT(del(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1)))), DEL(if(le(0, z0), min(0, z1), min(z0, z1)), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) MINSORT(cons(s(z0), cons(0, z1))) -> c11(MINSORT(del(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1)))), DEL(if(le(s(z0), 0), min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2)))), DEL(if(le(s(z0), s(z1)), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) MINSORT(cons(z0, cons(z1, z2))) -> c(DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c11(DEL(z0, cons(z0, nil))) MIN(z0, cons(z1, cons(y1, y2))) -> c14(MIN(z0, cons(y1, y2))) MIN(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c14(MIN(s(s(y0)), cons(s(s(y1)), y2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil))) MIN(z0, cons(z1, cons(y1, y2))) -> c15(MIN(z1, cons(y1, y2))) MIN(z0, cons(s(s(y0)), cons(s(s(y1)), y2))) -> c15(MIN(s(s(y0)), cons(s(s(y1)), y2))) MIN(z0, cons(z1, cons(y1, cons(y2, y3)))) -> c15(MIN(z1, cons(y1, cons(y2, y3)))) MIN(z0, cons(s(s(y0)), cons(y1, cons(s(s(y2)), y3)))) -> c15(MIN(s(s(y0)), cons(y1, cons(s(s(y2)), y3)))) DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c17(EQ(s(s(y0)), s(s(y1)))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) DEL(z0, cons(z1, cons(y1, y2))) -> c18(DEL(z0, cons(y1, y2))) DEL(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c18(DEL(s(s(y0)), cons(s(s(y1)), y2))) MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) S tuples: LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1)))) MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), if(le(z0, z2), min(z0, z3), min(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) MINSORT(cons(0, cons(z0, z1))) -> c11(MINSORT(del(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1)))), DEL(if(le(0, z0), min(0, z1), min(z0, z1)), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) MINSORT(cons(s(z0), cons(0, z1))) -> c11(MINSORT(del(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1)))), DEL(if(le(s(z0), 0), min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2)))), DEL(if(le(s(z0), s(z1)), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil))) MIN(z0, cons(z1, cons(y1, y2))) -> c14(MIN(z0, cons(y1, y2))) MIN(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c14(MIN(s(s(y0)), cons(s(s(y1)), y2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil))) MIN(z0, cons(z1, cons(y1, y2))) -> c15(MIN(z1, cons(y1, y2))) MIN(z0, cons(s(s(y0)), cons(s(s(y1)), y2))) -> c15(MIN(s(s(y0)), cons(s(s(y1)), y2))) MIN(z0, cons(z1, cons(y1, cons(y2, y3)))) -> c15(MIN(z1, cons(y1, cons(y2, y3)))) MIN(z0, cons(s(s(y0)), cons(y1, cons(s(s(y2)), y3)))) -> c15(MIN(s(s(y0)), cons(y1, cons(s(s(y2)), y3)))) DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c17(EQ(s(s(y0)), s(s(y1)))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) DEL(z0, cons(z1, cons(y1, y2))) -> c18(DEL(z0, cons(y1, y2))) DEL(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c18(DEL(s(s(y0)), cons(s(s(y1)), y2))) MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) K tuples: MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1))) MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2))) MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil))) MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2))) Defined Rule Symbols: del_2, min_2, if_3, le_2, eq_2 Defined Pair Symbols: MINSORT_1, LE_2, EQ_2, MIN_2, DEL_2 Compound Symbols: c11_1, c_1, c2_1, c6_1, c10_1, c13_1, c11_3, c11_2, c14_1, c15_1, c17_1, c18_1 ---------------------------------------- (159) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), min(z1, nil)), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) by MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) ---------------------------------------- (160) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if(eq(z0, z1), z2, cons(z1, del(z0, z2))) del(z0, nil) -> nil min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if(le(z0, z1), min(z0, z2), min(z1, z2)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 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) Tuples: MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1))) MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2))) MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1)))) MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), if(le(z0, z2), min(z0, z3), min(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) MINSORT(cons(0, cons(z0, z1))) -> c11(MINSORT(del(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1)))), DEL(if(le(0, z0), min(0, z1), min(z0, z1)), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) MINSORT(cons(s(z0), cons(0, z1))) -> c11(MINSORT(del(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1)))), DEL(if(le(s(z0), 0), min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2)))), DEL(if(le(s(z0), s(z1)), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) MINSORT(cons(z0, cons(z1, z2))) -> c(DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c11(DEL(z0, cons(z0, nil))) MIN(z0, cons(z1, cons(y1, y2))) -> c14(MIN(z0, cons(y1, y2))) MIN(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c14(MIN(s(s(y0)), cons(s(s(y1)), y2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil))) MIN(z0, cons(z1, cons(y1, y2))) -> c15(MIN(z1, cons(y1, y2))) MIN(z0, cons(s(s(y0)), cons(s(s(y1)), y2))) -> c15(MIN(s(s(y0)), cons(s(s(y1)), y2))) MIN(z0, cons(z1, cons(y1, cons(y2, y3)))) -> c15(MIN(z1, cons(y1, cons(y2, y3)))) MIN(z0, cons(s(s(y0)), cons(y1, cons(s(s(y2)), y3)))) -> c15(MIN(s(s(y0)), cons(y1, cons(s(s(y2)), y3)))) DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c17(EQ(s(s(y0)), s(s(y1)))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) DEL(z0, cons(z1, cons(y1, y2))) -> c18(DEL(z0, cons(y1, y2))) DEL(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c18(DEL(s(s(y0)), cons(s(s(y1)), y2))) MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) S tuples: LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1)))) MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), if(le(z0, z2), min(z0, z3), min(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) MINSORT(cons(0, cons(z0, z1))) -> c11(MINSORT(del(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1)))), DEL(if(le(0, z0), min(0, z1), min(z0, z1)), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) MINSORT(cons(s(z0), cons(0, z1))) -> c11(MINSORT(del(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1)))), DEL(if(le(s(z0), 0), min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2)))), DEL(if(le(s(z0), s(z1)), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil))) MIN(z0, cons(z1, cons(y1, y2))) -> c14(MIN(z0, cons(y1, y2))) MIN(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c14(MIN(s(s(y0)), cons(s(s(y1)), y2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil))) MIN(z0, cons(z1, cons(y1, y2))) -> c15(MIN(z1, cons(y1, y2))) MIN(z0, cons(s(s(y0)), cons(s(s(y1)), y2))) -> c15(MIN(s(s(y0)), cons(s(s(y1)), y2))) MIN(z0, cons(z1, cons(y1, cons(y2, y3)))) -> c15(MIN(z1, cons(y1, cons(y2, y3)))) MIN(z0, cons(s(s(y0)), cons(y1, cons(s(s(y2)), y3)))) -> c15(MIN(s(s(y0)), cons(y1, cons(s(s(y2)), y3)))) DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c17(EQ(s(s(y0)), s(s(y1)))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) DEL(z0, cons(z1, cons(y1, y2))) -> c18(DEL(z0, cons(y1, y2))) DEL(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c18(DEL(s(s(y0)), cons(s(s(y1)), y2))) MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) K tuples: MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1))) MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2))) MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil))) MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2))) Defined Rule Symbols: del_2, min_2, if_3, le_2, eq_2 Defined Pair Symbols: MINSORT_1, LE_2, EQ_2, MIN_2, DEL_2 Compound Symbols: c11_1, c_1, c2_1, c6_1, c10_1, c13_1, c11_3, c11_2, c14_1, c15_1, c17_1, c18_1 ---------------------------------------- (161) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), if(le(z0, z2), min(z0, z3), min(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) by MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), if(le(z0, z2), min(z0, z3), min(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) ---------------------------------------- (162) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if(eq(z0, z1), z2, cons(z1, del(z0, z2))) del(z0, nil) -> nil min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if(le(z0, z1), min(z0, z2), min(z1, z2)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 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) Tuples: MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1))) MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2))) MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1)))) MINSORT(cons(0, cons(z0, z1))) -> c11(MINSORT(del(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1)))), DEL(if(le(0, z0), min(0, z1), min(z0, z1)), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) MINSORT(cons(s(z0), cons(0, z1))) -> c11(MINSORT(del(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1)))), DEL(if(le(s(z0), 0), min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2)))), DEL(if(le(s(z0), s(z1)), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) MINSORT(cons(z0, cons(z1, z2))) -> c(DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c11(DEL(z0, cons(z0, nil))) MIN(z0, cons(z1, cons(y1, y2))) -> c14(MIN(z0, cons(y1, y2))) MIN(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c14(MIN(s(s(y0)), cons(s(s(y1)), y2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil))) MIN(z0, cons(z1, cons(y1, y2))) -> c15(MIN(z1, cons(y1, y2))) MIN(z0, cons(s(s(y0)), cons(s(s(y1)), y2))) -> c15(MIN(s(s(y0)), cons(s(s(y1)), y2))) MIN(z0, cons(z1, cons(y1, cons(y2, y3)))) -> c15(MIN(z1, cons(y1, cons(y2, y3)))) MIN(z0, cons(s(s(y0)), cons(y1, cons(s(s(y2)), y3)))) -> c15(MIN(s(s(y0)), cons(y1, cons(s(s(y2)), y3)))) DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c17(EQ(s(s(y0)), s(s(y1)))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) DEL(z0, cons(z1, cons(y1, y2))) -> c18(DEL(z0, cons(y1, y2))) DEL(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c18(DEL(s(s(y0)), cons(s(s(y1)), y2))) MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), if(le(z0, z2), min(z0, z3), min(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) S tuples: LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1)))) MINSORT(cons(0, cons(z0, z1))) -> c11(MINSORT(del(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1)))), DEL(if(le(0, z0), min(0, z1), min(z0, z1)), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) MINSORT(cons(s(z0), cons(0, z1))) -> c11(MINSORT(del(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1)))), DEL(if(le(s(z0), 0), min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2)))), DEL(if(le(s(z0), s(z1)), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil))) MIN(z0, cons(z1, cons(y1, y2))) -> c14(MIN(z0, cons(y1, y2))) MIN(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c14(MIN(s(s(y0)), cons(s(s(y1)), y2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil))) MIN(z0, cons(z1, cons(y1, y2))) -> c15(MIN(z1, cons(y1, y2))) MIN(z0, cons(s(s(y0)), cons(s(s(y1)), y2))) -> c15(MIN(s(s(y0)), cons(s(s(y1)), y2))) MIN(z0, cons(z1, cons(y1, cons(y2, y3)))) -> c15(MIN(z1, cons(y1, cons(y2, y3)))) MIN(z0, cons(s(s(y0)), cons(y1, cons(s(s(y2)), y3)))) -> c15(MIN(s(s(y0)), cons(y1, cons(s(s(y2)), y3)))) DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c17(EQ(s(s(y0)), s(s(y1)))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) DEL(z0, cons(z1, cons(y1, y2))) -> c18(DEL(z0, cons(y1, y2))) DEL(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c18(DEL(s(s(y0)), cons(s(s(y1)), y2))) MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), if(le(z0, z2), min(z0, z3), min(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) K tuples: MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1))) MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2))) MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil))) MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2))) Defined Rule Symbols: del_2, min_2, if_3, le_2, eq_2 Defined Pair Symbols: MINSORT_1, LE_2, EQ_2, MIN_2, DEL_2 Compound Symbols: c11_1, c_1, c2_1, c6_1, c10_1, c13_1, c11_3, c11_2, c14_1, c15_1, c17_1, c18_1 ---------------------------------------- (163) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace MINSORT(cons(0, cons(z0, z1))) -> c11(MINSORT(del(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1)))), DEL(if(le(0, z0), min(0, z1), min(z0, z1)), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) by MINSORT(cons(0, cons(z0, z1))) -> c11(MINSORT(del(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1)))), DEL(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) ---------------------------------------- (164) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if(eq(z0, z1), z2, cons(z1, del(z0, z2))) del(z0, nil) -> nil min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if(le(z0, z1), min(z0, z2), min(z1, z2)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 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) Tuples: MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1))) MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2))) MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1)))) MINSORT(cons(s(z0), cons(0, z1))) -> c11(MINSORT(del(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1)))), DEL(if(le(s(z0), 0), min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2)))), DEL(if(le(s(z0), s(z1)), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) MINSORT(cons(z0, cons(z1, z2))) -> c(DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c11(DEL(z0, cons(z0, nil))) MIN(z0, cons(z1, cons(y1, y2))) -> c14(MIN(z0, cons(y1, y2))) MIN(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c14(MIN(s(s(y0)), cons(s(s(y1)), y2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil))) MIN(z0, cons(z1, cons(y1, y2))) -> c15(MIN(z1, cons(y1, y2))) MIN(z0, cons(s(s(y0)), cons(s(s(y1)), y2))) -> c15(MIN(s(s(y0)), cons(s(s(y1)), y2))) MIN(z0, cons(z1, cons(y1, cons(y2, y3)))) -> c15(MIN(z1, cons(y1, cons(y2, y3)))) MIN(z0, cons(s(s(y0)), cons(y1, cons(s(s(y2)), y3)))) -> c15(MIN(s(s(y0)), cons(y1, cons(s(s(y2)), y3)))) DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c17(EQ(s(s(y0)), s(s(y1)))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) DEL(z0, cons(z1, cons(y1, y2))) -> c18(DEL(z0, cons(y1, y2))) DEL(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c18(DEL(s(s(y0)), cons(s(s(y1)), y2))) MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), if(le(z0, z2), min(z0, z3), min(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) MINSORT(cons(0, cons(z0, z1))) -> c11(MINSORT(del(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1)))), DEL(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) S tuples: LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1)))) MINSORT(cons(s(z0), cons(0, z1))) -> c11(MINSORT(del(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1)))), DEL(if(le(s(z0), 0), min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2)))), DEL(if(le(s(z0), s(z1)), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil))) MIN(z0, cons(z1, cons(y1, y2))) -> c14(MIN(z0, cons(y1, y2))) MIN(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c14(MIN(s(s(y0)), cons(s(s(y1)), y2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil))) MIN(z0, cons(z1, cons(y1, y2))) -> c15(MIN(z1, cons(y1, y2))) MIN(z0, cons(s(s(y0)), cons(s(s(y1)), y2))) -> c15(MIN(s(s(y0)), cons(s(s(y1)), y2))) MIN(z0, cons(z1, cons(y1, cons(y2, y3)))) -> c15(MIN(z1, cons(y1, cons(y2, y3)))) MIN(z0, cons(s(s(y0)), cons(y1, cons(s(s(y2)), y3)))) -> c15(MIN(s(s(y0)), cons(y1, cons(s(s(y2)), y3)))) DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c17(EQ(s(s(y0)), s(s(y1)))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) DEL(z0, cons(z1, cons(y1, y2))) -> c18(DEL(z0, cons(y1, y2))) DEL(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c18(DEL(s(s(y0)), cons(s(s(y1)), y2))) MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), if(le(z0, z2), min(z0, z3), min(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) MINSORT(cons(0, cons(z0, z1))) -> c11(MINSORT(del(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1)))), DEL(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) K tuples: MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1))) MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2))) MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil))) MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2))) Defined Rule Symbols: del_2, min_2, if_3, le_2, eq_2 Defined Pair Symbols: MINSORT_1, LE_2, EQ_2, MIN_2, DEL_2 Compound Symbols: c11_1, c_1, c2_1, c6_1, c10_1, c13_1, c11_3, c11_2, c14_1, c15_1, c17_1, c18_1 ---------------------------------------- (165) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace MINSORT(cons(s(z0), cons(0, z1))) -> c11(MINSORT(del(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1)))), DEL(if(le(s(z0), 0), min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) by MINSORT(cons(s(z0), cons(0, z1))) -> c11(MINSORT(del(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1)))), DEL(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) ---------------------------------------- (166) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if(eq(z0, z1), z2, cons(z1, del(z0, z2))) del(z0, nil) -> nil min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if(le(z0, z1), min(z0, z2), min(z1, z2)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 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) Tuples: MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1))) MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2))) MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1)))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2)))), DEL(if(le(s(z0), s(z1)), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) MINSORT(cons(z0, cons(z1, z2))) -> c(DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c11(DEL(z0, cons(z0, nil))) MIN(z0, cons(z1, cons(y1, y2))) -> c14(MIN(z0, cons(y1, y2))) MIN(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c14(MIN(s(s(y0)), cons(s(s(y1)), y2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil))) MIN(z0, cons(z1, cons(y1, y2))) -> c15(MIN(z1, cons(y1, y2))) MIN(z0, cons(s(s(y0)), cons(s(s(y1)), y2))) -> c15(MIN(s(s(y0)), cons(s(s(y1)), y2))) MIN(z0, cons(z1, cons(y1, cons(y2, y3)))) -> c15(MIN(z1, cons(y1, cons(y2, y3)))) MIN(z0, cons(s(s(y0)), cons(y1, cons(s(s(y2)), y3)))) -> c15(MIN(s(s(y0)), cons(y1, cons(s(s(y2)), y3)))) DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c17(EQ(s(s(y0)), s(s(y1)))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) DEL(z0, cons(z1, cons(y1, y2))) -> c18(DEL(z0, cons(y1, y2))) DEL(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c18(DEL(s(s(y0)), cons(s(s(y1)), y2))) MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), if(le(z0, z2), min(z0, z3), min(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) MINSORT(cons(0, cons(z0, z1))) -> c11(MINSORT(del(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1)))), DEL(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) MINSORT(cons(s(z0), cons(0, z1))) -> c11(MINSORT(del(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1)))), DEL(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) S tuples: LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1)))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2)))), DEL(if(le(s(z0), s(z1)), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil))) MIN(z0, cons(z1, cons(y1, y2))) -> c14(MIN(z0, cons(y1, y2))) MIN(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c14(MIN(s(s(y0)), cons(s(s(y1)), y2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil))) MIN(z0, cons(z1, cons(y1, y2))) -> c15(MIN(z1, cons(y1, y2))) MIN(z0, cons(s(s(y0)), cons(s(s(y1)), y2))) -> c15(MIN(s(s(y0)), cons(s(s(y1)), y2))) MIN(z0, cons(z1, cons(y1, cons(y2, y3)))) -> c15(MIN(z1, cons(y1, cons(y2, y3)))) MIN(z0, cons(s(s(y0)), cons(y1, cons(s(s(y2)), y3)))) -> c15(MIN(s(s(y0)), cons(y1, cons(s(s(y2)), y3)))) DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c17(EQ(s(s(y0)), s(s(y1)))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) DEL(z0, cons(z1, cons(y1, y2))) -> c18(DEL(z0, cons(y1, y2))) DEL(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c18(DEL(s(s(y0)), cons(s(s(y1)), y2))) MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), if(le(z0, z2), min(z0, z3), min(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) MINSORT(cons(0, cons(z0, z1))) -> c11(MINSORT(del(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1)))), DEL(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) MINSORT(cons(s(z0), cons(0, z1))) -> c11(MINSORT(del(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1)))), DEL(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) K tuples: MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1))) MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2))) MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil))) MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2))) Defined Rule Symbols: del_2, min_2, if_3, le_2, eq_2 Defined Pair Symbols: MINSORT_1, LE_2, EQ_2, MIN_2, DEL_2 Compound Symbols: c11_1, c_1, c2_1, c6_1, c10_1, c13_1, c11_3, c11_2, c14_1, c15_1, c17_1, c18_1 ---------------------------------------- (167) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace MINSORT(cons(s(z0), cons(s(z1), z2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2)))), DEL(if(le(s(z0), s(z1)), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) by MINSORT(cons(s(z0), cons(s(z1), z2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2)))), DEL(if(le(z0, z1), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) ---------------------------------------- (168) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if(eq(z0, z1), z2, cons(z1, del(z0, z2))) del(z0, nil) -> nil min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if(le(z0, z1), min(z0, z2), min(z1, z2)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 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) Tuples: MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1))) MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2))) MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1)))) MINSORT(cons(z0, cons(z1, z2))) -> c(DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c11(DEL(z0, cons(z0, nil))) MIN(z0, cons(z1, cons(y1, y2))) -> c14(MIN(z0, cons(y1, y2))) MIN(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c14(MIN(s(s(y0)), cons(s(s(y1)), y2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil))) MIN(z0, cons(z1, cons(y1, y2))) -> c15(MIN(z1, cons(y1, y2))) MIN(z0, cons(s(s(y0)), cons(s(s(y1)), y2))) -> c15(MIN(s(s(y0)), cons(s(s(y1)), y2))) MIN(z0, cons(z1, cons(y1, cons(y2, y3)))) -> c15(MIN(z1, cons(y1, cons(y2, y3)))) MIN(z0, cons(s(s(y0)), cons(y1, cons(s(s(y2)), y3)))) -> c15(MIN(s(s(y0)), cons(y1, cons(s(s(y2)), y3)))) DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c17(EQ(s(s(y0)), s(s(y1)))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) DEL(z0, cons(z1, cons(y1, y2))) -> c18(DEL(z0, cons(y1, y2))) DEL(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c18(DEL(s(s(y0)), cons(s(s(y1)), y2))) MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), if(le(z0, z2), min(z0, z3), min(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) MINSORT(cons(0, cons(z0, z1))) -> c11(MINSORT(del(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1)))), DEL(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) MINSORT(cons(s(z0), cons(0, z1))) -> c11(MINSORT(del(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1)))), DEL(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2)))), DEL(if(le(z0, z1), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) S tuples: LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1)))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil))) MIN(z0, cons(z1, cons(y1, y2))) -> c14(MIN(z0, cons(y1, y2))) MIN(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c14(MIN(s(s(y0)), cons(s(s(y1)), y2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil))) MIN(z0, cons(z1, cons(y1, y2))) -> c15(MIN(z1, cons(y1, y2))) MIN(z0, cons(s(s(y0)), cons(s(s(y1)), y2))) -> c15(MIN(s(s(y0)), cons(s(s(y1)), y2))) MIN(z0, cons(z1, cons(y1, cons(y2, y3)))) -> c15(MIN(z1, cons(y1, cons(y2, y3)))) MIN(z0, cons(s(s(y0)), cons(y1, cons(s(s(y2)), y3)))) -> c15(MIN(s(s(y0)), cons(y1, cons(s(s(y2)), y3)))) DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c17(EQ(s(s(y0)), s(s(y1)))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) DEL(z0, cons(z1, cons(y1, y2))) -> c18(DEL(z0, cons(y1, y2))) DEL(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c18(DEL(s(s(y0)), cons(s(s(y1)), y2))) MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), if(le(z0, z2), min(z0, z3), min(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) MINSORT(cons(0, cons(z0, z1))) -> c11(MINSORT(del(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1)))), DEL(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) MINSORT(cons(s(z0), cons(0, z1))) -> c11(MINSORT(del(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1)))), DEL(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2)))), DEL(if(le(z0, z1), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) K tuples: MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1))) MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2))) MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil))) MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2))) Defined Rule Symbols: del_2, min_2, if_3, le_2, eq_2 Defined Pair Symbols: MINSORT_1, LE_2, EQ_2, MIN_2, DEL_2 Compound Symbols: c11_1, c_1, c2_1, c6_1, c10_1, c13_1, c11_2, c14_1, c11_3, c15_1, c17_1, c18_1 ---------------------------------------- (169) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, del(z0, nil)))), DEL(z0, cons(z0, nil))) by MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil))) ---------------------------------------- (170) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if(eq(z0, z1), z2, cons(z1, del(z0, z2))) del(z0, nil) -> nil min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if(le(z0, z1), min(z0, z2), min(z1, z2)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 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) Tuples: MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1))) MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2))) MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1)))) MINSORT(cons(z0, cons(z1, z2))) -> c(DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c11(DEL(z0, cons(z0, nil))) MIN(z0, cons(z1, cons(y1, y2))) -> c14(MIN(z0, cons(y1, y2))) MIN(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c14(MIN(s(s(y0)), cons(s(s(y1)), y2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil))) MIN(z0, cons(z1, cons(y1, y2))) -> c15(MIN(z1, cons(y1, y2))) MIN(z0, cons(s(s(y0)), cons(s(s(y1)), y2))) -> c15(MIN(s(s(y0)), cons(s(s(y1)), y2))) MIN(z0, cons(z1, cons(y1, cons(y2, y3)))) -> c15(MIN(z1, cons(y1, cons(y2, y3)))) MIN(z0, cons(s(s(y0)), cons(y1, cons(s(s(y2)), y3)))) -> c15(MIN(s(s(y0)), cons(y1, cons(s(s(y2)), y3)))) DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c17(EQ(s(s(y0)), s(s(y1)))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) DEL(z0, cons(z1, cons(y1, y2))) -> c18(DEL(z0, cons(y1, y2))) DEL(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c18(DEL(s(s(y0)), cons(s(s(y1)), y2))) MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), if(le(z0, z2), min(z0, z3), min(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) MINSORT(cons(0, cons(z0, z1))) -> c11(MINSORT(del(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1)))), DEL(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) MINSORT(cons(s(z0), cons(0, z1))) -> c11(MINSORT(del(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1)))), DEL(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2)))), DEL(if(le(z0, z1), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) S tuples: LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1)))) MIN(z0, cons(z1, cons(y1, y2))) -> c14(MIN(z0, cons(y1, y2))) MIN(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c14(MIN(s(s(y0)), cons(s(s(y1)), y2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil))) MIN(z0, cons(z1, cons(y1, y2))) -> c15(MIN(z1, cons(y1, y2))) MIN(z0, cons(s(s(y0)), cons(s(s(y1)), y2))) -> c15(MIN(s(s(y0)), cons(s(s(y1)), y2))) MIN(z0, cons(z1, cons(y1, cons(y2, y3)))) -> c15(MIN(z1, cons(y1, cons(y2, y3)))) MIN(z0, cons(s(s(y0)), cons(y1, cons(s(s(y2)), y3)))) -> c15(MIN(s(s(y0)), cons(y1, cons(s(s(y2)), y3)))) DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c17(EQ(s(s(y0)), s(s(y1)))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) DEL(z0, cons(z1, cons(y1, y2))) -> c18(DEL(z0, cons(y1, y2))) DEL(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c18(DEL(s(s(y0)), cons(s(s(y1)), y2))) MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), if(le(z0, z2), min(z0, z3), min(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) MINSORT(cons(0, cons(z0, z1))) -> c11(MINSORT(del(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1)))), DEL(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) MINSORT(cons(s(z0), cons(0, z1))) -> c11(MINSORT(del(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1)))), DEL(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2)))), DEL(if(le(z0, z1), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) K tuples: MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1))) MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2))) MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil))) MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2))) Defined Rule Symbols: del_2, min_2, if_3, le_2, eq_2 Defined Pair Symbols: MINSORT_1, LE_2, EQ_2, MIN_2, DEL_2 Compound Symbols: c11_1, c_1, c2_1, c6_1, c10_1, c13_1, c14_1, c11_3, c11_2, c15_1, c17_1, c18_1 ---------------------------------------- (171) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace MINSORT(cons(x0, cons(x1, x2))) -> c(MIN(x0, cons(x1, x2))) by MINSORT(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c(MIN(s(s(y0)), cons(s(s(y1)), z2))) MINSORT(cons(z0, cons(z1, cons(y2, y3)))) -> c(MIN(z0, cons(z1, cons(y2, y3)))) MINSORT(cons(s(s(y0)), cons(z1, cons(s(s(y2)), y3)))) -> c(MIN(s(s(y0)), cons(z1, cons(s(s(y2)), y3)))) MINSORT(cons(z0, cons(s(s(y1)), cons(s(s(y2)), y3)))) -> c(MIN(z0, cons(s(s(y1)), cons(s(s(y2)), y3)))) MINSORT(cons(z0, cons(z1, cons(y2, cons(y3, y4))))) -> c(MIN(z0, cons(z1, cons(y2, cons(y3, y4))))) MINSORT(cons(z0, cons(s(s(y1)), cons(y2, cons(s(s(y3)), y4))))) -> c(MIN(z0, cons(s(s(y1)), cons(y2, cons(s(s(y3)), y4))))) ---------------------------------------- (172) Obligation: Complexity Dependency Tuples Problem Rules: del(z0, cons(z1, z2)) -> if(eq(z0, z1), z2, cons(z1, del(z0, z2))) del(z0, nil) -> nil min(z0, nil) -> z0 min(z0, cons(z1, z2)) -> if(le(z0, z1), min(z0, z2), min(z1, z2)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 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) Tuples: MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2))) MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1)))) MINSORT(cons(z0, cons(z1, z2))) -> c(DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2)))) MINSORT(cons(z0, nil)) -> c11(DEL(z0, cons(z0, nil))) MIN(z0, cons(z1, cons(y1, y2))) -> c14(MIN(z0, cons(y1, y2))) MIN(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c14(MIN(s(s(y0)), cons(s(s(y1)), y2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil))) MIN(z0, cons(z1, cons(y1, y2))) -> c15(MIN(z1, cons(y1, y2))) MIN(z0, cons(s(s(y0)), cons(s(s(y1)), y2))) -> c15(MIN(s(s(y0)), cons(s(s(y1)), y2))) MIN(z0, cons(z1, cons(y1, cons(y2, y3)))) -> c15(MIN(z1, cons(y1, cons(y2, y3)))) MIN(z0, cons(s(s(y0)), cons(y1, cons(s(s(y2)), y3)))) -> c15(MIN(s(s(y0)), cons(y1, cons(s(s(y2)), y3)))) DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c17(EQ(s(s(y0)), s(s(y1)))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) DEL(z0, cons(z1, cons(y1, y2))) -> c18(DEL(z0, cons(y1, y2))) DEL(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c18(DEL(s(s(y0)), cons(s(s(y1)), y2))) MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), if(le(z0, z2), min(z0, z3), min(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) MINSORT(cons(0, cons(z0, z1))) -> c11(MINSORT(del(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1)))), DEL(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) MINSORT(cons(s(z0), cons(0, z1))) -> c11(MINSORT(del(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1)))), DEL(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2)))), DEL(if(le(z0, z1), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) MINSORT(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c(MIN(s(s(y0)), cons(s(s(y1)), z2))) MINSORT(cons(z0, cons(z1, cons(y2, y3)))) -> c(MIN(z0, cons(z1, cons(y2, y3)))) MINSORT(cons(s(s(y0)), cons(z1, cons(s(s(y2)), y3)))) -> c(MIN(s(s(y0)), cons(z1, cons(s(s(y2)), y3)))) MINSORT(cons(z0, cons(s(s(y1)), cons(s(s(y2)), y3)))) -> c(MIN(z0, cons(s(s(y1)), cons(s(s(y2)), y3)))) MINSORT(cons(z0, cons(z1, cons(y2, cons(y3, y4))))) -> c(MIN(z0, cons(z1, cons(y2, cons(y3, y4))))) MINSORT(cons(z0, cons(s(s(y1)), cons(y2, cons(s(s(y3)), y4))))) -> c(MIN(z0, cons(s(s(y1)), cons(y2, cons(s(s(y3)), y4))))) S tuples: LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) EQ(s(s(y0)), s(s(y1))) -> c6(EQ(s(y0), s(y1))) MIN(s(s(y0)), cons(s(s(y1)), z2)) -> c13(LE(s(s(y0)), s(s(y1)))) MIN(z0, cons(z1, cons(y1, y2))) -> c14(MIN(z0, cons(y1, y2))) MIN(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c14(MIN(s(s(y0)), cons(s(s(y1)), y2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(min(z0, cons(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(min(z0, cons(z1, z2)), z1), z2, cons(z1, del(min(z0, cons(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(z0, z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil))) MINSORT(cons(z0, nil)) -> c11(MINSORT(if(eq(min(z0, nil), z0), nil, cons(z0, nil))), DEL(z0, cons(z0, nil))) MIN(z0, cons(z1, cons(y1, y2))) -> c15(MIN(z1, cons(y1, y2))) MIN(z0, cons(s(s(y0)), cons(s(s(y1)), y2))) -> c15(MIN(s(s(y0)), cons(s(s(y1)), y2))) MIN(z0, cons(z1, cons(y1, cons(y2, y3)))) -> c15(MIN(z1, cons(y1, cons(y2, y3)))) MIN(z0, cons(s(s(y0)), cons(y1, cons(s(s(y2)), y3)))) -> c15(MIN(s(s(y0)), cons(y1, cons(s(s(y2)), y3)))) DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c17(EQ(s(s(y0)), s(s(y1)))) MINSORT(cons(z0, cons(z1, z2))) -> c11(MINSORT(if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z0), cons(z1, z2), cons(z0, if(eq(if(le(z0, z1), min(z0, z2), min(z1, z2)), z1), z2, cons(z1, del(if(le(z0, z1), min(z0, z2), min(z1, z2)), z2)))))), DEL(if(le(z0, z1), min(z0, z2), min(z1, z2)), cons(z0, cons(z1, z2))), MIN(z0, cons(z1, z2))) DEL(z0, cons(z1, cons(y1, y2))) -> c18(DEL(z0, cons(y1, y2))) DEL(s(s(y0)), cons(z1, cons(s(s(y1)), y2))) -> c18(DEL(s(s(y0)), cons(s(s(y1)), y2))) MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) MINSORT(cons(z0, cons(z1, nil))) -> c11(MINSORT(del(if(le(z0, z1), z0, min(z1, nil)), cons(z0, cons(z1, nil)))), DEL(if(le(z0, z1), min(z0, nil), z1), cons(z0, cons(z1, nil))), MIN(z0, cons(z1, nil))) MINSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c11(MINSORT(del(if(le(z0, z1), if(le(z0, z2), min(z0, z3), min(z2, z3)), min(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3))))), DEL(if(le(z0, z1), min(z0, cons(z2, z3)), if(le(z1, z2), min(z1, z3), min(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), MIN(z0, cons(z1, cons(z2, z3)))) MINSORT(cons(0, cons(z0, z1))) -> c11(MINSORT(del(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1)))), DEL(if(true, min(0, z1), min(z0, z1)), cons(0, cons(z0, z1))), MIN(0, cons(z0, z1))) MINSORT(cons(s(z0), cons(0, z1))) -> c11(MINSORT(del(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1)))), DEL(if(false, min(s(z0), z1), min(0, z1)), cons(s(z0), cons(0, z1))), MIN(s(z0), cons(0, z1))) MINSORT(cons(s(z0), cons(s(z1), z2))) -> c11(MINSORT(del(if(le(z0, z1), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2)))), DEL(if(le(z0, z1), min(s(z0), z2), min(s(z1), z2)), cons(s(z0), cons(s(z1), z2))), MIN(s(z0), cons(s(z1), z2))) K tuples: MINSORT(cons(x0, x1)) -> c11(DEL(min(x0, x1), cons(x0, x1))) MINSORT(cons(x0, cons(x1, x2))) -> c(DEL(min(x0, cons(x1, x2)), cons(x0, cons(x1, x2)))) MINSORT(cons(x0, nil)) -> c11(DEL(min(x0, nil), cons(x0, nil))) MINSORT(cons(z0, cons(y1, y2))) -> c10(MIN(z0, cons(y1, y2))) MINSORT(cons(s(s(y0)), cons(s(s(y1)), z2))) -> c(MIN(s(s(y0)), cons(s(s(y1)), z2))) MINSORT(cons(z0, cons(z1, cons(y2, y3)))) -> c(MIN(z0, cons(z1, cons(y2, y3)))) MINSORT(cons(s(s(y0)), cons(z1, cons(s(s(y2)), y3)))) -> c(MIN(s(s(y0)), cons(z1, cons(s(s(y2)), y3)))) MINSORT(cons(z0, cons(s(s(y1)), cons(s(s(y2)), y3)))) -> c(MIN(z0, cons(s(s(y1)), cons(s(s(y2)), y3)))) MINSORT(cons(z0, cons(z1, cons(y2, cons(y3, y4))))) -> c(MIN(z0, cons(z1, cons(y2, cons(y3, y4))))) MINSORT(cons(z0, cons(s(s(y1)), cons(y2, cons(s(s(y3)), y4))))) -> c(MIN(z0, cons(s(s(y1)), cons(y2, cons(s(s(y3)), y4))))) Defined Rule Symbols: del_2, min_2, if_3, le_2, eq_2 Defined Pair Symbols: MINSORT_1, LE_2, EQ_2, MIN_2, DEL_2 Compound Symbols: c11_1, c2_1, c6_1, c10_1, c13_1, c_1, c14_1, c11_3, c11_2, c15_1, c17_1, c18_1